Subgroups — #21462598

Order 7808.

452 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16840841	C₂	1	2	yes	maximal, complemented
2	#16840809	C₂	1	2	yes	maximal, complemented
3	#16840808	C₂	1	2	yes	maximal, complemented
4	#16840805	C₂	1	4	yes	maximal, complemented
5	C₂×#14519847	C₂	1	1	yes	maximal, characteristic, complemented
6	#16840759	C₂	1	2	yes	maximal, complemented
7	#16840725	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₆₁	C₂	1	1	yes	maximal, characteristic
9	#14519865	C₂²	1	4	yes	complemented
10	#14519864	—	2	4	no	—
11	#14519864	C₂²	1	4	yes	complemented
12	C₂×#11758213	—	2	2	no	—
13	C₂×#11758213	C₂²	1	2	yes	—
14	D₆₁⋊M₁₆	—	2	2	no	—
15	D₆₁⋊M₁₆	C₂²	1	1	yes	characteristic
16	#14519856	C₂²	1	1	yes	characteristic
17	#14519847	C₂²	1	2	yes	complemented
18	#14519847	C₂²	1	2	yes	complemented
19	#14519842	C₂²	1	4	yes	complemented
20	#14519841	C₂²	1	4	yes	complemented
21	#14519831	C₂²	1	2	yes	complemented
22	C₈×#6923	C₂²	1	2	yes	complemented
23	#41×D₆₁	C₂²	1	2	yes	—
24	#14519800	C₂²	1	1	yes	characteristic
25	#14519799	C₄	1	1	yes	characteristic, complemented
26	#14519798	C₂²	1	2	yes	—
27	#14519797	C₄	1	2	yes	complemented
28	#14519796	C₂²	1	1	yes	characteristic
29	Q₁₆×D₆₁	C₄	1	1	yes	characteristic, complemented
30	SD₁₆×D₆₁	C₄	1	2	yes	complemented
31	D₈×D₆₁	C₄	1	1	yes	characteristic, complemented
32	C₂×C₈×D₆₁	C₂²	1	1	yes	characteristic
33	#135×C₆₁	C₄	1	1	yes	characteristic, complemented
34	#41×C₆₁	C₂×C₄	1	2	yes	—
35	#11758226	C₂³	1	2	yes	—
36	Q₈×D₆₁	C₂×C₄	1	2	yes	—
37	#11758224	—	2	2	no	—
38	#11758224	C₂³	1	2	yes	—
39	D₄×D₆₁	—	2	2	no	—
40	D₄×D₆₁	C₂×C₄	1	2	yes	—
41	#11758222	—	2	2	no	—
42	#11758222	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₆₁	—	2	2	no	—
44	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
45	C₂²⋊#6923	—	2	1	no	—
46	#11758217	—	2	4	no	—
47	#11758217	D₄	1	2	yes	—
48	C₂×#92420	—	2	2	no	—
49	C₂×#92420	—	2	4	no	—
50	C₄⋊#6923	—	2	1	no	—
51	C₄×#6923	C₂³	1	2	yes	—
52	#11758213	—	2	2	no	—
53	#11758213	—	2	4	no	—
54	#11758213	D₄	1	2	yes	—
55	#11758213	C₂³	1	4	yes	—
56	D₆₁⋊C₈	—	2	4	no	—
57	D₆₁⋊C₈	C₂³	1	2	yes	—
58	Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
59	SD₁₆×C₆₁	C₂×C₄	1	2	yes	—
60	D₈×C₆₁	C₂×C₄	1	1	yes	characteristic
61	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
62	#11758202	C₂×C₄	1	2	yes	—
63	#11758201	C₂×C₄	1	2	yes	—
64	#11758200	C₂×C₄	1	2	yes	—
65	#11758199	C₂×C₄	1	2	yes	—
66	C₂×#92418	C₂×C₄	1	1	yes	characteristic
67	Dic₂₄₄	C₂×C₄	1	1	yes	characteristic
68	D₄₈₈	C₂×C₄	1	1	yes	characteristic
69	#11758190	C₂×C₄	1	2	yes	—
70	C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
71	C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
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	#41	1	1	yes	characteristic
