Subgroups — C₂×C₆₇×#2814 #30715902

Order 17152.

2674 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#123×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2814	C₂	1	16	yes	maximal, direct product factor
4	C₂×#398×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#398×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#397×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#397×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#396×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂×#395×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#394×C₆₇	C₂	1	2	yes	maximal
11	C₂×#393×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#380×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#123×C₆₇	—	2	8	no	—
17	C₂×#123×C₆₇	C₂²	1	8	yes	—
18	C₂×C₂²⋊Q₈×C₆₇	—	2	8	no	—
19	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	8	yes	complemented
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
25	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
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	1	yes	characteristic
30	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
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	#398×C₆₇	C₂²	1	16	yes	complemented
35	#398×C₆₇	C₂²	1	8	yes	complemented
36	#397×C₆₇	C₂²	1	16	yes	complemented
37	#397×C₆₇	C₂²	1	8	yes	complemented
38	#396×C₆₇	C₂²	1	16	yes	complemented
39	#395×C₆₇	C₂²	1	8	yes	complemented
40	#394×C₆₇	C₂²	1	16	yes	—
41	#393×C₆₇	C₂²	1	8	yes	complemented
42	#380×C₆₇	C₂²	1	8	yes	complemented
43	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×#95×C₆₇	C₂²	1	2	yes	—
46	C₂×#95×C₆₇	C₂²	1	2	yes	—
47	C₂×#95×C₆₇	C₂²	1	2	yes	—
48	C₂×#95×C₆₇	C₂²	1	2	yes	—
49	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
50	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×Q₈×C₆₇	—	2	4	no	—
52	C₂²×Q₈×C₆₇	—	4	4	no	—
53	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
54	C₂²×D₄×C₆₇	—	2	4	no	—
55	C₂²×D₄×C₆₇	—	4	4	no	—
56	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
57	C₂³×C₄×C₆₇	—	4	4	no	—
58	C₂³×C₄×C₆₇	D₄	1	8	yes	—
59	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
61	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
62	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
63	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
64	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
69	#123×C₆₇	—	2	32	no	—
70	C₂²⋊Q₈×C₆₇	—	2	32	no	—
71	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
72	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
73	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
75	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
76	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
77	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
83	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
84	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
85	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
86	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
87	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
88	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
89	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
90	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
91	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
