Subgroups — #30597624

Order 15616.

554 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462625	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519842	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×#6923	C₂	1	1	yes	maximal, characteristic, complemented
4	#21461021	C₂	1	1	yes	maximal, characteristic, complemented
5	#21460651	C₂	1	1	yes	maximal, characteristic, complemented
6	#21460646	C₂	1	1	yes	maximal, characteristic, complemented
7	#21444200	C₂	1	1	yes	maximal, characteristic
8	Q₈×#92420	C₄	1	2	yes	complemented
9	#16840790	—	2	2	no	—
10	#16840790	C₄	1	2	yes	complemented
11	#16840781	—	2	1	no	—
12	#16840781	C₂²	1	1	yes	characteristic
13	C₂×#14519842	C₄	1	2	yes	—
14	C₄×#11758213	C₂²	1	1	yes	characteristic
15	C₄×D₆₁⋊C₈	—	2	1	no	—
16	C₄×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₈×#6923	—	2	2	no	—
18	C₄²×#6923	C₂²	1	1	yes	characteristic
19	#16840632	C₄	1	2	yes	—
20	#92420⋊C₈	—	2	1	no	—
21	C₈×#92420	—	2	1	no	—
22	C₄×#11758226	C₂²	1	1	yes	characteristic
23	#16839814	C₂²	1	1	yes	characteristic
24	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
25	Q₈⋊#92418	C₄	1	1	yes	characteristic, complemented
26	#16839491	C₄	1	1	yes	characteristic, complemented
27	#16839489	C₄	1	1	yes	characteristic, complemented
28	C₆₁×Q₈⋊C₈	C₄	1	1	yes	characteristic, complemented
29	C₂×#11758213	C₂×C₄	1	2	yes	—
30	C₂×D₆₁⋊C₈	—	2	2	no	—
31	C₂×D₆₁⋊C₈	D₄	1	2	yes	—
32	C₂×C₄×#6923	—	2	1	no	—
33	C₂×C₄×#6923	—	2	1	no	—
34	C₂×C₄×#6923	C₂×C₄	1	2	yes	—
35	C₄⋊#92420	—	2	2	no	—
36	C₄⋊#92420	C₂×C₄	1	2	yes	—
37	#14519842	C₈	1	8	yes	complemented
38	#14519834	—	2	1	no	—
39	#14519834	—	2	2	no	—
40	#14519832	—	2	1	no	—
41	C₄×#92420	—	2	2	no	—
42	C₄×#92420	—	2	1	no	—
43	C₄×#92420	—	2	1	no	—
44	C₄×#92420	C₂×C₄	1	2	yes	—
45	C₄×#92420	D₄	1	1	yes	characteristic
46	C₄×#92420	D₄	1	1	yes	characteristic
47	C₈×#6923	—	2	8	no	—
48	C₂×#11758226	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₆₁	—	2	1	no	—
50	Q₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
51	C₄×D₂₄₄	—	2	1	no	—
52	C₄×D₂₄₄	C₂×C₄	1	1	yes	characteristic
53	#14519739	—	2	1	no	—
54	#14519739	C₂×C₄	1	1	yes	characteristic
55	#14519735	—	2	1	no	—
56	#14519735	C₂×C₄	1	1	yes	characteristic
57	C₄²×D₆₁	—	2	1	no	—
58	C₄²×D₆₁	C₂³	1	1	yes	characteristic
59	C₄⋊#92418	C₂×C₄	1	1	yes	characteristic
60	#14519689	—	2	1	no	—
61	C₄×#92418	C₂×C₄	1	1	yes	characteristic
62	C₈×Dic₆₁	—	2	1	no	—
63	C₈×Dic₆₁	—	2	1	no	—
64	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
65	C₄⋊C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
67	C₂²×#6923	—	2	2	no	—
68	C₂×Q₈×C₆₁	C₄²	1	1	yes	characteristic
69	#11758226	—	2	2	no	—
70	#11758226	C₂×C₈	1	4	yes	—
71	C₂×D₂₄₄	—	2	1	no	—
72	C₂×D₂₄₄	C₄²	1	1	yes	characteristic
73	C₂×C₄×D₆₁	—	2	1	no	—
