Subgroups — #21462573

Order 7808.

492 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#6923	C₂	1	2	yes	maximal, complemented
2	#16840803	C₂	1	2	yes	maximal, complemented
3	#16840802	C₂	1	2	yes	maximal, complemented
4	#16840801	C₂	1	4	yes	maximal, complemented
5	#16840772	C₂	1	1	yes	maximal, characteristic, complemented
6	#16840752	C₂	1	2	yes	maximal, complemented
7	C₂×#14519831	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₆₁	C₂	1	1	yes	maximal, characteristic
9	C₂×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
10	Q₈×#6923	C₂²	1	4	yes	—
11	D₄×#6923	—	2	4	no	—
12	D₄×#6923	C₂²	1	4	yes	—
13	#14519856	—	2	2	no	—
14	#14519856	C₂²	1	2	yes	—
15	C₂×C₄×#6923	—	2	2	no	—
16	C₂×C₄×#6923	C₂²	1	1	yes	characteristic
17	#14519846	C₂²	1	2	yes	—
18	#14519845	C₂²	1	2	yes	—
19	#14519840	C₂²	1	4	yes	—
20	#14519839	C₂²	1	4	yes	—
21	#14519831	C₂²	1	2	yes	—
22	#14519831	C₂²	1	2	yes	—
23	#41×D₆₁	C₂²	1	2	yes	—
24	#14519800	C₄	1	1	yes	characteristic, complemented
25	#14519799	C₄	1	1	yes	characteristic, complemented
26	#14519798	C₄	1	2	yes	complemented
27	#14519797	C₄	1	2	yes	complemented
28	#14519796	C₄	1	1	yes	characteristic, complemented
29	Q₁₆×D₆₁	C₂²	1	1	yes	characteristic
30	SD₁₆×D₆₁	C₂²	1	2	yes	—
31	D₈×D₆₁	C₂²	1	1	yes	characteristic
32	C₂×C₈×D₆₁	C₂²	1	1	yes	characteristic
33	#135×C₆₁	C₄	1	1	yes	characteristic, complemented
34	C₂²×#6923	—	2	4	no	—
35	C₂²×#6923	D₄	1	2	yes	—
36	#41×C₆₁	C₂×C₄	1	2	yes	complemented
37	#11758226	C₂×C₄	1	2	yes	complemented
38	Q₈×D₆₁	C₂³	1	2	yes	—
39	#11758224	—	2	2	no	—
40	#11758224	C₂×C₄	1	2	yes	complemented
41	D₄×D₆₁	—	2	2	no	—
42	D₄×D₆₁	C₂³	1	2	yes	—
43	#11758222	—	2	2	no	—
44	#11758222	C₂×C₄	1	2	yes	complemented
45	C₂×C₄×D₆₁	—	2	2	no	—
46	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
47	C₂²⋊#6923	—	2	2	no	—
48	C₂²⋊#6923	—	2	4	no	—
49	C₂×#92420	—	2	1	no	—
50	C₄⋊#6923	—	2	4	no	—
51	C₄⋊#6923	C₂³	1	4	yes	—
52	C₄×#6923	—	2	4	no	—
53	C₄×#6923	—	2	2	no	—
54	C₄×#6923	D₄	1	2	yes	—
55	C₄×#6923	C₂³	1	2	yes	—
56	D₆₁⋊C₈	—	2	1	no	—
57	D₆₁⋊C₈	C₂³	1	2	yes	—
58	Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
59	SD₁₆×C₆₁	C₂×C₄	1	2	yes	complemented
60	D₈×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
61	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic, complemented
62	#11758202	C₂×C₄	1	2	yes	complemented
63	#11758201	C₂×C₄	1	2	yes	complemented
64	#11758200	C₂×C₄	1	2	yes	complemented
65	#11758199	C₂×C₄	1	2	yes	complemented
66	C₂×#92418	C₂×C₄	1	1	yes	characteristic, complemented
67	Dic₂₄₄	C₂×C₄	1	1	yes	characteristic, complemented
68	D₄₈₈	C₂×C₄	1	1	yes	characteristic, complemented
69	#11758190	C₂×C₄	1	2	yes	complemented
70	C₈×D₆₁	C₂×C₄	1	1	yes	characteristic, complemented
71	C₈×D₆₁	C₂×C₄	1	1	yes	characteristic, complemented
72	C₈×D₆₁	C₂³	1	1	yes	characteristic
73	C₈×D₆₁	C₂³	1	1	yes	characteristic
74	C₂²×D₆₁	—	2	2	no	—
75	C₂²×D₆₁	C₂×D₄	1	1	yes	characteristic
76	C₂×#6923	—	2	4	no	—
77	C₂×#6923	—	4	2	no	—
78	C₂×#6923	—	4	8	no	—
79	C₂×#6923	C₂×D₄	1	4	yes	—
80	Q₈×C₆₁	C₂²×C₄	1	2	yes	—
