Subgroups — #21454665

Order 7808.

470 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14519807	C₂	1	1	yes	maximal, characteristic
2	#16840245	C₂	1	1	yes	maximal, characteristic
3	#16840239	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839866	C₂	1	2	yes	maximal, complemented
5	#16839859	C₂	1	2	yes	maximal, complemented
6	#16839854	C₂	1	2	yes	maximal, complemented
7	#16839853	C₂	1	2	yes	maximal, complemented
8	#16839805	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14519702	C₂	1	1	yes	maximal, characteristic, complemented
10	#16839717	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#416	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₁₂₂	C₂²	1	1	yes	characteristic
13	#14519807	C₄	1	8	yes	complemented
14	C₂×#11758202	C₂²	1	2	yes	—
15	C₂×#11758200	C₂²	1	2	yes	—
16	C₂×#11758194	C₂²	1	1	yes	characteristic
17	Q₈×Dic₆₁	C₂²	1	2	yes	—
18	C₄×Dic₁₂₂	—	2	1	no	—
19	C₄×Dic₁₂₂	—	2	1	no	—
20	#14519742	C₂²	1	2	yes	—
21	#14519742	C₂²	1	2	yes	—
22	D₄×Dic₆₁	—	2	2	no	—
23	D₄×Dic₆₁	C₂²	1	2	yes	—
24	#14519736	—	2	1	no	—
25	#14519736	C₂²	1	1	yes	characteristic
26	#14519733	—	2	2	no	—
27	C₂×C₄⋊Dic₆₁	—	2	1	no	—
28	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
29	#14519710	C₂²	1	2	yes	—
30	#14519708	C₂²	1	2	yes	—
31	#14519702	C₂²	1	2	yes	—
32	#14519702	C₂²	1	2	yes	—
33	#14519687	C₂²	1	2	yes	—
34	C₈×Dic₆₁	C₂²	1	2	yes	—
35	C₂×#41×C₆₁	C₂²	1	1	yes	characteristic
36	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
37	#117×C₆₁	C₂²	1	1	yes	characteristic
38	#103×C₆₁	C₂²	1	1	yes	characteristic
39	#103×C₆₁	C₂²	1	1	yes	characteristic
40	#102×C₆₁	C₂²	1	1	yes	characteristic
41	#102×C₆₁	C₂²	1	1	yes	characteristic
42	#41×C₆₁	—	2	2	no	—
43	#41×C₆₁	C₂×C₄	1	4	yes	—
44	C₂×Q₈×C₆₁	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₆₁	—	2	1	no	—
46	C₂×D₄×C₆₁	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	—	2	1	no	—
48	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
49	C₂²×Dic₆₁	—	2	1	no	—
50	C₂²×Dic₆₁	—	2	1	no	—
51	C₂²×Dic₆₁	—	2	2	no	—
52	C₂×Dic₁₂₂	—	2	1	no	—
53	C₂×Dic₁₂₂	—	2	2	no	—
54	C₂×Dic₁₂₂	C₂×C₄	1	4	yes	complemented
55	C₂×Dic₁₂₂	C₂³	1	2	yes	—
56	C₂×C₈×C₆₁	—	2	2	no	—
57	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
61	C₂²⋊C₄×C₆₁	—	2	1	no	—
62	C₄²×C₆₁	—	2	1	no	—
63	C₂²⋊Dic₆₁	—	2	2	no	—
64	C₂²⋊Dic₆₁	D₄	1	2	yes	—
65	#11758202	C₂×C₄	1	8	yes	complemented
66	#11758200	C₂×C₄	1	8	yes	complemented
67	C₄⋊Dic₆₁	—	2	1	no	—
68	C₄⋊Dic₆₁	—	2	2	no	—
69	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
70	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
71	C₄⋊Dic₆₁	C₂³	1	2	yes	—
72	#11758196	—	2	2	no	—
73	#11758196	—	2	2	no	—
74	C₄×Dic₆₁	—	2	2	no	—
75	C₄×Dic₆₁	—	2	2	no	—
76	C₄×Dic₆₁	—	2	2	no	—
77	C₄×Dic₆₁	C₂³	1	2	yes	—
78	#11758194	C₂×C₄	1	4	yes	complemented
79	C₂×#92418	C₂³	1	2	yes	—
80	C₂³×C₆₁	—	2	1	no	—