92	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
93	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
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	#95×C₆₇	C₂³	1	4	yes	—
100	#95×C₆₇	C₂³	1	4	yes	—
101	#95×C₆₇	C₂³	1	8	yes	—
102	#95×C₆₇	C₂³	1	8	yes	—
103	#95×C₆₇	C₂³	1	8	yes	—
104	#95×C₆₇	C₂³	1	8	yes	—
105	C₂⁴×C₆₇	—	2	2	no	—
106	C₂⁴×C₆₇	—	2	4	no	—
107	C₂⁴×C₆₇	—	2	4	no	—
108	C₂⁴×C₆₇	—	2	2	no	—
109	C₂⁴×C₆₇	—	2	2	no	—
110	C₂⁴×C₆₇	—	2	2	no	—
111	C₂⁴×C₆₇	—	2	4	no	—
112	C₂⁴×C₆₇	—	2	4	no	—
113	C₂⁴×C₆₇	—	2	2	no	—
114	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
115	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
116	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
117	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
118	C₂×Q₈×C₆₇	—	2	8	no	—
119	C₂×Q₈×C₆₇	—	4	8	no	—
120	C₂×Q₈×C₆₇	—	4	16	no	—
121	C₂×D₄×C₆₇	—	2	8	no	—
122	C₂×D₄×C₆₇	—	4	8	no	—
123	C₂×D₄×C₆₇	—	4	16	no	—
124	C₂²×C₄×C₆₇	—	2	4	no	—
125	C₂²×C₄×C₆₇	—	2	4	no	—
126	C₂²×C₄×C₆₇	—	2	2	no	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	—	2	4	no	—
129	C₂²×C₄×C₆₇	—	2	4	no	—
130	C₂²×C₄×C₆₇	—	2	2	no	—
131	C₂²×C₄×C₆₇	—	2	4	no	—
132	C₂²×C₄×C₆₇	—	2	4	no	—
133	C₂²×C₄×C₆₇	—	2	4	no	—
134	C₂²×C₄×C₆₇	—	2	4	no	—
135	C₂²×C₄×C₆₇	—	2	4	no	—
136	C₂²×C₄×C₆₇	—	2	2	no	—
137	C₂²×C₄×C₆₇	—	2	4	no	—
138	C₂²×C₄×C₆₇	—	2	4	no	—
139	C₂²×C₄×C₆₇	—	2	4	no	—
140	C₂²×C₄×C₆₇	—	2	4	no	—
141	C₂²×C₄×C₆₇	—	2	2	no	—
142	C₂²×C₄×C₆₇	—	2	4	no	—
143	C₂²×C₄×C₆₇	—	2	4	no	—
144	C₂²×C₄×C₆₇	—	2	4	no	—
145	C₂²×C₄×C₆₇	—	4	16	no	—
146	C₂²×C₄×C₆₇	—	4	8	no	—
147	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
148	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
149	C₂²×C₄×C₆₇	C₂×D₄	1	16	yes	—
150	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
154	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
155	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
156	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
158	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
159	C₄⋊C₄×C₆₇	—	2	8	no	—
160	C₄⋊C₄×C₆₇	—	2	16	no	—
161	C₄⋊C₄×C₆₇	—	2	8	no	—
162	C₄⋊C₄×C₆₇	—	2	8	no	—
163	C₄⋊C₄×C₆₇	—	4	8	no	—
164	C₄⋊C₄×C₆₇	—	4	8	no	—
165	C₄⋊C₄×C₆₇	—	4	8	no	—
166	C₂²⋊C₄×C₆₇	—	2	16	no	—
167	C₂²⋊C₄×C₆₇	—	2	8	no	—
168	C₂²⋊C₄×C₆₇	—	2	8	no	—
169	C₂²⋊C₄×C₆₇	—	2	8	no	—
170	C₂²⋊C₄×C₆₇	—	4	8	no	—
171	C₂²⋊C₄×C₆₇	—	4	8	no	—
172	C₂²⋊C₄×C₆₇	—	4	8	no	—
173	C₂³×C₆₇	—	2	8	no	—
174	C₂³×C₆₇	—	2	8	no	—
175	C₂³×C₆₇	—	2	8	no	—
176	C₂³×C₆₇	—	2	2	no	—
177	C₂³×C₆₇	—	2	4	no	—
178	C₂³×C₆₇	—	2	4	no	—
179	C₂³×C₆₇	—	2	2	no	—
180	C₂³×C₆₇	—	2	2	no	—
181	C₂³×C₆₇	—	2	2	no	—
182	C₂³×C₆₇	—	2	4	no	—
183	C₂³×C₆₇	—	2	4	no	—
184	C₂³×C₆₇	—	2	2	no	—
185	C₂³×C₆₇	—	4	8	no	—
186	C₂³×C₆₇	—	4	8	no	—
187	C₂³×C₆₇	—	4	4	no	—
188	C₂³×C₆₇	—	4	4	no	—
189	C₂³×C₆₇	—	4	4	no	—
190	C₂³×C₆₇	—	4	8	no	—
191	C₂³×C₆₇	—	4	8	no	—
192	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
193	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
194	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
195	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
196	C₂³×C₆₇	2₋¹⁺⁴	1	4	yes	—
197	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
198	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
199	Q₈×C₆₇	—	4	16	no	—
200	D₄×C₆₇	—	4	16	no	—