81	D₄×C₆₁	—	2	2	no	—
82	D₄×C₆₁	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₆₁	—	2	2	no	—
84	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
85	C₂²⋊D₆₁	—	2	2	no	—
86	C₂²⋊D₆₁	—	2	2	no	—
87	C₂²⋊D₆₁	—	2	2	no	—
88	C₂×Dic₆₁	—	2	2	no	—
89	C₂×Dic₆₁	#41	1	1	yes	characteristic
90	D₂₄₄	—	2	2	no	—
91	D₂₄₄	C₂²×C₄	1	2	yes	—
92	C₄×D₆₁	—	2	2	no	—
93	C₄×D₆₁	—	2	2	no	—
94	C₄×D₆₁	—	2	2	no	—
95	C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
96	C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
97	C₄×D₆₁	#41	1	1	yes	characteristic
98	C₄×D₆₁	C₂⁴	1	1	yes	characteristic
99	Dic₁₂₂	—	2	2	no	—
100	Dic₁₂₂	C₂²×C₄	1	2	yes	—
101	#92420	—	2	2	no	—
102	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
103	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
104	#92418	C₂²×C₄	1	1	yes	characteristic
105	#92418	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₆₁	—	2	2	no	—
107	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
108	C₂×D₆₁	—	2	2	no	—
109	C₂×D₆₁	—	2	2	no	—
110	C₂×D₆₁	—	4	4	no	—
111	C₂×D₆₁	C₄×D₄	1	1	yes	characteristic
112	C₂×D₆₁	C₂²×D₄	1	1	yes	characteristic
113	#6923	—	2	4	no	—
114	#6923	—	4	4	no	—
115	C₄×C₆₁	—	2	2	no	—
116	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
117	C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
118	Dic₆₁	—	2	2	no	—
119	Dic₆₁	C₄×D₄	1	1	yes	characteristic
120	Dic₆₁	C₂×#41	1	1	yes	characteristic
121	#2943	—	61	1	no	maximal, Sylow
122	C₂×C₆₁	—	2	1	no	—
123	C₂×C₆₁	—	4	2	no	—
124	C₂×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
125	D₆₁	—	2	1	no	—
126	D₆₁	—	4	2	no	—
127	D₆₁	#575	1	2	yes	—
128	C₂×#135	—	61	1	no	—
129	C₄×#41	—	61	2	no	—
130	#442	—	61	2	no	—
131	#441	—	61	2	no	—
132	#440	—	61	4	no	—
133	#427	—	61	1	no	—
134	#416	—	61	2	no	—
135	C₂×#97	—	61	1	no	—
136	C₆₁	#2943	1	1	yes	characteristic, complemented, Sylow
137	C₂×#41	—	61	2	no	—
138	#135	—	61	2	no	—
139	#135	—	61	1	no	—
140	#135	—	61	2	no	—
141	#135	—	61	1	no	—
142	#135	—	61	1	no	—
143	#135	#6923	1	1	yes	characteristic, complemented
144	C₂×Q₁₆	—	61	1	no	—
145	C₂×SD₁₆	—	61	2	no	—
146	C₂×D₈	—	61	1	no	—
147	C₂²×C₈	—	61	1	no	—
148	C₂²×C₈	—	61	1	no	—
149	C₄×Q₈	—	61	4	no	—
150	C₄×D₄	—	61	4	no	—
151	C₄×D₄	—	122	4	no	—
152	#117	—	61	2	no	—
153	#117	—	122	2	no	—
154	C₂×C₄²	—	61	1	no	—
155	C₂×C₄²	—	122	2	no	—
156	#107	—	61	2	no	—
157	#106	—	61	2	no	—
158	#103	—	61	4	no	—
159	#102	—	61	4	no	—
160	#97	—	61	2	no	—
161	#97	—	61	2	no	—
162	#41	—	61	2	no	—
163	#41	—	61	2	no	—
164	#41	—	61	2	no	—
165	#41	—	122	2	no	—
166	#41	—	122	2	no	—
167	#41	C₂×#6923	1	2	yes	complemented
168	C₂×Q₈	—	61	2	no	—
169	C₂×D₄	—	61	2	no	—
170	C₂×D₄	—	122	2	no	—
171	C₂²×C₄	—	61	1	no	—
172	C₂²×C₄	—	61	2	no	—
173	C₂²×C₄	—	122	2	no	—
174	C₂²×C₄	—	122	4	no	—
175	Q₁₆	—	61	2	no	—
176	Q₁₆	—	61	1	no	—
177	Q₁₆	C₂×#6923	1	1	yes	characteristic, complemented
178	SD₁₆	—	61	2	no	—
179	SD₁₆	—	61	2	no	—
180	SD₁₆	—	61	2	no	—
181	SD₁₆	C₂×#6923	1	2	yes	complemented
182	D₈	—	61	2	no	—
183	D₈	—	61	1	no	—
184	D₈	C₂×#6923	1	1	yes	characteristic, complemented
185	C₂×C₈	—	61	1	no	—
