Subgroups — #21455895

Order 7808.

390 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840090	C₂	1	1	yes	maximal, characteristic, complemented
3	#16840073	C₂	1	1	yes	maximal, characteristic
4	#16840066	C₂	1	2	yes	maximal, complemented
5	#16839981	C₂	1	2	yes	maximal, complemented
6	#16839917	C₂	1	1	yes	maximal, characteristic, complemented
7	#16839900	C₂	1	1	yes	maximal, characteristic
8	C₄×#11758202	C₂	1	2	yes	maximal, complemented
9	C₄×#11758200	C₂	1	2	yes	maximal, complemented
10	#16839815	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#529	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₁₂₂	C₂²	1	1	yes	characteristic
13	C₂×#11758202	C₂²	1	2	yes	—
14	C₂×#11758200	C₂²	1	2	yes	—
15	C₄×Dic₁₂₂	—	2	1	no	—
16	C₄×Dic₁₂₂	—	2	1	no	—
17	C₄×Dic₁₂₂	C₂²	1	2	yes	—
18	C₄×Dic₁₂₂	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
20	C₂×#11758196	—	2	1	no	—
21	C₂×C₄×Dic₆₁	—	2	1	no	—
22	#14519710	C₂²	1	2	yes	—
23	#14519708	C₂²	1	2	yes	—
24	C₄⋊#92418	C₂²	1	2	yes	—
25	#14519701	C₂²	1	2	yes	—
26	#14519700	C₂²	1	2	yes	—
27	#14519700	C₂²	1	2	yes	—
28	#14519698	C₂²	1	2	yes	—
29	C₂²⋊#92418	C₂²	1	1	yes	characteristic
30	C₂²⋊#92418	C₂²	1	1	yes	characteristic
31	C₄×#92418	C₂²	1	2	yes	—
32	#126×C₆₁	—	2	1	no	—
33	#125×C₆₁	C₂²	1	1	yes	characteristic
34	#124×C₆₁	C₂²	1	1	yes	characteristic
35	#123×C₆₁	—	2	1	no	—
36	C₂²⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
37	#121×C₆₁	C₂²	1	1	yes	characteristic
38	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
39	C₄×D₄×C₆₁	—	2	1	no	—
40	C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
41	#117×C₆₁	—	2	1	no	—
42	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
43	C₂×Q₈×C₆₁	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₆₁	—	2	1	no	—
45	C₂×D₄×C₆₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	—	2	1	no	—
47	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
50	C₂²×Dic₆₁	—	2	1	no	—
51	C₂²×Dic₆₁	—	2	1	no	—
52	C₂×Dic₁₂₂	—	2	1	no	—
53	C₂×Dic₁₂₂	—	2	4	no	—
54	C₂×Dic₁₂₂	C₂³	1	2	yes	—
55	C₄⋊C₄×C₆₁	—	2	1	no	—
56	C₄⋊C₄×C₆₁	—	2	1	no	—
57	C₄⋊C₄×C₆₁	—	2	1	no	—
58	C₄⋊C₄×C₆₁	—	2	1	no	—
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	C₂²⋊C₄×C₆₁	—	2	1	no	—
63	C₂²⋊C₄×C₆₁	—	2	1	no	—
64	C₂²⋊C₄×C₆₁	—	2	1	no	—
65	C₂²⋊C₄×C₆₁	—	2	1	no	—
66	C₄²×C₆₁	—	2	1	no	—
67	C₄²×C₆₁	D₄	1	1	yes	characteristic
68	C₄²×C₆₁	D₄	1	1	yes	characteristic
69	C₄²×C₆₁	C₂³	1	1	yes	characteristic
70	C₄²×C₆₁	C₂³	1	1	yes	characteristic
71	#11758202	—	2	4	no	—
72	#11758200	—	2	4	no	—
73	C₄⋊Dic₆₁	—	2	1	no	—
74	C₄⋊Dic₆₁	C₂³	1	2	yes	—
75	#11758196	—	2	2	no	—
