Subgroups — C₆₇×#22301 #30701552

Order 17152.

654 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#125×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1940	C₂	1	2	yes	maximal
3	C₆₇×#1939	C₂	1	2	yes	maximal
4	C₆₇×#1939	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1939	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1917	C₂	1	2	yes	maximal
7	C₆₇×#1916	C₂	1	2	yes	maximal
8	C₆₇×#1842	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1770	C₂	1	1	yes	maximal, characteristic
10	C₈×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1744	C₂	1	1	yes	maximal, characteristic
12	C₂×#125×C₆₇	C₄	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
19	C₂×#97×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	#389×C₆₇	C₄	1	1	yes	characteristic
24	#389×C₆₇	C₄	1	1	yes	characteristic
25	#387×C₆₇	C₄	1	2	yes	—
26	#387×C₆₇	C₄	1	2	yes	—
27	#382×C₆₇	C₄	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
30	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
32	C₄³×C₆₇	C₂²	1	1	yes	characteristic
33	#336×C₆₇	C₂²	1	1	yes	characteristic
34	#336×C₆₇	C₂²	1	1	yes	characteristic
35	#336×C₆₇	C₂²	1	1	yes	characteristic
36	#336×C₆₇	C₂²	1	1	yes	characteristic
37	#336×C₆₇	C₂²	1	2	yes	—
38	#336×C₆₇	C₂²	1	2	yes	—
39	#336×C₆₇	C₂²	1	2	yes	—
40	#336×C₆₇	C₂²	1	2	yes	—
41	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
43	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
45	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	#125×C₆₇	—	2	4	no	—
48	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
63	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
64	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
65	C₄⋊C₈×C₆₇	—	2	2	no	—
66	C₄⋊C₈×C₆₇	—	2	2	no	—
67	C₄⋊C₈×C₆₇	—	4	2	no	—
68	C₄⋊C₈×C₆₇	—	4	2	no	—
69	C₄⋊C₈×C₆₇	—	4	2	no	—
70	C₄⋊C₈×C₆₇	—	4	2	no	—
71	#97×C₆₇	—	4	2	no	—
72	C₄×C₈×C₆₇	—	2	2	no	—
73	C₄×C₈×C₆₇	—	4	2	no	—
74	C₄×C₈×C₆₇	Q₈	1	4	yes	—
75	#95×C₆₇	C₂×C₄	1	2	yes	—
76	#95×C₆₇	C₂×C₄	1	2	yes	—
77	#95×C₆₇	C₂×C₄	1	2	yes	—
78	#95×C₆₇	C₂×C₄	1	2	yes	—
79	#95×C₆₇	C₂×C₄	1	2	yes	—
80	#95×C₆₇	C₂×C₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
92	C₂×C₈×C₆₇	—	2	1	no	—
93	C₂×C₈×C₆₇	—	2	1	no	—
94	C₂×C₈×C₆₇	—	2	1	no	—
95	C₂×C₈×C₆₇	—	2	1	no	—
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	—	2	2	no	—
98	C₂×C₈×C₆₇	—	2	1	no	—
99	C₂×C₈×C₆₇	—	2	1	no	—
100	C₂×C₈×C₆₇	—	2	2	no	—
101	C₂×C₈×C₆₇	—	2	1	no	—
102	C₂×C₈×C₆₇	—	2	1	no	—
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	2	1	no	—
107	C₂×C₈×C₆₇	—	2	1	no	—
108	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
109	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
110	C₄⋊C₄×C₆₇	—	2	4	no	—
111	C₄⋊C₄×C₆₇	—	2	4	no	—
112	C₄⋊C₄×C₆₇	—	2	2	no	—
113	C₄⋊C₄×C₆₇	—	2	4	no	—
114	C₄⋊C₄×C₆₇	—	2	4	no	—
115	C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	M₁₆	1	4	yes	—
118	C₄²×C₆₇	—	2	2	no	—
119	C₄²×C₆₇	—	2	2	no	—
120	C₄²×C₆₇	—	2	2	no	—
121	C₄²×C₆₇	—	2	2	no	—
122	C₄²×C₆₇	—	2	1	no	—
123	C₄²×C₆₇	—	2	1	no	—
124	C₄²×C₆₇	—	2	2	no	—
125	C₄²×C₆₇	—	2	2	no	—
126	C₄²×C₆₇	—	2	1	no	—
127	C₄²×C₆₇	—	2	1	no	—
128	C₄²×C₆₇	—	2	2	no	—
129	C₄²×C₆₇	—	2	2	no	—
130	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
131	C₄²×C₆₇	#41	1	2	yes	—
132	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	2	1	no	—
135	C₂×C₄×C₆₇	—	2	1	no	—
136	C₂×C₄×C₆₇	—	2	1	no	—
137	C₂×C₄×C₆₇	—	2	1	no	—
138	C₂×C₄×C₆₇	—	2	1	no	—
139	C₂×C₄×C₆₇	—	2	1	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	1	no	—
142	C₂×C₄×C₆₇	—	2	1	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	1	no	—
147	C₂×C₄×C₆₇	—	2	1	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	1	no	—
150	C₂×C₄×C₆₇	—	2	1	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	2	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
157	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
158	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
159	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
160	C₂×C₄×C₆₇	#131	1	2	yes	—
161	C₂×C₄×C₆₇	#131	1	2	yes	—
162	C₂×C₄×C₆₇	#131	1	2	yes	—
163	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
166	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
167	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
