Subgroups — #21455670

Order 7808.

376 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16840390	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840069	C₂	1	2	yes	maximal, complemented
3	#16840062	C₂	1	2	yes	maximal, complemented
4	#16840057	C₂	1	2	yes	maximal, complemented
5	#16840042	C₂	1	2	yes	maximal, complemented
6	#16839802	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14519710	C₂	1	1	yes	maximal, characteristic, complemented
8	#16839789	C₂	1	1	yes	maximal, characteristic, complemented
9	#16839763	C₂	1	1	yes	maximal, characteristic
10	#16839758	C₂	1	1	yes	maximal, characteristic
11	C₆₁×#520	C₂	1	1	yes	maximal, characteristic
12	C₂×#11758194	C₂²	1	1	yes	characteristic
13	C₂²×#92418	C₂²	1	1	yes	characteristic
14	#14519777	C₂²	1	2	yes	—
15	#14519776	—	2	1	no	—
16	#14519769	C₂²	1	2	yes	—
17	#14519768	—	2	1	no	—
18	#14519754	—	2	2	no	—
19	#14519736	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
21	#14519710	C₂²	1	2	yes	—
22	#14519710	C₂²	1	2	yes	—
23	#14519710	C₂²	1	2	yes	—
24	#14519708	C₂²	1	2	yes	—
25	C₄⋊#92418	C₂²	1	2	yes	—
26	C₄⋊#92418	C₂²	1	2	yes	—
27	#14519701	C₂²	1	1	yes	characteristic
28	#14519701	C₂²	1	1	yes	characteristic
29	#14519701	C₂²	1	2	yes	—
30	#14519698	C₂²	1	1	yes	characteristic
31	#14519698	C₂²	1	1	yes	characteristic
32	#14519698	C₂²	1	2	yes	—
33	C₂×#41×C₆₁	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₆₁	C₂²	1	2	yes	—
35	C₄×D₄×C₆₁	—	2	2	no	—
36	C₄×D₄×C₆₁	C₂²	1	2	yes	—
37	#117×C₆₁	—	2	1	no	—
38	#117×C₆₁	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
40	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
41	#41×C₆₁	—	2	2	no	—
42	#41×C₆₁	Q₈	1	4	yes	complemented
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	—	2	1	no	—
48	C₂²×C₄×C₆₁	—	2	2	no	—
49	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
50	C₂²×Dic₆₁	—	2	1	no	—
51	C₂×Dic₁₂₂	—	2	2	no	—
52	C₄⋊C₄×C₆₁	—	2	1	no	—
53	C₄⋊C₄×C₆₁	—	2	2	no	—
54	C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
55	C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
56	C₄⋊C₄×C₆₁	C₂³	1	2	yes	—
57	C₂²⋊C₄×C₆₁	—	2	2	no	—
58	C₂²⋊C₄×C₆₁	D₄	1	2	yes	—
59	C₄²×C₆₁	—	2	2	no	—
60	C₄²×C₆₁	C₂³	1	2	yes	—
61	C₂²⋊Dic₆₁	—	2	1	no	—
62	C₄⋊Dic₆₁	—	2	1	no	—
63	C₄⋊Dic₆₁	—	2	2	no	—
64	C₄⋊Dic₆₁	C₂³	1	1	yes	characteristic
65	C₄⋊Dic₆₁	C₂³	1	2	yes	—
66	C₄⋊Dic₆₁	C₂³	1	1	yes	characteristic
67	#11758196	—	2	2	no	—
68	#11758196	—	2	2	no	—
69	C₄×Dic₆₁	—	2	1	no	—
70	#11758194	—	2	2	no	—
71	C₂×#92418	—	2	2	no	—
72	C₂×#92418	C₂³	1	2	yes	—
73	C₂×#92418	C₂³	1	1	yes	characteristic
74	C₂×#92418	C₂³	1	1	yes	characteristic
75	C₂³×C₆₁	—	2	1	no	—
76	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
77	Q₈×C₆₁	—	2	1	no	—
78	Q₈×C₆₁	C₂×Q₈	1	2	yes	—
79	D₄×C₆₁	—	2	4	no	—
80	D₄×C₆₁	—	2	1	no	—
81	D₄×C₆₁	C₂×Q₈	1	2	yes	—
82	C₂×C₄×C₆₁	—	2	1	no	—
83	C₂×C₄×C₆₁	—	2	4	no	—
84	C₂×C₄×C₆₁	—	2	2	no	—
85	C₂×C₄×C₆₁	—	8	2	no	—
86	C₂×C₄×C₆₁	—	8	1	no	—
87	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
91	C₂×C₄×C₆₁	#41	1	2	yes	—
92	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
93	C₂×Dic₆₁	—	2	1	no	—
94	C₂×Dic₆₁	—	2	1	no	—
95	C₂×Dic₆₁	—	2	2	no	—
96	C₂×Dic₆₁	—	8	1	no	—
97	Dic₁₂₂	—	8	2	no	—
98	#92418	—	2	1	no	—
99	#92418	—	2	1	no	—
100	#92418	—	2	2	no	—
101	C₂²×C₆₁	—	2	1	no	—
102	C₂²×C₆₁	—	2	1	no	—
103	C₂²×C₆₁	—	2	2	no	—
104	C₂²×C₆₁	—	4	1	no	—
105	C₂²×C₆₁	—	4	1	no	—
106	C₂²×C₆₁	C₂²⋊Q₈	1	2	yes	—
107	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₆₁	—	2	2	no	—
109	C₄×C₆₁	—	2	2	no	—
110	C₄×C₆₁	—	2	2	no	—
111	C₄×C₆₁	—	4	2	no	—
112	C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
113	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
114	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
115	Dic₆₁	—	4	1	no	—
116	Dic₆₁	—	4	2	no	—
117	Dic₆₁	—	4	1	no	—
118	#3083	—	61	1	no	maximal, Sylow
119	C₂×C₆₁	—	2	2	no	—