76	C₂×#6923	—	2	2	no	—
77	Q₈×C₆₁	C₂²×C₄	1	2	yes	—
78	D₄×C₆₁	—	2	2	no	—
79	D₄×C₆₁	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₆₁	—	2	2	no	—
81	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
82	C₂²⋊D₆₁	—	2	2	no	—
83	C₂²⋊D₆₁	—	2	2	no	—
84	C₂²⋊D₆₁	—	2	2	no	—
85	C₂×Dic₆₁	—	2	2	no	—
86	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
87	D₂₄₄	—	2	2	no	—
88	D₂₄₄	C₂²×C₄	1	2	yes	—
89	C₄×D₆₁	—	2	2	no	—
90	C₄×D₆₁	—	2	2	no	—
91	C₄×D₆₁	—	2	2	no	—
92	C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
93	C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
94	C₄×D₆₁	#41	1	1	yes	characteristic
95	C₄×D₆₁	C₂⁴	1	1	yes	characteristic
96	Dic₁₂₂	—	2	2	no	—
97	Dic₁₂₂	C₂²×C₄	1	2	yes	—
98	#92420	—	2	4	no	—
99	#92420	C₂×D₄	1	4	yes	—
100	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
101	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
102	#92418	C₂²×C₄	1	1	yes	characteristic
103	#92418	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₆₁	—	2	2	no	—
105	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
106	C₂×D₆₁	—	2	2	no	—
107	C₂×D₆₁	—	4	1	no	—
108	C₂×D₆₁	—	8	2	no	—
109	C₂×D₆₁	C₄×D₄	1	1	yes	characteristic
110	C₂×D₆₁	C₂×#41	1	1	yes	characteristic
111	#6923	—	4	2	no	—
112	C₄×C₆₁	—	2	2	no	—
113	C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
114	C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
115	Dic₆₁	—	2	2	no	—
116	Dic₆₁	C₄×D₄	1	1	yes	characteristic
117	Dic₆₁	C₂²×D₄	1	1	yes	characteristic
118	#2957	—	61	1	no	maximal, Sylow
119	C₂×C₆₁	—	2	1	no	—
120	C₂×C₆₁	—	4	2	no	—
121	C₂×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
122	D₆₁	—	2	1	no	—
123	D₆₁	—	2	1	no	—
124	D₆₁	—	4	2	no	—
125	C₂×#135	—	61	1	no	—
126	#568	—	61	2	no	—
127	#444	—	61	2	no	—
128	#444	—	61	2	no	—
129	#443	—	61	4	no	—
130	C₂×#108	—	61	1	no	—
131	#421	—	61	2	no	—
132	#405	—	61	1	no	—
133	C₆₁	#2957	1	1	yes	characteristic, complemented, Sylow
134	C₂×#41	—	61	2	no	—
135	#135	—	61	2	no	—
136	#135	—	61	1	no	—
137	#135	—	61	2	no	—
138	#135	—	61	1	no	—
139	#135	—	61	1	no	—
140	#135	#6923	1	1	yes	characteristic, complemented
141	C₂×Q₁₆	—	61	1	no	—
142	C₂×SD₁₆	—	61	2	no	—
143	C₂×D₈	—	61	1	no	—
144	#131	—	61	4	no	—
145	#131	—	61	4	no	—
146	#131	—	122	4	no	—
147	C₂×M₁₆	—	61	1	no	—
148	C₂×M₁₆	—	61	2	no	—
149	C₂×M₁₆	—	122	2	no	—
150	C₂×M₁₆	—	122	2	no	—
151	C₂²×C₈	—	61	1	no	—
152	#117	—	61	1	no	—
153	#108	—	61	2	no	—
154	#108	—	61	2	no	—
155	#104	—	61	4	no	—
156	#104	—	61	4	no	—
157	#97	—	61	2	no	—
158	C₄×C₈	—	61	2	no	—
159	#41	—	61	2	no	—
160	#41	—	61	2	no	—
161	#41	—	61	2	no	—
162	#41	—	122	2	no	—
163	#41	—	122	2	no	—
164	#41	C₂×#6923	1	2	yes	—
165	C₂×Q₈	—	61	2	no	—
166	C₂×D₄	—	61	2	no	—
167	C₂×D₄	—	122	2	no	—
168	C₂²×C₄	—	61	1	no	—
