Subgroups — C₂×C₆₇×#2629 #30713863

Order 17152.

2614 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#117×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2629	C₂	1	16	yes	maximal, direct product factor
4	C₂×#398×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂×#396×C₆₇	C₂	1	4	yes	maximal
6	C₂×#389×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#384×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#380×C₆₇	C₂	1	2	yes	maximal
9	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#125×C₆₇	—	2	8	no	—
11	C₂×#125×C₆₇	C₂²	1	8	yes	complemented
12	C₂×#117×C₆₇	—	2	8	no	—
13	C₂×#117×C₆₇	C₂²	1	8	yes	—
14	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
17	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
20	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
22	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	#398×C₆₇	C₂²	1	32	yes	complemented
25	#396×C₆₇	C₂²	1	32	yes	—
26	#389×C₆₇	C₂²	1	16	yes	complemented
27	#384×C₆₇	C₂²	1	8	yes	complemented
28	#380×C₆₇	C₂²	1	16	yes	—
29	C₂×#95×C₆₇	C₂²	1	4	yes	—
30	C₂×#95×C₆₇	C₂²	1	4	yes	—
31	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
32	C₂³×C₄×C₆₇	—	2	4	no	—
33	C₂³×C₄×C₆₇	—	2	4	no	—
34	C₂³×C₄×C₆₇	Q₈	1	8	yes	—
35	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
36	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
37	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
38	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
39	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
40	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	#125×C₆₇	—	2	32	no	—
43	#117×C₆₇	—	2	32	no	—
44	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	16	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
47	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
48	C₂×C₄⋊C₄×C₆₇	—	4	8	no	—
49	C₂×C₄⋊C₄×C₆₇	—	4	8	no	—
50	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
51	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
52	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
53	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
54	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
55	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
56	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
57	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
58	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
59	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
60	C₂×C₄²×C₆₇	—	2	4	no	—
61	C₂×C₄²×C₆₇	D₄	1	8	yes	—
62	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
63	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
64	#95×C₆₇	C₂³	1	16	yes	—
65	#95×C₆₇	C₂³	1	16	yes	—
66	C₂⁴×C₆₇	—	2	2	no	—
67	C₂⁴×C₆₇	—	2	2	no	—
68	C₂⁴×C₆₇	—	2	4	no	—
69	C₂⁴×C₆₇	—	2	4	no	—
70	C₂⁴×C₆₇	C₂×Q₈	1	4	yes	—
71	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
72	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	—	2	2	no	—
74	C₂²×C₄×C₆₇	—	2	2	no	—
75	C₂²×C₄×C₆₇	—	2	4	no	—
76	C₂²×C₄×C₆₇	—	2	4	no	—
77	C₂²×C₄×C₆₇	—	2	32	no	—
78	C₂²×C₄×C₆₇	—	2	8	no	—
79	C₂²×C₄×C₆₇	—	2	8	no	—
80	C₂²×C₄×C₆₇	—	2	4	no	—
81	C₂²×C₄×C₆₇	—	2	8	no	—
82	C₂²×C₄×C₆₇	—	2	8	no	—
83	C₂²×C₄×C₆₇	—	2	8	no	—
84	C₂²×C₄×C₆₇	—	2	8	no	—
85	C₂²×C₄×C₆₇	—	2	4	no	—
86	C₂²×C₄×C₆₇	—	2	8	no	—
87	C₂²×C₄×C₆₇	—	2	4	no	—
88	C₂²×C₄×C₆₇	—	2	8	no	—
89	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
90	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
91	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
92	C₂²×C₄×C₆₇	C₂×Q₈	1	16	yes	—
93	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
