Subgroups — C₂×C₆₇×#2545 #30714151

Order 17152.

3310 conjugacy classes of subgroups.

The subgroup lattice has 464 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	1	yes	maximal, characteristic
3	C₂²×#124×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2545	C₂	1	16	yes	maximal, direct product factor
6	C₂×#396×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₂×#391×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#390×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#388×C₆₇	C₂	1	4	yes	maximal, complemented
10	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₆₇	—	2	8	no	—
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	8	yes	complemented
16	C₂×#124×C₆₇	—	2	8	no	—
17	C₂×#124×C₆₇	—	2	8	no	—
18	C₂×#124×C₆₇	C₂²	1	8	yes	—
19	C₂×#124×C₆₇	C₂²	1	8	yes	—
20	C₂×C₄×Q₈×C₆₇	—	2	8	no	—
21	C₂×C₄×Q₈×C₆₇	C₂²	1	8	yes	complemented
22	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
27	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
28	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
29	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
30	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
31	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
32	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
33	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
34	#396×C₆₇	C₂²	1	32	yes	complemented
35	#391×C₆₇	C₂²	1	8	yes	complemented
36	#390×C₆₇	C₂²	1	8	yes	—
37	#388×C₆₇	C₂²	1	32	yes	complemented
38	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	complemented
39	C₂×#95×C₆₇	C₂²	1	2	yes	—
40	C₂×#95×C₆₇	C₂²	1	4	yes	—
41	C₂⁵×C₆₇	C₂³	1	2	yes	—
42	C₂²×Q₈×C₆₇	—	2	4	no	—
43	C₂²×Q₈×C₆₇	—	2	4	no	—
44	C₂²×Q₈×C₆₇	—	2	4	no	—
45	C₂²×Q₈×C₆₇	—	2	4	no	—
46	C₂²×Q₈×C₆₇	—	2	4	no	—
47	C₂²×Q₈×C₆₇	D₄	1	8	yes	—
48	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
49	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
50	C₂²×D₄×C₆₇	—	2	4	no	—
51	C₂²×D₄×C₆₇	—	2	4	no	—
52	C₂²×D₄×C₆₇	—	2	4	no	—
53	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
54	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
55	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
56	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
57	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
59	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
60	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
61	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₄⋊Q₈×C₆₇	—	2	32	no	—
63	#124×C₆₇	—	2	32	no	—
64	#124×C₆₇	—	2	32	no	—
65	C₄×Q₈×C₆₇	—	2	32	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
68	C₂×C₄⋊C₄×C₆₇	—	2	16	no	—
69	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
72	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
73	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
75	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
76	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
77	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
78	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
79	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
80	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
81	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
82	C₂×C₄²×C₆₇	—	2	8	no	—
83	C₂×C₄²×C₆₇	—	2	4	no	—
84	C₂×C₄²×C₆₇	—	2	4	no	—
85	C₂×C₄²×C₆₇	D₄	1	8	yes	—
86	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
87	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
88	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
89	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
90	#95×C₆₇	C₂³	1	8	yes	—
