Subgroups — C₆₇×#22716 #30701967

Order 17152.

906 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2352	C₂	1	2	yes	maximal
3	C₆₇×#1949	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1854	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1852	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1835	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1809	C₂	1	2	yes	maximal, complemented
8	C₄×#107×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1736	C₂	1	2	yes	maximal, complemented
10	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#117×C₆₇	C₂²	1	2	yes	—
13	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
14	#436×C₆₇	C₄	1	8	yes	complemented
15	C₂×#107×C₆₇	C₂²	1	2	yes	—
16	C₂×#107×C₆₇	C₂²	1	2	yes	—
17	C₂×#106×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	#388×C₆₇	—	2	2	no	—
22	#387×C₆₇	—	2	2	no	—
23	#386×C₆₇	—	2	2	no	—
24	#382×C₆₇	C₂²	1	2	yes	—
25	#380×C₆₇	C₂²	1	2	yes	—
26	#380×C₆₇	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
28	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
29	#340×C₆₇	C₂²	1	1	yes	characteristic
30	#340×C₆₇	C₂²	1	2	yes	—
31	#340×C₆₇	C₂²	1	2	yes	—
32	#340×C₆₇	C₂²	1	2	yes	—
33	#340×C₆₇	C₂²	1	1	yes	characteristic
34	#340×C₆₇	C₂²	1	1	yes	characteristic
35	#340×C₆₇	C₂²	1	1	yes	characteristic
36	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₆₇	—	2	2	no	—
38	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
39	C₂×M₁₆×C₆₇	—	4	2	no	—
40	C₂×M₁₆×C₆₇	—	4	2	no	—
41	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
44	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
46	C₄×D₄×C₆₇	Q₈	1	4	yes	—
47	#117×C₆₇	—	2	4	no	—
48	#117×C₆₇	—	4	2	no	—
49	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
50	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
52	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
57	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
59	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
61	C₂×C₄²×C₆₇	—	2	2	no	—
62	C₂×C₄²×C₆₇	—	2	2	no	—
63	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
64	#107×C₆₇	—	2	4	no	—
65	#107×C₆₇	—	4	2	no	—
66	#106×C₆₇	—	4	2	no	—
67	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	complemented
68	C₂²⋊C₈×C₆₇	C₂×C₄	1	8	yes	complemented
69	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
70	#95×C₆₇	—	2	2	no	—
71	#95×C₆₇	—	2	2	no	—
72	#95×C₆₇	—	2	2	no	—
73	#95×C₆₇	—	2	2	no	—
74	#95×C₆₇	—	2	2	no	—
75	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
76	C₂×D₄×C₆₇	—	4	2	no	—
77	C₂×D₄×C₆₇	C₄⋊C₄	1	4	yes	—
78	C₂²×C₄×C₆₇	—	2	2	no	—
79	C₂²×C₄×C₆₇	—	2	2	no	—
80	C₂²×C₄×C₆₇	—	2	2	no	—
81	C₂²×C₄×C₆₇	—	2	2	no	—
82	C₂²×C₄×C₆₇	—	2	2	no	—
83	C₂²×C₄×C₆₇	—	2	2	no	—
84	C₂²×C₄×C₆₇	—	2	2	no	—
85	C₂²×C₄×C₆₇	—	2	2	no	—
86	C₂²×C₄×C₆₇	—	4	4	no	—
87	C₂²×C₄×C₆₇	—	4	4	no	—
88	C₂²×C₄×C₆₇	—	8	2	no	—
89	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
90	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
94	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
95	M₁₆×C₆₇	—	4	8	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	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
103	C₄⋊C₄×C₆₇	—	2	2	no	—