76	#11758196	—	2	2	no	—
77	C₄×Dic₆₁	—	2	2	no	—
78	C₄×Dic₆₁	—	2	2	no	—
79	C₂×#92418	C₂³	1	2	yes	—
80	C₂×#92418	C₂³	1	2	yes	—
81	C₂³×C₆₁	—	2	1	no	—
82	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
83	Q₈×C₆₁	—	2	2	no	—
84	D₄×C₆₁	—	2	2	no	—
85	D₄×C₆₁	—	4	2	no	—
86	C₂×C₄×C₆₁	—	2	2	no	—
87	C₂×C₄×C₆₁	—	2	2	no	—
88	C₂×C₄×C₆₁	—	2	2	no	—
89	C₂×C₄×C₆₁	—	2	1	no	—
90	C₂×C₄×C₆₁	—	2	1	no	—
91	C₂×C₄×C₆₁	—	2	1	no	—
92	C₂×C₄×C₆₁	—	4	2	no	—
93	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
99	C₂×Dic₆₁	—	2	4	no	—
100	C₂×Dic₆₁	—	2	2	no	—
101	C₂×Dic₆₁	—	2	2	no	—
102	C₂×Dic₆₁	—	4	2	no	—
103	Dic₁₂₂	—	2	4	no	—
104	Dic₁₂₂	—	2	2	no	—
105	Dic₁₂₂	#41	1	4	yes	—
106	#92418	—	2	2	no	—
107	#92418	—	2	2	no	—
108	C₂²×C₆₁	—	2	1	no	—
109	C₂²×C₆₁	—	2	1	no	—
110	C₂²×C₆₁	—	4	1	no	—
111	C₂²×C₆₁	—	4	1	no	—
112	C₂²×C₆₁	—	4	1	no	—
113	C₂²×C₆₁	#135	1	2	yes	—
114	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₆₁	—	2	1	no	—
116	C₄×C₆₁	—	2	1	no	—
117	C₄×C₆₁	—	2	1	no	—
118	C₄×C₆₁	—	2	1	no	—
119	C₄×C₆₁	—	4	1	no	—
120	C₄×C₆₁	—	4	1	no	—
121	C₄×C₆₁	—	4	1	no	—
122	C₄×C₆₁	#137	1	2	yes	—
123	C₄×C₆₁	C₂×#41	1	2	yes	—
124	Dic₆₁	—	2	4	no	—
125	Dic₆₁	—	4	2	no	—
126	#3108	—	61	1	no	maximal, Sylow
127	C₂×C₆₁	—	2	2	no	—
128	C₂×C₆₁	—	8	1	no	—
129	C₂×C₆₁	#525	1	1	yes	characteristic
130	C₂×C₆₁	C₂×#135	1	1	yes	characteristic
131	C₂×C₆₁	C₂×#137	1	1	yes	characteristic
132	#529	D₆₁	1	1	yes	characteristic
133	C₂×C₄×Q₈	—	61	1	no	—
134	#483	—	61	1	no	—
135	#480	—	61	1	no	—
136	#479	—	61	2	no	—
137	#464	—	61	2	no	—
138	#451	—	61	1	no	—
139	#448	—	61	1	no	—
140	C₄×Q₁₆	—	61	2	no	—
141	C₄×SD₁₆	—	61	2	no	—
142	#431	—	61	1	no	—
143	C₆₁	#3108	1	1	yes	characteristic, complemented, Sylow
144	C₂²×Q₈	—	61	1	no	—
145	C₂×Q₁₆	—	61	2	no	—
146	C₂×SD₁₆	—	61	2	no	—
147	#126	—	2	1	no	—
148	#125	C₂×D₆₁	1	1	yes	characteristic
149	#124	C₂×D₆₁	1	1	yes	characteristic
150	#123	—	2	1	no	—
151	C₂²⋊Q₈	C₂×D₆₁	1	1	yes	characteristic
152	#121	C₂×D₆₁	1	1	yes	characteristic
153	C₄×Q₈	—	61	2	no	—
154	C₄×Q₈	—	61	2	no	—
155	C₄×Q₈	—	122	1	no	—
156	C₄×Q₈	—	122	1	no	—
157	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
158	C₄×D₄	—	2	1	no	—
159	C₄×D₄	C₂×D₆₁	1	1	yes	characteristic
160	#117	—	2	1	no	—
161	C₂×C₄⋊C₄	—	61	1	no	—
162	C₂×C₄⋊C₄	—	122	1	no	—
163	C₂×C₄²	—	122	1	no	—
164	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
165	#107	—	61	2	no	—
166	#106	—	61	2	no	—
167	C₄⋊C₈	—	61	2	no	—
168	#103	—	61	2	no	—
169	#103	—	61	2	no	—
170	#103	—	61	2	no	—
171	#102	—	61	2	no	—
172	C₂²⋊C₈	—	61	1	no	—
173	C₂²⋊C₈	—	61	1	no	—
174	C₄×C₈	—	61	2	no	—
175	C₂×Q₈	—	61	2	no	—
