Subgroups — C₂×C₆₇×#2491 #30714010

Order 17152.

3678 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#124×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂²×#121×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2491	C₂	1	16	yes	maximal, direct product factor
5	C₂×#394×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#393×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#390×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#390×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#388×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂×#380×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
14	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
16	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Q₈×C₆₇	—	2	8	no	—
18	C₂×C₄⋊Q₈×C₆₇	C₂²	1	8	yes	complemented
19	C₂×#124×C₆₇	—	2	16	no	—
20	C₂×#124×C₆₇	C₂²	1	16	yes	complemented
21	C₂×#121×C₆₇	—	2	8	no	—
22	C₂×#121×C₆₇	C₂²	1	8	yes	—
23	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
28	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
29	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
30	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
31	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
32	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
33	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
34	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
35	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
36	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
37	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
38	#394×C₆₇	C₂²	1	16	yes	complemented
39	#393×C₆₇	C₂²	1	16	yes	complemented
40	#390×C₆₇	C₂²	1	8	yes	complemented
41	#390×C₆₇	C₂²	1	8	yes	complemented
42	#388×C₆₇	C₂²	1	16	yes	complemented
43	#380×C₆₇	C₂²	1	8	yes	complemented
44	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	complemented
45	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	complemented
46	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
48	C₂×#95×C₆₇	C₂²	1	2	yes	—
49	C₂×#95×C₆₇	C₂²	1	2	yes	—
50	C₂⁵×C₆₇	C₂³	1	2	yes	—
51	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
52	C₂²×Q₈×C₆₇	—	2	8	no	—
53	C₂²×Q₈×C₆₇	—	2	8	no	—
54	C₂²×Q₈×C₆₇	—	4	4	no	—
55	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
56	C₂²×D₄×C₆₇	—	2	8	no	—
57	C₂²×D₄×C₆₇	—	2	8	no	—
58	C₂²×D₄×C₆₇	—	2	4	no	—
59	C₂²×D₄×C₆₇	—	2	4	no	—
60	C₂²×D₄×C₆₇	—	2	4	no	—
61	C₂²×D₄×C₆₇	—	4	4	no	—
62	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
63	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
64	C₂³×C₄×C₆₇	—	2	4	no	—
65	C₂³×C₄×C₆₇	—	2	4	no	—
66	C₂³×C₄×C₆₇	—	2	4	no	—
67	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
68	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
69	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
71	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
75	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
77	C₄⋊Q₈×C₆₇	—	2	32	no	—
78	#124×C₆₇	—	2	64	no	—
79	#121×C₆₇	—	2	32	no	—
80	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
81	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
82	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
83	C₂×C₄⋊C₄×C₆₇	D₄	1	8	yes	—
84	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
85	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
86	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
88	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
89	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