94	C₂²×C₄×C₆₇	#41	1	4	yes	—
95	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
96	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
97	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
98	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
99	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
101	C₄⋊C₄×C₆₇	—	2	16	no	—
102	C₄⋊C₄×C₆₇	—	2	32	no	—
103	C₄⋊C₄×C₆₇	—	2	16	no	—
104	C₄⋊C₄×C₆₇	—	2	16	no	—
105	C₄⋊C₄×C₆₇	—	4	16	no	—
106	C₄⋊C₄×C₆₇	—	4	16	no	—
107	C₂²⋊C₄×C₆₇	—	2	16	no	—
108	C₂²⋊C₄×C₆₇	—	4	16	no	—
109	C₄²×C₆₇	—	2	8	no	—
110	C₄²×C₆₇	C₂×D₄	1	16	yes	—
111	C₂³×C₆₇	—	2	8	no	—
112	C₂³×C₆₇	—	2	16	no	—
113	C₂³×C₆₇	—	2	2	no	—
114	C₂³×C₆₇	—	2	2	no	—
115	C₂³×C₆₇	—	2	4	no	—
116	C₂³×C₆₇	—	2	4	no	—
117	C₂³×C₆₇	—	4	8	no	—
118	C₂³×C₆₇	—	4	4	no	—
119	C₂³×C₆₇	—	4	4	no	—
120	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
121	C₂³×C₆₇	C₂²×Q₈	1	4	yes	—
122	C₂³×C₆₇	C₂²×Q₈	1	2	yes	—
123	C₂³×C₆₇	C₂×#41	1	2	yes	—
124	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
125	C₂³×C₆₇	2₋¹⁺⁴	1	4	yes	—
126	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
127	C₂×C₄×C₆₇	—	2	8	no	—
128	C₂×C₄×C₆₇	—	2	16	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	2	4	no	—
132	C₂×C₄×C₆₇	—	2	4	no	—
133	C₂×C₄×C₆₇	—	2	32	no	—
134	C₂×C₄×C₆₇	—	2	4	no	—
135	C₂×C₄×C₆₇	—	2	8	no	—
136	C₂×C₄×C₆₇	—	2	8	no	—
137	C₂×C₄×C₆₇	—	2	8	no	—
138	C₂×C₄×C₆₇	—	2	8	no	—
139	C₂×C₄×C₆₇	—	2	4	no	—
140	C₂×C₄×C₆₇	—	2	8	no	—
141	C₂×C₄×C₆₇	—	2	4	no	—
142	C₂×C₄×C₆₇	—	2	8	no	—
143	C₂×C₄×C₆₇	—	4	8	no	—
144	C₂×C₄×C₆₇	—	4	16	no	—
145	C₂×C₄×C₆₇	—	4	16	no	—
146	C₂×C₄×C₆₇	C₂²⋊Q₈	1	16	yes	—
147	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
148	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
149	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
150	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
151	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
152	C₂²×C₆₇	—	2	4	no	—
153	C₂²×C₆₇	—	2	8	no	—
154	C₂²×C₆₇	—	4	16	no	—
155	C₂²×C₆₇	—	4	4	no	—
156	C₂²×C₆₇	—	4	2	no	—
157	C₂²×C₆₇	—	4	2	no	—
158	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	4	yes	—
159	C₂²×C₆₇	#537	1	8	yes	—
160	C₂²×C₆₇	#540	1	4	yes	—
161	C₂²×C₆₇	#541	1	4	yes	—
162	C₂²×C₆₇	#544	1	8	yes	—
163	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
164	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
165	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
166	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
167	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
168	C₄×C₆₇	—	2	4	no	—
169	C₄×C₆₇	—	2	8	no	—
170	C₄×C₆₇	—	4	4	no	—
171	C₄×C₆₇	—	4	8	no	—
172	C₄×C₆₇	—	4	8	no	—
173	C₄×C₆₇	C₂×C₂²⋊Q₈	1	8	yes	—
174	C₂×#2629	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
175	C₂×C₆₇	—	4	4	no	—
176	C₂×C₆₇	#2629	1	8	yes	direct product factor
177	C₂×C₆₇	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
178	C₂×C₆₇	C₂×#537	1	2	yes	—
179	C₂×C₆₇	C₂×#540	1	1	yes	characteristic
180	C₂×C₆₇	C₂×#541	1	1	yes	characteristic
181	C₂×C₆₇	C₂×#544	1	2	yes	—
182	C₂²×#125	C₂×C₆₇	1	1	yes	characteristic, complemented
183	C₂²×#117	C₂×C₆₇	1	1	yes	characteristic
184	#2629	C₂×C₆₇	1	16	yes	direct product factor
185	C₂×#398	C₂×C₆₇	1	4	yes	complemented
186	C₂×#396	C₂×C₆₇	1	4	yes	—
187	C₂×#389	C₂×C₆₇	1	2	yes	complemented
188	C₂×#384	C₂×C₆₇	1	1	yes	characteristic, complemented
189	C₂×#380	C₂×C₆₇	1	2	yes	—
190	C₆₇	C₂×#2629	1	1	yes	characteristic, direct product factor, Sylow
191	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
192	C₂×#125	—	2	8	no	—