91	#95×C₆₇	C₂³	1	16	yes	—
92	C₂⁴×C₆₇	—	2	4	no	—
93	C₂⁴×C₆₇	—	2	4	no	—
94	C₂⁴×C₆₇	—	2	8	no	—
95	C₂⁴×C₆₇	—	2	4	no	—
96	C₂⁴×C₆₇	—	2	8	no	—
97	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
98	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
99	C₂×Q₈×C₆₇	—	2	32	no	—
100	C₂×Q₈×C₆₇	—	2	8	no	—
101	C₂×Q₈×C₆₇	—	2	8	no	—
102	C₂×Q₈×C₆₇	—	2	8	no	—
103	C₂×Q₈×C₆₇	—	2	8	no	—
104	C₂×Q₈×C₆₇	—	2	8	no	—
105	C₂×Q₈×C₆₇	—	4	16	no	—
106	C₂×Q₈×C₆₇	C₂×D₄	1	16	yes	—
107	C₂×D₄×C₆₇	—	2	8	no	—
108	C₂×D₄×C₆₇	—	2	8	no	—
109	C₂×D₄×C₆₇	—	2	8	no	—
110	C₂×D₄×C₆₇	—	4	16	no	—
111	C₂²×C₄×C₆₇	—	2	4	no	—
112	C₂²×C₄×C₆₇	—	2	4	no	—
113	C₂²×C₄×C₆₇	—	2	4	no	—
114	C₂²×C₄×C₆₇	—	2	4	no	—
115	C₂²×C₄×C₆₇	—	2	4	no	—
116	C₂²×C₄×C₆₇	—	2	4	no	—
117	C₂²×C₄×C₆₇	—	2	4	no	—
118	C₂²×C₄×C₆₇	—	2	4	no	—
119	C₂²×C₄×C₆₇	—	2	4	no	—
120	C₂²×C₄×C₆₇	—	2	4	no	—
121	C₂²×C₄×C₆₇	—	2	8	no	—
122	C₂²×C₄×C₆₇	—	2	4	no	—
123	C₂²×C₄×C₆₇	—	2	4	no	—
124	C₂²×C₄×C₆₇	—	2	4	no	—
125	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
126	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
127	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
128	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
129	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
130	C₂²×C₄×C₆₇	#41	1	8	yes	—
131	C₂²×C₄×C₆₇	#41	1	4	yes	—
132	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
133	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
134	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
135	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
136	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
137	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
138	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
139	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
140	C₄⋊C₄×C₆₇	—	2	16	no	—
141	C₄⋊C₄×C₆₇	—	2	8	no	—
142	C₄⋊C₄×C₆₇	—	2	32	no	—
143	C₄⋊C₄×C₆₇	—	2	16	no	—
144	C₂²⋊C₄×C₆₇	—	2	32	no	—
145	C₂²⋊C₄×C₆₇	—	2	16	no	—
146	C₂²⋊C₄×C₆₇	—	2	16	no	—
147	C₂²⋊C₄×C₆₇	—	2	32	no	—
148	C₄²×C₆₇	—	2	16	no	—
149	C₄²×C₆₇	—	2	8	no	—
150	C₄²×C₆₇	—	2	8	no	—
151	C₄²×C₆₇	C₂×D₄	1	16	yes	—
152	C₂³×C₆₇	—	2	4	no	—
153	C₂³×C₆₇	—	2	4	no	—
154	C₂³×C₆₇	—	2	8	no	—
155	C₂³×C₆₇	—	2	4	no	—
156	C₂³×C₆₇	—	2	8	no	—
157	C₂³×C₆₇	—	4	8	no	—
158	C₂³×C₆₇	—	4	8	no	—
159	C₂³×C₆₇	—	4	8	no	—
160	C₂³×C₆₇	—	4	16	no	—
161	C₂³×C₆₇	—	4	16	no	—
162	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
163	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
164	C₂³×C₆₇	C₂×#41	1	2	yes	—
165	C₂³×C₆₇	C₂×#41	1	4	yes	—
166	C₂³×C₆₇	2₋¹⁺⁴	1	4	yes	—
167	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
168	Q₈×C₆₇	—	2	32	no	—
169	Q₈×C₆₇	—	4	16	no	—
170	D₄×C₆₇	—	4	16	no	—
171	C₂×C₄×C₆₇	—	2	16	no	—
172	C₂×C₄×C₆₇	—	2	8	no	—
173	C₂×C₄×C₆₇	—	2	8	no	—
174	C₂×C₄×C₆₇	—	2	16	no	—
175	C₂×C₄×C₆₇	—	2	4	no	—
176	C₂×C₄×C₆₇	—	2	4	no	—
177	C₂×C₄×C₆₇	—	2	4	no	—
178	C₂×C₄×C₆₇	—	2	4	no	—
179	C₂×C₄×C₆₇	—	2	4	no	—
180	C₂×C₄×C₆₇	—	2	4	no	—
181	C₂×C₄×C₆₇	—	2	4	no	—
182	C₂×C₄×C₆₇	—	2	4	no	—
183	C₂×C₄×C₆₇	—	2	4	no	—
184	C₂×C₄×C₆₇	—	2	4	no	—
185	C₂×C₄×C₆₇	—	2	8	no	—
186	C₂×C₄×C₆₇	—	2	4	no	—
187	C₂×C₄×C₆₇	—	2	4	no	—
188	C₂×C₄×C₆₇	—	2	4	no	—
189	C₂×C₄×C₆₇	—	4	8	no	—
190	C₂×C₄×C₆₇	—	4	8	no	—
191	C₂×C₄×C₆₇	—	4	8	no	—
192	C₂×C₄×C₆₇	#121	1	16	yes	—
193	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
194	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
195	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
196	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
197	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
198	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
