Subgroups — #30773566

Order 17152.

382 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×(C₂×D₆₇)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×(C₂×D₆₇)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863171	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×(C₂×D₆₇)⋊C₄	C₂	1	1	yes	maximal, characteristic
6	#22558286	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₆₇	C₂²	1	1	yes	characteristic
9	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×(C₂×D₆₇)⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
13	C₄×(C₂×D₆₇)⋊C₄	C₄	1	1	yes	characteristic
14	C₄⋊(C₆₇⋊C₁₆)	—	2	2	no	—
15	(C₂×D₆₇)⋊C₁₆	—	2	2	no	—
16	(C₂×D₆₇)⋊C₁₆	—	2	2	no	—
17	#16863171	—	2	2	no	—
18	#16863151	C₄	1	1	yes	characteristic
19	#16863150	C₄	1	1	yes	characteristic
20	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×D₆₇	—	2	1	no	—
27	C₂×C₈×D₆₇	—	2	1	no	—
28	C₂×C₈×D₆₇	—	2	1	no	—
29	C₂×C₈×D₆₇	—	2	1	no	—
30	C₂×C₈×D₆₇	—	2	1	no	—
31	C₂×C₈×D₆₇	—	2	1	no	—
32	C₂×(C₂×D₆₇)⋊C₄	C₈	1	1	yes	characteristic
33	C₂×(C₂×D₆₇)⋊C₄	C₈	1	1	yes	characteristic
34	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₆₇⋊C₁₆	—	2	1	no	—
36	C₂×C₆₇⋊C₁₆	—	2	1	no	—
37	C₂×C₆₇⋊C₁₆	—	2	1	no	—
38	C₂×C₆₇⋊C₁₆	—	2	1	no	—
39	C₂×C₆₇⋊C₁₆	—	2	1	no	—
40	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
41	(C₂×D₆₇)⋊C₈	—	2	2	no	—
42	C₈×Dic₆₇	—	2	1	no	—
43	C₈×Dic₆₇	—	2	1	no	—
44	C₆₇⋊#95	C₈	1	1	yes	characteristic
45	C₆₇⋊#95	C₈	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₁₆×C₆₇	—	2	1	no	—
50	C₂×C₁₆×C₆₇	—	2	1	no	—
51	C₂×C₁₆×C₆₇	—	2	1	no	—
52	C₂×C₁₆×C₆₇	—	2	1	no	—
53	C₂×C₁₆×C₆₇	—	2	1	no	—
54	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
55	C₄×C₈×C₆₇	—	2	1	no	—
56	C₄×C₈×C₆₇	—	2	1	no	—
57	C₂³×D₆₇	C₂×C₈	1	1	yes	characteristic
58	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
59	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
60	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
61	C₂×C₄×D₆₇	—	2	1	no	—
62	C₂×C₄×D₆₇	—	2	1	no	—
63	C₂×C₄×D₆₇	—	2	1	no	—
64	C₂×C₄×D₆₇	—	2	1	no	—
65	C₂×C₄×D₆₇	—	2	1	no	—
66	C₂×C₄×D₆₇	—	2	1	no	—
67	C₂×C₆₇⋊C₈	—	2	1	no	—
68	C₂×C₆₇⋊C₈	—	2	1	no	—
69	C₂×C₆₇⋊C₈	—	2	1	no	—
70	C₂×C₆₇⋊C₈	—	2	1	no	—
71	C₂×C₆₇⋊C₈	—	2	1	no	—
72	C₂×C₆₇⋊C₈	M₁₆	1	2	yes	—
73	C₈×D₆₇	—	4	1	no	—
74	C₈×D₆₇	—	4	1	no	—
75	C₈×D₆₇	—	4	1	no	—
76	C₈×D₆₇	—	4	1	no	—
77	(C₂×D₆₇)⋊C₄	—	2	2	no	—
78	(C₂×D₆₇)⋊C₄	—	2	2	no	—
79	C₄×Dic₆₇	—	2	1	no	—
80	C₄×Dic₆₇	—	2	1	no	—
81	C₆₇⋊C₁₆	—	2	2	no	—
82	C₆₇⋊C₁₆	—	4	1	no	—
83	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₆₇	—	2	1	no	—
87	C₂×C₈×C₆₇	—	2	1	no	—
88	C₂×C₈×C₆₇	—	2	1	no	—