193	C₂×#125	C₂²×C₆₇	1	8	yes	complemented
194	C₂×#117	—	2	8	no	—
195	C₂×#117	C₂²×C₆₇	1	8	yes	—
196	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
197	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
198	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
199	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
200	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
201	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
202	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
203	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
204	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
205	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
206	#398	C₂²×C₆₇	1	32	yes	complemented
207	#396	C₂²×C₆₇	1	32	yes	—
208	#389	C₂²×C₆₇	1	16	yes	complemented
209	#384	C₂²×C₆₇	1	8	yes	complemented
210	#380	C₂²×C₆₇	1	16	yes	—
211	C₂×#95	C₂²×C₆₇	1	4	yes	—
212	C₂×#95	C₂²×C₆₇	1	4	yes	—
213	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
214	C₂³×C₄	—	2	4	no	—
215	C₂³×C₄	—	2	4	no	—
216	C₂³×C₄	Q₈×C₆₇	1	8	yes	—
217	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
218	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
219	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
220	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
221	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
222	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
223	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
224	#125	—	2	32	no	—
225	#117	—	2	32	no	—
226	C₂×C₄⋊C₄	—	2	8	no	—
227	C₂×C₄⋊C₄	—	2	16	no	—
228	C₂×C₄⋊C₄	—	2	8	no	—
229	C₂×C₄⋊C₄	—	2	8	no	—
230	C₂×C₄⋊C₄	—	4	8	no	—
231	C₂×C₄⋊C₄	—	4	8	no	—
232	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
233	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
234	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
235	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
236	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
237	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
238	C₂×C₂²⋊C₄	—	2	8	no	—
239	C₂×C₂²⋊C₄	—	4	8	no	—
240	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
241	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
242	C₂×C₄²	—	2	4	no	—
243	C₂×C₄²	D₄×C₆₇	1	8	yes	—
244	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
245	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
246	#95	C₂³×C₆₇	1	16	yes	—
247	#95	C₂³×C₆₇	1	16	yes	—
248	C₂⁴	—	2	2	no	—
249	C₂⁴	—	2	2	no	—
250	C₂⁴	—	2	4	no	—
251	C₂⁴	—	2	4	no	—
252	C₂⁴	C₂×Q₈×C₆₇	1	4	yes	—
253	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
254	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	4	no	—
258	C₂²×C₄	—	2	4	no	—
259	C₂²×C₄	—	2	32	no	—
260	C₂²×C₄	—	2	8	no	—
261	C₂²×C₄	—	2	8	no	—
262	C₂²×C₄	—	2	4	no	—
263	C₂²×C₄	—	2	8	no	—
264	C₂²×C₄	—	2	8	no	—
265	C₂²×C₄	—	2	8	no	—
266	C₂²×C₄	—	2	8	no	—
267	C₂²×C₄	—	2	4	no	—
268	C₂²×C₄	—	2	8	no	—
269	C₂²×C₄	—	2	8	no	—
270	C₂²×C₄	—	2	4	no	—
271	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
272	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
273	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
274	C₂²×C₄	C₂×Q₈×C₆₇	1	16	yes	—
275	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
276	C₂²×C₄	#41×C₆₇	1	4	yes	—
277	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
278	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
279	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
280	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
281	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