104	C₄⋊C₄×C₆₇	—	2	2	no	—
105	C₄⋊C₄×C₆₇	—	2	2	no	—
106	C₄⋊C₄×C₆₇	—	2	2	no	—
107	C₄⋊C₄×C₆₇	—	2	2	no	—
108	C₄⋊C₄×C₆₇	—	2	2	no	—
109	C₄⋊C₄×C₆₇	—	2	2	no	—
110	C₄⋊C₄×C₆₇	—	4	4	no	—
111	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
112	C₄⋊C₄×C₆₇	#41	1	4	yes	—
113	C₂²⋊C₄×C₆₇	—	4	4	no	—
114	C₂²⋊C₄×C₆₇	—	4	4	no	—
115	C₂²⋊C₄×C₆₇	—	8	2	no	—
116	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	8	yes	—
117	C₄²×C₆₇	—	4	4	no	—
118	C₄²×C₆₇	—	8	2	no	—
119	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
120	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
121	C₂³×C₆₇	—	2	2	no	—
122	C₂³×C₆₇	—	2	2	no	—
123	C₂³×C₆₇	—	2	2	no	—
124	C₂³×C₆₇	—	2	2	no	—
125	C₂³×C₆₇	—	2	2	no	—
126	C₂³×C₆₇	—	2	2	no	—
127	C₂³×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
128	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
129	D₄×C₆₇	—	4	4	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	2	2	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	4	no	—
136	C₂×C₄×C₆₇	—	2	4	no	—
137	C₂×C₄×C₆₇	—	2	4	no	—
138	C₂×C₄×C₆₇	—	2	4	no	—
139	C₂×C₄×C₆₇	—	2	4	no	—
140	C₂×C₄×C₆₇	—	4	2	no	—
141	C₂×C₄×C₆₇	—	4	2	no	—
142	C₂×C₄×C₆₇	—	4	2	no	—
143	C₂×C₄×C₆₇	—	4	2	no	—
144	C₂×C₄×C₆₇	—	4	2	no	—
145	C₂×C₄×C₆₇	—	4	2	no	—
146	C₂×C₄×C₆₇	—	4	2	no	—
147	C₂×C₄×C₆₇	—	4	2	no	—
148	C₂×C₄×C₆₇	—	4	2	no	—
149	C₂×C₄×C₆₇	—	4	2	no	—
150	C₂×C₄×C₆₇	—	4	2	no	—
151	C₂×C₄×C₆₇	—	8	8	no	—
152	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
153	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
154	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
155	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
158	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
159	C₈×C₆₇	—	2	2	no	—
160	C₈×C₆₇	—	2	4	no	—
161	C₈×C₆₇	—	2	2	no	—
162	C₂²×C₆₇	—	2	4	no	—
163	C₂²×C₆₇	—	2	4	no	—
164	C₂²×C₆₇	—	2	4	no	—
165	C₂²×C₆₇	—	4	2	no	—
166	C₂²×C₆₇	—	4	2	no	—
167	C₂²×C₆₇	—	4	2	no	—
168	C₂²×C₆₇	—	4	2	no	—
169	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
170	C₂²×C₆₇	#546	1	2	yes	—
171	C₂²×C₆₇	#575	1	2	yes	—
172	C₂²×C₆₇	#576	1	1	yes	characteristic
173	C₂²×C₆₇	#578	1	1	yes	characteristic
174	C₄×C₆₇	—	2	4	no	—
175	C₄×C₆₇	—	2	2	no	—
176	C₄×C₆₇	—	2	2	no	—
177	C₄×C₆₇	—	4	4	no	—
178	C₄×C₆₇	—	8	2	no	—
179	C₄×C₆₇	—	8	2	no	—
180	C₄×C₆₇	—	8	2	no	—
181	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
182	C₄×C₆₇	#518	1	1	yes	characteristic
183	C₄×C₆₇	#554	1	2	yes	—
184	#22716	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
185	C₂×C₆₇	—	4	4	no	—
186	C₂×C₆₇	#2349	1	1	yes	characteristic
187	C₂×C₆₇	#2911	1	1	yes	characteristic
188	C₂×C₆₇	#2913	1	1	yes	characteristic
189	C₂×C₆₇	#3322	1	2	yes	—
190	C₂×C₆₇	#3327	1	2	yes	—
191	C₂×#436	C₂×C₆₇	1	1	yes	characteristic
192	#2352	C₂×C₆₇	1	2	yes	—
193	#1949	C₂×C₆₇	1	2	yes	complemented
194	#1854	C₂×C₆₇	1	2	yes	complemented
195	#1852	C₂×C₆₇	1	2	yes	complemented
196	#1835	C₂×C₆₇	1	1	yes	characteristic, complemented
197	#1809	C₂×C₆₇	1	2	yes	complemented
198	C₄×#107	C₂×C₆₇	1	1	yes	characteristic, complemented
199	#1736	C₂×C₆₇	1	2	yes	complemented
200	C₆₇	#22716	1	1	yes	characteristic, direct product factor, Sylow
201	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