89	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
92	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
93	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
94	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
95	C₄²×C₆₇	—	2	1	no	—
96	C₄²×C₆₇	—	2	1	no	—
97	C₁₆×C₆₇	—	2	2	no	—
98	C₁₆×C₆₇	—	4	1	no	—
99	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
100	C₂²×D₆₇	—	2	1	no	—
101	C₂²×D₆₇	—	2	1	no	—
102	C₂²×D₆₇	—	2	1	no	—
103	C₂²×D₆₇	—	2	1	no	—
104	C₂²×D₆₇	—	2	1	no	—
105	C₂²×D₆₇	—	2	1	no	—
106	C₂²×D₆₇	M₃₂	1	2	yes	—
107	C₂×C₄×C₆₇	—	2	1	no	—
108	C₂×C₄×C₆₇	—	2	1	no	—
109	C₂×C₄×C₆₇	—	2	1	no	—
110	C₂×C₄×C₆₇	—	2	1	no	—
111	C₂×C₄×C₆₇	—	2	1	no	—
112	C₂×C₄×C₆₇	M₃₂	1	2	yes	—
113	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
117	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
118	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
119	C₂×Dic₆₇	—	2	1	no	—
120	C₂×Dic₆₇	—	2	1	no	—
121	C₂×Dic₆₇	—	2	1	no	—
122	C₂×Dic₆₇	—	2	1	no	—
123	C₂×Dic₆₇	—	2	1	no	—
124	C₂×Dic₆₇	—	2	1	no	—
125	C₂×Dic₆₇	—	2	1	no	—
126	C₂×Dic₆₇	—	2	1	no	—
127	C₂×Dic₆₇	—	2	1	no	—
128	C₄×D₆₇	—	4	1	no	—
129	C₄×D₆₇	—	4	1	no	—
130	C₄×D₆₇	—	4	1	no	—
131	C₄×D₆₇	—	4	1	no	—
132	C₈×C₆₇	—	4	1	no	—
133	C₈×C₆₇	#121	1	1	yes	characteristic
134	C₈×C₆₇	#123	1	1	yes	characteristic
135	C₈×C₆₇	#124	1	1	yes	characteristic
136	C₈×C₆₇	#126	1	1	yes	characteristic
137	C₆₇⋊C₈	—	2	2	no	—
138	C₆₇⋊C₈	—	4	1	no	—
139	C₂²×C₆₇	#431	1	1	yes	characteristic
140	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
141	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
142	C₂²×C₆₇	C₂×M₃₂	1	1	yes	characteristic
143	C₂²×C₆₇	C₂×M₃₂	1	1	yes	characteristic
144	C₂²×C₆₇	#504	1	1	yes	characteristic
145	C₂²×C₆₇	#504	1	1	yes	characteristic
146	C₂×D₆₇	—	2	2	no	—
147	C₂×D₆₇	—	2	2	no	—
148	C₂×D₆₇	—	2	2	no	—
149	C₂×D₆₇	—	4	1	no	—
150	C₂×D₆₇	—	4	1	no	—
151	C₂×D₆₇	—	4	1	no	—
152	C₂×D₆₇	—	4	1	no	—
153	C₄×C₆₇	—	2	2	no	—
154	C₄×C₆₇	—	4	1	no	—
155	C₄×C₆₇	#388	1	1	yes	characteristic
156	C₄×C₆₇	#433	1	1	yes	characteristic
157	C₄×C₆₇	#435	1	1	yes	characteristic
158	C₄×C₆₇	#436	1	1	yes	characteristic
159	Dic₆₇	—	4	1	no	—
160	Dic₆₇	—	4	1	no	—
161	Dic₆₇	—	4	1	no	—
162	#13121	—	67	1	no	maximal, Sylow
163	C₂×C₆₇	#1923	1	1	yes	characteristic
164	C₂×C₆₇	#2163	1	1	yes	characteristic
165	C₂×C₆₇	#2164	1	1	yes	characteristic
166	C₂×C₆₇	#2168	1	1	yes	characteristic
167	C₂×C₆₇	#2168	1	1	yes	characteristic
168	C₂×C₆₇	#2168	1	1	yes	characteristic
169	C₂×C₆₇	#2169	1	1	yes	characteristic
170	D₆₇	—	4	2	no	—
171	C₂×C₄⋊C₁₆	—	67	1	no	—
172	C₂×C₂²⋊C₁₆	—	67	1	no	—
173	C₂×C₂²⋊C₁₆	—	67	1	no	—