282	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
283	C₄⋊C₄	—	2	16	no	—
284	C₄⋊C₄	—	2	32	no	—
285	C₄⋊C₄	—	2	16	no	—
286	C₄⋊C₄	—	2	16	no	—
287	C₄⋊C₄	—	4	16	no	—
288	C₄⋊C₄	—	4	16	no	—
289	C₂²⋊C₄	—	2	16	no	—
290	C₂²⋊C₄	—	4	16	no	—
291	C₄²	—	2	8	no	—
292	C₄²	C₂×D₄×C₆₇	1	16	yes	—
293	C₂³	—	2	8	no	—
294	C₂³	—	2	16	no	—
295	C₂³	—	2	2	no	—
296	C₂³	—	2	2	no	—
297	C₂³	—	2	4	no	—
298	C₂³	—	2	4	no	—
299	C₂³	—	4	8	no	—
300	C₂³	—	4	4	no	—
301	C₂³	—	4	4	no	—
302	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
303	C₂³	C₂²×Q₈×C₆₇	1	4	yes	—
304	C₂³	C₂²×Q₈×C₆₇	1	2	yes	—
305	C₂³	C₂×#41×C₆₇	1	2	yes	—
306	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
307	C₂³	2₋¹⁺⁴×C₆₇	1	4	yes	—
308	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
309	C₂×C₄	—	2	8	no	—
310	C₂×C₄	—	2	16	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	2	32	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	8	no	—
318	C₂×C₄	—	2	8	no	—
319	C₂×C₄	—	2	8	no	—
320	C₂×C₄	—	2	8	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	2	8	no	—
323	C₂×C₄	—	2	4	no	—
324	C₂×C₄	—	2	8	no	—
325	C₂×C₄	—	4	8	no	—
326	C₂×C₄	—	4	16	no	—
327	C₂×C₄	—	4	16	no	—
328	C₂×C₄	C₂²⋊Q₈×C₆₇	1	16	yes	—
329	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
330	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
331	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
332	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
333	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
334	C₂²	—	2	4	no	—
335	C₂²	—	2	8	no	—
336	C₂²	—	4	16	no	—
337	C₂²	—	4	4	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	2	no	—
340	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	4	yes	—
341	C₂²	#537×C₆₇	1	8	yes	—
342	C₂²	#540×C₆₇	1	4	yes	—
343	C₂²	#541×C₆₇	1	4	yes	—
344	C₂²	#544×C₆₇	1	8	yes	—
345	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
346	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
347	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
348	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
349	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
350	C₄	—	2	4	no	—
351	C₄	—	2	8	no	—
352	C₄	—	4	4	no	—
353	C₄	—	4	8	no	—
354	C₄	—	4	8	no	—
355	C₄	C₂×C₂²⋊Q₈×C₆₇	1	8	yes	—
356	C₂	—	4	4	no	—
357	C₂	C₆₇×#2629	1	8	yes	direct product factor
358	C₂	C₂²×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
359	C₂	C₂×#537×C₆₇	1	2	yes	—
360	C₂	C₂×#540×C₆₇	1	1	yes	characteristic
361	C₂	C₂×#541×C₆₇	1	1	yes	characteristic
362	C₂	C₂×#544×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂⁵×C₆₇	C₂³
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×#125×C₆₇	C₂	semidirect
3. C₆₇×#2629	C₂	direct (N × H)
4. C₂×#398×C₆₇	C₂	semidirect
6. C₂×#389×C₆₇	C₂	semidirect
7. C₂×#384×C₆₇	C₂	semidirect
11. C₂×#125×C₆₇	C₂²	semidirect
24. #398×C₆₇	C₂²	semidirect
26. #389×C₆₇	C₂²	semidirect
27. #384×C₆₇	C₂²	semidirect
174. C₂×#2629	C₆₇	direct (N × H)
176. C₂×C₆₇	#2629	direct (N × H)
182. C₂²×#125	C₂×C₆₇	semidirect
184. #2629	C₂×C₆₇	direct (N × H)
185. C₂×#398	C₂×C₆₇	semidirect
187. C₂×#389	C₂×C₆₇	semidirect
188. C₂×#384	C₂×C₆₇	semidirect
190. C₆₇	C₂×#2629	direct (N × H)
193. C₂×#125	C₂²×C₆₇	semidirect
206. #398	C₂²×C₆₇	semidirect
208. #389	C₂²×C₆₇	semidirect
209. #384	C₂²×C₆₇	semidirect
357. C₂	C₆₇×#2629	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×#125×C₆₇	2
C₂²×#117×C₆₇	2
C₆₇×#2629	2
C₂×#398×C₆₇	2
C₂×#396×C₆₇	2
C₂×#389×C₆₇	2
C₂×#384×C₆₇	2
C₂×#380×C₆₇	2
C₂×#2629	67

Subgroups — C2×C67×#2629 #30713863

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2629 #30713863