201	C₂×C₄×C₆₇	—	2	8	no	—
202	C₂×C₄×C₆₇	—	2	8	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	4	no	—
205	C₂×C₄×C₆₇	—	2	2	no	—
206	C₂×C₄×C₆₇	—	2	2	no	—
207	C₂×C₄×C₆₇	—	2	4	no	—
208	C₂×C₄×C₆₇	—	2	4	no	—
209	C₂×C₄×C₆₇	—	2	2	no	—
210	C₂×C₄×C₆₇	—	2	4	no	—
211	C₂×C₄×C₆₇	—	2	4	no	—
212	C₂×C₄×C₆₇	—	2	4	no	—
213	C₂×C₄×C₆₇	—	2	4	no	—
214	C₂×C₄×C₆₇	—	2	4	no	—
215	C₂×C₄×C₆₇	—	2	2	no	—
216	C₂×C₄×C₆₇	—	2	4	no	—
217	C₂×C₄×C₆₇	—	2	4	no	—
218	C₂×C₄×C₆₇	—	2	4	no	—
219	C₂×C₄×C₆₇	—	2	4	no	—
220	C₂×C₄×C₆₇	—	2	2	no	—
221	C₂×C₄×C₆₇	—	2	4	no	—
222	C₂×C₄×C₆₇	—	2	4	no	—
223	C₂×C₄×C₆₇	—	2	4	no	—
224	C₂×C₄×C₆₇	—	4	16	no	—
225	C₂×C₄×C₆₇	—	4	4	no	—
226	C₂×C₄×C₆₇	—	4	8	no	—
227	C₂×C₄×C₆₇	—	4	4	no	—
228	C₂×C₄×C₆₇	—	4	4	no	—
229	C₂×C₄×C₆₇	—	4	8	no	—
230	C₂×C₄×C₆₇	—	4	8	no	—
231	C₂×C₄×C₆₇	—	4	8	no	—
232	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
233	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
234	C₂²×C₆₇	—	2	4	no	—
235	C₂²×C₆₇	—	2	4	no	—
236	C₂²×C₆₇	—	2	4	no	—
237	C₂²×C₆₇	—	4	4	no	—
238	C₂²×C₆₇	—	4	4	no	—
239	C₂²×C₆₇	—	4	16	no	—
240	C₂²×C₆₇	—	4	2	no	—
241	C₂²×C₆₇	—	4	2	no	—
242	C₂²×C₆₇	—	4	2	no	—
243	C₂²×C₆₇	—	4	4	no	—
244	C₂²×C₆₇	—	4	4	no	—
245	C₂²×C₆₇	—	8	8	no	—
246	C₂²×C₆₇	#534	1	4	yes	—
247	C₂²×C₆₇	#536	1	4	yes	—
248	C₂²×C₆₇	#537	1	4	yes	—
249	C₂²×C₆₇	#562	1	8	yes	—
250	C₂²×C₆₇	#563	1	8	yes	—
251	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
252	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
253	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
254	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
255	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
256	C₄×C₆₇	—	2	4	no	—
257	C₄×C₆₇	—	2	4	no	—
258	C₄×C₆₇	—	4	2	no	—
259	C₄×C₆₇	—	4	4	no	—
260	C₄×C₆₇	—	4	2	no	—
261	C₄×C₆₇	—	4	2	no	—
262	C₄×C₆₇	—	4	4	no	—
263	C₄×C₆₇	—	4	4	no	—
264	C₄×C₆₇	—	4	4	no	—
265	C₂×#2814	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
266	C₂×C₆₇	—	4	4	no	—
267	C₂×C₆₇	—	8	2	no	—
268	C₂×C₆₇	#2814	1	8	yes	direct product factor
269	C₂×C₆₇	C₂×#534	1	1	yes	characteristic
270	C₂×C₆₇	C₂×#536	1	1	yes	characteristic
271	C₂×C₆₇	C₂×#537	1	1	yes	characteristic
272	C₂×C₆₇	C₂×#562	1	2	yes	—
273	C₂×C₆₇	C₂×#563	1	2	yes	—
274	C₂²×#123	C₂×C₆₇	1	1	yes	characteristic
275	C₂²×C₂²⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
276	#2814	C₂×C₆₇	1	16	yes	direct product factor
277	C₂×#398	C₂×C₆₇	1	1	yes	characteristic, complemented
278	C₂×#398	C₂×C₆₇	1	2	yes	complemented
279	C₂×#397	C₂×C₆₇	1	1	yes	characteristic, complemented
280	C₂×#397	C₂×C₆₇	1	2	yes	complemented
281	C₂×#396	C₂×C₆₇	1	2	yes	complemented
282	C₂×#395	C₂×C₆₇	1	1	yes	characteristic, complemented
283	C₂×#394	C₂×C₆₇	1	2	yes	—
284	C₂×#393	C₂×C₆₇	1	1	yes	characteristic, complemented
285	C₂×#380	C₂×C₆₇	1	1	yes	characteristic, complemented
286	C₆₇	C₂×#2814	1	1	yes	characteristic, direct product factor, Sylow
287	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
288	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
289	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
290	C₂×#123	—	2	8	no	—
291	C₂×#123	C₂²×C₆₇	1	8	yes	—
292	C₂×C₂²⋊Q₈	—	2	8	no	—
293	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	8	yes	complemented
294	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
