Subgroups — C₆₇×#21788 #30701039

Order 17152.

1782 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#432×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1955	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1954	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1779	C₂	1	4	yes	maximal, complemented
7	C₂³×D₄×C₆₇	C₄	1	1	yes	characteristic
8	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
9	C₂×C₄×D₄×C₆₇	—	2	4	no	—
10	C₂×C₄×D₄×C₆₇	—	2	4	no	—
11	C₂×C₄×D₄×C₆₇	C₄	1	2	yes	—
12	C₂×C₄×D₄×C₆₇	C₄	1	2	yes	—
13	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
16	C₂²×C₂²⋊C₄×C₆₇	C₄	1	2	yes	—
17	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	#432×C₆₇	—	2	4	no	—
19	#431×C₆₇	—	2	4	no	—
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
22	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₈×C₆₇	C₂²	1	16	yes	complemented
25	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
26	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
27	C₂⁵×C₆₇	C₂×C₄	1	2	yes	—
28	C₂²×D₄×C₆₇	—	2	8	no	—
29	C₂²×D₄×C₆₇	—	2	4	no	—
30	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
31	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
32	C₂³×C₄×C₆₇	—	2	4	no	—
33	C₂³×C₄×C₆₇	—	2	4	no	—
34	C₂³×C₄×C₆₇	—	4	8	no	—
35	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
38	C₂³×C₄×C₆₇	C₂×C₄	1	4	yes	—
39	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
40	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
42	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
43	C₄×D₄×C₆₇	—	2	8	no	—
44	C₄×D₄×C₆₇	—	2	8	no	—
45	C₄×D₄×C₆₇	—	2	4	no	—
46	C₄×D₄×C₆₇	—	2	4	no	—
47	C₄×D₄×C₆₇	—	2	4	no	—
48	C₄×D₄×C₆₇	—	2	4	no	—
49	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
50	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
53	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
54	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
55	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₆₇	D₄	1	4	yes	—
58	C₂×C₄²×C₆₇	D₄	1	4	yes	—
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₆₇	—	2	4	no	—
62	C₄⋊C₈×C₆₇	—	2	4	no	—
63	C₂²⋊C₈×C₆₇	—	2	4	no	—
64	C₂²⋊C₈×C₆₇	—	2	4	no	—
65	C₂²⋊C₈×C₆₇	—	4	16	no	—
66	#97×C₆₇	—	2	4	no	—
67	C₄×C₈×C₆₇	—	2	4	no	—
68	C₂⁴×C₆₇	—	2	4	no	—
69	C₂⁴×C₆₇	—	2	8	no	—
70	C₂⁴×C₆₇	—	2	8	no	—
71	C₂⁴×C₆₇	—	4	4	no	—
72	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
73	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×D₄×C₆₇	—	2	32	no	—
75	C₂×D₄×C₆₇	—	2	8	no	—
76	C₂×D₄×C₆₇	—	2	4	no	—
77	C₂×D₄×C₆₇	—	2	4	no	—
78	C₂×D₄×C₆₇	—	2	4	no	—
79	C₂×D₄×C₆₇	M₁₆	1	8	yes	—
80	C₂²×C₄×C₆₇	—	2	4	no	—
81	C₂²×C₄×C₆₇	—	2	4	no	—
82	C₂²×C₄×C₆₇	—	2	4	no	—
83	C₂²×C₄×C₆₇	—	2	4	no	—
84	C₂²×C₄×C₆₇	—	2	8	no	—
85	C₂²×C₄×C₆₇	—	2	4	no	—
86	C₂²×C₄×C₆₇	—	2	4	no	—
87	C₂²×C₄×C₆₇	—	2	16	no	—
88	C₂²×C₄×C₆₇	—	4	8	no	—
89	C₂²×C₄×C₆₇	—	4	4	no	—
90	C₂²×C₄×C₆₇	—	4	16	no	—
91	C₂²×C₄×C₆₇	—	4	16	no	—
92	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
93	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
94	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
95	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
103	C₂×C₈×C₆₇	—	2	4	no	—
104	C₂×C₈×C₆₇	—	2	4	no	—
105	C₂×C₈×C₆₇	—	2	8	no	—
106	C₂×C₈×C₆₇	—	2	8	no	—
107	C₄⋊C₄×C₆₇	—	2	4	no	—
108	C₄⋊C₄×C₆₇	—	2	4	no	—
109	C₂²⋊C₄×C₆₇	—	2	8	no	—
110	C₂²⋊C₄×C₆₇	—	2	8	no	—