90	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
91	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
92	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
93	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
94	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
95	C₂×C₂²⋊C₄×C₆₇	D₄	1	8	yes	complemented
96	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
97	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
98	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
99	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
100	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
101	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
102	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
103	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
104	C₂×C₄²×C₆₇	—	2	8	no	—
105	C₂×C₄²×C₆₇	—	2	4	no	—
106	C₂×C₄²×C₆₇	—	2	4	no	—
107	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
108	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
109	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
110	#95×C₆₇	C₂³	1	4	yes	—
111	#95×C₆₇	C₂³	1	4	yes	—
112	#95×C₆₇	C₂³	1	8	yes	—
113	#95×C₆₇	C₂³	1	8	yes	—
114	C₂⁴×C₆₇	—	2	2	no	—
115	C₂⁴×C₆₇	—	2	2	no	—
116	C₂⁴×C₆₇	—	2	2	no	—
117	C₂⁴×C₆₇	—	2	2	no	—
118	C₂⁴×C₆₇	—	2	2	no	—
119	C₂⁴×C₆₇	—	2	2	no	—
120	C₂⁴×C₆₇	—	2	4	no	—
121	C₂⁴×C₆₇	—	2	4	no	—
122	C₂⁴×C₆₇	—	2	4	no	—
123	C₂⁴×C₆₇	—	2	4	no	—
124	C₂⁴×C₆₇	—	2	4	no	—
125	C₂⁴×C₆₇	—	2	4	no	—
126	C₂⁴×C₆₇	—	2	4	no	—
127	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
128	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
129	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
130	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
131	C₂×Q₈×C₆₇	—	2	16	no	—
132	C₂×Q₈×C₆₇	—	2	16	no	—
133	C₂×Q₈×C₆₇	—	4	8	no	—
134	C₂×Q₈×C₆₇	—	4	32	no	—
135	C₂×D₄×C₆₇	—	2	16	no	—
136	C₂×D₄×C₆₇	—	2	16	no	—
137	C₂×D₄×C₆₇	—	2	8	no	—
138	C₂×D₄×C₆₇	—	2	8	no	—
139	C₂×D₄×C₆₇	—	2	8	no	—
140	C₂×D₄×C₆₇	—	4	8	no	—
141	C₂×D₄×C₆₇	—	4	32	no	—
142	C₂×D₄×C₆₇	—	4	16	no	—
143	C₂²×C₄×C₆₇	—	2	2	no	—
144	C₂²×C₄×C₆₇	—	2	4	no	—
145	C₂²×C₄×C₆₇	—	2	2	no	—
146	C₂²×C₄×C₆₇	—	2	2	no	—
147	C₂²×C₄×C₆₇	—	2	4	no	—
148	C₂²×C₄×C₆₇	—	2	2	no	—
149	C₂²×C₄×C₆₇	—	2	2	no	—
150	C₂²×C₄×C₆₇	—	2	2	no	—
151	C₂²×C₄×C₆₇	—	2	4	no	—
152	C₂²×C₄×C₆₇	—	2	2	no	—
153	C₂²×C₄×C₆₇	—	2	4	no	—
154	C₂²×C₄×C₆₇	—	2	4	no	—
155	C₂²×C₄×C₆₇	—	2	2	no	—
156	C₂²×C₄×C₆₇	—	2	4	no	—
157	C₂²×C₄×C₆₇	—	2	2	no	—
158	C₂²×C₄×C₆₇	—	2	4	no	—
159	C₂²×C₄×C₆₇	—	2	2	no	—
160	C₂²×C₄×C₆₇	—	2	2	no	—
161	C₂²×C₄×C₆₇	—	2	2	no	—
162	C₂²×C₄×C₆₇	—	2	4	no	—
163	C₂²×C₄×C₆₇	—	2	8	no	—
164	C₂²×C₄×C₆₇	—	2	8	no	—
165	C₂²×C₄×C₆₇	—	2	8	no	—
166	C₂²×C₄×C₆₇	—	4	16	no	—
167	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
168	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
169	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
170	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
171	C₂²×C₄×C₆₇	#41	1	4	yes	—
172	C₂²×C₄×C₆₇	#41	1	4	yes	—
173	C₂²×C₄×C₆₇	#41	1	4	yes	—
174	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
175	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
177	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
179	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
181	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
183	C₄⋊C₄×C₆₇	—	2	16	no	—
184	C₄⋊C₄×C₆₇	—	2	8	no	—
185	C₄⋊C₄×C₆₇	—	2	8	no	—
186	C₄⋊C₄×C₆₇	C₂×D₄	1	16	yes	—
187	C₂²⋊C₄×C₆₇	—	2	16	no	—
188	C₂²⋊C₄×C₆₇	—	2	16	no	—
189	C₂²⋊C₄×C₆₇	—	2	16	no	—
190	C₂²⋊C₄×C₆₇	—	2	16	no	—
191	C₂²⋊C₄×C₆₇	—	2	8	no	—
192	C₂²⋊C₄×C₆₇	—	2	16	no	—
