Subgroups — #21458222

Order 7808.

366 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16840458	C₂	1	2	yes	maximal, complemented
2	#16840164	C₂	1	1	yes	maximal, characteristic, complemented
3	#16840087	C₂	1	2	yes	maximal, complemented
4	#16840081	C₂	1	2	yes	maximal, complemented
5	#16840063	C₂	1	2	yes	maximal, complemented
6	#16839998	C₂	1	1	yes	maximal, characteristic, complemented
7	#16839998	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₄₈₈	C₂	1	1	yes	maximal, characteristic, complemented
9	#16839800	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14519711	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄₈₈	C₂²	1	1	yes	characteristic
13	#14519772	—	2	2	no	—
14	#14519769	C₂²	1	2	yes	—
15	#14519759	—	2	2	no	—
16	#14519751	C₂²	1	2	yes	—
17	#14519751	C₂²	1	2	yes	—
18	#14519749	—	2	2	no	—
19	C₄×D₂₄₄	C₂²	1	2	yes	—
20	#14519737	—	2	2	no	—
21	#14519736	C₂²	1	2	yes	—
22	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
23	#14519711	C₂²	1	1	yes	characteristic
24	#14519711	C₂²	1	1	yes	characteristic
25	#14519711	C₂²	1	1	yes	characteristic
26	#14519711	C₂²	1	2	yes	—
27	#14519709	C₂²	1	2	yes	—
28	#14519709	C₂²	1	2	yes	—
29	#14519699	C₂²	1	2	yes	—
30	#14519699	C₂²	1	2	yes	—
31	#14519699	C₂²	1	2	yes	—
32	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
33	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
34	C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
36	C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
37	C₂²⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
39	C₂×D₄×C₆₁	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
42	C₂×C₂²⋊D₆₁	—	2	2	no	—
43	C₂×C₂²⋊D₆₁	—	2	2	no	—
44	C₂²×Dic₆₁	—	2	2	no	—
45	C₂×D₂₄₄	C₂³	1	2	yes	—
46	C₂×C₄×D₆₁	—	2	2	no	—
47	C₂×C₈×C₆₁	—	2	2	no	—
48	C₂×C₈×C₆₁	—	2	2	no	—
49	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
54	C₂²⋊C₄×C₆₁	D₄	1	1	yes	characteristic
55	C₂²⋊C₄×C₆₁	D₄	1	1	yes	characteristic
56	C₄²×C₆₁	C₂³	1	1	yes	characteristic
57	C₂²⋊Dic₆₁	—	2	2	no	—
58	#11758198	—	2	2	no	—
59	#11758198	—	2	2	no	—
60	#11758198	—	2	2	no	—
61	C₄⋊Dic₆₁	—	2	2	no	—
62	C₄⋊Dic₆₁	C₂³	1	2	yes	—
63	C₄⋊Dic₆₁	C₂³	1	2	yes	—
64	C₄⋊Dic₆₁	C₂³	1	2	yes	—
65	#11758196	—	2	2	no	—
66	#11758196	—	2	2	no	—
67	C₄×Dic₆₁	—	2	2	no	—
68	D₄₈₈	—	2	2	no	—
69	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
70	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×D₆₁	—	2	2	no	—
72	D₄×C₆₁	—	2	2	no	—
73	C₂×C₄×C₆₁	—	2	1	no	—
74	C₂×C₄×C₆₁	—	2	1	no	—
75	C₂×C₄×C₆₁	—	2	1	no	—
76	C₂×C₄×C₆₁	—	2	1	no	—
77	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
82	C₂²⋊D₆₁	—	4	4	no	—
83	C₂²⋊D₆₁	—	8	2	no	—
84	C₂×Dic₆₁	—	2	2	no	—
85	C₂×Dic₆₁	—	2	2	no	—
86	C₂×Dic₆₁	—	2	2	no	—
87	C₂×Dic₆₁	—	8	2	no	—
88	D₂₄₄	—	2	2	no	—
89	D₂₄₄	—	2	2	no	—
90	C₄×D₆₁	—	8	2	no	—
91	C₈×C₆₁	—	2	1	no	—
92	C₈×C₆₁	—	2	1	no	—
93	C₈×C₆₁	—	2	1	no	—
94	C₈×C₆₁	#41	1	2	yes	—
95	C₂²×C₆₁	—	2	1	no	—
96	C₂²×C₆₁	—	2	1	no	—
97	C₂²×C₆₁	—	2	1	no	—
98	C₂²×C₆₁	—	2	1	no	—
99	C₂²×C₆₁	—	2	1	no	—
100	C₂²×C₆₁	#135	1	2	yes	—
101	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
102	C₂×D₆₁	—	4	2	no	—
103	C₂×D₆₁	—	4	2	no	—
104	C₂×D₆₁	—	4	2	no	—
105	C₄×C₆₁	—	2	1	no	—
106	C₄×C₆₁	—	2	1	no	—
107	C₄×C₆₁	—	2	1	no	—
108	C₄×C₆₁	—	2	1	no	—
109	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
110	C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
111	Dic₆₁	—	4	2	no	—
112	Dic₆₁	—	4	2	no	—
113	Dic₆₁	—	4	2	no	—
114	#3339	—	61	1	no	maximal, Sylow
115	C₂×C₆₁	—	2	2	no	—
116	C₂×C₆₁	—	4	1	no	—
117	C₂×C₆₁	#547	1	1	yes	characteristic
118	C₂×C₆₁	C₂×#135	1	1	yes	characteristic
119	C₂×C₆₁	#576	1	1	yes	characteristic
120	D₆₁	—	8	2	no	—
121	#553	—	61	2	no	—
122	#499	—	61	1	no	—
123	#482	—	61	2	no	—