176	C₂×Q₈	—	122	4	no	—
177	C₂×Q₈	—	122	1	no	—
178	C₂×Q₈	C₂²×D₆₁	1	1	yes	characteristic
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	C₂²×D₆₁	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	122	1	no	—
183	C₂²×C₄	—	122	1	no	—
184	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
185	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
186	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
187	Q₁₆	—	122	4	no	—
188	SD₁₆	—	122	4	no	—
189	C₂×C₈	—	61	2	no	—
190	C₂×C₈	—	61	2	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	61	2	no	—
196	C₄⋊C₄	—	122	1	no	—
197	C₄⋊C₄	—	122	2	no	—
198	C₄⋊C₄	—	122	2	no	—
199	C₄⋊C₄	C₂²×D₆₁	1	1	yes	characteristic
200	C₄⋊C₄	C₂²×D₆₁	1	1	yes	characteristic
201	C₂²⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	122	2	no	—
208	C₄²	—	122	2	no	—
209	C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
210	C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
211	C₄²	C₂²×D₆₁	1	1	yes	characteristic
212	C₄²	C₂²×D₆₁	1	1	yes	characteristic
213	C₂³	—	2	1	no	—
214	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
215	Q₈	—	2	2	no	—
216	Q₈	—	61	4	no	—
217	Q₈	—	122	4	no	—
218	Q₈	—	122	2	no	—
219	D₄	—	2	2	no	—
220	D₄	—	4	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	122	4	no	—
229	C₂×C₄	—	122	2	no	—
230	C₂×C₄	—	122	2	no	—
231	C₂×C₄	—	244	2	no	—
232	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
233	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
235	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
237	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
238	C₈	—	122	2	no	—
239	C₈	—	122	2	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	—	4	1	no	—
245	C₂²	#14519797	1	2	yes	—
246	C₂²	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	4	1	no	—
252	C₄	—	4	1	no	—
253	C₄	—	4	1	no	—
254	C₄	—	122	4	no	—
255	C₄	—	244	2	no	—
256	C₄	#14519807	1	2	yes	—
257	C₄	#41×D₆₁	1	2	yes	—
258	C₂	—	2	2	no	—
259	C₂	—	8	1	no	—
260	C₂	#16840269	1	1	yes	characteristic
261	C₂	C₂×#14519797	1	1	yes	characteristic
262	C₂	C₂×#14519807	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#529	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#529	D₆₁
hypocenter	C₆₁	#3108
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #529

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×Dic₁₂₂	C₂	semidirect
2. #16840090	C₂	semidirect
4. #16840066	C₂	semidirect
5. #16839981	C₂	semidirect
6. #16839917	C₂	semidirect
8. C₄×#11758202	C₂	semidirect
9. C₄×#11758200	C₂	semidirect
10. #16839815	C₂	semidirect
143. C₆₁	#3108	semidirect