Subgroups — #29698861

Order 15616.

750 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
2	#21448778	C₂	1	2	yes	maximal
3	#21446584	C₂	1	4	yes	maximal
4	#21446580	C₂	1	2	yes	maximal
5	#21445982	C₂	1	2	yes	maximal
6	#107×Dic₆₁	C₂	1	2	yes	maximal
7	C₄×#14519711	C₂	1	1	yes	maximal, characteristic
8	C₆₁×#1845	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
10	C₈×Dic₁₂₂	C₄	1	8	yes	complemented
11	C₂×#14519711	C₂²	1	1	yes	characteristic
12	C₂×#14519711	C₂²	1	1	yes	characteristic
13	C₂×#14519708	C₂²	1	4	yes	—
14	C₂×C₄⋊#92418	C₂²	1	1	yes	characteristic
15	C₂×#14519706	C₂²	1	2	yes	—
16	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
17	#16839667	—	2	2	no	—
18	#16839643	—	2	4	no	—
19	#16839627	C₂²	1	4	yes	—
20	C₄⋊C₄×Dic₆₁	—	2	2	no	—
21	C₄⋊C₄×Dic₆₁	C₂²	1	4	yes	—
22	C₄×C₄⋊Dic₆₁	C₂²	1	2	yes	—
23	#16839525	C₂²	1	2	yes	—
24	#16839524	C₂²	1	4	yes	—
25	C₂×#107×C₆₁	C₂²	1	2	yes	—
26	C₂×#107×C₆₁	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
28	C₆₁×#384	C₂²	1	2	yes	—
29	C₂²×Dic₁₂₂	D₄	1	1	yes	characteristic
30	C₂²×#92418	C₂³	1	2	yes	—
31	C₄×Dic₁₂₂	C₂×C₄	1	4	yes	—
32	C₄×Dic₁₂₂	Q₈	1	4	yes	—
33	C₂×C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₆₁	C₂³	1	2	yes	—
35	C₂×C₄⋊Dic₆₁	C₂³	1	2	yes	—
36	C₂×#11758196	D₄	1	2	yes	—
37	C₂×C₄×Dic₆₁	—	2	2	no	—
38	C₂×C₄×Dic₆₁	—	2	4	no	—
39	C₂×C₄×Dic₆₁	C₂³	1	2	yes	—
40	#14519711	—	2	2	no	—
41	#14519711	Q₈	1	4	yes	complemented
42	#14519708	—	4	4	no	—
43	C₄⋊#92418	C₂×C₄	1	4	yes	—
44	#14519706	C₂×C₄	1	8	yes	—
45	C₈×Dic₆₁	C₂×C₄	1	8	yes	—
46	#14519684	—	2	2	no	—
47	#14519684	—	2	4	no	—
48	#14519684	—	2	4	no	—
49	#14519684	—	2	4	no	—
50	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
53	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
54	C₂×C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
55	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
56	#107×C₆₁	—	2	4	no	—
57	#107×C₆₁	—	4	2	no	—
58	C₄×C₈×C₆₁	C₂×C₄	1	4	yes	—
59	C₂²×C₄×C₆₁	—	2	4	no	—
60	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
63	C₂²×Dic₆₁	—	2	2	no	—
64	C₂²×Dic₆₁	—	2	2	no	—
65	C₂²×Dic₆₁	C₂×D₄	1	2	yes	—
66	C₂²×Dic₆₁	C₂×D₄	1	2	yes	—
67	C₂×Dic₁₂₂	—	2	2	no	—
68	C₂×Dic₁₂₂	—	2	2	no	—
69	C₂×Dic₁₂₂	C₄⋊C₄	1	4	yes	—
70	C₂×C₈×C₆₁	—	2	1	no	—
71	C₂×C₈×C₆₁	—	2	1	no	—
72	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
75	C₂×C₈×C₆₁	#41	1	2	yes	—
76	C₄⋊C₄×C₆₁	—	2	4	no	—
77	C₄⋊C₄×C₆₁	—	2	4	no	—
78	C₄⋊C₄×C₆₁	—	4	4	no	—
79	C₄⋊C₄×C₆₁	—	8	2	no	—
80	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
81	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
82	C₄⋊Dic₆₁	—	2	2	no	—
83	C₄⋊Dic₆₁	—	2	2	no	—
84	C₄⋊Dic₆₁	—	2	2	no	—
85	C₄⋊Dic₆₁	C₄⋊C₄	1	4	yes	—
86	C₄⋊Dic₆₁	C₂×Q₈	1	4	yes	—
87	#11758196	C₄⋊C₄	1	8	yes	—
88	C₄×Dic₆₁	—	4	4	no	—
89	C₄×Dic₆₁	—	8	4	no	—
90	C₄×Dic₆₁	C₂²×C₄	1	4	yes	—
91	C₄×Dic₆₁	C₂×Q₈	1	4	yes	—
92	C₂×#92418	—	2	2	no	—
93	C₂×#92418	—	2	2	no	—
94	C₂×#92418	C₂²×C₄	1	4	yes	—
95	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₆₁	—	2	1	no	—
97	C₂×C₄×C₆₁	—	2	1	no	—
98	C₂×C₄×C₆₁	—	4	4	no	—
99	C₂×C₄×C₆₁	—	4	4	no	—
100	C₂×C₄×C₆₁	—	4	4	no	—
101	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₆₁	#135	1	2	yes	—