295	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
296	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
297	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
298	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
299	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
300	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
301	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
302	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
303	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
304	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
305	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
306	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
307	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
308	#398	C₂²×C₆₇	1	8	yes	complemented
309	#398	C₂²×C₆₇	1	16	yes	complemented
310	#397	C₂²×C₆₇	1	8	yes	complemented
311	#397	C₂²×C₆₇	1	16	yes	complemented
312	#396	C₂²×C₆₇	1	16	yes	complemented
313	#395	C₂²×C₆₇	1	8	yes	complemented
314	#394	C₂²×C₆₇	1	16	yes	—
315	#393	C₂²×C₆₇	1	8	yes	complemented
316	#380	C₂²×C₆₇	1	8	yes	complemented
317	C₂×#95	C₂²×C₆₇	1	2	yes	—
318	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
319	C₂×#95	C₂²×C₆₇	1	2	yes	—
320	C₂×#95	C₂²×C₆₇	1	2	yes	—
321	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
322	C₂×#95	C₂²×C₆₇	1	2	yes	—
323	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
324	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
325	C₂²×Q₈	—	2	4	no	—
326	C₂²×Q₈	—	4	4	no	—
327	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
328	C₂²×D₄	—	2	4	no	—
329	C₂²×D₄	—	4	4	no	—
330	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
331	C₂³×C₄	—	4	4	no	—
332	C₂³×C₄	D₄×C₆₇	1	8	yes	—
333	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
334	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
335	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
336	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
337	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
338	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
339	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
340	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
341	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
342	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
343	#123	—	2	32	no	—
344	C₂²⋊Q₈	—	2	32	no	—
345	C₂×C₄⋊C₄	—	2	4	no	—
346	C₂×C₄⋊C₄	—	2	8	no	—
347	C₂×C₄⋊C₄	—	2	4	no	—
348	C₂×C₄⋊C₄	—	2	4	no	—
349	C₂×C₄⋊C₄	—	4	4	no	—
350	C₂×C₄⋊C₄	—	4	4	no	—
351	C₂×C₄⋊C₄	—	4	4	no	—
352	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
353	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
354	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
355	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
356	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
357	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
358	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
359	C₂×C₂²⋊C₄	—	2	8	no	—
360	C₂×C₂²⋊C₄	—	2	4	no	—
361	C₂×C₂²⋊C₄	—	2	4	no	—
362	C₂×C₂²⋊C₄	—	2	4	no	—
363	C₂×C₂²⋊C₄	—	4	4	no	—
364	C₂×C₂²⋊C₄	—	4	4	no	—
365	C₂×C₂²⋊C₄	—	4	4	no	—
366	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
367	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
