Subgroups — #30748624

Order 17152.

516 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560361	C₂	1	1	yes	maximal, characteristic
2	#22560361	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863142	C₂	1	2	yes	maximal
4	#22558092	C₂	1	2	yes	maximal
5	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
8	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
12	#16863142	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
15	#391×C₆₇	—	2	1	no	—
16	#389×C₆₇	—	2	1	no	—
17	#389×C₆₇	C₄	1	2	yes	—
18	#384×C₆₇	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
20	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
21	C₄³×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₆₇⋊C₈	—	2	2	no	—
23	C₂²×C₆₇⋊C₈	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
27	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
29	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
30	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
31	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
32	C₂²×Q₈×C₆₇	—	2	1	no	—
33	C₄⋊Q₈×C₆₇	—	2	1	no	—
34	C₄⋊Q₈×C₆₇	—	2	1	no	—
35	C₄⋊Q₈×C₆₇	D₄	1	2	yes	—
36	C₄⋊Q₈×C₆₇	Q₈	1	2	yes	—
37	C₄×Q₈×C₆₇	—	4	2	no	—
38	C₄×Q₈×C₆₇	—	4	2	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
43	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₆₇	—	2	2	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	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
53	#95×C₆₇	—	2	2	no	—
54	#95×C₆₇	—	2	2	no	—
55	C₂×C₆₇⋊C₈	—	4	2	no	—
56	C₂×C₆₇⋊C₈	—	4	2	no	—
57	C₂×C₆₇⋊C₈	—	4	2	no	—
58	C₂×C₆₇⋊C₈	—	4	2	no	—
59	C₂×C₆₇⋊C₈	—	4	2	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₂×Q₈×C₆₇	—	2	2	no	—
62	C₂×Q₈×C₆₇	—	2	2	no	—
63	C₂×Q₈×C₆₇	—	4	1	no	—
64	C₂×Q₈×C₆₇	—	4	1	no	—
65	C₂×Q₈×C₆₇	—	4	1	no	—
66	C₂×Q₈×C₆₇	—	4	1	no	—
67	C₂²×C₄×C₆₇	—	2	2	no	—
68	C₂²×C₄×C₆₇	—	2	2	no	—
69	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₄⋊C₄×C₆₇	—	2	2	no	—
77	C₄⋊C₄×C₆₇	—	2	2	no	—
78	C₄⋊C₄×C₆₇	—	2	2	no	—
79	C₄⋊C₄×C₆₇	—	4	2	no	—
80	C₄⋊C₄×C₆₇	—	4	4	no	—
81	C₄⋊C₄×C₆₇	—	4	1	no	—
82	C₄⋊C₄×C₆₇	—	4	1	no	—
83	C₄⋊C₄×C₆₇	—	4	1	no	—
84	C₄⋊C₄×C₆₇	—	4	1	no	—
85	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
86	C₄²×C₆₇	—	2	2	no	—
87	C₄²×C₆₇	—	2	1	no	—
88	C₄²×C₆₇	—	2	1	no	—
89	C₄²×C₆₇	—	2	1	no	—
90	C₄²×C₆₇	—	2	1	no	—
91	C₄²×C₆₇	—	4	1	no	—
92	C₄²×C₆₇	—	4	4	no	—
93	C₄²×C₆₇	D₈	1	2	yes	—
94	C₄²×C₆₇	SD₁₆	1	2	yes	—
95	C₄²×C₆₇	SD₁₆	1	2	yes	—
96	C₄²×C₆₇	Q₁₆	1	2	yes	—
97	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
99	C₄²×C₆₇	#41	1	1	yes	characteristic
100	C₄²×C₆₇	#41	1	1	yes	characteristic
101	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	Q₈×C₆₇	—	4	2	no	—
103	Q₈×C₆₇	—	4	2	no	—
104	Q₈×C₆₇	—	4	2	no	—
105	Q₈×C₆₇	—	4	2	no	—
106	C₂×C₄×C₆₇	—	2	1	no	—
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	—	2	4	no	—
113	C₂×C₄×C₆₇	—	2	1	no	—
114	C₂×C₄×C₆₇	—	2	1	no	—
115	C₂×C₄×C₆₇	—	2	1	no	—
116	C₂×C₄×C₆₇	—	4	2	no	—
117	C₂×C₄×C₆₇	—	4	2	no	—
118	C₂×C₄×C₆₇	—	4	2	no	—
119	C₂×C₄×C₆₇	—	4	2	no	—
120	C₂×C₄×C₆₇	—	4	2	no	—
121	C₂×C₄×C₆₇	#101	1	2	yes	—
122	C₂×C₄×C₆₇	#102	1	2	yes	—
123	C₂×C₄×C₆₇	#102	1	2	yes	—
124	C₂×C₄×C₆₇	#103	1	2	yes	—
125	C₂×C₄×C₆₇	#103	1	2	yes	—
126	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
128	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