111	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
112	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
113	C₄²×C₆₇	C₂×D₄	1	2	yes	—
114	C₄²×C₆₇	C₂×D₄	1	2	yes	—
115	C₄²×C₆₇	C₂×D₄	1	2	yes	—
116	C₄²×C₆₇	C₂×D₄	1	2	yes	—
117	C₂³×C₆₇	—	2	2	no	—
118	C₂³×C₆₇	—	2	2	no	—
119	C₂³×C₆₇	—	2	16	no	—
120	C₂³×C₆₇	—	2	4	no	—
121	C₂³×C₆₇	—	2	8	no	—
122	C₂³×C₆₇	—	2	4	no	—
123	C₂³×C₆₇	—	2	8	no	—
124	C₂³×C₆₇	—	4	32	no	—
125	C₂³×C₆₇	—	4	8	no	—
126	C₂³×C₆₇	—	4	8	no	—
127	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
128	C₂³×C₆₇	C₂×M₁₆	1	8	yes	—
129	C₂³×C₆₇	#131	1	4	yes	—
130	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
131	D₄×C₆₇	—	2	16	no	—
132	D₄×C₆₇	—	2	16	no	—
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	2	2	no	—
137	C₂×C₄×C₆₇	—	2	8	no	—
138	C₂×C₄×C₆₇	—	2	4	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	16	no	—
144	C₂×C₄×C₆₇	—	4	8	no	—
145	C₂×C₄×C₆₇	—	4	8	no	—
146	C₂×C₄×C₆₇	—	4	8	no	—
147	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
148	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
149	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
150	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
151	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
152	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
153	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
156	C₈×C₆₇	—	4	4	no	—
157	C₈×C₆₇	—	4	4	no	—
158	C₂²×C₆₇	—	2	2	no	—
159	C₂²×C₆₇	—	2	2	no	—
160	C₂²×C₆₇	—	2	8	no	—
161	C₂²×C₆₇	—	2	8	no	—
162	C₂²×C₆₇	—	2	16	no	—
163	C₂²×C₆₇	—	4	32	no	—
164	C₂²×C₆₇	—	4	4	no	—
165	C₂²×C₆₇	—	4	4	no	—
166	C₂²×C₆₇	—	4	8	no	—
167	C₂²×C₆₇	#408	1	4	yes	—
168	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
169	C₂²×C₆₇	#518	1	2	yes	—
170	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
171	C₂²×C₆₇	C₂×#131	1	2	yes	—
172	C₄×C₆₇	—	2	2	no	—
173	C₄×C₆₇	—	2	2	no	—
174	C₄×C₆₇	—	2	2	no	—
175	C₄×C₆₇	—	2	2	no	—
176	C₄×C₆₇	—	4	4	no	—
177	C₄×C₆₇	#407	1	4	yes	—
178	C₄×C₆₇	#535	1	2	yes	—
179	C₄×C₆₇	#535	1	2	yes	—
180	#21788	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
181	C₂×C₆₇	—	2	4	no	—
182	C₂×C₆₇	—	4	8	no	—
183	C₂×C₆₇	#2309	1	1	yes	characteristic
184	C₂×C₆₇	C₂×#407	1	1	yes	characteristic
185	C₂×C₆₇	C₂×#408	1	1	yes	characteristic
186	C₂×C₆₇	#2974	1	4	yes	—
187	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic
188	C₂×#432	C₂×C₆₇	1	1	yes	characteristic, complemented
189	C₂×#431	C₂×C₆₇	1	1	yes	characteristic, complemented
190	#1955	C₂×C₆₇	1	4	yes	complemented
191	#1954	C₂×C₆₇	1	4	yes	complemented
192	#1779	C₂×C₆₇	1	4	yes	complemented
193	C₆₇	#21788	1	1	yes	characteristic, direct product factor, Sylow
194	C₂³×D₄	C₄×C₆₇	1	1	yes	characteristic
195	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
196	C₂×C₄×D₄	—	2	4	no	—
197	C₂×C₄×D₄	—	2	4	no	—
198	C₂×C₄×D₄	C₄×C₆₇	1	2	yes	—
199	C₂×C₄×D₄	C₄×C₆₇	1	2	yes	—
200	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
201	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
202	C₂²×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
203	C₂²×C₂²⋊C₄	C₄×C₆₇	1	2	yes	—
204	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
205	#432	—	2	4	no	—
206	#431	—	2	4	no	—
207	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
208	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
