Subgroups — C₆₇×#22603 #30701854

Order 17152.

924 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	C₂×#432×C₆₇	C₂	1	2	yes	maximal
2	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1939	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1917	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1817	C₂	1	2	yes	maximal
6	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1770	C₂	1	2	yes	maximal, complemented
8	C₄×C₂²⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
11	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
13	#432×C₆₇	—	2	8	no	—
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
16	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#97×C₆₇	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
23	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#95×C₆₇	C₄	1	2	yes	—
28	#336×C₆₇	C₂²	1	2	yes	—
29	#336×C₆₇	C₂²	1	4	yes	—
30	#336×C₆₇	C₂²	1	2	yes	—
31	#336×C₆₇	C₂²	1	4	yes	—
32	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
33	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
39	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	—	2	4	no	—
49	C₂×C₄²×C₆₇	—	2	4	no	—
50	C₂×C₄²×C₆₇	—	2	4	no	—
51	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
57	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
58	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₆₇	—	2	8	no	—
60	C₄⋊C₈×C₆₇	—	2	4	no	—
61	C₂²⋊C₈×C₆₇	—	2	4	no	—
62	C₂²⋊C₈×C₆₇	—	2	2	no	—
63	C₂²⋊C₈×C₆₇	D₄	1	4	yes	—
64	#97×C₆₇	—	2	8	no	—
65	C₄×C₈×C₆₇	—	2	4	no	—
66	#95×C₆₇	C₂×C₄	1	2	yes	—
67	#95×C₆₇	C₂×C₄	1	2	yes	—
68	#95×C₆₇	C₂×C₄	1	2	yes	—
69	#95×C₆₇	C₂×C₄	1	2	yes	—
70	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	—	2	2	no	—
72	C₂²×C₄×C₆₇	—	2	2	no	—
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	4	no	—
78	C₂²×C₄×C₆₇	—	2	2	no	—
79	C₂²×C₄×C₆₇	—	2	4	no	—
80	C₂²×C₄×C₆₇	—	2	2	no	—
81	C₂²×C₄×C₆₇	—	2	4	no	—
82	C₂²×C₄×C₆₇	—	4	2	no	—
83	C₂²×C₄×C₆₇	—	4	1	no	—
84	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
85	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
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	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₆₇	#41	1	2	yes	—
96	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
97	C₂×C₈×C₆₇	—	2	2	no	—
98	C₂×C₈×C₆₇	—	2	2	no	—
99	C₂×C₈×C₆₇	—	2	4	no	—
100	C₂×C₈×C₆₇	—	2	4	no	—
101	C₂×C₈×C₆₇	—	2	2	no	—
102	C₂×C₈×C₆₇	—	2	2	no	—
103	C₂×C₈×C₆₇	—	2	4	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
106	C₄⋊C₄×C₆₇	—	2	4	no	—
107	C₄⋊C₄×C₆₇	—	2	4	no	—
108	C₄⋊C₄×C₆₇	—	2	2	no	—
109	C₄⋊C₄×C₆₇	—	2	2	no	—
110	C₄⋊C₄×C₆₇	—	2	2	no	—
111	C₄⋊C₄×C₆₇	M₁₆	1	4	yes	—
112	C₄²×C₆₇	—	2	2	no	—
113	C₄²×C₆₇	—	2	2	no	—
114	C₄²×C₆₇	—	2	4	no	—
115	C₄²×C₆₇	—	2	4	no	—
116	C₄²×C₆₇	—	2	2	no	—
117	C₄²×C₆₇	—	2	2	no	—
118	C₄²×C₆₇	—	2	2	no	—
119	C₄²×C₆₇	—	2	2	no	—
120	C₄²×C₆₇	—	2	2	no	—
121	C₄²×C₆₇	#41	1	4	yes	—
122	C₂³×C₆₇	—	2	1	no	—
123	C₂³×C₆₇	—	2	1	no	—
124	C₂³×C₆₇	—	2	1	no	—
125	C₂³×C₆₇	—	2	1	no	—
126	C₂³×C₆₇	C₄×D₄	1	2	yes	—
127	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
128	C₂³×C₆₇	#131	1	2	yes	—
129	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	2	1	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	2	1	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	2	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	1	no	—
142	C₂×C₄×C₆₇	—	2	2	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	2	no	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	1	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	1	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	4	4	no	—
154	C₂×C₄×C₆₇	—	4	8	no	—
155	C₂×C₄×C₆₇	—	4	8	no	—
156	C₂×C₄×C₆₇	#117	1	2	yes	—
157	C₂×C₄×C₆₇	#117	1	2	yes	—
158	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
159	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
160	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
161	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
162	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
163	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
165	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
166	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
167	C₈×C₆₇	—	2	4	no	—
168	C₈×C₆₇	—	4	4	no	—
169	C₈×C₆₇	—	4	2	no	—
170	C₂²×C₆₇	—	2	2	no	—
171	C₂²×C₆₇	—	2	2	no	—
172	C₂²×C₆₇	—	2	2	no	—
173	C₂²×C₆₇	—	2	2	no	—