104	C₂×C₄×C₆₁	#135	1	2	yes	—
105	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂×C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
111	C₂×Dic₆₁	—	2	2	no	—
112	C₂×Dic₆₁	—	2	2	no	—
113	C₂×Dic₆₁	—	2	2	no	—
114	C₂×Dic₆₁	—	2	2	no	—
115	C₂×Dic₆₁	—	2	4	no	—
116	C₂×Dic₆₁	—	4	2	no	—
117	C₂×Dic₆₁	—	4	2	no	—
118	C₂×Dic₆₁	—	4	2	no	—
119	C₂×Dic₆₁	—	4	2	no	—
120	C₂×Dic₆₁	—	4	2	no	—
121	C₂×Dic₆₁	C₂×C₄⋊C₄	1	4	yes	—
122	C₂×Dic₆₁	C₂×C₄⋊C₄	1	4	yes	—
123	Dic₁₂₂	—	2	8	no	—
124	C₈×C₆₁	—	2	2	no	—
125	C₈×C₆₁	C₄×Q₈	1	4	yes	—
126	#92418	—	2	4	no	—
127	#22727	—	61	1	no	maximal, Sylow
128	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₆₁	D₄×Q₈	1	1	yes	characteristic
130	C₂²×C₆₁	#550	1	1	yes	characteristic
131	C₂²×C₆₁	C₂×#135	1	1	yes	characteristic
132	C₂²×C₆₁	C₂×#135	1	1	yes	characteristic
133	C₂²×C₆₁	#576	1	1	yes	characteristic
134	C₂²×C₆₁	#578	1	1	yes	characteristic
135	C₄×C₆₁	—	2	2	no	—
136	C₄×C₆₁	—	8	4	no	—
137	C₄×C₆₁	#427	1	4	yes	—
138	C₄×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
139	C₄×C₆₁	#520	1	1	yes	characteristic
140	C₄×C₆₁	#557	1	1	yes	characteristic
141	C₄×C₆₁	Q₈²	1	1	yes	characteristic
142	Dic₆₁	—	2	4	no	—
143	Dic₆₁	—	2	4	no	—
144	Dic₆₁	—	4	4	no	—
145	Dic₆₁	—	8	2	no	—
146	C₂×C₈×Q₈	—	61	1	no	—
147	#2355	—	61	2	no	—
148	#1949	—	61	4	no	—
149	#1948	—	61	2	no	—
150	#1846	—	61	2	no	—
151	#1845	D₆₁	1	1	yes	characteristic
152	C₄×#107	—	61	2	no	—
153	C₄×#107	—	61	1	no	—
154	C₂×C₆₁	Q₈×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₆₁	C₂×#427	1	1	yes	characteristic
156	C₂×C₆₁	#2913	1	1	yes	characteristic
157	C₂×C₆₁	#3380	1	1	yes	characteristic
158	C₂×C₆₁	#3383	1	1	yes	characteristic
159	C₂×C₆₁	#3393	1	1	yes	characteristic
160	C₂×C₆₁	#3396	1	1	yes	characteristic
161	C₂×C₄×Q₈	—	61	1	no	—
162	C₈×Q₈	—	61	8	no	—
163	C₂×#107	—	61	1	no	—
164	C₂×#107	—	61	1	no	—
165	C₂×#107	C₂×D₆₁	1	2	yes	—
166	C₂×#107	C₂×D₆₁	1	2	yes	—
167	C₂×#106	—	61	4	no	—
168	C₂×C₄⋊C₈	—	61	2	no	—
169	C₂×C₄⋊C₈	—	61	1	no	—
170	C₂×C₄×C₈	—	61	2	no	—
171	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
172	#391	—	122	2	no	—
173	#387	—	122	4	no	—
174	#384	C₂×D₆₁	1	2	yes	—
175	#382	—	61	4	no	—
176	C₄×C₄⋊C₄	—	61	4	no	—
177	C₄×C₄⋊C₄	—	61	2	no	—
178	C₄×C₄⋊C₄	—	122	2	no	—
179	#340	—	61	2	no	—
180	#340	—	61	4	no	—
181	C₆₁	#22727	1	1	yes	characteristic, complemented, Sylow
182	C₂²×Q₈	—	61	1	no	—
183	C₂²×C₈	—	61	2	no	—
184	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
186	C₄×Q₈	—	61	4	no	—
187	C₄×Q₈	—	61	4	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	61	1	no	—
191	C₂×C₄⋊C₄	—	61	2	no	—
192	C₂×C₄⋊C₄	—	61	2	no	—
193	C₂×C₄⋊C₄	—	61	2	no	—
194	C₂×C₄⋊C₄	C₂²×D₆₁	1	4	yes	—
195	C₂×C₄²	—	61	2	no	—
196	C₂×C₄²	—	122	4	no	—
197	C₂×C₄²	—	122	2	no	—
198	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
199	#107	—	2	4	no	—
200	#107	—	4	2	no	—
201	#107	—	61	4	no	—
202	#107	—	122	2	no	—
203	#106	—	244	4	no	—
204	C₄⋊C₈	—	61	8	no	—
205	C₄⋊C₈	—	61	4	no	—
206	C₄×C₈	—	61	8	no	—
207	C₄×C₈	C₄×D₆₁	1	4	yes	—
208	#95	—	122	4	no	—
209	#95	—	122	4	no	—
210	#95	—	122	4	no	—
211	#95	—	122	2	no	—
212	C₂×Q₈	—	61	4	no	—
213	C₂×Q₈	—	122	2	no	—