193	C₂²⋊C₄×C₆₇	—	4	8	no	—
194	C₂²⋊C₄×C₆₇	C₂×D₄	1	16	yes	complemented
195	C₄²×C₆₇	—	2	16	no	—
196	C₄²×C₆₇	—	2	8	no	—
197	C₄²×C₆₇	—	2	8	no	—
198	C₂³×C₆₇	—	2	8	no	—
199	C₂³×C₆₇	—	2	8	no	—
200	C₂³×C₆₇	—	2	8	no	—
201	C₂³×C₆₇	—	2	2	no	—
202	C₂³×C₆₇	—	2	2	no	—
203	C₂³×C₆₇	—	2	2	no	—
204	C₂³×C₆₇	—	2	2	no	—
205	C₂³×C₆₇	—	2	2	no	—
206	C₂³×C₆₇	—	2	2	no	—
207	C₂³×C₆₇	—	2	4	no	—
208	C₂³×C₆₇	—	2	4	no	—
209	C₂³×C₆₇	—	2	4	no	—
210	C₂³×C₆₇	—	2	4	no	—
211	C₂³×C₆₇	—	2	4	no	—
212	C₂³×C₆₇	—	2	4	no	—
213	C₂³×C₆₇	—	2	4	no	—
214	C₂³×C₆₇	—	4	4	no	—
215	C₂³×C₆₇	—	4	4	no	—
216	C₂³×C₆₇	—	4	4	no	—
217	C₂³×C₆₇	—	4	4	no	—
218	C₂³×C₆₇	—	4	8	no	—
219	C₂³×C₆₇	—	4	8	no	—
220	C₂³×C₆₇	—	4	8	no	—
221	C₂³×C₆₇	—	4	8	no	—
222	C₂³×C₆₇	—	4	8	no	—
223	C₂³×C₆₇	—	4	8	no	—
224	C₂³×C₆₇	—	4	8	no	—
225	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
226	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
227	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
228	C₂³×C₆₇	C₂×#41	1	2	yes	—
229	C₂³×C₆₇	C₂×#41	1	2	yes	—
230	C₂³×C₆₇	C₂×#41	1	2	yes	—
231	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
232	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
233	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
234	Q₈×C₆₇	—	4	32	no	—
235	D₄×C₆₇	—	4	32	no	—
236	D₄×C₆₇	—	4	16	no	—
237	C₂×C₄×C₆₇	—	2	16	no	—
238	C₂×C₄×C₆₇	—	2	8	no	—
239	C₂×C₄×C₆₇	—	2	8	no	—
240	C₂×C₄×C₆₇	—	2	8	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	4	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	2	2	no	—
245	C₂×C₄×C₆₇	—	2	4	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	2	no	—
249	C₂×C₄×C₆₇	—	2	4	no	—
250	C₂×C₄×C₆₇	—	2	2	no	—
251	C₂×C₄×C₆₇	—	2	4	no	—
252	C₂×C₄×C₆₇	—	2	4	no	—
253	C₂×C₄×C₆₇	—	2	2	no	—
254	C₂×C₄×C₆₇	—	2	4	no	—
255	C₂×C₄×C₆₇	—	2	2	no	—
256	C₂×C₄×C₆₇	—	2	4	no	—
257	C₂×C₄×C₆₇	—	2	2	no	—
258	C₂×C₄×C₆₇	—	2	2	no	—
259	C₂×C₄×C₆₇	—	2	2	no	—
260	C₂×C₄×C₆₇	—	2	4	no	—
261	C₂×C₄×C₆₇	—	4	4	no	—
262	C₂×C₄×C₆₇	—	4	8	no	—
263	C₂×C₄×C₆₇	—	4	8	no	—
264	C₂×C₄×C₆₇	—	4	4	no	—
265	C₂×C₄×C₆₇	—	4	16	no	—
266	C₂×C₄×C₆₇	#124	1	16	yes	complemented
267	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
268	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
269	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
270	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
271	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
272	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
273	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
274	C₂²×C₆₇	—	2	4	no	—
275	C₂²×C₆₇	—	2	4	no	—
276	C₂²×C₆₇	—	2	4	no	—
277	C₂²×C₆₇	—	4	2	no	—
278	C₂²×C₆₇	—	4	2	no	—
279	C₂²×C₆₇	—	4	2	no	—
280	C₂²×C₆₇	—	4	2	no	—
281	C₂²×C₆₇	—	4	16	no	—
282	C₂²×C₆₇	—	4	4	no	—
283	C₂²×C₆₇	—	4	4	no	—
284	C₂²×C₆₇	—	4	4	no	—
285	C₂²×C₆₇	—	4	4	no	—
286	C₂²×C₆₇	—	4	4	no	—
287	C₂²×C₆₇	—	4	4	no	—
288	C₂²×C₆₇	—	4	4	no	—
289	C₂²×C₆₇	—	8	16	no	—
290	C₂²×C₆₇	C₂×#124	1	4	yes	—
291	C₂²×C₆₇	#532	1	4	yes	—
292	C₂²×C₆₇	#542	1	4	yes	—
293	C₂²×C₆₇	D₄²	1	4	yes	—
294	C₂²×C₆₇	#547	1	4	yes	—
295	C₂²×C₆₇	#548	1	4	yes	—
296	C₂²×C₆₇	#555	1	4	yes	—
297	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
298	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
299	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
300	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
301	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
302	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
303	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
304	C₄×C₆₇	—	2	8	no	—
305	C₄×C₆₇	—	2	4	no	—
