Subgroups — C₆₇×#22643 #30701894

Order 17152.

780 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1939	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1924	C₂	1	4	yes	maximal
4	C₆₇×#1839	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1770	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1767	C₂	1	2	yes	maximal, complemented
8	C₄×C₂²⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#117×C₆₇	C₄	1	1	yes	characteristic
10	C₂×#117×C₆₇	C₄	1	1	yes	characteristic
11	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×#97×C₆₇	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
18	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊C₄×C₆₇	C₄	1	2	yes	—
23	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
24	#376×C₆₇	C₄	1	2	yes	—
25	C₄³×C₆₇	C₂²	1	2	yes	—
26	#336×C₆₇	C₂²	1	1	yes	characteristic
27	#336×C₆₇	C₂²	1	1	yes	characteristic
28	#336×C₆₇	C₂²	1	2	yes	—
29	#336×C₆₇	C₂²	1	4	yes	—
30	#336×C₆₇	C₂²	1	4	yes	—
31	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
36	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
37	#117×C₆₇	—	2	4	no	—
38	#117×C₆₇	—	2	4	no	—
39	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
44	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₆₇	—	2	4	no	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
54	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₆₇	—	2	8	no	—
57	C₄⋊C₈×C₆₇	—	2	4	no	—
58	C₂²⋊C₈×C₆₇	—	2	4	no	—
59	C₂²⋊C₈×C₆₇	—	4	2	no	—
60	#97×C₆₇	—	4	4	no	—
61	C₄×C₈×C₆₇	—	2	4	no	—
62	#95×C₆₇	C₂×C₄	1	2	yes	—
63	#95×C₆₇	C₂×C₄	1	2	yes	—
64	#95×C₆₇	C₂×C₄	1	2	yes	—
65	#95×C₆₇	C₂×C₄	1	2	yes	—
66	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	—	2	2	no	—
68	C₂²×C₄×C₆₇	—	2	2	no	—
69	C₂²×C₄×C₆₇	—	2	2	no	—
70	C₂²×C₄×C₆₇	—	2	2	no	—
71	C₂²×C₄×C₆₇	—	2	2	no	—
72	C₂²×C₄×C₆₇	—	2	2	no	—
73	C₂²×C₄×C₆₇	—	2	1	no	—
74	C₂²×C₄×C₆₇	—	2	1	no	—
75	C₂²×C₄×C₆₇	—	4	1	no	—
76	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
87	C₂×C₈×C₆₇	—	2	2	no	—
88	C₂×C₈×C₆₇	—	2	2	no	—
89	C₂×C₈×C₆₇	—	2	4	no	—
90	C₂×C₈×C₆₇	—	2	4	no	—
91	C₂×C₈×C₆₇	—	2	2	no	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	—	2	4	no	—
94	C₂×C₈×C₆₇	—	2	2	no	—
95	C₂×C₈×C₆₇	#41	1	4	yes	—
96	C₄⋊C₄×C₆₇	—	2	2	no	—
97	C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₄⋊C₄×C₆₇	M₁₆	1	4	yes	—
100	C₂²⋊C₄×C₆₇	—	2	4	no	—
101	C₂²⋊C₄×C₆₇	—	2	4	no	—
102	C₄²×C₆₇	—	2	2	no	—
103	C₄²×C₆₇	—	2	2	no	—
104	C₄²×C₆₇	—	2	4	no	—
105	C₄²×C₆₇	—	2	4	no	—
106	C₄²×C₆₇	—	2	4	no	—
107	C₄²×C₆₇	—	2	4	no	—
108	C₄²×C₆₇	—	2	2	no	—
109	C₄²×C₆₇	—	2	2	no	—
110	C₄²×C₆₇	—	2	2	no	—
111	C₄²×C₆₇	—	4	2	no	—
112	C₄²×C₆₇	#41	1	4	yes	—
113	C₂³×C₆₇	—	2	1	no	—
114	C₂³×C₆₇	—	2	1	no	—
115	C₂³×C₆₇	—	2	1	no	—
116	C₂³×C₆₇	—	2	1	no	—
117	C₂³×C₆₇	—	2	1	no	—
118	C₂³×C₆₇	—	2	1	no	—
119	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
120	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	—	2	1	no	—
122	C₂×C₄×C₆₇	—	2	1	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	1	no	—
125	C₂×C₄×C₆₇	—	2	1	no	—
126	C₂×C₄×C₆₇	—	2	1	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	1	no	—
129	C₂×C₄×C₆₇	—	2	8	no	—
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	2	1	no	—
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	2	1	no	—
136	C₂×C₄×C₆₇	—	2	2	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	4	4	no	—
143	C₂×C₄×C₆₇	—	4	2	no	—
144	C₂×C₄×C₆₇	—	4	2	no	—
145	C₂×C₄×C₆₇	#117	1	2	yes	—
146	C₂×C₄×C₆₇	#117	1	2	yes	—
147	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
148	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
149	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
150	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
151	C₂×C₄×C₆₇	#131	1	2	yes	—
152	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
153	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
154	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
156	C₈×C₆₇	—	2	4	no	—
157	C₈×C₆₇	—	4	4	no	—
158	C₈×C₆₇	—	4	2	no	—
159	C₂²×C₆₇	—	2	2	no	—
160	C₂²×C₆₇	—	2	2	no	—
