Subgroups — C₆₇×#22318 #30701569

Order 17152.

672 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	C₆₇×#2396	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1984	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1923	C₂	1	2	yes	maximal
4	C₆₇×#1923	C₂	1	2	yes	maximal
5	C₆₇×#1917	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1916	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1818	C₂	1	1	yes	maximal, characteristic
8	C₈×C₄⋊C₄×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1750	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	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	1	yes	characteristic
16	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
21	#400×C₆₇	C₄	1	1	yes	characteristic
22	#399×C₆₇	C₄	1	1	yes	characteristic
23	#398×C₆₇	C₄	1	2	yes	—
24	#396×C₆₇	C₄	1	2	yes	—
25	#395×C₆₇	C₄	1	1	yes	characteristic
26	#393×C₆₇	C₄	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
30	#336×C₆₇	C₂²	1	2	yes	—
31	#336×C₆₇	C₂²	1	2	yes	—
32	#336×C₆₇	C₂²	1	2	yes	—
33	#336×C₆₇	C₂²	1	2	yes	—
34	#336×C₆₇	C₂²	1	2	yes	—
35	#336×C₆₇	C₂²	1	2	yes	—
36	#336×C₆₇	C₂²	1	2	yes	—
37	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
41	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
43	C₄×Q₈×C₆₇	—	4	1	no	—
44	C₄×Q₈×C₆₇	—	4	1	no	—
45	#117×C₆₇	—	4	1	no	—
46	#117×C₆₇	—	4	1	no	—
47	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
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	2	yes	—
53	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
55	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
61	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
62	C₄⋊C₈×C₆₇	—	2	4	no	—
63	C₄⋊C₈×C₆₇	—	4	2	no	—
64	C₂²⋊C₈×C₆₇	—	2	4	no	—
65	C₂²⋊C₈×C₆₇	—	4	1	no	—
66	C₂²⋊C₈×C₆₇	—	4	1	no	—
67	#97×C₆₇	—	4	1	no	—
68	C₄×C₈×C₆₇	—	2	4	no	—
69	C₄×C₈×C₆₇	—	4	1	no	—
70	C₄×C₈×C₆₇	—	4	2	no	—
71	#95×C₆₇	C₂×C₄	1	2	yes	—
72	#95×C₆₇	C₂×C₄	1	2	yes	—
73	#95×C₆₇	C₂×C₄	1	2	yes	—
74	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
75	#95×C₆₇	C₂×C₄	1	1	yes	characteristic
76	#95×C₆₇	C₂×C₄	1	2	yes	—
77	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×Q₈×C₆₇	—	2	4	no	—
79	C₂×Q₈×C₆₇	—	4	1	no	—
80	C₂²×C₄×C₆₇	—	2	2	no	—
81	C₂²×C₄×C₆₇	—	2	1	no	—
82	C₂²×C₄×C₆₇	—	2	1	no	—
83	C₂²×C₄×C₆₇	—	2	2	no	—
84	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
90	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
94	C₂×C₈×C₆₇	—	2	2	no	—
95	C₂×C₈×C₆₇	—	2	2	no	—
96	C₂×C₈×C₆₇	—	2	2	no	—
97	C₂×C₈×C₆₇	—	2	2	no	—
98	C₂×C₈×C₆₇	—	2	2	no	—
99	C₂×C₈×C₆₇	—	2	2	no	—
100	C₂×C₈×C₆₇	—	2	2	no	—
101	C₂×C₈×C₆₇	—	2	2	no	—
102	C₂×C₈×C₆₇	—	2	2	no	—
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	—	2	2	no	—
105	C₂×C₈×C₆₇	#41	1	4	yes	—
106	C₄⋊C₄×C₆₇	—	2	4	no	—
107	C₄⋊C₄×C₆₇	—	2	4	no	—
108	C₄⋊C₄×C₆₇	—	2	4	no	—
109	C₄⋊C₄×C₆₇	—	4	1	no	—
110	C₄⋊C₄×C₆₇	—	4	1	no	—
111	C₄⋊C₄×C₆₇	—	4	2	no	—
112	C₄⋊C₄×C₆₇	—	4	1	no	—
113	C₂²⋊C₄×C₆₇	—	2	4	no	—
114	C₂²⋊C₄×C₆₇	—	2	4	no	—
115	C₂²⋊C₄×C₆₇	—	4	1	no	—
116	C₂²⋊C₄×C₆₇	—	4	1	no	—
117	C₄²×C₆₇	—	2	2	no	—
118	C₄²×C₆₇	—	2	2	no	—
119	C₄²×C₆₇	—	2	2	no	—
120	C₄²×C₆₇	—	2	1	no	—
121	C₄²×C₆₇	—	2	1	no	—
122	C₄²×C₆₇	—	2	1	no	—
123	C₄²×C₆₇	—	2	1	no	—
124	C₄²×C₆₇	—	2	2	no	—
125	C₂³×C₆₇	—	2	1	no	—
126	C₂³×C₆₇	—	2	1	no	—
127	C₂³×C₆₇	—	2	1	no	—
128	C₂³×C₆₇	—	2	2	no	—
129	C₂³×C₆₇	—	2	2	no	—
130	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
131	Q₈×C₆₇	—	4	4	no	—
132	C₂×C₄×C₆₇	—	2	2	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	1	no	—
138	C₂×C₄×C₆₇	—	2	1	no	—
139	C₂×C₄×C₆₇	—	2	1	no	—
140	C₂×C₄×C₆₇	—	2	1	no	—
141	C₂×C₄×C₆₇	—	2	1	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	2	1	no	—
144	C₂×C₄×C₆₇	—	2	1	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	2	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	4	2	no	—