174	C₂²×C₆₇	—	2	2	no	—
175	C₂²×C₆₇	—	2	2	no	—
176	C₂²×C₆₇	#436	1	4	yes	—
177	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
178	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
179	C₂²×C₆₇	#519	1	1	yes	characteristic
180	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
181	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
182	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
183	C₂²×C₆₇	#568	1	1	yes	characteristic
184	C₄×C₆₇	—	2	2	no	—
185	C₄×C₆₇	—	2	2	no	—
186	C₄×C₆₇	—	2	2	no	—
187	C₄×C₆₇	—	2	4	no	—
188	C₄×C₆₇	—	4	1	no	—
189	C₄×C₆₇	—	4	2	no	—
190	C₄×C₆₇	—	4	2	no	—
191	C₄×C₆₇	—	4	2	no	—
192	C₄×C₆₇	#432	1	4	yes	complemented
193	C₄×C₆₇	#547	1	2	yes	—
194	C₄×C₆₇	#552	1	2	yes	—
195	#22603	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂×C₆₇	—	2	4	no	—
197	C₂×C₆₇	#2344	1	1	yes	characteristic
198	C₂×C₆₇	C₂×#432	1	1	yes	characteristic
199	C₂×C₆₇	#2924	1	1	yes	characteristic
200	C₂×C₆₇	C₂×#436	1	1	yes	characteristic
201	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
202	C₂×C₆₇	#2983	1	1	yes	characteristic
203	C₂×C₆₇	#2984	1	1	yes	characteristic
204	C₂×#432	C₂×C₆₇	1	2	yes	—
205	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
206	#1939	C₂×C₆₇	1	2	yes	complemented
207	#1917	C₂×C₆₇	1	4	yes	complemented
208	#1817	C₂×C₆₇	1	2	yes	—
209	#1797	C₂×C₆₇	1	1	yes	characteristic
210	#1770	C₂×C₆₇	1	2	yes	complemented
211	C₄×C₂²⋊C₈	C₂×C₆₇	1	1	yes	characteristic
212	C₆₇	#22603	1	1	yes	characteristic, direct product factor, Sylow
213	C₂²×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
214	C₂²×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
215	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
216	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
217	#432	—	2	8	no	—
218	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
219	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
220	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
221	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
222	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
223	C₂×#97	C₂²×C₆₇	1	4	yes	—
224	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
225	C₄×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
226	C₄×C₄⋊C₄	C₄×C₆₇	1	2	yes	—
227	C₄×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
228	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
229	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
230	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
231	C₂×#95	C₄×C₆₇	1	2	yes	—
232	#336	C₂²×C₆₇	1	2	yes	—
233	#336	C₂²×C₆₇	1	4	yes	—
234	#336	C₂²×C₆₇	1	4	yes	—
235	#336	C₂²×C₆₇	1	2	yes	—
236	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
237	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
238	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
239	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
240	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
241	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
242	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
243	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
244	C₂×C₄⋊C₄	—	2	4	no	—
245	C₂×C₄⋊C₄	—	2	4	no	—
246	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
247	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
248	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
249	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
250	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
251	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
252	C₂×C₄²	—	2	4	no	—
253	C₂×C₄²	—	2	4	no	—
254	C₂×C₄²	—	2	4	no	—
255	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
256	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
257	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
258	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
259	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
260	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
261	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
262	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
263	C₄⋊C₈	—	2	8	no	—
264	C₄⋊C₈	—	2	4	no	—
265	C₂²⋊C₈	—	2	4	no	—
266	C₂²⋊C₈	—	2	2	no	—
267	C₂²⋊C₈	D₄×C₆₇	1	4	yes	—
268	#97	—	2	8	no	—
269	C₄×C₈	—	2	4	no	—
270	#95	C₂×C₄×C₆₇	1	2	yes	—
271	#95	C₂×C₄×C₆₇	1	2	yes	—
272	#95	C₂×C₄×C₆₇	1	2	yes	—
273	#95	C₂×C₄×C₆₇	1	2	yes	—