74	C₂×C₄×D₆₁	—	2	1	no	—
75	C₂×C₄×D₆₁	—	2	1	no	—
76	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×D₆₁	#41	1	1	yes	characteristic
79	C₂×#92420	—	2	2	no	—
80	C₂×#92420	—	2	2	no	—
81	C₂×#92420	C₂²⋊C₄	1	2	yes	—
82	C₂×#92420	C₂²⋊C₄	1	2	yes	—
83	C₄×#6923	—	2	4	no	—
84	C₄×#6923	—	2	2	no	—
85	C₄×#6923	—	2	2	no	—
86	C₄×#6923	—	2	2	no	—
87	C₄×#6923	C₂×C₈	1	4	yes	complemented
88	#11758213	—	2	2	no	—
89	#11758213	C₂×C₈	1	4	yes	complemented
90	D₆₁⋊C₈	—	2	4	no	—
91	D₆₁⋊C₈	—	4	4	no	—
92	C₂×C₈×C₆₁	—	2	1	no	—
93	C₂×C₈×C₆₁	—	2	1	no	—
94	C₄⋊C₄×C₆₁	—	2	1	no	—
95	C₄⋊C₄×C₆₁	C₄²	1	1	yes	characteristic
96	C₄²×C₆₁	—	2	1	no	—
97	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
98	#11758198	—	2	1	no	—
99	#11758198	—	2	1	no	—
100	#11758198	—	2	1	no	—
101	C₄⋊Dic₆₁	—	2	1	no	—
102	C₄⋊Dic₆₁	C₄²	1	1	yes	characteristic
103	C₄×Dic₆₁	—	2	1	no	—
104	C₄×Dic₆₁	—	2	1	no	—
105	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
106	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
107	C₄×Dic₆₁	#41	1	1	yes	characteristic
108	C₂×#92418	—	2	1	no	—
109	C₂×#92418	—	2	1	no	—
110	C₈×D₆₁	—	2	4	no	—
111	C₂²×D₆₁	—	2	1	no	—
112	C₂²×D₆₁	C₄×D₄	1	1	yes	characteristic
113	C₂×#6923	—	2	4	no	—
114	C₂×#6923	—	2	4	no	—
115	C₂×#6923	—	2	4	no	—
116	Q₈×C₆₁	—	2	1	no	—
117	Q₈×C₆₁	C₄×C₈	1	2	yes	complemented
118	C₂×C₄×C₆₁	—	2	1	no	—
119	C₂×C₄×C₆₁	—	2	1	no	—
120	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
121	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
123	C₂×Dic₆₁	—	2	1	no	—
124	C₂×Dic₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×Dic₆₁	#117	1	1	yes	characteristic
126	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
127	D₂₄₄	—	2	4	no	—
128	D₂₄₄	—	2	1	no	—
129	D₂₄₄	C₄×C₈	1	2	yes	complemented
130	C₄×D₆₁	—	2	1	no	—
131	C₄×D₆₁	—	2	1	no	—
132	C₄×D₆₁	—	2	2	no	—
133	C₄×D₆₁	—	2	4	no	—
134	C₄×D₆₁	—	4	1	no	—
135	C₄×D₆₁	—	4	1	no	—
136	C₄×D₆₁	—	4	2	no	—
137	C₄×D₆₁	C₂²×C₈	1	2	yes	—
138	C₄×D₆₁	#131	1	2	yes	—
139	#92420	—	2	2	no	—
140	#92420	—	2	4	no	—
141	#92420	—	4	2	no	—
142	#92420	#104	1	4	yes	complemented
143	C₈×C₆₁	—	2	2	no	—
144	C₈×C₆₁	—	4	1	no	—
145	#92418	—	2	2	no	—
146	#92418	—	4	1	no	—
147	C₈×#104	—	61	1	no	maximal, Sylow
148	C₂²×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×D₆₁	—	2	1	no	—
150	C₂×D₆₁	—	2	1	no	—
151	C₂×D₆₁	—	2	2	no	—
152	C₂×D₆₁	—	4	1	no	—
153	C₂×D₆₁	—	4	1	no	—
154	C₂×D₆₁	C₈×D₄	1	2	yes	—
155	#6923	—	2	8	no	—
156	C₄×C₆₁	—	2	2	no	—
157	C₄×C₆₁	—	2	1	no	—
158	C₄×C₆₁	—	4	1	no	—
159	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
160	C₄×C₆₁	#405	1	1	yes	characteristic
161	C₄×C₆₁	#443	1	1	yes	characteristic
162	C₄×C₆₁	#443	1	1	yes	characteristic
163	Dic₆₁	—	2	1	no	—
164	Dic₆₁	—	4	1	no	—
165	Dic₆₁	C₂×#104	1	2	yes	—
166	Dic₆₁	C₈×D₄	1	2	yes	—
167	C₈×#41	—	61	1	no	—