209	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
210	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
211	C₂×C₂²⋊C₈	C₂²×C₆₇	1	16	yes	complemented
212	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
213	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
214	C₂⁵	C₂×C₄×C₆₇	1	2	yes	—
215	C₂²×D₄	—	2	8	no	—
216	C₂²×D₄	—	2	4	no	—
217	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
218	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
219	C₂³×C₄	—	2	4	no	—
220	C₂³×C₄	—	2	4	no	—
221	C₂³×C₄	—	4	8	no	—
222	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
223	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
224	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
225	C₂³×C₄	C₂×C₄×C₆₇	1	4	yes	—
226	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
227	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
228	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
229	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
230	C₄×D₄	—	2	8	no	—
231	C₄×D₄	—	2	8	no	—
232	C₄×D₄	—	2	4	no	—
233	C₄×D₄	—	2	4	no	—
234	C₄×D₄	—	2	4	no	—
235	C₄×D₄	—	2	4	no	—
236	C₂×C₄⋊C₄	—	2	4	no	—
237	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
238	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
239	C₂×C₂²⋊C₄	—	2	8	no	—
240	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
241	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
242	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
243	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
244	C₂×C₄²	D₄×C₆₇	1	4	yes	—
245	C₂×C₄²	D₄×C₆₇	1	4	yes	—
246	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
247	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
248	C₄⋊C₈	—	2	4	no	—
249	C₄⋊C₈	—	2	4	no	—
250	C₂²⋊C₈	—	2	4	no	—
251	C₂²⋊C₈	—	2	4	no	—
252	C₂²⋊C₈	—	4	16	no	—
253	#97	—	2	4	no	—
254	C₄×C₈	—	2	4	no	—
255	C₂⁴	—	2	4	no	—
256	C₂⁴	—	2	8	no	—
257	C₂⁴	—	2	8	no	—
258	C₂⁴	—	4	4	no	—
259	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
260	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
261	C₂×D₄	—	2	32	no	—
262	C₂×D₄	—	2	8	no	—
263	C₂×D₄	—	2	4	no	—
264	C₂×D₄	—	2	4	no	—
265	C₂×D₄	—	2	4	no	—
266	C₂×D₄	M₁₆×C₆₇	1	8	yes	—
267	C₂²×C₄	—	2	4	no	—
268	C₂²×C₄	—	2	4	no	—
269	C₂²×C₄	—	2	4	no	—
270	C₂²×C₄	—	2	4	no	—
271	C₂²×C₄	—	2	8	no	—
272	C₂²×C₄	—	2	4	no	—
273	C₂²×C₄	—	2	4	no	—
274	C₂²×C₄	—	2	16	no	—
275	C₂²×C₄	—	4	8	no	—
276	C₂²×C₄	—	4	4	no	—
277	C₂²×C₄	—	4	16	no	—
278	C₂²×C₄	—	4	16	no	—
279	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
280	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
281	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
282	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
283	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
284	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
285	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
286	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
287	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
288	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
289	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
290	C₂×C₈	—	2	4	no	—
291	C₂×C₈	—	2	4	no	—
292	C₂×C₈	—	2	8	no	—
293	C₂×C₈	—	2	8	no	—
294	C₄⋊C₄	—	2	4	no	—
295	C₄⋊C₄	—	2	4	no	—
296	C₂²⋊C₄	—	2	8	no	—
297	C₂²⋊C₄	—	2	8	no	—
298	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