306	C₄×C₆₇	—	2	4	no	—
307	C₄×C₆₇	—	2	4	no	—
308	C₄×C₆₇	—	4	2	no	—
309	C₄×C₆₇	—	4	4	no	—
310	C₄×C₆₇	—	4	4	no	—
311	C₄×C₆₇	—	4	2	no	—
312	C₄×C₆₇	C₂×#124	1	8	yes	complemented
313	C₂×#2491	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
314	C₂×C₆₇	—	4	4	no	—
315	C₂×C₆₇	—	8	4	no	—
316	C₂×C₆₇	#2491	1	8	yes	direct product factor
317	C₂×C₆₇	C₂²×#124	1	1	yes	characteristic
318	C₂×C₆₇	C₂×#532	1	1	yes	characteristic
319	C₂×C₆₇	C₂×#542	1	1	yes	characteristic
320	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
321	C₂×C₆₇	C₂×#547	1	1	yes	characteristic
322	C₂×C₆₇	C₂×#548	1	1	yes	characteristic
323	C₂×C₆₇	C₂×#555	1	1	yes	characteristic
324	C₂²×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
325	C₂²×#124	C₂×C₆₇	1	2	yes	complemented
326	C₂²×#121	C₂×C₆₇	1	1	yes	characteristic
327	#2491	C₂×C₆₇	1	16	yes	direct product factor
328	C₂×#394	C₂×C₆₇	1	2	yes	complemented
329	C₂×#393	C₂×C₆₇	1	2	yes	complemented
330	C₂×#390	C₂×C₆₇	1	1	yes	characteristic, complemented
331	C₂×#390	C₂×C₆₇	1	1	yes	characteristic, complemented
332	C₂×#388	C₂×C₆₇	1	2	yes	complemented
333	C₂×#380	C₂×C₆₇	1	1	yes	characteristic, complemented
334	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
335	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
336	C₆₇	C₂×#2491	1	1	yes	characteristic, direct product factor, Sylow
337	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
338	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
339	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
340	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
341	C₂×C₄⋊Q₈	—	2	8	no	—
342	C₂×C₄⋊Q₈	C₂²×C₆₇	1	8	yes	complemented
343	C₂×#124	—	2	16	no	—
344	C₂×#124	C₂²×C₆₇	1	16	yes	complemented
345	C₂×#121	—	2	8	no	—
346	C₂×#121	C₂²×C₆₇	1	8	yes	—
347	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
348	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
349	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
350	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
351	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
352	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
353	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
354	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
355	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
356	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
357	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
358	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
359	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
360	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
361	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
362	#394	C₂²×C₆₇	1	16	yes	complemented
363	#393	C₂²×C₆₇	1	16	yes	complemented
364	#390	C₂²×C₆₇	1	8	yes	complemented
365	#390	C₂²×C₆₇	1	8	yes	complemented
366	#388	C₂²×C₆₇	1	16	yes	complemented
367	#380	C₂²×C₆₇	1	8	yes	complemented
368	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	complemented
369	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	complemented
370	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
371	C₂×#95	C₂²×C₆₇	1	2	yes	—
372	C₂×#95	C₂²×C₆₇	1	2	yes	—
373	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
374	C₂⁵	C₂³×C₆₇	1	2	yes	—
375	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
376	C₂²×Q₈	—	2	8	no	—
377	C₂²×Q₈	—	2	8	no	—
378	C₂²×Q₈	—	4	4	no	—
379	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
380	C₂²×D₄	—	2	8	no	—
381	C₂²×D₄	—	2	8	no	—
382	C₂²×D₄	—	2	4	no	—
383	C₂²×D₄	—	2	4	no	—
384	C₂²×D₄	—	2	4	no	—