168	C₄×#104	—	61	1	no	—
169	C₄²×C₈	—	61	1	no	—
170	#1476	—	61	1	no	—
171	C₈×M₁₆	—	61	1	no	—
172	#1278	—	61	1	no	—
173	#1276	—	61	1	no	—
174	C₂×C₆₁	C₈×C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₆₁	C₄×#104	1	1	yes	characteristic
176	C₂×C₆₁	#1764	1	1	yes	characteristic
177	D₆₁	—	2	2	no	—
178	D₆₁	—	4	2	no	—
179	C₄×#41	—	61	1	no	—
180	C₈×Q₈	—	61	2	no	—
181	C₈×D₄	—	61	2	no	—
182	C₈×D₄	—	122	2	no	—
183	#431	—	61	1	no	—
184	#431	—	61	1	no	—
185	#431	—	122	1	no	—
186	C₂×#104	—	61	2	no	—
187	C₄×M₁₆	—	61	1	no	—
188	C₂×C₄×C₈	—	61	1	no	—
189	C₂×C₄×C₈	—	61	1	no	—
190	C₂×C₄×C₈	—	122	1	no	—
191	C₂×C₄×C₈	—	122	2	no	—
192	C₄³	—	61	1	no	—
193	#339	—	61	2	no	—
194	Q₈⋊C₈	—	61	1	no	—
195	Q₈⋊C₈	#6923	1	1	yes	characteristic, complemented
196	D₄⋊C₈	—	61	1	no	—
197	D₄⋊C₈	—	61	1	no	—
198	#322	—	122	1	no	—
199	C₈²	—	122	1	no	—
200	C₆₁	C₈×#104	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	—	61	1	no	—
202	C₂×M₁₆	—	61	2	no	—
203	C₂²×C₈	—	61	2	no	—
204	C₂²×C₈	—	122	1	no	—
205	C₂²×C₈	—	122	2	no	—
206	C₄×Q₈	—	61	1	no	—
207	C₄×Q₈	C₂×#6923	1	1	yes	characteristic
208	C₄×D₄	—	61	1	no	—
209	C₄×D₄	—	61	1	no	—
210	C₄×D₄	—	122	1	no	—
211	C₄×D₄	—	122	1	no	—
212	#117	—	61	1	no	—
213	#117	—	122	1	no	—
214	C₂×C₄²	—	61	1	no	—
215	C₂×C₄²	—	61	2	no	—
216	C₂×C₄²	—	122	1	no	—
217	C₂×C₄²	—	122	1	no	—
218	C₂×C₄²	—	122	1	no	—
219	C₄⋊C₈	—	61	2	no	—
220	C₄⋊C₈	—	61	1	no	—
221	C₄⋊C₈	—	122	2	no	—
222	C₄⋊C₈	C₂×#6923	1	1	yes	characteristic
223	#104	—	61	8	no	—
224	C₂²⋊C₈	—	122	1	no	—
225	C₂²⋊C₈	—	122	2	no	—
226	C₂²⋊C₈	—	122	1	no	—
227	#97	—	122	1	no	—
228	C₄×C₈	—	61	2	no	—
229	C₄×C₈	—	61	1	no	—
230	C₄×C₈	—	61	1	no	—
231	C₄×C₈	—	61	1	no	—
232	C₄×C₈	—	122	1	no	—
233	C₄×C₈	—	122	1	no	—
234	C₄×C₈	—	122	1	no	—
235	C₄×C₈	—	122	1	no	—
236	C₄×C₈	—	122	2	no	—
237	C₄×C₈	—	122	8	no	—
238	C₄×C₈	C₂×#6923	1	1	yes	characteristic
239	#41	—	61	4	no	—
240	#41	—	122	2	no	—
241	C₂×Q₈	C₄×#6923	1	1	yes	characteristic
242	C₂×D₄	—	61	1	no	—
243	C₂×D₄	—	122	1	no	—
244	C₂²×C₄	—	61	1	no	—
245	C₂²×C₄	—	61	1	no	—
246	C₂²×C₄	—	61	1	no	—
247	C₂²×C₄	—	122	1	no	—
248	C₂²×C₄	—	122	1	no	—
249	C₂²×C₄	—	122	2	no	—
250	C₂²×C₄	—	122	1	no	—
251	M₁₆	—	61	4	no	—
252	M₁₆	—	122	2	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	61	2	no	—
256	C₂×C₈	—	61	2	no	—
257	C₂×C₈	—	122	4	no	—
258	C₂×C₈	—	122	4	no	—
259	C₂×C₈	—	122	2	no	—
260	C₂×C₈	—	122	1	no	—
261	C₂×C₈	—	122	2	no	—
262	C₂×C₈	—	122	1	no	—
263	C₂×C₈	—	244	4	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	61	1	no	—
266	C₄⋊C₄	—	122	1	no	—
267	C₄⋊C₄	C₄×#6923	1	1	yes	characteristic
268	C₂²⋊C₄	—	122	1	no	—
269	C₂²⋊C₄	—	122	1	no	—
270	C₂²⋊C₄	—	122	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	61	1	no	—
273	C₄²	—	61	1	no	—
274	C₄²	—	61	1	no	—
275	C₄²	—	61	4	no	—
276	C₄²	—	122	1	no	—