120	C₂×C₆₁	—	4	2	no	—
121	C₂×C₆₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₆₁	#576	1	1	yes	characteristic
123	C₂×C₆₁	#578	1	1	yes	characteristic
124	#544	—	61	1	no	—
125	#520	D₆₁	1	1	yes	characteristic
126	#479	—	61	2	no	—
127	#478	—	61	2	no	—
128	#477	—	61	2	no	—
129	#474	—	61	2	no	—
130	#428	—	61	1	no	—
131	C₂×#107	—	61	1	no	—
132	#424	—	61	1	no	—
133	#417	—	61	1	no	—
134	#416	—	61	1	no	—
135	C₆₁	#3083	1	1	yes	characteristic, complemented, Sylow
136	C₂×#41	C₂×D₆₁	1	1	yes	characteristic
137	C₂×M₁₆	—	61	1	no	—
138	C₂²×C₈	—	61	1	no	—
139	C₄⋊Q₈	—	61	2	no	—
140	C₄⋊Q₈	—	122	1	no	—
141	#125	—	61	2	no	—
142	#125	—	122	1	no	—
143	C₂²⋊Q₈	—	122	2	no	—
144	C₄×Q₈	C₂×D₆₁	1	2	yes	—
145	C₄×D₄	—	2	2	no	—
146	C₄×D₄	C₂×D₆₁	1	2	yes	—
147	#117	—	2	1	no	—
148	#117	—	61	1	no	—
149	#117	C₂×D₆₁	1	1	yes	characteristic
150	C₂×C₄⋊C₄	—	2	1	no	—
151	C₂×C₄⋊C₄	—	61	1	no	—
152	C₂×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
153	#107	—	61	2	no	—
154	#107	—	61	2	no	—
155	#107	—	61	2	no	—
156	#106	—	61	2	no	—
157	C₄⋊C₈	—	61	2	no	—
158	C₄⋊C₈	—	61	2	no	—
159	#103	—	61	1	no	—
160	#103	—	61	1	no	—
161	#103	—	61	2	no	—
162	#102	—	61	1	no	—
163	#102	—	61	1	no	—
164	#102	—	61	2	no	—
165	#41	—	2	2	no	—
166	#41	Dic₁₂₂	1	4	yes	complemented
167	C₂×Q₈	—	122	2	no	—
168	C₂×Q₈	C₂²×D₆₁	1	1	yes	characteristic
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	C₂²×D₆₁	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	122	1	no	—
175	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
176	M₁₆	—	122	2	no	—
177	C₂×C₈	—	61	2	no	—
178	C₂×C₈	—	61	1	no	—
179	C₂×C₈	—	61	1	no	—
180	C₂×C₈	—	122	2	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	61	1	no	—
184	C₄⋊C₄	—	61	2	no	—
185	C₄⋊C₄	—	61	1	no	—
186	C₄⋊C₄	—	122	1	no	—
187	C₄⋊C₄	—	122	2	no	—
188	C₄⋊C₄	—	122	2	no	—
189	C₄⋊C₄	—	122	2	no	—
190	C₄⋊C₄	C₂²⋊D₆₁	1	1	yes	characteristic
191	C₄⋊C₄	C₂²⋊D₆₁	1	1	yes	characteristic
192	C₄⋊C₄	C₂²×D₆₁	1	2	yes	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	122	1	no	—
195	C₂²⋊C₄	C₂²⋊D₆₁	1	2	yes	—
196	C₄²	—	2	2	no	—
197	C₄²	—	122	1	no	—
198	C₄²	C₂²×D₆₁	1	2	yes	—
199	C₂³	—	2	1	no	—
200	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
201	Q₈	—	2	1	no	—
202	Q₈	—	488	2	no	—
203	Q₈	C₂×Dic₁₂₂	1	2	yes	—
204	D₄	—	2	4	no	—
205	D₄	—	2	1	no	—
206	D₄	C₂×Dic₁₂₂	1	2	yes	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	—	8	1	no	—
212	C₂×C₄	—	122	1	no	—
213	C₂×C₄	—	122	1	no	—
214	C₂×C₄	—	122	2	no	—
215	C₂×C₄	—	488	1	no	—
216	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
217	C₂×C₄	#11758222	1	2	yes	—
218	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
219	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
220	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
221	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
222	C₈	—	122	1	no	—
223	C₈	—	122	1	no	—
224	C₈	—	122	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	1	no	—
230	C₂²	#14519756	1	2	yes	—
231	C₂²	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	—	244	2	no	—
237	C₄	—	244	1	no	—
238	C₄	—	244	1	no	—
239	C₄	#14519756	1	2	yes	—
240	C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
241	C₄	C₂×#11758222	1	1	yes	characteristic
242	C₂	—	2	2	no	—
243	C₂	—	4	2	no	—
244	C₂	C₂×#14519756	1	1	yes	characteristic
245	C₂	#16840565	1	1	yes	characteristic
246	C₂	#16840577	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#520	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#520	D₆₁
hypocenter	C₆₁	#3083
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
1. #16840390	C₂	semidirect
2. #16840069	C₂	semidirect
3. #16840062	C₂	semidirect
4. #16840057	C₂	semidirect
5. #16840042	C₂	semidirect
6. #16839802	C₂	semidirect
7. C₂×#14519710	C₂	semidirect
8. #16839789	C₂	semidirect
42. #41×C₆₁	Q₈	semidirect
135. C₆₁	#3083	semidirect
166. #41	Dic₁₂₂	semidirect