202	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
203	C₂×#117	C₂²×C₆₇	1	2	yes	—
204	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
205	#436	C₄×C₆₇	1	8	yes	complemented
206	C₂×#107	C₂²×C₆₇	1	2	yes	—
207	C₂×#107	C₂²×C₆₇	1	2	yes	—
208	C₂×#106	C₂²×C₆₇	1	2	yes	—
209	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
210	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
211	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
212	#388	—	2	2	no	—
213	#387	—	2	2	no	—
214	#386	—	2	2	no	—
215	#382	C₂²×C₆₇	1	2	yes	—
216	#380	C₂²×C₆₇	1	2	yes	—
217	#380	C₂²×C₆₇	1	2	yes	—
218	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
219	C₄×C₂²⋊C₄	—	2	2	no	—
220	#340	C₂²×C₆₇	1	1	yes	characteristic
221	#340	C₂²×C₆₇	1	2	yes	—
222	#340	C₂²×C₆₇	1	1	yes	characteristic
223	#340	C₂²×C₆₇	1	1	yes	characteristic
224	#340	C₂²×C₆₇	1	2	yes	—
225	#340	C₂²×C₆₇	1	2	yes	—
226	#340	C₂²×C₆₇	1	1	yes	characteristic
227	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
228	C₂³×C₄	—	2	2	no	—
229	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
230	C₂×M₁₆	—	4	2	no	—
231	C₂×M₁₆	—	4	2	no	—
232	C₂×M₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
233	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
234	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
235	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
236	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
237	C₄×D₄	Q₈×C₆₇	1	4	yes	—
238	#117	—	2	4	no	—
239	#117	—	4	2	no	—
240	C₂×C₄⋊C₄	—	2	4	no	—
241	C₂×C₄⋊C₄	—	2	2	no	—
242	C₂×C₄⋊C₄	—	4	2	no	—
243	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
244	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
245	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
246	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
247	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
248	C₂×C₂²⋊C₄	—	2	2	no	—
249	C₂×C₂²⋊C₄	—	2	2	no	—
250	C₂×C₂²⋊C₄	—	2	2	no	—
251	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
252	C₂×C₄²	—	2	2	no	—
253	C₂×C₄²	—	2	2	no	—
254	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
255	#107	—	2	4	no	—
256	#107	—	4	2	no	—
257	#106	—	4	2	no	—
258	C₄⋊C₈	C₂×C₄×C₆₇	1	4	yes	complemented
259	C₂²⋊C₈	C₂×C₄×C₆₇	1	8	yes	complemented
260	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	complemented
261	#95	—	2	2	no	—
262	#95	—	2	2	no	—
263	#95	—	2	2	no	—
264	#95	—	2	2	no	—
265	#95	—	2	2	no	—
266	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
267	C₂×D₄	—	4	2	no	—
268	C₂×D₄	C₄⋊C₄×C₆₇	1	4	yes	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	4	4	no	—
278	C₂²×C₄	—	4	4	no	—
279	C₂²×C₄	—	8	2	no	—
280	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
281	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
282	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
283	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
284	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
285	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
286	M₁₆	—	4	8	no	—
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	C₂²×C₄×C₆₇	1	2	yes	—
292	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