129	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
130	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
136	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
139	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
140	C₆₇⋊C₈	—	8	2	no	—
141	C₆₇⋊C₈	—	8	2	no	—
142	C₂²×C₆₇	#380	1	1	yes	characteristic
143	C₂²×C₆₇	C₂×#101	1	1	yes	characteristic
144	C₂²×C₆₇	#413	1	1	yes	characteristic
145	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
146	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
147	C₂²×C₆₇	#417	1	1	yes	characteristic
148	C₂²×C₆₇	#417	1	1	yes	characteristic
149	C₄×C₆₇	—	2	1	no	—
150	C₄×C₆₇	—	2	1	no	—
151	C₄×C₆₇	—	2	2	no	—
152	C₄×C₆₇	—	2	2	no	—
153	C₄×C₆₇	—	4	4	no	—
154	C₄×C₆₇	—	4	1	no	—
155	C₄×C₆₇	—	8	2	no	—
156	C₄×C₆₇	#332	1	4	yes	complemented
157	C₄×C₆₇	#459	1	1	yes	characteristic
158	C₄×C₆₇	#460	1	1	yes	characteristic
159	C₄×C₆₇	#461	1	1	yes	characteristic
160	C₄×C₆₇	#462	1	1	yes	characteristic
161	C₄×C₆₇	#474	1	1	yes	characteristic
162	C₄×C₆₇	#475	1	1	yes	characteristic
163	C₄×C₆₇	#476	1	1	yes	characteristic
164	C₄×C₆₇	#477	1	1	yes	characteristic
165	#8586	—	67	1	no	maximal, Sylow
166	C₂×C₆₇	C₂×#332	1	1	yes	characteristic
167	C₂×C₆₇	#1547	1	1	yes	characteristic
168	C₂×C₆₇	#1550	1	1	yes	characteristic
169	C₂×C₆₇	#1561	1	1	yes	characteristic
170	C₂×C₆₇	#1801	1	1	yes	characteristic
171	C₂×C₆₇	#1802	1	1	yes	characteristic
172	C₂×C₆₇	#1803	1	1	yes	characteristic
173	C₄×C₄⋊Q₈	D₆₇	1	1	yes	characteristic
174	#1842	—	67	1	no	—
175	#1842	—	67	1	no	—
176	C₂×#332	—	67	2	no	—
177	#1296	—	67	2	no	—
178	C₆₇	#8586	1	1	yes	characteristic, complemented, Sylow
179	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
180	C₂×C₄×Q₈	—	2	1	no	—
181	C₂×C₄⋊C₈	—	67	2	no	—
182	C₂×C₄⋊C₈	—	67	2	no	—
183	C₂×C₄⋊C₈	—	134	1	no	—
184	C₂×C₄⋊C₈	—	134	1	no	—
185	C₂×C₄⋊C₈	—	134	1	no	—
186	C₂×C₄⋊C₈	—	134	1	no	—
187	#391	—	2	1	no	—
188	#389	—	2	1	no	—
189	#389	Dic₆₇	1	2	yes	—
190	#384	C₂×D₆₇	1	1	yes	characteristic
191	C₄×C₄⋊C₄	—	2	1	no	—
192	C₄×C₄⋊C₄	Dic₆₇	1	2	yes	—
193	C₄³	C₂×D₆₇	1	1	yes	characteristic
194	#332	—	134	4	no	—
195	C₂²×Q₈	—	2	1	no	—
196	C₂²×C₈	—	134	2	no	—
197	C₂²×C₈	—	134	2	no	—
198	C₄⋊Q₈	—	2	1	no	—
199	C₄⋊Q₈	—	2	1	no	—
200	C₄⋊Q₈	Dic₁₃₄	1	2	yes	—
201	C₄⋊Q₈	D₂₆₈	1	2	yes	—
202	C₄×Q₈	—	4	2	no	—
203	C₄×Q₈	—	4	2	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
209	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
212	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
213	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
214	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
215	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
216	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
217	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
218	C₄⋊C₈	—	268	2	no	—
219	C₄⋊C₈	—	268	2	no	—
220	C₄⋊C₈	—	268	2	no	—
221	C₄⋊C₈	—	268	2	no	—
222	C₄⋊C₈	—	134	2	no	—
223	C₄⋊C₈	—	134	2	no	—
224	C₄⋊C₈	—	134	2	no	—
225	C₄⋊C₈	—	134	2	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	4	1	no	—
231	C₂×Q₈	—	4	1	no	—
232	C₂×Q₈	—	4	1	no	—
233	C₂×Q₈	—	4	1	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
237	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
238	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
239	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