174	C₂×#346	—	67	1	no	—
175	C₈×C₂²⋊C₄	—	67	1	no	—
176	#1374	—	67	1	no	—
177	#1374	D₆₇	1	1	yes	characteristic, complemented
178	C₆₇	#13121	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₈	—	67	1	no	—
180	C₂²×C₁₆	—	67	1	no	—
181	C₂²×C₁₆	—	67	1	no	—
182	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
183	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
184	C₂×C₂²⋊C₈	—	67	1	no	—
185	C₂×C₄×C₈	—	67	1	no	—
186	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
187	C₄×C₂²⋊C₄	—	67	1	no	—
188	C₄⋊C₁₆	—	134	2	no	—
189	C₂²⋊C₁₆	—	134	2	no	—
190	C₂²⋊C₁₆	—	134	2	no	—
191	#346	—	134	2	no	—
192	#336	—	67	1	no	—
193	#336	—	67	1	no	—
194	C₂³×C₄	—	67	1	no	—
195	C₂²×C₈	—	67	1	no	—
196	C₂²×C₈	—	67	1	no	—
197	C₂²×C₈	—	134	1	no	—
198	C₂²×C₈	—	134	1	no	—
199	C₂²×C₈	—	134	1	no	—
200	C₂²×C₈	—	134	1	no	—
201	C₂²×C₈	—	134	1	no	—
202	C₂²×C₈	—	134	1	no	—
203	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
205	C₂×C₂²⋊C₄	—	67	1	no	—
206	C₂×C₂²⋊C₄	—	67	1	no	—
207	C₂×C₄²	—	67	1	no	—
208	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
209	C₂×C₁₆	—	2	1	no	—
210	C₂×C₁₆	—	2	1	no	—
211	C₂×C₁₆	—	2	1	no	—
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	—	67	2	no	—
215	C₂×C₁₆	—	134	1	no	—
216	C₂×C₁₆	—	134	1	no	—
217	C₂×C₁₆	—	134	1	no	—
218	C₂×C₁₆	—	134	1	no	—
219	C₂×C₁₆	—	134	1	no	—
220	C₂×C₁₆	D₂₆₈	1	2	yes	—
221	C₂²⋊C₈	—	134	2	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	—	134	1	no	—
225	C₄×C₈	—	134	1	no	—
226	#95	—	67	1	no	—
227	#95	—	67	1	no	—
228	C₂⁴	—	67	1	no	—
229	C₂²×C₄	—	67	1	no	—
230	C₂²×C₄	—	67	1	no	—
231	C₂²×C₄	—	67	1	no	—
232	C₂²×C₄	—	134	1	no	—
233	C₂²×C₄	—	134	1	no	—
234	C₂²×C₄	—	134	1	no	—
235	C₂²×C₄	—	134	1	no	—
236	C₂²×C₄	—	134	1	no	—
237	C₂²×C₄	—	134	1	no	—
238	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
239	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	268	1	no	—
245	C₂×C₈	—	268	1	no	—
246	C₂×C₈	—	268	1	no	—
247	C₂×C₈	—	268	1	no	—
248	C₂×C₈	—	67	2	no	—
249	C₂×C₈	—	134	1	no	—
250	C₂×C₈	—	134	1	no	—
251	C₂×C₈	—	134	1	no	—
252	C₂×C₈	—	134	1	no	—
253	C₂×C₈	—	134	1	no	—
254	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
255	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
256	C₂×C₈	#11761627	1	1	yes	characteristic
257	C₂×C₈	#11761627	1	1	yes	characteristic
258	C₂×C₈	#11761628	1	1	yes	characteristic
259	C₂×C₈	#11761629	1	1	yes	characteristic
260	C₂²⋊C₄	—	134	2	no	—
261	C₂²⋊C₄	—	134	2	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	134	1	no	—
265	C₄²	—	134	1	no	—
266	C₁₆	—	2	2	no	—
267	C₁₆	—	4	1	no	—
268	C₁₆	—	268	1	no	—
269	C₁₆	—	134	2	no	—
270	C₂³	—	67	2	no	—