199	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
200	C₂²×C₆₇	—	4	4	no	—
201	C₂²×C₆₇	—	4	4	no	—
202	C₂²×C₆₇	—	4	4	no	—
203	C₂²×C₆₇	—	4	8	no	—
204	C₂²×C₆₇	—	4	8	no	—
205	C₂²×C₆₇	—	8	16	no	—
206	C₂²×C₆₇	C₂×#121	1	4	yes	—
207	C₂²×C₆₇	#532	1	4	yes	—
208	C₂²×C₆₇	#536	1	4	yes	—
209	C₂²×C₆₇	#548	1	8	yes	—
210	C₂²×C₆₇	#556	1	8	yes	—
211	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
212	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
213	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
214	C₂²×C₆₇	C₂²×#41	1	2	yes	—
215	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
216	C₄×C₆₇	—	2	8	no	—
217	C₄×C₆₇	—	2	4	no	—
218	C₄×C₆₇	—	2	4	no	—
219	C₄×C₆₇	—	2	8	no	—
220	C₄×C₆₇	—	4	4	no	—
221	C₄×C₆₇	—	4	4	no	—
222	C₄×C₆₇	—	4	4	no	—
223	C₄×C₆₇	C₂×#121	1	8	yes	—
224	C₂×#2545	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
225	C₂×C₆₇	—	8	4	no	—
226	C₂×C₆₇	#2545	1	8	yes	direct product factor
227	C₂×C₆₇	C₂²×#121	1	1	yes	characteristic
228	C₂×C₆₇	C₂×#532	1	1	yes	characteristic
229	C₂×C₆₇	C₂×#536	1	1	yes	characteristic
230	C₂×C₆₇	C₂×#548	1	2	yes	—
231	C₂×C₆₇	C₂×#556	1	2	yes	—
232	C₂²×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
233	C₂²×#124	C₂×C₆₇	1	1	yes	characteristic
234	C₂²×#124	C₂×C₆₇	1	1	yes	characteristic
235	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
236	#2545	C₂×C₆₇	1	16	yes	direct product factor
237	C₂×#396	C₂×C₆₇	1	4	yes	complemented
238	C₂×#391	C₂×C₆₇	1	1	yes	characteristic, complemented
239	C₂×#390	C₂×C₆₇	1	1	yes	characteristic
240	C₂×#388	C₂×C₆₇	1	4	yes	complemented
241	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
242	C₆₇	C₂×#2545	1	1	yes	characteristic, direct product factor, Sylow
243	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
244	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
245	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
246	C₂×C₄⋊Q₈	—	2	8	no	—
247	C₂×C₄⋊Q₈	C₂²×C₆₇	1	8	yes	complemented
248	C₂×#124	—	2	8	no	—
249	C₂×#124	—	2	8	no	—
250	C₂×#124	C₂²×C₆₇	1	8	yes	—
251	C₂×#124	C₂²×C₆₇	1	8	yes	—
252	C₂×C₄×Q₈	—	2	8	no	—
253	C₂×C₄×Q₈	C₂²×C₆₇	1	8	yes	complemented
254	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
255	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
256	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
257	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
258	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
259	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
260	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
261	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
262	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
263	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
264	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
265	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
266	#396	C₂²×C₆₇	1	32	yes	complemented
267	#391	C₂²×C₆₇	1	8	yes	complemented
268	#390	C₂²×C₆₇	1	8	yes	—
269	#388	C₂²×C₆₇	1	32	yes	complemented
270	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	complemented
271	C₂×#95	C₂²×C₆₇	1	4	yes	—
272	C₂×#95	C₂²×C₆₇	1	2	yes	—
273	C₂⁵	C₂³×C₆₇	1	2	yes	—
274	C₂²×Q₈	—	2	4	no	—
275	C₂²×Q₈	—	2	4	no	—
276	C₂²×Q₈	—	2	4	no	—
277	C₂²×Q₈	—	2	4	no	—
278	C₂²×Q₈	—	2	4	no	—
279	C₂²×Q₈	D₄×C₆₇	1	8	yes	—
280	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
281	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
282	C₂²×D₄	—	2	4	no	—
283	C₂²×D₄	—	2	4	no	—
284	C₂²×D₄	—	2	4	no	—
285	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
286	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