161	C₂²×C₆₇	—	2	2	no	—
162	C₂²×C₆₇	—	4	1	no	—
163	C₂²×C₆₇	—	4	1	no	—
164	C₂²×C₆₇	—	4	1	no	—
165	C₂²×C₆₇	—	4	1	no	—
166	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
167	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
168	C₂²×C₆₇	#520	1	1	yes	characteristic
169	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
170	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
171	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
172	C₂²×C₆₇	#568	1	1	yes	characteristic
173	C₄×C₆₇	—	2	2	no	—
174	C₄×C₆₇	—	2	2	no	—
175	C₄×C₆₇	—	2	2	no	—
176	C₄×C₆₇	—	2	4	no	—
177	C₄×C₆₇	—	4	1	no	—
178	C₄×C₆₇	—	4	2	no	—
179	C₄×C₆₇	—	4	2	no	—
180	C₄×C₆₇	—	4	2	no	—
181	C₄×C₆₇	#432	1	4	yes	—
182	C₄×C₆₇	#553	1	2	yes	—
183	C₄×C₆₇	#556	1	2	yes	—
184	#22643	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
185	C₂×C₆₇	—	4	2	no	—
186	C₂×C₆₇	#2347	1	1	yes	characteristic
187	C₂×C₆₇	C₂×#432	1	1	yes	characteristic
188	C₂×C₆₇	#2924	1	1	yes	characteristic
189	C₂×C₆₇	#2965	1	1	yes	characteristic
190	C₂×C₆₇	#2970	1	1	yes	characteristic
191	C₂×C₆₇	#2989	1	1	yes	characteristic
192	C₂×C₆₇	#2991	1	1	yes	characteristic
193	C₄×#117	C₂×C₆₇	1	1	yes	characteristic
194	#1939	C₂×C₆₇	1	2	yes	complemented
195	#1924	C₂×C₆₇	1	4	yes	—
196	#1839	C₂×C₆₇	1	2	yes	complemented
197	#1797	C₂×C₆₇	1	1	yes	characteristic
198	#1770	C₂×C₆₇	1	2	yes	complemented
199	#1767	C₂×C₆₇	1	2	yes	complemented
200	C₄×C₂²⋊C₈	C₂×C₆₇	1	1	yes	characteristic
201	C₆₇	#22643	1	1	yes	characteristic, direct product factor, Sylow
202	C₂×#117	C₄×C₆₇	1	1	yes	characteristic
203	C₂×#117	C₄×C₆₇	1	1	yes	characteristic
204	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
205	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
206	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
207	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
208	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
209	C₂×#97	C₂²×C₆₇	1	4	yes	—
210	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
211	C₄×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
212	C₄×C₄⋊C₄	C₄×C₆₇	1	1	yes	characteristic
213	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
214	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
215	C₄×C₂²⋊C₄	C₄×C₆₇	1	2	yes	—
216	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
217	#376	C₄×C₆₇	1	2	yes	—
218	C₄³	C₂²×C₆₇	1	2	yes	—
219	#336	C₂²×C₆₇	1	2	yes	—
220	#336	C₂²×C₆₇	1	1	yes	characteristic
221	#336	C₂²×C₆₇	1	4	yes	—
222	#336	C₂²×C₆₇	1	4	yes	—
223	#336	C₂²×C₆₇	1	1	yes	characteristic
224	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
225	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
226	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
227	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
228	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
229	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
230	#117	—	2	4	no	—
231	#117	—	2	4	no	—
232	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
233	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
234	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
235	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
236	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
237	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
238	C₂×C₄²	—	2	4	no	—
239	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
240	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
241	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
242	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
243	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
244	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
245	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
246	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
247	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
248	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
249	C₄⋊C₈	—	2	8	no	—
250	C₄⋊C₈	—	2	4	no	—
251	C₂²⋊C₈	—	2	4	no	—
252	C₂²⋊C₈	—	4	2	no	—