168	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
169	C₈×C₆₇	—	2	2	no	—
170	C₈×C₆₇	—	2	2	no	—
171	C₈×C₆₇	—	4	2	no	—
172	C₈×C₆₇	—	4	2	no	—
173	C₈×C₆₇	—	4	1	no	—
174	C₈×C₆₇	—	4	1	no	—
175	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
176	C₂²×C₆₇	#520	1	1	yes	characteristic
177	C₂²×C₆₇	#520	1	1	yes	characteristic
178	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
179	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
180	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
181	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
182	C₄×C₆₇	—	2	2	no	—
183	C₄×C₆₇	—	2	4	no	—
184	C₄×C₆₇	—	2	2	no	—
185	C₄×C₆₇	—	2	2	no	—
186	C₄×C₆₇	—	4	1	no	—
187	C₄×C₆₇	—	4	1	no	—
188	C₄×C₆₇	—	4	2	no	—
189	C₄×C₆₇	—	4	2	no	—
190	C₄×C₆₇	—	4	2	no	—
191	C₄×C₆₇	#446	1	4	yes	—
192	C₄×C₆₇	#540	1	1	yes	characteristic
193	C₄×C₆₇	#541	1	1	yes	characteristic
194	C₄×C₆₇	#544	1	1	yes	characteristic
195	C₄×C₆₇	#544	1	1	yes	characteristic
196	#22301	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂×C₆₇	#2330	1	1	yes	characteristic
198	C₂×C₆₇	C₂×#446	1	1	yes	characteristic
199	C₂×C₆₇	#2960	1	1	yes	characteristic
200	C₂×C₆₇	#2990	1	1	yes	characteristic
201	C₂×C₆₇	#2991	1	1	yes	characteristic
202	C₂×C₆₇	#2994	1	1	yes	characteristic
203	C₂×C₆₇	#2994	1	1	yes	characteristic
204	C₄×#125	C₂×C₆₇	1	1	yes	characteristic
205	#1940	C₂×C₆₇	1	2	yes	—
206	#1939	C₂×C₆₇	1	1	yes	characteristic
207	#1939	C₂×C₆₇	1	1	yes	characteristic
208	#1939	C₂×C₆₇	1	2	yes	—
209	#1917	C₂×C₆₇	1	2	yes	—
210	#1916	C₂×C₆₇	1	2	yes	—
211	#1842	C₂×C₆₇	1	1	yes	characteristic
212	#1770	C₂×C₆₇	1	1	yes	characteristic
213	C₈×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
214	#1744	C₂×C₆₇	1	1	yes	characteristic
215	C₆₇	#22301	1	1	yes	characteristic, direct product factor, Sylow
216	C₂×#125	C₄×C₆₇	1	1	yes	characteristic
217	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
218	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
219	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
220	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
221	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
222	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
223	C₂×#97	C₂²×C₆₇	1	2	yes	—
224	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
225	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
226	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
227	#389	C₄×C₆₇	1	1	yes	characteristic
228	#389	C₄×C₆₇	1	1	yes	characteristic
229	#387	C₄×C₆₇	1	2	yes	—
230	#387	C₄×C₆₇	1	2	yes	—
231	#382	C₄×C₆₇	1	1	yes	characteristic
232	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
233	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
234	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
235	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
236	C₄³	C₂²×C₆₇	1	1	yes	characteristic
237	#336	C₂²×C₆₇	1	1	yes	characteristic
238	#336	C₂²×C₆₇	1	2	yes	—
239	#336	C₂²×C₆₇	1	2	yes	—
240	#336	C₂²×C₆₇	1	2	yes	—
241	#336	C₂²×C₆₇	1	1	yes	characteristic
242	#336	C₂²×C₆₇	1	2	yes	—
243	#336	C₂²×C₆₇	1	1	yes	characteristic
244	#336	C₂²×C₆₇	1	1	yes	characteristic
245	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
246	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
247	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
248	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
249	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
250	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
251	#125	—	2	4	no	—
252	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
253	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
254	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
255	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
256	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
257	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
258	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
259	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
260	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
261	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
262	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
263	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