287	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
288	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
289	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
290	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
291	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
292	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
293	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
294	C₄⋊Q₈	—	2	32	no	—
295	#124	—	2	32	no	—
296	#124	—	2	32	no	—
297	C₄×Q₈	—	2	32	no	—
298	C₂×C₄⋊C₄	—	2	8	no	—
299	C₂×C₄⋊C₄	—	2	4	no	—
300	C₂×C₄⋊C₄	—	2	16	no	—
301	C₂×C₄⋊C₄	—	2	8	no	—
302	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
303	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
304	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
305	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
306	C₂×C₂²⋊C₄	—	2	16	no	—
307	C₂×C₂²⋊C₄	—	2	8	no	—
308	C₂×C₂²⋊C₄	—	2	8	no	—
309	C₂×C₂²⋊C₄	—	2	16	no	—
310	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
311	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
312	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
313	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
314	C₂×C₄²	—	2	8	no	—
315	C₂×C₄²	—	2	4	no	—
316	C₂×C₄²	—	2	4	no	—
317	C₂×C₄²	D₄×C₆₇	1	8	yes	—
318	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
319	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
320	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
321	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
322	#95	C₂³×C₆₇	1	8	yes	—
323	#95	C₂³×C₆₇	1	16	yes	—
324	C₂⁴	—	2	4	no	—
325	C₂⁴	—	2	4	no	—
326	C₂⁴	—	2	8	no	—
327	C₂⁴	—	2	4	no	—
328	C₂⁴	—	2	8	no	—
329	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
330	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
331	C₂×Q₈	—	2	32	no	—
332	C₂×Q₈	—	2	8	no	—
333	C₂×Q₈	—	2	8	no	—
334	C₂×Q₈	—	2	8	no	—
335	C₂×Q₈	—	2	8	no	—
336	C₂×Q₈	—	2	8	no	—
337	C₂×Q₈	—	4	16	no	—
338	C₂×Q₈	C₂×D₄×C₆₇	1	16	yes	—
339	C₂×D₄	—	2	8	no	—
340	C₂×D₄	—	2	8	no	—
341	C₂×D₄	—	2	8	no	—
342	C₂×D₄	—	4	16	no	—
343	C₂²×C₄	—	2	4	no	—
344	C₂²×C₄	—	2	4	no	—
345	C₂²×C₄	—	2	4	no	—
346	C₂²×C₄	—	2	4	no	—
347	C₂²×C₄	—	2	4	no	—
348	C₂²×C₄	—	2	4	no	—
349	C₂²×C₄	—	2	4	no	—
350	C₂²×C₄	—	2	4	no	—
351	C₂²×C₄	—	2	4	no	—
352	C₂²×C₄	—	2	4	no	—
353	C₂²×C₄	—	2	8	no	—
354	C₂²×C₄	—	2	4	no	—
355	C₂²×C₄	—	2	4	no	—
356	C₂²×C₄	—	2	4	no	—
357	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
358	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
359	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
360	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
361	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
362	C₂²×C₄	#41×C₆₇	1	8	yes	—
363	C₂²×C₄	#41×C₆₇	1	4	yes	—
364	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
365	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
366	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
367	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
368	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
369	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
370	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
371	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
372	C₄⋊C₄	—	2	16	no	—
373	C₄⋊C₄	—	2	8	no	—
374	C₄⋊C₄	—	2	32	no	—
375	C₄⋊C₄	—	2	16	no	—
376	C₂²⋊C₄	—	2	32	no	—
377	C₂²⋊C₄	—	2	16	no	—
378	C₂²⋊C₄	—	2	16	no	—
379	C₂²⋊C₄	—	2	32	no	—
380	C₄²	—	2	16	no	—
381	C₄²	—	2	8	no	—
382	C₄²	—	2	8	no	—
383	C₄²	C₂×D₄×C₆₇	1	16	yes	—
384	C₂³	—	2	4	no	—
385	C₂³	—	2	4	no	—
386	C₂³	—	2	8	no	—
387	C₂³	—	2	4	no	—
388	C₂³	—	2	8	no	—