274	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	2	2	no	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	2	4	no	—
280	C₂²×C₄	—	2	4	no	—
281	C₂²×C₄	—	2	4	no	—
282	C₂²×C₄	—	2	2	no	—
283	C₂²×C₄	—	2	4	no	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	2	4	no	—
286	C₂²×C₄	—	4	2	no	—
287	C₂²×C₄	—	4	1	no	—
288	C₂²×C₄	M₁₆×C₆₇	1	4	yes	—
289	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
290	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
291	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
292	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
293	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
294	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
295	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
296	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
297	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
298	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
299	C₂²×C₄	#41×C₆₇	1	2	yes	—
300	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	4	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	4	no	—
307	C₂×C₈	—	2	4	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
310	C₄⋊C₄	—	2	4	no	—
311	C₄⋊C₄	—	2	4	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	M₁₆×C₆₇	1	4	yes	—
316	C₄²	—	2	2	no	—
317	C₄²	—	2	2	no	—
318	C₄²	—	2	4	no	—
319	C₄²	—	2	4	no	—
320	C₄²	—	2	2	no	—
321	C₄²	—	2	2	no	—
322	C₄²	—	2	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	#41×C₆₇	1	4	yes	—
326	C₂³	—	2	1	no	—
327	C₂³	—	2	1	no	—
328	C₂³	—	2	1	no	—
329	C₂³	—	2	1	no	—
330	C₂³	C₄×D₄×C₆₇	1	2	yes	—
331	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
332	C₂³	#131×C₆₇	1	2	yes	—
333	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	4	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	2	8	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	4	4	no	—
358	C₂×C₄	—	4	8	no	—
359	C₂×C₄	—	4	8	no	—
360	C₂×C₄	#117×C₆₇	1	2	yes	—
361	C₂×C₄	#117×C₆₇	1	2	yes	—
362	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
363	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
364	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
365	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
366	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
367	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
368	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
369	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
370	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
371	C₈	—	2	4	no	—
372	C₈	—	4	4	no	—
373	C₈	—	4	2	no	—
374	C₂²	—	2	2	no	—
375	C₂²	—	2	2	no	—
376	C₂²	—	2	2	no	—
377	C₂²	—	2	2	no	—
378	C₂²	—	2	2	no	—
379	C₂²	—	2	2	no	—
380	C₂²	#436×C₆₇	1	4	yes	—
381	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
382	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
383	C₂²	#519×C₆₇	1	1	yes	characteristic
384	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
385	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
386	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
387	C₂²	#568×C₆₇	1	1	yes	characteristic
388	C₄	—	2	2	no	—
389	C₄	—	2	2	no	—
390	C₄	—	2	2	no	—
391	C₄	—	2	4	no	—
392	C₄	—	4	1	no	—
393	C₄	—	4	2	no	—
394	C₄	—	4	2	no	—
395	C₄	—	4	2	no	—
396	C₄	#432×C₆₇	1	4	yes	complemented
397	C₄	#547×C₆₇	1	2	yes	—
398	C₄	#552×C₆₇	1	2	yes	—
399	C₂	—	2	4	no	—
400	C₂	C₆₇×#2344	1	1	yes	characteristic
401	C₂	C₂×#432×C₆₇	1	1	yes	characteristic
402	C₂	C₆₇×#2924	1	1	yes	characteristic
403	C₂	C₂×#436×C₆₇	1	1	yes	characteristic
404	C₂	D₄×M₁₆×C₆₇	1	1	yes	characteristic
405	C₂	C₆₇×#2983	1	1	yes	characteristic
406	C₂	C₆₇×#2984	1	1	yes	characteristic

	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	C₂⁴×C₆₇	C₂²×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
3. C₆₇×#1939	C₂	semidirect
4. C₆₇×#1917	C₂	semidirect
7. C₆₇×#1770	C₂	semidirect
192. C₄×C₆₇	#432	semidirect
195. #22603	C₆₇	direct (N × H)
206. #1939	C₂×C₆₇	semidirect
207. #1917	C₂×C₆₇	semidirect
210. #1770	C₂×C₆₇	semidirect
212. C₆₇	#22603	direct (N × H)
396. C₄	#432×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#432×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₆₇×#1939	2
C₆₇×#1917	2
C₆₇×#1817	2
C₆₇×#1797	2
C₆₇×#1770	2
C₄×C₂²⋊C₈×C₆₇	2
#22603	67

Subgroups — C67×#22603 #30701854

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22603 #30701854