240	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
243	C₂×C₈	—	268	2	no	—
244	C₂×C₈	—	268	2	no	—
245	C₂×C₈	—	268	2	no	—
246	C₂×C₈	—	268	2	no	—
247	C₂×C₈	—	268	2	no	—
248	C₂×C₈	—	268	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₄⋊C₄	—	4	1	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	C₄⋊Dic₆₇	1	4	yes	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	4	1	no	—
265	C₄²	—	4	4	no	—
266	C₄²	#11761613	1	2	yes	—
267	C₄²	#11761615	1	2	yes	—
268	C₄²	#11761616	1	2	yes	—
269	C₄²	#11761618	1	2	yes	—
270	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
271	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
272	C₄²	#11761628	1	1	yes	characteristic
273	C₄²	#11761628	1	1	yes	characteristic
274	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
275	Q₈	—	4	2	no	—
276	Q₈	—	4	2	no	—
277	Q₈	—	4	2	no	—
278	Q₈	—	4	2	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	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	#14523962	1	2	yes	—
295	C₂×C₄	#14523964	1	2	yes	—
296	C₂×C₄	#14523964	1	2	yes	—
297	C₂×C₄	#14523967	1	2	yes	—
298	C₂×C₄	#14523967	1	2	yes	—
299	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
300	C₂×C₄	#14524002	1	1	yes	characteristic
301	C₂×C₄	#14524017	1	1	yes	characteristic
302	C₂×C₄	#14524017	1	1	yes	characteristic
303	C₂×C₄	#14524022	1	1	yes	characteristic
304	C₂×C₄	#14524022	1	1	yes	characteristic
305	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
306	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
307	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
308	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
309	C₂×C₄	#14524068	1	1	yes	characteristic
310	C₂×C₄	#14524068	1	1	yes	characteristic
311	C₂×C₄	#14524073	1	1	yes	characteristic
312	C₂×C₄	#14524073	1	1	yes	characteristic
313	C₈	—	536	2	no	—
314	C₈	—	536	2	no	—
315	C₂²	#16863256	1	1	yes	characteristic
316	C₂²	C₂×#14523962	1	1	yes	characteristic
317	C₂²	#16863380	1	1	yes	characteristic
318	C₂²	C₂×#14523964	1	1	yes	characteristic
319	C₂²	C₂×#14523967	1	1	yes	characteristic
320	C₂²	#16863397	1	1	yes	characteristic
321	C₂²	#16863397	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	4	4	no	—
327	C₄	—	4	1	no	—
328	C₄	—	8	2	no	—
329	C₄	#16863142	1	4	yes	complemented
330	C₄	#16863591	1	1	yes	characteristic
331	C₄	#16863596	1	1	yes	characteristic
332	C₄	#16863601	1	1	yes	characteristic
333	C₄	#16863606	1	1	yes	characteristic
334	C₄	#16863676	1	1	yes	characteristic
335	C₄	#16863681	1	1	yes	characteristic
336	C₄	#16863686	1	1	yes	characteristic
337	C₄	#16863691	1	1	yes	characteristic
338	C₂	C₂×#16863142	1	1	yes	characteristic
339	C₂	#22558886	1	1	yes	characteristic
340	C₂	#22558899	1	1	yes	characteristic
341	C₂	#22558947	1	1	yes	characteristic
342	C₂	#22560205	1	1	yes	characteristic
343	C₂	#22560209	1	1	yes	characteristic
344	C₂	#22560214	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×C₄⋊Q₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×C₄⋊Q₈	D₆₇
hypocenter	C₆₇	#8586
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
156. C₄×C₆₇	#332	semidirect
178. C₆₇	#8586	semidirect
329. C₄	#16863142	semidirect

Maximal subgroups

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

subgroup	index
#22560361	2
C₂×#16863142	2
#22558092	2
C₄×C₄⋊Q₈×C₆₇	2
#8586	67