385	C₂²×D₄	—	4	4	no	—
386	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
387	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
388	C₂³×C₄	—	2	4	no	—
389	C₂³×C₄	—	2	4	no	—
390	C₂³×C₄	—	2	4	no	—
391	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
392	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
393	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
394	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
395	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
396	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
397	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
398	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
399	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
400	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
401	C₄⋊Q₈	—	2	32	no	—
402	#124	—	2	64	no	—
403	#121	—	2	32	no	—
404	C₂×C₄⋊C₄	—	2	8	no	—
405	C₂×C₄⋊C₄	—	2	4	no	—
406	C₂×C₄⋊C₄	—	2	4	no	—
407	C₂×C₄⋊C₄	D₄×C₆₇	1	8	yes	—
408	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
409	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
410	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
411	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
412	C₂×C₂²⋊C₄	—	2	8	no	—
413	C₂×C₂²⋊C₄	—	2	8	no	—
414	C₂×C₂²⋊C₄	—	2	8	no	—
415	C₂×C₂²⋊C₄	—	2	8	no	—
416	C₂×C₂²⋊C₄	—	2	4	no	—
417	C₂×C₂²⋊C₄	—	2	8	no	—
418	C₂×C₂²⋊C₄	—	4	4	no	—
419	C₂×C₂²⋊C₄	D₄×C₆₇	1	8	yes	complemented
420	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
421	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
422	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
423	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
424	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
425	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
426	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
427	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
428	C₂×C₄²	—	2	8	no	—
429	C₂×C₄²	—	2	4	no	—
430	C₂×C₄²	—	2	4	no	—
431	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
432	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
433	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
434	#95	C₂³×C₆₇	1	4	yes	—
435	#95	C₂³×C₆₇	1	4	yes	—
436	#95	C₂³×C₆₇	1	8	yes	—
437	#95	C₂³×C₆₇	1	8	yes	—
438	C₂⁴	—	2	2	no	—
439	C₂⁴	—	2	2	no	—
440	C₂⁴	—	2	2	no	—
441	C₂⁴	—	2	2	no	—
442	C₂⁴	—	2	2	no	—
443	C₂⁴	—	2	2	no	—
444	C₂⁴	—	2	4	no	—
445	C₂⁴	—	2	4	no	—
446	C₂⁴	—	2	4	no	—
447	C₂⁴	—	2	4	no	—
448	C₂⁴	—	2	4	no	—
449	C₂⁴	—	2	4	no	—
450	C₂⁴	—	2	4	no	—
451	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
452	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
453	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
454	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
455	C₂×Q₈	—	2	16	no	—
456	C₂×Q₈	—	2	16	no	—
457	C₂×Q₈	—	4	8	no	—
458	C₂×Q₈	—	4	32	no	—
459	C₂×D₄	—	2	16	no	—
460	C₂×D₄	—	2	16	no	—
461	C₂×D₄	—	2	8	no	—
462	C₂×D₄	—	2	8	no	—
463	C₂×D₄	—	2	8	no	—
464	C₂×D₄	—	4	8	no	—
465	C₂×D₄	—	4	32	no	—
466	C₂×D₄	—	4	16	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	4	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	2	4	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	2	2	no	—
474	C₂²×C₄	—	2	2	no	—
475	C₂²×C₄	—	2	4	no	—
476	C₂²×C₄	—	2	2	no	—
477	C₂²×C₄	—	2	4	no	—
478	C₂²×C₄	—	2	4	no	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	2	4	no	—
481	C₂²×C₄	—	2	2	no	—
482	C₂²×C₄	—	2	4	no	—
483	C₂²×C₄	—	2	2	no	—
484	C₂²×C₄	—	2	2	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	2	4	no	—
487	C₂²×C₄	—	2	8	no	—