271	C₂³	—	134	1	no	—
272	C₂³	—	134	1	no	—
273	C₂³	—	134	1	no	—
274	C₂³	—	134	1	no	—
275	C₂³	—	134	1	no	—
276	C₂³	—	134	1	no	—
277	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	268	1	no	—
284	C₂×C₄	—	268	1	no	—
285	C₂×C₄	—	268	1	no	—
286	C₂×C₄	—	268	1	no	—
287	C₂×C₄	—	134	1	no	—
288	C₂×C₄	—	134	1	no	—
289	C₂×C₄	—	134	1	no	—
290	C₂×C₄	—	134	1	no	—
291	C₂×C₄	—	134	1	no	—
292	C₂×C₄	—	134	1	no	—
293	C₂×C₄	—	134	1	no	—
294	C₂×C₄	—	134	1	no	—
295	C₂×C₄	—	134	1	no	—
296	C₂×C₄	#14523982	1	2	yes	—
297	C₂×C₄	#14524001	1	1	yes	characteristic
298	C₂×C₄	(C₂²⋊D₆₇)⋊C₄	1	1	yes	characteristic
299	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
300	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
301	C₂×C₄	#14524049	1	1	yes	characteristic
302	C₂×C₄	#14524051	1	1	yes	characteristic
303	C₈	—	4	1	no	—
304	C₈	—	268	1	no	—
305	C₈	—	134	2	no	—
306	C₈	#14524018	1	1	yes	characteristic
307	C₈	#14524027	1	1	yes	characteristic
308	C₈	#14524029	1	1	yes	characteristic
309	C₈	#14524038	1	1	yes	characteristic
310	C₂²	—	268	1	no	—
311	C₂²	—	268	1	no	—
312	C₂²	—	268	1	no	—
313	C₂²	—	268	1	no	—
314	C₂²	—	134	2	no	—
315	C₂²	—	134	2	no	—
316	C₂²	—	134	2	no	—
317	C₂²	#16863448	1	1	yes	characteristic
318	C₂²	(C₂²⋊D₆₇)⋊C₈	1	1	yes	characteristic
319	C₂²	C₈×D₂₆₈	1	1	yes	characteristic
320	C₂²	M₃₂×D₆₇	1	1	yes	characteristic
321	C₂²	C₂×#14523982	1	1	yes	characteristic
322	C₂²	#16863812	1	1	yes	characteristic
323	C₂²	#16863814	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	4	1	no	—
326	C₄	—	268	1	no	—
327	C₄	—	268	1	no	—
328	C₄	—	268	1	no	—
329	C₄	#16863288	1	1	yes	characteristic
330	C₄	#16863456	1	1	yes	characteristic
331	C₄	#16863469	1	1	yes	characteristic
332	C₄	#16863471	1	1	yes	characteristic
333	C₂	—	268	2	no	—
334	C₂	#22560816	1	1	yes	characteristic
335	C₂	#22562011	1	1	yes	characteristic
336	C₂	#22562015	1	1	yes	characteristic
337	C₂	#22562031	1	1	yes	characteristic
338	C₂	#22562036	1	1	yes	characteristic
339	C₂	#22562037	1	1	yes	characteristic
340	C₂	#22562042	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator, socle	C₂³×C₆₇	C₂²×C₈
Fitting	C₆₇×#1374	C₂
prime radical	C₂³×D₆₇	C₂×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1374	D₆₇
hypocenter	C₆₇	#13121
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	semidirect
4. C₂×#16863171	C₂	semidirect
6. #22558286	C₂	semidirect
7. C₆₇×#1374	C₂	semidirect
177. #1374	D₆₇	semidirect
178. C₆₇	#13121	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₆₇⋊C₁₆)	2
C₂×(C₂×D₆₇)⋊C₁₆	2
C₂×#16863171	2
C₈×(C₂×D₆₇)⋊C₄	2
#22558286	2
C₆₇×#1374	2
#13121	67