277	C₄²	—	122	1	no	—
278	C₄²	—	122	2	no	—
279	C₄²	—	122	4	no	—
280	C₄²	—	122	2	no	—
281	C₄²	—	122	2	no	—
282	C₄²	C₂²×#6923	1	1	yes	characteristic
283	C₂³	—	61	1	no	—
284	C₂³	—	122	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	C₈×#6923	1	2	yes	complemented
287	D₄	—	61	2	no	—
288	D₄	—	122	4	no	—
289	D₄	—	122	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	61	1	no	—
293	C₂×C₄	—	61	1	no	—
294	C₂×C₄	—	61	2	no	—
295	C₂×C₄	—	61	2	no	—
296	C₂×C₄	—	61	1	no	—
297	C₂×C₄	—	122	2	no	—
298	C₂×C₄	—	122	4	no	—
299	C₂×C₄	—	122	4	no	—
300	C₂×C₄	—	122	4	no	—
301	C₂×C₄	—	122	4	no	—
302	C₂×C₄	—	122	1	no	—
303	C₂×C₄	—	122	1	no	—
304	C₂×C₄	—	122	1	no	—
305	C₂×C₄	—	244	2	no	—
306	C₂×C₄	—	244	1	no	—
307	C₂×C₄	—	244	1	no	—
308	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
309	C₂×C₄	D₄×#6923	1	1	yes	characteristic
310	C₂×C₄	D₄×#6923	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	4	1	no	—
313	C₈	—	61	4	no	—
314	C₈	—	122	2	no	—
315	C₈	—	122	2	no	—
316	C₈	—	122	4	no	—
317	C₈	—	244	2	no	—
318	C₈	—	244	1	no	—
319	C₂²	—	61	2	no	—
320	C₂²	—	122	1	no	—
321	C₂²	—	122	1	no	—
322	C₂²	—	122	2	no	—
323	C₂²	—	244	1	no	—
324	C₂²	—	244	1	no	—
325	C₂²	C₂²⋊C₄×#6923	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	61	2	no	—
330	C₄	—	61	2	no	—
331	C₄	—	122	8	no	—
332	C₄	—	122	1	no	—
333	C₄	—	244	1	no	—
334	C₄	C₂×C₈×#6923	1	1	yes	characteristic
335	C₄	#16840728	1	1	yes	characteristic
336	C₄	#16840805	1	1	yes	characteristic
337	C₄	#16840806	1	1	yes	characteristic
338	C₂	—	122	2	no	—
339	C₂	—	244	2	no	—
340	C₂	#21461554	1	1	yes	characteristic
341	C₂	#104×#6923	1	1	yes	characteristic
342	C₂	#21461596	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6923
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₄²	C₂²×#6923
Fitting	C₆₁×Q₈⋊C₈	C₄
socle	C₂²×C₆₁	C₄×C₂²⋊C₄
prime radical	C₂×#11758226	C₂×C₄
cosocle	C₄²×D₆₁	C₂³
hypercenter	Q₈⋊C₈	#6923
hypocenter	C₆₁	C₈×#104
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ Q₈⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₄×C₈×#6923	C₂	semidirect
4. #21461021	C₂	semidirect
5. #21460651	C₂	semidirect
6. #21460646	C₂	semidirect
8. Q₈×#92420	C₄	semidirect
10. #16840790	C₄	semidirect
25. Q₈⋊#92418	C₄	semidirect
26. #16839491	C₄	semidirect
27. #16839489	C₄	semidirect
28. C₆₁×Q₈⋊C₈	C₄	semidirect
37. #14519842	C₈	semidirect
87. C₄×#6923	C₂×C₈	semidirect
89. #11758213	C₂×C₈	semidirect
117. Q₈×C₆₁	C₄×C₈	semidirect
129. D₂₄₄	C₄×C₈	semidirect
142. #92420	#104	semidirect
195. Q₈⋊C₈	#6923	semidirect
200. C₆₁	C₈×#104	semidirect
286. Q₈	C₈×#6923	semidirect

Maximal subgroups

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

subgroup	index
#21462625	2
C₄×#14519842	2
C₄×C₈×#6923	2
#21461021	2
#21460651	2
#21460646	2
#21444200	2
C₈×#104	61