488	C₂²×C₄	—	2	8	no	—
489	C₂²×C₄	—	2	8	no	—
490	C₂²×C₄	—	4	16	no	—
491	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
492	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
493	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
494	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
495	C₂²×C₄	#41×C₆₇	1	4	yes	—
496	C₂²×C₄	#41×C₆₇	1	4	yes	—
497	C₂²×C₄	#41×C₆₇	1	4	yes	—
498	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
499	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
500	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
501	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
502	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
503	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
504	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
505	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
506	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
507	C₄⋊C₄	—	2	16	no	—
508	C₄⋊C₄	—	2	8	no	—
509	C₄⋊C₄	—	2	8	no	—
510	C₄⋊C₄	C₂×D₄×C₆₇	1	16	yes	—
511	C₂²⋊C₄	—	2	16	no	—
512	C₂²⋊C₄	—	2	16	no	—
513	C₂²⋊C₄	—	2	16	no	—
514	C₂²⋊C₄	—	2	16	no	—
515	C₂²⋊C₄	—	2	8	no	—
516	C₂²⋊C₄	—	2	16	no	—
517	C₂²⋊C₄	—	4	8	no	—
518	C₂²⋊C₄	C₂×D₄×C₆₇	1	16	yes	complemented
519	C₄²	—	2	16	no	—
520	C₄²	—	2	8	no	—
521	C₄²	—	2	8	no	—
522	C₂³	—	2	8	no	—
523	C₂³	—	2	8	no	—
524	C₂³	—	2	8	no	—
525	C₂³	—	2	2	no	—
526	C₂³	—	2	2	no	—
527	C₂³	—	2	2	no	—
528	C₂³	—	2	2	no	—
529	C₂³	—	2	2	no	—
530	C₂³	—	2	2	no	—
531	C₂³	—	2	4	no	—
532	C₂³	—	2	4	no	—
533	C₂³	—	2	4	no	—
534	C₂³	—	2	4	no	—
535	C₂³	—	2	4	no	—
536	C₂³	—	2	4	no	—
537	C₂³	—	2	4	no	—
538	C₂³	—	4	4	no	—
539	C₂³	—	4	4	no	—
540	C₂³	—	4	4	no	—
541	C₂³	—	4	4	no	—
542	C₂³	—	4	8	no	—
543	C₂³	—	4	8	no	—
544	C₂³	—	4	8	no	—
545	C₂³	—	4	8	no	—
546	C₂³	—	4	8	no	—
547	C₂³	—	4	8	no	—
548	C₂³	—	4	8	no	—
549	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
550	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
551	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
552	C₂³	C₂×#41×C₆₇	1	2	yes	—
553	C₂³	C₂×#41×C₆₇	1	2	yes	—
554	C₂³	C₂×#41×C₆₇	1	2	yes	—
555	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
556	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
557	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
558	Q₈	—	4	32	no	—
559	D₄	—	4	32	no	—
560	D₄	—	4	16	no	—
561	C₂×C₄	—	2	16	no	—
562	C₂×C₄	—	2	8	no	—
563	C₂×C₄	—	2	8	no	—
564	C₂×C₄	—	2	8	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	4	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	4	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	4	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	4	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	4	no	—
585	C₂×C₄	—	4	4	no	—
586	C₂×C₄	—	4	8	no	—
587	C₂×C₄	—	4	8	no	—
588	C₂×C₄	—	4	4	no	—
589	C₂×C₄	—	4	16	no	—
590	C₂×C₄	#124×C₆₇	1	16	yes	complemented
591	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
592	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
593	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
594	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
595	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
596	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
597	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
598	C₂²	—	2	4	no	—
599	C₂²	—	2	4	no	—
600	C₂²	—	2	4	no	—
601	C₂²	—	4	2	no	—
602	C₂²	—	4	2	no	—
603	C₂²	—	4	2	no	—
604	C₂²	—	4	2	no	—
605	C₂²	—	4	16	no	—
606	C₂²	—	4	4	no	—
607	C₂²	—	4	4	no	—
608	C₂²	—	4	4	no	—
609	C₂²	—	4	4	no	—
610	C₂²	—	4	4	no	—
611	C₂²	—	4	4	no	—