368	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
369	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
370	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
371	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
372	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
373	#95	C₂³×C₆₇	1	4	yes	—
374	#95	C₂³×C₆₇	1	4	yes	—
375	#95	C₂³×C₆₇	1	8	yes	—
376	#95	C₂³×C₆₇	1	8	yes	—
377	#95	C₂³×C₆₇	1	8	yes	—
378	#95	C₂³×C₆₇	1	8	yes	—
379	C₂⁴	—	2	2	no	—
380	C₂⁴	—	2	4	no	—
381	C₂⁴	—	2	4	no	—
382	C₂⁴	—	2	2	no	—
383	C₂⁴	—	2	2	no	—
384	C₂⁴	—	2	2	no	—
385	C₂⁴	—	2	4	no	—
386	C₂⁴	—	2	4	no	—
387	C₂⁴	—	2	2	no	—
388	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
389	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
390	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
391	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
392	C₂×Q₈	—	2	8	no	—
393	C₂×Q₈	—	4	8	no	—
394	C₂×Q₈	—	4	16	no	—
395	C₂×D₄	—	2	8	no	—
396	C₂×D₄	—	4	8	no	—
397	C₂×D₄	—	4	16	no	—
398	C₂²×C₄	—	2	4	no	—
399	C₂²×C₄	—	2	4	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	4	no	—
403	C₂²×C₄	—	2	4	no	—
404	C₂²×C₄	—	2	2	no	—
405	C₂²×C₄	—	2	4	no	—
406	C₂²×C₄	—	2	4	no	—
407	C₂²×C₄	—	2	4	no	—
408	C₂²×C₄	—	2	4	no	—
409	C₂²×C₄	—	2	4	no	—
410	C₂²×C₄	—	2	2	no	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	4	16	no	—
420	C₂²×C₄	—	4	8	no	—
421	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
422	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
423	C₂²×C₄	C₂×D₄×C₆₇	1	16	yes	—
424	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
425	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
426	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
427	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
428	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
429	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
430	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
431	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
432	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
433	C₄⋊C₄	—	2	8	no	—
434	C₄⋊C₄	—	2	16	no	—
435	C₄⋊C₄	—	2	8	no	—
436	C₄⋊C₄	—	2	8	no	—
437	C₄⋊C₄	—	4	8	no	—
438	C₄⋊C₄	—	4	8	no	—
439	C₄⋊C₄	—	4	8	no	—
440	C₂²⋊C₄	—	2	16	no	—
441	C₂²⋊C₄	—	2	8	no	—
442	C₂²⋊C₄	—	2	8	no	—
443	C₂²⋊C₄	—	2	8	no	—
444	C₂²⋊C₄	—	4	8	no	—
445	C₂²⋊C₄	—	4	8	no	—
446	C₂²⋊C₄	—	4	8	no	—
447	C₂³	—	2	8	no	—
448	C₂³	—	2	8	no	—
449	C₂³	—	2	8	no	—
450	C₂³	—	2	2	no	—
451	C₂³	—	2	4	no	—
452	C₂³	—	2	4	no	—
453	C₂³	—	2	2	no	—
454	C₂³	—	2	2	no	—
455	C₂³	—	2	2	no	—
456	C₂³	—	2	4	no	—
457	C₂³	—	2	4	no	—
458	C₂³	—	2	2	no	—
459	C₂³	—	4	8	no	—
460	C₂³	—	4	8	no	—
461	C₂³	—	4	4	no	—
462	C₂³	—	4	4	no	—
463	C₂³	—	4	4	no	—
464	C₂³	—	4	8	no	—
465	C₂³	—	4	8	no	—
466	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
467	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
468	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
469	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
470	C₂³	2₋¹⁺⁴×C₆₇	1	4	yes	—
471	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
472	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
473	Q₈	—	4	16	no	—
474	D₄	—	4	16	no	—
475	C₂×C₄	—	2	8	no	—