124	#481	—	61	2	no	—
125	#478	—	61	2	no	—
126	#466	—	61	1	no	—
127	#466	—	61	1	no	—
128	C₄×D₈	—	61	1	no	—
129	C₈×D₄	D₆₁	1	1	yes	characteristic, complemented
130	#427	—	61	1	no	—
131	C₂×#107	—	61	1	no	—
132	C₆₁	#3339	1	1	yes	characteristic, complemented, Sylow
133	C₂×D₈	—	61	1	no	—
134	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
135	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
136	#126	—	122	2	no	—
137	#125	—	61	2	no	—
138	#123	—	122	2	no	—
139	#121	—	61	2	no	—
140	#121	—	61	2	no	—
141	#121	—	122	2	no	—
142	C₄×D₄	—	61	2	no	—
143	C₄×D₄	—	122	2	no	—
144	C₄×D₄	C₂×D₆₁	1	1	yes	characteristic
145	#117	—	61	2	no	—
146	C₂×C₄⋊C₄	—	61	2	no	—
147	#107	—	61	1	no	—
148	#107	—	61	1	no	—
149	#107	—	61	2	no	—
150	#107	—	61	1	no	—
151	#106	—	61	2	no	—
152	#106	—	61	2	no	—
153	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
154	#102	—	61	2	no	—
155	#102	—	61	2	no	—
156	#102	—	61	2	no	—
157	C₂²⋊C₈	C₂×D₆₁	1	1	yes	characteristic
158	C₂²⋊C₈	C₂×D₆₁	1	1	yes	characteristic
159	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
160	C₂×D₄	—	61	2	no	—
161	C₂×D₄	—	122	2	no	—
162	C₂×D₄	—	122	2	no	—
163	C₂×D₄	D₂₄₄	1	1	yes	characteristic
164	C₂²×C₄	—	122	2	no	—
165	C₂²×C₄	—	122	2	no	—
166	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
167	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
168	D₈	—	122	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
174	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
175	C₄⋊C₄	—	61	2	no	—
176	C₄⋊C₄	—	61	2	no	—
177	C₄⋊C₄	—	61	2	no	—
178	C₄⋊C₄	—	122	2	no	—
179	C₄⋊C₄	—	122	2	no	—
180	C₄⋊C₄	—	122	2	no	—
181	C₄⋊C₄	D₂₄₄	1	1	yes	characteristic
182	C₂²⋊C₄	—	122	2	no	—
183	C₂²⋊C₄	—	122	2	no	—
184	C₂²⋊C₄	—	122	2	no	—
185	C₂²⋊C₄	—	122	2	no	—
186	C₂²⋊C₄	D₂₄₄	1	1	yes	characteristic
187	C₂²⋊C₄	D₂₄₄	1	1	yes	characteristic
188	C₄²	—	122	2	no	—
189	C₄²	C₂²×D₆₁	1	1	yes	characteristic
190	C₂³	—	122	2	no	—
191	C₂³	C₂×D₂₄₄	1	1	yes	characteristic
192	C₂³	C₂×D₂₄₄	1	1	yes	characteristic
193	D₄	—	2	2	no	—
194	D₄	—	122	2	no	—
195	D₄	—	122	2	no	—
196	D₄	—	488	2	no	—
197	D₄	—	244	4	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	122	2	no	—
203	C₂×C₄	—	122	2	no	—
204	C₂×C₄	—	122	2	no	—
205	C₂×C₄	—	488	2	no	—
206	C₂×C₄	—	488	2	no	—
207	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
208	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
209	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
210	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
211	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
212	C₈	—	2	1	no	—
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	#11758224	1	2	yes	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	244	2	no	—
222	C₂²	—	244	2	no	—
223	C₂²	—	244	2	no	—
224	C₂²	#14519799	1	2	yes	—
225	C₂²	C₂²×D₂₄₄	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	244	2	no	—
231	C₄	—	244	2	no	—
232	C₄	—	244	2	no	—
233	C₄	C₂×#11758224	1	1	yes	characteristic
234	C₄	#14519824	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	—	4	1	no	—
237	C₂	—	488	2	no	—
238	C₂	#16840416	1	1	yes	characteristic
239	C₂	C₂×#14519799	1	1	yes	characteristic
240	C₂	#16840567	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₈×D₄×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	C₈×D₄	D₆₁
hypocenter	C₆₁	#3339
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #16840458	C₂	semidirect
2. #16840164	C₂	semidirect
3. #16840087	C₂	semidirect
4. #16840081	C₂	semidirect
5. #16840063	C₂	semidirect
6. #16839998	C₂	semidirect
7. #16839998	C₂	semidirect
8. C₄×D₄₈₈	C₂	semidirect
9. #16839800	C₂	semidirect
10. C₂×#14519711	C₂	semidirect
11. C₈×D₄×C₆₁	C₂	semidirect
129. C₈×D₄	D₆₁	semidirect
132. C₆₁	#3339	semidirect