253	#97	—	4	4	no	—
254	C₄×C₈	—	2	4	no	—
255	#95	C₂×C₄×C₆₇	1	2	yes	—
256	#95	C₂×C₄×C₆₇	1	2	yes	—
257	#95	C₂×C₄×C₆₇	1	2	yes	—
258	#95	C₂×C₄×C₆₇	1	2	yes	—
259	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	1	no	—
267	C₂²×C₄	—	2	1	no	—
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	M₁₆×C₆₇	1	4	yes	—
270	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
271	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
272	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
273	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
274	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
275	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
276	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
277	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
278	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
279	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	4	no	—
285	C₂×C₈	—	2	4	no	—
286	C₂×C₈	—	2	4	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	#41×C₆₇	1	4	yes	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	M₁₆×C₆₇	1	4	yes	—
293	C₂²⋊C₄	—	2	4	no	—
294	C₂²⋊C₄	—	2	4	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	4	no	—
298	C₄²	—	2	4	no	—
299	C₄²	—	2	4	no	—
300	C₄²	—	2	4	no	—
301	C₄²	—	2	2	no	—
302	C₄²	—	2	2	no	—
303	C₄²	—	2	2	no	—
304	C₄²	—	4	2	no	—
305	C₄²	#41×C₆₇	1	4	yes	—
306	C₂³	—	2	1	no	—
307	C₂³	—	2	1	no	—
308	C₂³	—	2	1	no	—
309	C₂³	—	2	1	no	—
310	C₂³	—	2	1	no	—
311	C₂³	—	2	1	no	—
312	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
313	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	8	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	4	4	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	#117×C₆₇	1	2	yes	—
339	C₂×C₄	#117×C₆₇	1	2	yes	—
340	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
341	C₂×C₄	C₂×M₁₆×C₆₇	1	4	yes	—
342	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
343	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
344	C₂×C₄	#131×C₆₇	1	2	yes	—
345	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
346	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
347	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
348	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
349	C₈	—	2	4	no	—
350	C₈	—	4	4	no	—
351	C₈	—	4	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	2	2	no	—
354	C₂²	—	2	2	no	—
355	C₂²	—	4	1	no	—
356	C₂²	—	4	1	no	—
357	C₂²	—	4	1	no	—
358	C₂²	—	4	1	no	—
359	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
360	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
361	C₂²	#520×C₆₇	1	1	yes	characteristic
362	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
363	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
364	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
365	C₂²	#568×C₆₇	1	1	yes	characteristic
366	C₄	—	2	2	no	—
367	C₄	—	2	2	no	—
368	C₄	—	2	2	no	—
369	C₄	—	2	4	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	2	no	—
372	C₄	—	4	2	no	—
373	C₄	—	4	2	no	—
374	C₄	#432×C₆₇	1	4	yes	—
375	C₄	#553×C₆₇	1	2	yes	—
376	C₄	#556×C₆₇	1	2	yes	—
377	C₂	—	4	2	no	—
378	C₂	C₆₇×#2347	1	1	yes	characteristic
379	C₂	C₂×#432×C₆₇	1	1	yes	characteristic
380	C₂	C₆₇×#2924	1	1	yes	characteristic
381	C₂	C₆₇×#2965	1	1	yes	characteristic
382	C₂	C₆₇×#2970	1	1	yes	characteristic
383	C₂	C₆₇×#2989	1	1	yes	characteristic
384	C₂	C₆₇×#2991	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
2. C₆₇×#1939	C₂	semidirect
4. C₆₇×#1839	C₂	semidirect
6. C₆₇×#1770	C₂	semidirect
7. C₆₇×#1767	C₂	semidirect
184. #22643	C₆₇	direct (N × H)
194. #1939	C₂×C₆₇	semidirect
196. #1839	C₂×C₆₇	semidirect
198. #1770	C₂×C₆₇	semidirect
199. #1767	C₂×C₆₇	semidirect
201. C₆₇	#22643	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×#117×C₆₇	2
C₆₇×#1939	2
C₆₇×#1924	2
C₆₇×#1839	2
C₆₇×#1797	2
C₆₇×#1770	2
C₆₇×#1767	2
C₄×C₂²⋊C₈×C₆₇	2
#22643	67

Subgroups — C67×#22643 #30701894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22643 #30701894