389	C₂³	—	4	8	no	—
390	C₂³	—	4	8	no	—
391	C₂³	—	4	8	no	—
392	C₂³	—	4	16	no	—
393	C₂³	—	4	16	no	—
394	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
395	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
396	C₂³	C₂×#41×C₆₇	1	2	yes	—
397	C₂³	C₂×#41×C₆₇	1	4	yes	—
398	C₂³	2₋¹⁺⁴×C₆₇	1	4	yes	—
399	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
400	Q₈	—	2	32	no	—
401	Q₈	—	4	16	no	—
402	D₄	—	4	16	no	—
403	C₂×C₄	—	2	16	no	—
404	C₂×C₄	—	2	8	no	—
405	C₂×C₄	—	2	8	no	—
406	C₂×C₄	—	2	16	no	—
407	C₂×C₄	—	2	4	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	8	no	—
418	C₂×C₄	—	2	4	no	—
419	C₂×C₄	—	2	4	no	—
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	4	8	no	—
422	C₂×C₄	—	4	8	no	—
423	C₂×C₄	—	4	8	no	—
424	C₂×C₄	#121×C₆₇	1	16	yes	—
425	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
426	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
427	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
428	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
429	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
430	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
431	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
432	C₂²	—	4	4	no	—
433	C₂²	—	4	4	no	—
434	C₂²	—	4	4	no	—
435	C₂²	—	4	8	no	—
436	C₂²	—	4	8	no	—
437	C₂²	—	8	16	no	—
438	C₂²	C₂×#121×C₆₇	1	4	yes	—
439	C₂²	#532×C₆₇	1	4	yes	—
440	C₂²	#536×C₆₇	1	4	yes	—
441	C₂²	#548×C₆₇	1	8	yes	—
442	C₂²	#556×C₆₇	1	8	yes	—
443	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
444	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
445	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
446	C₂²	C₂²×#41×C₆₇	1	2	yes	—
447	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
448	C₄	—	2	8	no	—
449	C₄	—	2	4	no	—
450	C₄	—	2	4	no	—
451	C₄	—	2	8	no	—
452	C₄	—	4	4	no	—
453	C₄	—	4	4	no	—
454	C₄	—	4	4	no	—
455	C₄	C₂×#121×C₆₇	1	8	yes	—
456	C₂	—	8	4	no	—
457	C₂	C₆₇×#2545	1	8	yes	direct product factor
458	C₂	C₂²×#121×C₆₇	1	1	yes	characteristic
459	C₂	C₂×#532×C₆₇	1	1	yes	characteristic
460	C₂	C₂×#536×C₆₇	1	1	yes	characteristic
461	C₂	C₂×#548×C₆₇	1	2	yes	—
462	C₂	C₂×#556×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
1. C₂²×C₄⋊Q₈×C₆₇	C₂	semidirect
4. C₂²×C₄×Q₈×C₆₇	C₂	semidirect
5. C₆₇×#2545	C₂	direct (N × H)
6. C₂×#396×C₆₇	C₂	semidirect
7. C₂×#391×C₆₇	C₂	semidirect
9. C₂×#388×C₆₇	C₂	semidirect
10. C₂×C₄×C₄⋊C₄×C₆₇	C₂	semidirect
15. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
21. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
34. #396×C₆₇	C₂²	semidirect
35. #391×C₆₇	C₂²	semidirect
37. #388×C₆₇	C₂²	semidirect
38. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
224. C₂×#2545	C₆₇	direct (N × H)
226. C₂×C₆₇	#2545	direct (N × H)
232. C₂²×C₄⋊Q₈	C₂×C₆₇	semidirect
235. C₂²×C₄×Q₈	C₂×C₆₇	semidirect
236. #2545	C₂×C₆₇	direct (N × H)
237. C₂×#396	C₂×C₆₇	semidirect
238. C₂×#391	C₂×C₆₇	semidirect
240. C₂×#388	C₂×C₆₇	semidirect
241. C₂×C₄×C₄⋊C₄	C₂×C₆₇	semidirect
242. C₆₇	C₂×#2545	direct (N × H)
247. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
253. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
266. #396	C₂²×C₆₇	semidirect
267. #391	C₂²×C₆₇	semidirect
269. #388	C₂²×C₆₇	semidirect
270. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
457. C₂	C₆₇×#2545	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×C₄×Q₈×C₆₇	2
C₆₇×#2545	2
C₂×#396×C₆₇	2
C₂×#391×C₆₇	2
C₂×#390×C₆₇	2
C₂×#388×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₂×#2545	67

Subgroups — C2×C67×#2545 #30714151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2545 #30714151