264	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
265	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
266	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
267	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
268	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
269	C₄⋊C₈	—	2	2	no	—
270	C₄⋊C₈	—	2	2	no	—
271	C₄⋊C₈	—	4	2	no	—
272	C₄⋊C₈	—	4	2	no	—
273	C₄⋊C₈	—	4	2	no	—
274	C₄⋊C₈	—	4	2	no	—
275	#97	—	4	2	no	—
276	C₄×C₈	—	2	2	no	—
277	C₄×C₈	—	4	2	no	—
278	C₄×C₈	Q₈×C₆₇	1	4	yes	—
279	#95	C₂×C₄×C₆₇	1	2	yes	—
280	#95	C₂×C₄×C₆₇	1	2	yes	—
281	#95	C₂×C₄×C₆₇	1	2	yes	—
282	#95	C₂×C₄×C₆₇	1	2	yes	—
283	#95	C₂×C₄×C₆₇	1	2	yes	—
284	#95	C₂×C₄×C₆₇	1	2	yes	—
285	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
286	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
287	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
288	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
289	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
290	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
291	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
292	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
293	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
294	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
295	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	1	no	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
313	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
314	C₄⋊C₄	—	2	4	no	—
315	C₄⋊C₄	—	2	4	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	2	4	no	—
318	C₄⋊C₄	—	2	4	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	M₁₆×C₆₇	1	4	yes	—
322	C₄²	—	2	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	—	2	2	no	—
326	C₄²	—	2	2	no	—
327	C₄²	—	2	2	no	—
328	C₄²	—	2	1	no	—
329	C₄²	—	2	1	no	—
330	C₄²	—	2	1	no	—
331	C₄²	—	2	1	no	—
332	C₄²	—	2	2	no	—
333	C₄²	—	2	2	no	—
334	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
335	C₄²	#41×C₆₇	1	2	yes	—
336	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
361	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
362	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
363	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
364	C₂×C₄	#131×C₆₇	1	2	yes	—
365	C₂×C₄	#131×C₆₇	1	2	yes	—
366	C₂×C₄	#131×C₆₇	1	2	yes	—
367	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
368	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
369	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
370	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
371	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
372	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
373	C₈	—	2	2	no	—
374	C₈	—	2	2	no	—
375	C₈	—	4	1	no	—
376	C₈	—	4	2	no	—
377	C₈	—	4	2	no	—
378	C₈	—	4	1	no	—
379	C₂²	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
380	C₂²	#520×C₆₇	1	1	yes	characteristic
381	C₂²	#520×C₆₇	1	1	yes	characteristic
382	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
383	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
384	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
385	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
386	C₄	—	2	2	no	—
387	C₄	—	2	4	no	—
388	C₄	—	2	2	no	—
389	C₄	—	2	2	no	—
390	C₄	—	4	1	no	—
391	C₄	—	4	1	no	—
392	C₄	—	4	2	no	—
393	C₄	—	4	2	no	—
394	C₄	—	4	2	no	—
395	C₄	#446×C₆₇	1	4	yes	—
396	C₄	#540×C₆₇	1	1	yes	characteristic
397	C₄	#541×C₆₇	1	1	yes	characteristic
398	C₄	#544×C₆₇	1	1	yes	characteristic
399	C₄	#544×C₆₇	1	1	yes	characteristic
400	C₂	C₆₇×#2330	1	1	yes	characteristic
401	C₂	C₂×#446×C₆₇	1	1	yes	characteristic
402	C₂	C₆₇×#2960	1	1	yes	characteristic
403	C₂	C₆₇×#2990	1	1	yes	characteristic
404	C₂	C₆₇×#2991	1	1	yes	characteristic
405	C₂	C₆₇×#2994	1	1	yes	characteristic
406	C₂	C₆₇×#2994	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle, prime radical	C₂³×C₆₇	C₂³×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
196. #22301	C₆₇	direct (N × H)
215. C₆₇	#22301	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#125×C₆₇	2
C₆₇×#1940	2
C₆₇×#1939	2
C₆₇×#1917	2
C₆₇×#1916	2
C₆₇×#1842	2
C₆₇×#1770	2
C₈×C₄⋊C₄×C₆₇	2
C₆₇×#1744	2
#22301	67

Subgroups — C67×#22301 #30701552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22301 #30701552