169	C₂²×C₄	—	122	2	no	—
170	Q₁₆	—	61	2	no	—
171	Q₁₆	—	61	1	no	—
172	Q₁₆	C₂×#6923	1	1	yes	characteristic
173	SD₁₆	—	61	2	no	—
174	SD₁₆	—	61	2	no	—
175	SD₁₆	—	61	2	no	—
176	SD₁₆	C₂×#6923	1	2	yes	—
177	D₈	—	61	2	no	—
178	D₈	—	61	1	no	—
179	D₈	C₂×#6923	1	1	yes	characteristic
180	M₁₆	—	61	2	no	—
181	M₁₆	—	61	2	no	—
182	M₁₆	—	61	4	no	—
183	M₁₆	—	122	4	no	—
184	M₁₆	—	122	4	no	—
185	M₁₆	—	122	2	no	—
186	C₂×C₈	—	61	2	no	—
187	C₂×C₈	—	61	1	no	—
188	C₂×C₈	—	61	1	no	—
189	C₂×C₈	—	61	1	no	—
190	C₂×C₈	—	61	1	no	—
191	C₂×C₈	—	61	1	no	—
192	C₂×C₈	—	122	4	no	—
193	C₂×C₈	—	122	4	no	—
194	C₂×C₈	—	122	2	no	—
195	C₂×C₈	C₂×#6923	1	1	yes	characteristic
196	C₄⋊C₄	—	122	1	no	—
197	C₂²⋊C₄	—	122	1	no	—
198	C₄²	—	61	2	no	—
199	C₂³	—	61	1	no	—
200	C₂³	—	122	2	no	—
201	Q₈	—	61	2	no	—
202	Q₈	—	122	2	no	—
203	Q₈	C₂²×#6923	1	2	yes	—
204	D₄	—	2	2	no	—
205	D₄	—	61	2	no	—
206	D₄	—	122	2	no	—
207	D₄	—	122	2	no	—
208	D₄	—	122	2	no	—
209	D₄	—	122	2	no	—
210	D₄	C₂²×#6923	1	2	yes	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	61	1	no	—
213	C₂×C₄	—	61	1	no	—
214	C₂×C₄	—	61	1	no	—
215	C₂×C₄	—	61	1	no	—
216	C₂×C₄	—	61	1	no	—
217	C₂×C₄	—	122	2	no	—
218	C₂×C₄	—	122	2	no	—
219	C₂×C₄	—	122	2	no	—
220	C₂×C₄	—	122	2	no	—
221	C₂×C₄	—	122	2	no	—
222	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
223	C₈	—	61	4	no	—
224	C₈	—	61	1	no	—
225	C₈	—	61	1	no	—
226	C₈	—	122	4	no	—
227	C₈	C₂²×#6923	1	1	yes	characteristic
228	C₈	C₂²×#6923	1	1	yes	characteristic
229	C₂²	—	2	2	no	—
230	C₂²	—	61	1	no	—
231	C₂²	—	61	1	no	—
232	C₂²	—	122	2	no	—
233	C₂²	—	488	2	no	—
234	C₂²	—	244	1	no	—
235	C₂²	D₄×#6923	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	61	1	no	—
238	C₄	—	61	1	no	—
239	C₄	—	122	2	no	—
240	C₄	—	244	2	no	—
241	C₄	D₄×#6923	1	1	yes	characteristic
242	C₄	C₂³×#6923	1	1	yes	characteristic
243	C₂	—	2	1	no	—
244	C₂	—	4	2	no	—
245	C₂	—	122	1	no	—
246	C₂	—	122	1	no	—
247	C₂	—	244	2	no	—
248	C₂	C₂×D₄×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×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	#2957
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #16840841	C₂	semidirect
2. #16840809	C₂	semidirect
3. #16840808	C₂	semidirect
4. #16840805	C₂	semidirect
5. C₂×#14519847	C₂	semidirect
6. #16840759	C₂	semidirect
7. #16840725	C₂	semidirect
9. #14519865	C₂²	semidirect
11. #14519864	C₂²	semidirect
17. #14519847	C₂²	semidirect
18. #14519847	C₂²	semidirect
19. #14519842	C₂²	semidirect
20. #14519841	C₂²	semidirect
21. #14519831	C₂²	semidirect
22. C₈×#6923	C₂²	semidirect
25. #14519799	C₄	semidirect
27. #14519797	C₄	semidirect
29. Q₁₆×D₆₁	C₄	semidirect
30. SD₁₆×D₆₁	C₄	semidirect
31. D₈×D₆₁	C₄	semidirect
33. #135×C₆₁	C₄	semidirect
133. C₆₁	#2957	semidirect
140. #135	#6923	semidirect