81	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
82	Q₈×C₆₁	—	2	1	no	—
83	Q₈×C₆₁	C₂²×C₄	1	2	yes	—
84	D₄×C₆₁	—	2	4	no	—
85	D₄×C₆₁	—	2	1	no	—
86	D₄×C₆₁	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₆₁	—	2	1	no	—
88	C₂×C₄×C₆₁	—	2	4	no	—
89	C₂×C₄×C₆₁	—	2	1	no	—
90	C₂×C₄×C₆₁	—	2	1	no	—
91	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₆₁	#41	1	2	yes	—
94	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
95	C₂×Dic₆₁	—	2	4	no	—
96	C₂×Dic₆₁	—	2	2	no	—
97	C₂×Dic₆₁	—	2	2	no	—
98	C₂×Dic₆₁	—	8	2	no	—
99	C₂×Dic₆₁	—	8	1	no	—
100	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
101	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
102	Dic₁₂₂	—	2	4	no	—
103	Dic₁₂₂	—	2	2	no	—
104	Dic₁₂₂	C₂²×C₄	1	4	yes	—
105	C₈×C₆₁	—	2	1	no	—
106	C₈×C₆₁	—	2	1	no	—
107	#92418	C₂²×C₄	1	4	yes	—
108	C₂²×C₆₁	—	2	1	no	—
109	C₂²×C₆₁	—	2	1	no	—
110	C₂²×C₆₁	—	2	2	no	—
111	C₂²×C₆₁	—	4	1	no	—
112	C₂²×C₆₁	—	4	1	no	—
113	C₂²×C₆₁	C₄×D₄	1	2	yes	—
114	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₆₁	—	2	2	no	—
116	C₄×C₆₁	—	4	1	no	—
117	C₄×C₆₁	—	4	1	no	—
118	C₄×C₆₁	C₄×D₄	1	2	yes	—
119	C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
120	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
121	Dic₆₁	—	2	4	no	—
122	Dic₆₁	—	2	2	no	—
123	Dic₆₁	—	2	2	no	—
124	Dic₆₁	—	4	2	no	—
125	#2947	—	61	1	no	maximal, Sylow
126	C₂×C₆₁	—	2	2	no	—
127	C₂×C₆₁	—	4	2	no	—
128	C₂×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
129	C₂×C₆₁	#577	1	1	yes	characteristic
130	C₂×C₆₁	#578	1	1	yes	characteristic
131	C₂×#137	—	61	1	no	—
132	#520	—	61	1	no	—
133	#519	—	61	1	no	—
134	#441	—	61	2	no	—
135	#440	—	61	2	no	—
136	C₄×Q₁₆	—	61	2	no	—
137	C₄×SD₁₆	—	61	2	no	—
138	#428	—	61	1	no	—
139	#416	D₆₁	1	1	yes	characteristic
140	C₂×#103	—	61	1	no	—
141	#405	—	61	1	no	—
142	C₆₁	#2947	1	1	yes	characteristic, complemented, Sylow
143	C₂×#41	C₂×D₆₁	1	1	yes	characteristic
144	C₂²×Q₈	—	61	1	no	—
145	#137	—	61	8	no	—
146	C₂×Q₁₆	—	61	2	no	—
147	C₂×SD₁₆	—	61	2	no	—
148	C₂×M₁₆	—	61	1	no	—
149	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
150	C₄×Q₈	—	61	2	no	—
151	C₄×Q₈	—	61	2	no	—
152	C₄×Q₈	—	61	2	no	—
153	C₄×Q₈	—	122	1	no	—
154	C₄×Q₈	—	122	1	no	—
155	C₄×D₄	—	61	2	no	—
156	C₄×D₄	—	122	2	no	—
157	#117	—	61	1	no	—
158	#117	—	122	2	no	—
159	#117	—	122	1	no	—
160	#117	C₂×D₆₁	1	1	yes	characteristic
161	C₂×C₄⋊C₄	—	61	1	no	—
162	C₂×C₄⋊C₄	—	122	1	no	—
163	#107	—	61	2	no	—
164	#106	—	61	2	no	—
165	#103	—	61	2	no	—
166	#103	—	61	2	no	—
167	#103	C₂×D₆₁	1	1	yes	characteristic
168	#103	C₂×D₆₁	1	1	yes	characteristic
169	#102	C₂×D₆₁	1	1	yes	characteristic
170	#102	C₂×D₆₁	1	1	yes	characteristic
171	#97	—	61	2	no	—
172	C₄×C₈	—	61	2	no	—
173	#41	—	2	2	no	—
174	#41	C₄×D₆₁	1	4	yes	—
175	C₂×Q₈	—	61	4	no	—
176	C₂×Q₈	—	61	2	no	—
177	C₂×Q₈	—	122	1	no	—
178	C₂×Q₈	—	122	2	no	—