154	C₂×C₄×C₆₇	—	4	2	no	—
155	C₂×C₄×C₆₇	#131	1	4	yes	—
156	C₂×C₄×C₆₇	#131	1	4	yes	—
157	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
158	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
160	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
161	C₈×C₆₇	—	2	4	no	—
162	C₈×C₆₇	—	4	2	no	—
163	C₈×C₆₇	—	4	2	no	—
164	C₈×C₆₇	—	4	2	no	—
165	C₂²×C₆₇	—	4	1	no	—
166	C₂²×C₆₇	—	4	2	no	—
167	C₂²×C₆₇	—	4	1	no	—
168	C₂²×C₆₇	—	4	1	no	—
169	C₂²×C₆₇	—	4	2	no	—
170	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
171	C₂²×C₆₇	#518	1	1	yes	characteristic
172	C₂²×C₆₇	#519	1	1	yes	characteristic
173	C₂²×C₆₇	C₂×#131	1	2	yes	—
174	C₂²×C₆₇	C₂×#131	1	2	yes	—
175	C₄×C₆₇	—	2	4	no	—
176	C₄×C₆₇	—	2	4	no	—
177	C₄×C₆₇	—	4	1	no	—
178	C₄×C₆₇	—	4	2	no	—
179	C₄×C₆₇	—	4	1	no	—
180	C₄×C₆₇	—	4	1	no	—
181	C₄×C₆₇	—	4	1	no	—
182	C₄×C₆₇	—	4	1	no	—
183	C₄×C₆₇	—	4	2	no	—
184	C₄×C₆₇	#539	1	2	yes	—
185	C₄×C₆₇	#542	1	2	yes	—
186	#22318	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
187	C₂×C₆₇	—	8	1	no	—
188	C₂×C₆₇	#2332	1	1	yes	characteristic
189	C₂×C₆₇	#2966	1	2	yes	—
190	C₂×C₆₇	#2977	1	1	yes	characteristic
191	C₂×C₆₇	#2979	1	1	yes	characteristic
192	C₂×C₆₇	#2986	1	2	yes	—
193	#2396	C₂×C₆₇	1	1	yes	characteristic
194	#1984	C₂×C₆₇	1	2	yes	complemented
195	#1923	C₂×C₆₇	1	2	yes	—
196	#1923	C₂×C₆₇	1	2	yes	—
197	#1917	C₂×C₆₇	1	2	yes	complemented
198	#1916	C₂×C₆₇	1	2	yes	complemented
199	#1818	C₂×C₆₇	1	1	yes	characteristic
200	C₈×C₄⋊C₄	C₂×C₆₇	1	2	yes	complemented
201	#1750	C₂×C₆₇	1	1	yes	characteristic
202	C₆₇	#22318	1	1	yes	characteristic, direct product factor, Sylow
203	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
204	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
205	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
206	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
207	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
208	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
209	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
210	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
211	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
212	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
213	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
214	#400	C₄×C₆₇	1	1	yes	characteristic
215	#399	C₄×C₆₇	1	1	yes	characteristic
216	#398	C₄×C₆₇	1	2	yes	—
217	#396	C₄×C₆₇	1	2	yes	—
218	#395	C₄×C₆₇	1	1	yes	characteristic
219	#393	C₄×C₆₇	1	1	yes	characteristic
220	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
221	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
222	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
223	#336	C₂²×C₆₇	1	2	yes	—
224	#336	C₂²×C₆₇	1	2	yes	—
225	#336	C₂²×C₆₇	1	2	yes	—
226	#336	C₂²×C₆₇	1	2	yes	—
227	#336	C₂²×C₆₇	1	2	yes	—
228	#336	C₂²×C₆₇	1	2	yes	—
229	#336	C₂²×C₆₇	1	2	yes	—
230	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
231	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
232	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
233	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
234	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
235	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
236	C₄×Q₈	—	4	1	no	—
237	C₄×Q₈	—	4	1	no	—
238	#117	—	4	1	no	—
239	#117	—	4	1	no	—
240	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
241	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
242	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
243	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
244	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
245	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
246	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
247	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
248	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