612	C₂²	—	4	4	no	—
613	C₂²	—	8	16	no	—
614	C₂²	C₂×#124×C₆₇	1	4	yes	—
615	C₂²	#532×C₆₇	1	4	yes	—
616	C₂²	#542×C₆₇	1	4	yes	—
617	C₂²	D₄²×C₆₇	1	4	yes	—
618	C₂²	#547×C₆₇	1	4	yes	—
619	C₂²	#548×C₆₇	1	4	yes	—
620	C₂²	#555×C₆₇	1	4	yes	—
621	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
622	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
623	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
624	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
625	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
626	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
627	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
628	C₄	—	2	8	no	—
629	C₄	—	2	4	no	—
630	C₄	—	2	4	no	—
631	C₄	—	2	4	no	—
632	C₄	—	4	2	no	—
633	C₄	—	4	4	no	—
634	C₄	—	4	4	no	—
635	C₄	—	4	2	no	—
636	C₄	C₂×#124×C₆₇	1	8	yes	complemented
637	C₂	—	4	4	no	—
638	C₂	—	8	4	no	—
639	C₂	C₆₇×#2491	1	8	yes	direct product factor
640	C₂	C₂²×#124×C₆₇	1	1	yes	characteristic
641	C₂	C₂×#532×C₆₇	1	1	yes	characteristic
642	C₂	C₂×#542×C₆₇	1	1	yes	characteristic
643	C₂	C₂×D₄²×C₆₇	1	1	yes	characteristic
644	C₂	C₂×#547×C₆₇	1	1	yes	characteristic
645	C₂	C₂×#548×C₆₇	1	1	yes	characteristic
646	C₂	C₂×#555×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂²×#121×C₆₇	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×C₄⋊Q₈×C₆₇	C₂	semidirect
2. C₂²×#124×C₆₇	C₂	semidirect
4. C₆₇×#2491	C₂	direct (N × H)
5. C₂×#394×C₆₇	C₂	semidirect
6. C₂×#393×C₆₇	C₂	semidirect
7. C₂×#390×C₆₇	C₂	semidirect
8. C₂×#390×C₆₇	C₂	semidirect
9. C₂×#388×C₆₇	C₂	semidirect
10. C₂×#380×C₆₇	C₂	semidirect
11. C₂×C₄×C₄⋊C₄×C₆₇	C₂	semidirect
12. C₂×C₄×C₂²⋊C₄×C₆₇	C₂	semidirect
18. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
20. C₂×#124×C₆₇	C₂²	semidirect
38. #394×C₆₇	C₂²	semidirect
39. #393×C₆₇	C₂²	semidirect
40. #390×C₆₇	C₂²	semidirect
41. #390×C₆₇	C₂²	semidirect
42. #388×C₆₇	C₂²	semidirect
43. #380×C₆₇	C₂²	semidirect
44. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
45. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
95. C₂×C₂²⋊C₄×C₆₇	D₄	semidirect
194. C₂²⋊C₄×C₆₇	C₂×D₄	semidirect
266. C₂×C₄×C₆₇	#124	semidirect
312. C₄×C₆₇	C₂×#124	semidirect
313. C₂×#2491	C₆₇	direct (N × H)
316. C₂×C₆₇	#2491	direct (N × H)
324. C₂²×C₄⋊Q₈	C₂×C₆₇	semidirect
325. C₂²×#124	C₂×C₆₇	semidirect
327. #2491	C₂×C₆₇	direct (N × H)
328. C₂×#394	C₂×C₆₇	semidirect
329. C₂×#393	C₂×C₆₇	semidirect
330. C₂×#390	C₂×C₆₇	semidirect
331. C₂×#390	C₂×C₆₇	semidirect
332. C₂×#388	C₂×C₆₇	semidirect
333. C₂×#380	C₂×C₆₇	semidirect
334. C₂×C₄×C₄⋊C₄	C₂×C₆₇	semidirect
335. C₂×C₄×C₂²⋊C₄	C₂×C₆₇	semidirect
336. C₆₇	C₂×#2491	direct (N × H)
342. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
344. C₂×#124	C₂²×C₆₇	semidirect
362. #394	C₂²×C₆₇	semidirect
363. #393	C₂²×C₆₇	semidirect
364. #390	C₂²×C₆₇	semidirect
365. #390	C₂²×C₆₇	semidirect
366. #388	C₂²×C₆₇	semidirect
367. #380	C₂²×C₆₇	semidirect
368. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
369. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
419. C₂×C₂²⋊C₄	D₄×C₆₇	semidirect
518. C₂²⋊C₄	C₂×D₄×C₆₇	semidirect
590. C₂×C₄	#124×C₆₇	semidirect
636. C₄	C₂×#124×C₆₇	semidirect
639. C₂	C₆₇×#2491	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₄⋊Q₈×C₆₇	2
C₂²×#124×C₆₇	2
C₂²×#121×C₆₇	2
C₆₇×#2491	2
C₂×#394×C₆₇	2
C₂×#393×C₆₇	2
C₂×#390×C₆₇	2
C₂×#388×C₆₇	2
C₂×#380×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₂×C₄×C₂²⋊C₄×C₆₇	2
C₂×#2491	67

Subgroups — C2×C67×#2491 #30714010

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2491 #30714010