293	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	4	4	no	—
302	C₄⋊C₄	C₄⋊C₄×C₆₇	1	4	yes	—
303	C₄⋊C₄	#41×C₆₇	1	4	yes	—
304	C₂²⋊C₄	—	4	4	no	—
305	C₂²⋊C₄	—	4	4	no	—
306	C₂²⋊C₄	—	8	2	no	—
307	C₂²⋊C₄	C₄⋊C₄×C₆₇	1	8	yes	—
308	C₄²	—	4	4	no	—
309	C₄²	—	8	2	no	—
310	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
311	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
312	C₂³	—	2	2	no	—
313	C₂³	—	2	2	no	—
314	C₂³	—	2	2	no	—
315	C₂³	—	2	2	no	—
316	C₂³	—	2	2	no	—
317	C₂³	—	2	2	no	—
318	C₂³	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
319	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
320	D₄	—	4	4	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	4	no	—
327	C₂×C₄	—	2	4	no	—
328	C₂×C₄	—	2	4	no	—
329	C₂×C₄	—	2	4	no	—
330	C₂×C₄	—	2	4	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	8	8	no	—
343	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
344	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
345	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
346	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
347	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
348	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
349	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
350	C₈	—	2	2	no	—
351	C₈	—	2	2	no	—
352	C₈	—	2	4	no	—
353	C₂²	—	2	4	no	—
354	C₂²	—	2	4	no	—
355	C₂²	—	2	4	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	4	2	no	—
359	C₂²	—	4	2	no	—
360	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
361	C₂²	#546×C₆₇	1	2	yes	—
362	C₂²	#575×C₆₇	1	2	yes	—
363	C₂²	#576×C₆₇	1	1	yes	characteristic
364	C₂²	#578×C₆₇	1	1	yes	characteristic
365	C₄	—	2	4	no	—
366	C₄	—	2	2	no	—
367	C₄	—	2	2	no	—
368	C₄	—	4	4	no	—
369	C₄	—	8	2	no	—
370	C₄	—	8	2	no	—
371	C₄	—	8	2	no	—
372	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
373	C₄	#518×C₆₇	1	1	yes	characteristic
374	C₄	#554×C₆₇	1	2	yes	—
375	C₂	—	4	4	no	—
376	C₂	C₆₇×#2349	1	1	yes	characteristic
377	C₂	C₆₇×#2911	1	1	yes	characteristic
378	C₂	C₆₇×#2913	1	1	yes	characteristic
379	C₂	C₆₇×#3322	1	2	yes	—
380	C₂	C₆₇×#3327	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1949	C₂	semidirect
4. C₆₇×#1854	C₂	semidirect
5. C₆₇×#1852	C₂	semidirect
6. C₆₇×#1835	C₂	semidirect
7. C₆₇×#1809	C₂	semidirect
8. C₄×#107×C₆₇	C₂	semidirect
9. C₆₇×#1736	C₂	semidirect
14. #436×C₆₇	C₄	semidirect
41. C₂×M₁₆×C₆₇	C₂×C₄	semidirect
67. C₄⋊C₈×C₆₇	C₂×C₄	semidirect
68. C₂²⋊C₈×C₆₇	C₂×C₄	semidirect
69. C₄×C₈×C₆₇	C₂×C₄	semidirect
184. #22716	C₆₇	direct (N × H)
193. #1949	C₂×C₆₇	semidirect
194. #1854	C₂×C₆₇	semidirect
195. #1852	C₂×C₆₇	semidirect
196. #1835	C₂×C₆₇	semidirect
197. #1809	C₂×C₆₇	semidirect
198. C₄×#107	C₂×C₆₇	semidirect
199. #1736	C₂×C₆₇	semidirect
200. C₆₇	#22716	direct (N × H)
205. #436	C₄×C₆₇	semidirect
232. C₂×M₁₆	C₂×C₄×C₆₇	semidirect
258. C₄⋊C₈	C₂×C₄×C₆₇	semidirect
259. C₂²⋊C₈	C₂×C₄×C₆₇	semidirect
260. C₄×C₈	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#436×C₆₇	2
C₆₇×#2352	2
C₆₇×#1949	2
C₆₇×#1854	2
C₆₇×#1852	2
C₆₇×#1835	2
C₆₇×#1809	2
C₄×#107×C₆₇	2
C₆₇×#1736	2
#22716	67

Subgroups — C67×#22716 #30701967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22716 #30701967