249	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
250	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
251	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
252	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
253	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
254	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
255	C₄⋊C₈	—	2	4	no	—
256	C₄⋊C₈	—	4	2	no	—
257	C₂²⋊C₈	—	2	4	no	—
258	C₂²⋊C₈	—	4	1	no	—
259	C₂²⋊C₈	—	4	1	no	—
260	#97	—	4	1	no	—
261	C₄×C₈	—	2	4	no	—
262	C₄×C₈	—	4	1	no	—
263	C₄×C₈	—	4	2	no	—
264	#95	C₂×C₄×C₆₇	1	2	yes	—
265	#95	C₂×C₄×C₆₇	1	2	yes	—
266	#95	C₂×C₄×C₆₇	1	2	yes	—
267	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
268	#95	C₂×C₄×C₆₇	1	1	yes	characteristic
269	#95	C₂×C₄×C₆₇	1	2	yes	—
270	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
271	C₂×Q₈	—	2	4	no	—
272	C₂×Q₈	—	4	1	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	1	no	—
275	C₂²×C₄	—	2	1	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
278	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
279	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
280	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
281	C₂²×C₄	C₂²×C₄×C₆₇	1	2	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	2	yes	—
285	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
286	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	#41×C₆₇	1	4	yes	—
299	C₄⋊C₄	—	2	4	no	—
300	C₄⋊C₄	—	2	4	no	—
301	C₄⋊C₄	—	2	4	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	4	2	no	—
305	C₄⋊C₄	—	4	1	no	—
306	C₂²⋊C₄	—	2	4	no	—
307	C₂²⋊C₄	—	2	4	no	—
308	C₂²⋊C₄	—	4	1	no	—
309	C₂²⋊C₄	—	4	1	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	2	1	no	—
313	C₄²	—	2	2	no	—
314	C₄²	—	2	1	no	—
315	C₄²	—	2	2	no	—
316	C₄²	—	2	1	no	—
317	C₄²	—	2	1	no	—
318	C₂³	—	2	1	no	—
319	C₂³	—	2	1	no	—
320	C₂³	—	2	1	no	—
321	C₂³	—	2	2	no	—
322	C₂³	—	2	2	no	—
323	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
324	Q₈	—	4	4	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	#131×C₆₇	1	4	yes	—
349	C₂×C₄	#131×C₆₇	1	4	yes	—
350	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
351	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
352	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
353	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
354	C₈	—	2	4	no	—
355	C₈	—	4	2	no	—
356	C₈	—	4	2	no	—
357	C₈	—	4	2	no	—
358	C₂²	—	4	1	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	4	1	no	—
361	C₂²	—	4	1	no	—
362	C₂²	—	4	2	no	—
363	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
364	C₂²	#518×C₆₇	1	1	yes	characteristic
365	C₂²	#519×C₆₇	1	1	yes	characteristic
366	C₂²	C₂×#131×C₆₇	1	2	yes	—
367	C₂²	C₂×#131×C₆₇	1	2	yes	—
368	C₄	—	2	4	no	—
369	C₄	—	2	4	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	2	no	—
372	C₄	—	4	1	no	—
373	C₄	—	4	1	no	—
374	C₄	—	4	1	no	—
375	C₄	—	4	1	no	—
376	C₄	—	4	2	no	—
377	C₄	#539×C₆₇	1	2	yes	—
378	C₄	#542×C₆₇	1	2	yes	—
379	C₂	—	8	1	no	—
380	C₂	C₆₇×#2332	1	1	yes	characteristic
381	C₂	C₆₇×#2966	1	2	yes	—
382	C₂	C₆₇×#2977	1	1	yes	characteristic
383	C₂	C₆₇×#2979	1	1	yes	characteristic
384	C₂	C₆₇×#2986	1	2	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	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×#1984	C₂	semidirect
5. C₆₇×#1917	C₂	semidirect
6. C₆₇×#1916	C₂	semidirect
8. C₈×C₄⋊C₄×C₆₇	C₂	semidirect
186. #22318	C₆₇	direct (N × H)
194. #1984	C₂×C₆₇	semidirect
197. #1917	C₂×C₆₇	semidirect
198. #1916	C₂×C₆₇	semidirect
200. C₈×C₄⋊C₄	C₂×C₆₇	semidirect
202. C₆₇	#22318	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2396	2
C₆₇×#1984	2
C₆₇×#1923	2
C₆₇×#1917	2
C₆₇×#1916	2
C₆₇×#1818	2
C₈×C₄⋊C₄×C₆₇	2
C₆₇×#1750	2
#22318	67

Subgroups — C67×#22318 #30701569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22318 #30701569