299	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
300	C₄²	C₂×D₄×C₆₇	1	2	yes	—
301	C₄²	C₂×D₄×C₆₇	1	2	yes	—
302	C₄²	C₂×D₄×C₆₇	1	2	yes	—
303	C₄²	C₂×D₄×C₆₇	1	2	yes	—
304	C₂³	—	2	2	no	—
305	C₂³	—	2	2	no	—
306	C₂³	—	2	16	no	—
307	C₂³	—	2	4	no	—
308	C₂³	—	2	8	no	—
309	C₂³	—	2	4	no	—
310	C₂³	—	2	8	no	—
311	C₂³	—	4	32	no	—
312	C₂³	—	4	8	no	—
313	C₂³	—	4	8	no	—
314	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
315	C₂³	C₂×M₁₆×C₆₇	1	8	yes	—
316	C₂³	#131×C₆₇	1	4	yes	—
317	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
318	D₄	—	2	16	no	—
319	D₄	—	2	16	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	8	no	—
325	C₂×C₄	—	2	4	no	—
326	C₂×C₄	—	2	4	no	—
327	C₂×C₄	—	2	8	no	—
328	C₂×C₄	—	2	4	no	—
329	C₂×C₄	—	2	8	no	—
330	C₂×C₄	—	4	16	no	—
331	C₂×C₄	—	4	8	no	—
332	C₂×C₄	—	4	8	no	—
333	C₂×C₄	—	4	8	no	—
334	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
335	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
336	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
337	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
338	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
339	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
340	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
341	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
342	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
343	C₈	—	4	4	no	—
344	C₈	—	4	4	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	2	2	no	—
347	C₂²	—	2	8	no	—
348	C₂²	—	2	8	no	—
349	C₂²	—	2	16	no	—
350	C₂²	—	4	32	no	—
351	C₂²	—	4	4	no	—
352	C₂²	—	4	4	no	—
353	C₂²	—	4	8	no	—
354	C₂²	#408×C₆₇	1	4	yes	—
355	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
356	C₂²	#518×C₆₇	1	2	yes	—
357	C₂²	C₂²×M₁₆×C₆₇	1	2	yes	—
358	C₂²	C₂×#131×C₆₇	1	2	yes	—
359	C₄	—	2	2	no	—
360	C₄	—	2	2	no	—
361	C₄	—	2	2	no	—
362	C₄	—	2	2	no	—
363	C₄	—	4	4	no	—
364	C₄	#407×C₆₇	1	4	yes	—
365	C₄	#535×C₆₇	1	2	yes	—
366	C₄	#535×C₆₇	1	2	yes	—
367	C₂	—	2	4	no	—
368	C₂	—	4	8	no	—
369	C₂	C₆₇×#2309	1	1	yes	characteristic
370	C₂	C₂×#407×C₆₇	1	1	yes	characteristic
371	C₂	C₂×#408×C₆₇	1	1	yes	characteristic
372	C₂	C₆₇×#2974	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂³×D₄×C₆₇	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#432×C₆₇	C₂	semidirect
3. C₂×#431×C₆₇	C₂	semidirect
4. C₆₇×#1955	C₂	semidirect
5. C₆₇×#1954	C₂	semidirect
6. C₆₇×#1779	C₂	semidirect
20. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
21. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
24. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
25. C₂×#97×C₆₇	C₂²	semidirect
26. C₂×C₄×C₈×C₆₇	C₂²	semidirect
180. #21788	C₆₇	direct (N × H)
188. C₂×#432	C₂×C₆₇	semidirect
189. C₂×#431	C₂×C₆₇	semidirect
190. #1955	C₂×C₆₇	semidirect
191. #1954	C₂×C₆₇	semidirect
192. #1779	C₂×C₆₇	semidirect
193. C₆₇	#21788	direct (N × H)
207. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
208. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
211. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
212. C₂×#97	C₂²×C₆₇	semidirect
213. C₂×C₄×C₈	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₄×C₆₇	2
C₂×#432×C₆₇	2
C₂×#431×C₆₇	2
C₆₇×#1955	2
C₆₇×#1954	2
C₆₇×#1779	2
#21788	67

Subgroups — C67×#21788 #30701039

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21788 #30701039