476	C₂×C₄	—	2	8	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	4	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	2	4	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	4	no	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	4	16	no	—
499	C₂×C₄	—	4	4	no	—
500	C₂×C₄	—	4	8	no	—
501	C₂×C₄	—	4	4	no	—
502	C₂×C₄	—	4	4	no	—
503	C₂×C₄	—	4	8	no	—
504	C₂×C₄	—	4	8	no	—
505	C₂×C₄	—	4	8	no	—
506	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
507	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
508	C₂²	—	2	4	no	—
509	C₂²	—	2	4	no	—
510	C₂²	—	2	4	no	—
511	C₂²	—	4	4	no	—
512	C₂²	—	4	4	no	—
513	C₂²	—	4	16	no	—
514	C₂²	—	4	4	no	—
515	C₂²	—	4	4	no	—
516	C₂²	—	4	2	no	—
517	C₂²	—	4	2	no	—
518	C₂²	—	4	2	no	—
519	C₂²	—	8	8	no	—
520	C₂²	#534×C₆₇	1	4	yes	—
521	C₂²	#536×C₆₇	1	4	yes	—
522	C₂²	#537×C₆₇	1	4	yes	—
523	C₂²	#562×C₆₇	1	8	yes	—
524	C₂²	#563×C₆₇	1	8	yes	—
525	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
526	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
527	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
528	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
529	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
530	C₄	—	2	4	no	—
531	C₄	—	2	4	no	—
532	C₄	—	4	4	no	—
533	C₄	—	4	4	no	—
534	C₄	—	4	2	no	—
535	C₄	—	4	4	no	—
536	C₄	—	4	2	no	—
537	C₄	—	4	2	no	—
538	C₄	—	4	4	no	—
539	C₂	—	4	4	no	—
540	C₂	—	8	2	no	—
541	C₂	C₆₇×#2814	1	8	yes	direct product factor
542	C₂	C₂×#534×C₆₇	1	1	yes	characteristic
543	C₂	C₂×#536×C₆₇	1	1	yes	characteristic
544	C₂	C₂×#537×C₆₇	1	1	yes	characteristic
545	C₂	C₂×#562×C₆₇	1	2	yes	—
546	C₂	C₂×#563×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂³×D₄×C₆₇	C₂²
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
2. C₂²×C₂²⋊Q₈×C₆₇	C₂	semidirect
3. C₆₇×#2814	C₂	direct (N × H)
4. C₂×#398×C₆₇	C₂	semidirect
5. C₂×#398×C₆₇	C₂	semidirect
6. C₂×#397×C₆₇	C₂	semidirect
7. C₂×#397×C₆₇	C₂	semidirect
8. C₂×#396×C₆₇	C₂	semidirect
9. C₂×#395×C₆₇	C₂	semidirect
11. C₂×#393×C₆₇	C₂	semidirect
12. C₂×#380×C₆₇	C₂	semidirect
19. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
34. #398×C₆₇	C₂²	semidirect
35. #398×C₆₇	C₂²	semidirect
36. #397×C₆₇	C₂²	semidirect
37. #397×C₆₇	C₂²	semidirect
38. #396×C₆₇	C₂²	semidirect
39. #395×C₆₇	C₂²	semidirect
41. #393×C₆₇	C₂²	semidirect
42. #380×C₆₇	C₂²	semidirect
265. C₂×#2814	C₆₇	direct (N × H)
268. C₂×C₆₇	#2814	direct (N × H)
275. C₂²×C₂²⋊Q₈	C₂×C₆₇	semidirect
276. #2814	C₂×C₆₇	direct (N × H)
277. C₂×#398	C₂×C₆₇	semidirect
278. C₂×#398	C₂×C₆₇	semidirect
279. C₂×#397	C₂×C₆₇	semidirect
280. C₂×#397	C₂×C₆₇	semidirect
281. C₂×#396	C₂×C₆₇	semidirect
282. C₂×#395	C₂×C₆₇	semidirect
284. C₂×#393	C₂×C₆₇	semidirect
285. C₂×#380	C₂×C₆₇	semidirect
286. C₆₇	C₂×#2814	direct (N × H)
293. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
308. #398	C₂²×C₆₇	semidirect
309. #398	C₂²×C₆₇	semidirect
310. #397	C₂²×C₆₇	semidirect
311. #397	C₂²×C₆₇	semidirect
312. #396	C₂²×C₆₇	semidirect
313. #395	C₂²×C₆₇	semidirect
315. #393	C₂²×C₆₇	semidirect
316. #380	C₂²×C₆₇	semidirect
541. C₂	C₆₇×#2814	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×#123×C₆₇	2
C₂²×C₂²⋊Q₈×C₆₇	2
C₆₇×#2814	2
C₂×#398×C₆₇	2
C₂×#397×C₆₇	2
C₂×#396×C₆₇	2
C₂×#395×C₆₇	2
C₂×#394×C₆₇	2
C₂×#393×C₆₇	2
C₂×#380×C₆₇	2
C₂×#2814	67

Subgroups — C2×C67×#2814 #30715902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2814 #30715902