214	C₂×Q₈	—	122	2	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	61	2	no	—
217	C₂²×C₄	—	61	2	no	—
218	C₂²×C₄	—	122	2	no	—
219	C₂²×C₄	—	122	2	no	—
220	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
221	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
222	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	61	4	no	—
226	C₂×C₈	—	122	2	no	—
227	C₂×C₈	—	122	2	no	—
228	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
229	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
230	C₂×C₈	#11758224	1	2	yes	—
231	C₂×C₈	Q₈×D₆₁	1	2	yes	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	4	4	no	—
235	C₄⋊C₄	—	8	2	no	—
236	C₄⋊C₄	—	61	8	no	—
237	C₄⋊C₄	—	61	4	no	—
238	C₄⋊C₄	—	61	4	no	—
239	C₄⋊C₄	—	122	2	no	—
240	C₄⋊C₄	—	122	2	no	—
241	C₄⋊C₄	—	122	2	no	—
242	C₄²	—	61	4	no	—
243	C₄²	—	61	4	no	—
244	C₄²	—	488	4	no	—
245	C₄²	—	244	4	no	—
246	C₄²	C₂×C₄×D₆₁	1	2	yes	—
247	C₄²	Q₈×D₆₁	1	2	yes	—
248	C₂³	C₂×D₄×D₆₁	1	1	yes	characteristic
249	Q₈	—	122	8	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	61	4	no	—
256	C₂×C₄	—	61	4	no	—
257	C₂×C₄	—	122	2	no	—
258	C₂×C₄	—	122	2	no	—
259	C₂×C₄	—	122	2	no	—
260	C₂×C₄	—	122	2	no	—
261	C₂×C₄	—	122	4	no	—
262	C₂×C₄	—	244	2	no	—
263	C₂×C₄	—	244	2	no	—
264	C₂×C₄	—	244	2	no	—
265	C₂×C₄	—	244	2	no	—
266	C₂×C₄	—	244	2	no	—
267	C₂×C₄	C₄⋊C₄×D₆₁	1	2	yes	—
268	C₂×C₄	C₄⋊C₄×D₆₁	1	2	yes	—
269	C₂×C₄	#14519796	1	2	yes	—
270	C₂×C₄	#14519800	1	2	yes	—
271	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
272	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
273	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
274	C₂×C₄	C₂×#11758224	1	1	yes	characteristic
275	C₂×C₄	#14519822	1	1	yes	characteristic
276	C₂×C₄	#14519825	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	122	4	no	—
279	C₈	#14519742	1	4	yes	—
280	C₂²	C₂×C₄⋊C₄×D₆₁	1	1	yes	characteristic
281	C₂²	#16840427	1	1	yes	characteristic
282	C₂²	#16840432	1	1	yes	characteristic
283	C₂²	C₂×#14519796	1	1	yes	characteristic
284	C₂²	C₂×#14519800	1	1	yes	characteristic
285	C₂²	#16840563	1	1	yes	characteristic
286	C₂²	#16840575	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	8	4	no	—
289	C₄	—	122	4	no	—
290	C₄	—	122	4	no	—
291	C₄	—	488	2	no	—
292	C₄	—	244	4	no	—
293	C₄	#16839799	1	4	yes	—
294	C₄	C₂×#14519742	1	1	yes	characteristic
295	C₄	#16840243	1	1	yes	characteristic
296	C₄	#16840487	1	1	yes	characteristic
297	C₄	#16840492	1	1	yes	characteristic
298	C₂	#21448731	1	1	yes	characteristic
299	C₂	C₂×#16839799	1	1	yes	characteristic
300	C₂	#21454415	1	1	yes	characteristic
301	C₂	#21458688	1	1	yes	characteristic
302	C₂	#21458712	1	1	yes	characteristic
303	C₂	#21458809	1	1	yes	characteristic
304	C₂	#21458833	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₆₁×#1845	C₂
socle, prime radical	C₂³×C₆₁	C₂²×D₄
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	#1845	D₆₁
hypocenter	C₆₁	#22727
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
10. C₈×Dic₁₂₂	C₄	semidirect
41. #14519711	Q₈	semidirect
181. C₆₁	#22727	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₂₂	2
#21448778	2
#21446584	2
#21446580	2
#21445982	2
#107×Dic₆₁	2
C₄×#14519711	2
C₆₁×#1845	2
#22727	61