179	C₂×Q₈	C₂²×D₆₁	1	1	yes	characteristic
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	C₂²×D₆₁	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	122	2	no	—
184	C₂²×C₄	—	122	1	no	—
185	C₂²×C₄	—	122	1	no	—
186	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
187	Q₁₆	—	61	8	no	—
188	SD₁₆	—	61	8	no	—
189	M₁₆	—	61	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	61	2	no	—
192	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
193	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
194	C₄⋊C₄	—	61	1	no	—
195	C₄⋊C₄	—	61	1	no	—
196	C₄⋊C₄	—	61	2	no	—
197	C₄⋊C₄	—	122	2	no	—
198	C₄⋊C₄	—	122	2	no	—
199	C₄⋊C₄	—	122	2	no	—
200	C₄⋊C₄	—	122	1	no	—
201	C₄⋊C₄	C₂²×D₆₁	1	1	yes	characteristic
202	C₄⋊C₄	C₂²×D₆₁	1	1	yes	characteristic
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	61	2	no	—
205	C₂²⋊C₄	—	122	2	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	61	2	no	—
208	C₄²	—	122	2	no	—
209	C₄²	—	122	2	no	—
210	C₄²	—	122	2	no	—
211	C₂³	—	2	1	no	—
212	C₂³	D₄×D₆₁	1	1	yes	characteristic
213	Q₈	—	2	1	no	—
214	Q₈	—	61	4	no	—
215	Q₈	—	122	2	no	—
216	Q₈	—	122	4	no	—
217	Q₈	C₂×C₄×D₆₁	1	2	yes	—
218	D₄	—	2	4	no	—
219	D₄	—	2	1	no	—
220	D₄	C₂×C₄×D₆₁	1	2	yes	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	61	2	no	—
226	C₂×C₄	—	61	2	no	—
227	C₂×C₄	—	122	2	no	—
228	C₂×C₄	—	122	4	no	—
229	C₂×C₄	—	122	2	no	—
230	C₂×C₄	—	488	2	no	—
231	C₂×C₄	—	488	1	no	—
232	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
233	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
234	C₂×C₄	#11758224	1	2	yes	—
235	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	61	4	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	#14519737	1	2	yes	—
245	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	4	1	no	—
248	C₄	—	4	1	no	—
249	C₄	—	122	4	no	—
250	C₄	—	122	2	no	—
251	C₄	—	122	2	no	—
252	C₄	—	244	2	no	—
253	C₄	#14519737	1	2	yes	—
254	C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
255	C₄	C₂×#11758224	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	4	2	no	—
258	C₂	C₂×#14519737	1	1	yes	characteristic
259	C₂	#16840569	1	1	yes	characteristic
260	C₂	#16840576	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₁
commutator	C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#416	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#416	D₆₁
hypocenter	C₆₁	#2947
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
3. #16840239	C₂	semidirect
4. #16839866	C₂	semidirect
5. #16839859	C₂	semidirect
6. #16839854	C₂	semidirect
7. #16839853	C₂	semidirect
8. #16839805	C₂	semidirect
9. C₂×#14519702	C₂	semidirect
10. #16839717	C₂	semidirect
13. #14519807	C₄	semidirect
54. C₂×Dic₁₂₂	C₂×C₄	semidirect
65. #11758202	C₂×C₄	semidirect
66. #11758200	C₂×C₄	semidirect
78. #11758194	C₂×C₄	semidirect
142. C₆₁	#2947	semidirect