186	C₂×C₈	—	61	1	no	—
187	C₂×C₈	—	61	1	no	—
188	C₂×C₈	—	61	1	no	—
189	C₂×C₈	—	61	1	no	—
190	C₂×C₈	—	61	2	no	—
191	C₂×C₈	—	122	1	no	—
192	C₂×C₈	—	122	1	no	—
193	C₂×C₈	C₂×#6923	1	1	yes	characteristic, complemented
194	C₄⋊C₄	—	61	4	no	—
195	C₄⋊C₄	—	122	4	no	—
196	C₂²⋊C₄	—	122	4	no	—
197	C₂²⋊C₄	—	122	2	no	—
198	C₄²	—	61	2	no	—
199	C₄²	—	61	2	no	—
200	C₄²	—	122	2	no	—
201	C₄²	—	122	4	no	—
202	C₂³	—	61	1	no	—
203	C₂³	—	122	2	no	—
204	Q₈	—	61	2	no	—
205	Q₈	—	122	2	no	—
206	Q₈	C₂²×#6923	1	2	yes	—
207	D₄	—	2	2	no	—
208	D₄	—	61	2	no	—
209	D₄	—	122	2	no	—
210	D₄	—	122	2	no	—
211	D₄	—	122	2	no	—
212	D₄	—	122	2	no	—
213	D₄	C₂²×#6923	1	2	yes	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	61	1	no	—
216	C₂×C₄	—	61	1	no	—
217	C₂×C₄	—	61	1	no	—
218	C₂×C₄	—	61	1	no	—
219	C₂×C₄	—	61	1	no	—
220	C₂×C₄	—	61	4	no	—
221	C₂×C₄	—	122	2	no	—
222	C₂×C₄	—	122	2	no	—
223	C₂×C₄	—	122	2	no	—
224	C₂×C₄	—	122	2	no	—
225	C₂×C₄	—	122	4	no	—
226	C₂×C₄	—	244	2	no	—
227	C₂×C₄	—	244	8	no	—
228	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
229	C₈	—	61	1	no	—
230	C₈	—	61	1	no	—
231	C₈	—	122	2	no	—
232	C₈	C₂²×#6923	1	1	yes	characteristic
233	C₈	C₂²×#6923	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	—	61	1	no	—
236	C₂²	—	61	1	no	—
237	C₂²	—	122	2	no	—
238	C₂²	—	122	2	no	—
239	C₂²	—	244	4	no	—
240	C₂²	D₄×#6923	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	61	1	no	—
243	C₄	—	61	1	no	—
244	C₄	—	122	4	no	—
245	C₄	—	122	2	no	—
246	C₄	—	244	4	no	—
247	C₄	D₄×#6923	1	1	yes	characteristic
248	C₄	C₂³×#6923	1	1	yes	characteristic
249	C₂	—	2	1	no	—
250	C₂	—	4	2	no	—
251	C₂	—	61	2	no	—
252	C₂	—	122	1	no	—
253	C₂	—	244	2	no	—
254	C₂	C₂×D₄×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×#6923
commutator	C₄×C₆₁	C₂³×C₄
Frattini	C₄	C₂³×#6923
Fitting	#135×C₆₁	C₄
socle	C₂×C₆₁	C₂×C₄×D₄
hypercenter	#135	#6923
hypocenter	C₆₁	#2943
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #41×#6923	C₂	semidirect
2. #16840803	C₂	semidirect
3. #16840802	C₂	semidirect
4. #16840801	C₂	semidirect
5. #16840772	C₂	semidirect
6. #16840752	C₂	semidirect
7. C₂×#14519831	C₂	semidirect
24. #14519800	C₄	semidirect
25. #14519799	C₄	semidirect
26. #14519798	C₄	semidirect
27. #14519797	C₄	semidirect
28. #14519796	C₄	semidirect
33. #135×C₆₁	C₄	semidirect
36. #41×C₆₁	C₂×C₄	semidirect
37. #11758226	C₂×C₄	semidirect
40. #11758224	C₂×C₄	semidirect
44. #11758222	C₂×C₄	semidirect
58. Q₁₆×C₆₁	C₂×C₄	semidirect
59. SD₁₆×C₆₁	C₂×C₄	semidirect
60. D₈×C₆₁	C₂×C₄	semidirect
61. C₂×C₈×C₆₁	C₂×C₄	semidirect
62. #11758202	C₂×C₄	semidirect
63. #11758201	C₂×C₄	semidirect
64. #11758200	C₂×C₄	semidirect
65. #11758199	C₂×C₄	semidirect
66. C₂×#92418	C₂×C₄	semidirect
67. Dic₂₄₄	C₂×C₄	semidirect
68. D₄₈₈	C₂×C₄	semidirect
69. #11758190	C₂×C₄	semidirect
70. C₈×D₆₁	C₂×C₄	semidirect
71. C₈×D₆₁	C₂×C₄	semidirect
136. C₆₁	#2943	semidirect
143. #135	#6923	semidirect
167. #41	C₂×#6923	semidirect
177. Q₁₆	C₂×#6923	semidirect
181. SD₁₆	C₂×#6923	semidirect
184. D₈	C₂×#6923	semidirect
193. C₂×C₈	C₂×#6923	semidirect