Subgroups — #21454751

Order 7808.

364 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16840315	C₂	1	1	yes	maximal, characteristic
2	#16839890	C₂	1	4	yes	maximal, complemented
3	C₈×Dic₁₂₂	C₂	1	4	yes	maximal, complemented
4	#16839790	C₂	1	1	yes	maximal, characteristic
5	#16839789	C₂	1	1	yes	maximal, characteristic
6	#16839785	C₂	1	1	yes	maximal, characteristic
7	#16839717	C₂	1	1	yes	maximal, characteristic
8	C₂×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
9	C₆₁×#405	C₂	1	1	yes	maximal, characteristic
10	C₂×#11758194	C₂²	1	1	yes	characteristic
11	C₂²×#92418	C₂²	1	1	yes	characteristic
12	#14519776	C₄	1	2	yes	—
13	#14519768	C₄	1	2	yes	—
14	#14519754	C₄	1	4	yes	—
15	C₄×Dic₁₂₂	C₂²	1	4	yes	—
16	#14519736	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
18	C₄⋊#92418	C₂²	1	4	yes	—
19	#14519706	C₂²	1	4	yes	—
20	#14519706	C₂²	1	4	yes	—
21	#14519687	C₂²	1	2	yes	—
22	C₈×Dic₆₁	C₂²	1	2	yes	—
23	C₈×Dic₆₁	C₂²	1	2	yes	—
24	C₈×Dic₆₁	C₂²	1	2	yes	—
25	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
26	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
27	#117×C₆₁	C₂²	1	1	yes	characteristic
28	#97×C₆₁	C₂²	1	2	yes	—
29	C₄×C₈×C₆₁	C₂²	1	2	yes	—
30	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
31	C₂²×Dic₆₁	C₂×C₄	1	2	yes	—
32	C₂×Dic₁₂₂	C₂×C₄	1	4	yes	—
33	M₁₆×C₆₁	—	2	2	no	—
34	C₂×C₈×C₆₁	Q₈	1	4	yes	complemented
35	C₂×C₈×C₆₁	C₂³	1	2	yes	—
36	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
39	C₂²⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
40	C₄²×C₆₁	C₂³	1	2	yes	—
41	C₂²⋊Dic₆₁	C₂×C₄	1	2	yes	—
42	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
43	C₄⋊Dic₆₁	C₂×C₄	1	4	yes	—
44	#11758196	C₂×C₄	1	4	yes	—
45	#11758196	C₂×C₄	1	4	yes	—
46	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
47	C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
48	C₄×Dic₆₁	C₂³	1	2	yes	—
49	C₄×Dic₆₁	C₂³	1	2	yes	—
50	#11758194	—	2	2	no	—
51	C₂×#92418	—	2	2	no	—
52	C₂×#92418	C₂³	1	2	yes	—
53	C₂×#92418	C₂³	1	2	yes	—
54	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₆₁	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₆₁	#41	1	2	yes	—
59	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
60	C₂×Dic₆₁	—	2	4	no	—
61	C₂×Dic₆₁	C₂²×C₄	1	2	yes	—
62	C₂×Dic₆₁	C₂²×C₄	1	4	yes	—
63	C₂×Dic₆₁	C₂²×C₄	1	2	yes	—
64	Dic₁₂₂	—	2	8	no	—
65	C₈×C₆₁	—	2	2	no	—
66	C₈×C₆₁	C₂×Q₈	1	2	yes	—
67	C₈×C₆₁	C₂×Q₈	1	2	yes	—
68	#92418	—	2	2	no	—
69	#92418	—	2	2	no	—
70	C₂²×C₆₁	—	2	2	no	—
71	C₂²×C₆₁	C₄×Q₈	1	2	yes	—
72	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
73	C₄×C₆₁	—	2	4	no	—
74	C₄×C₆₁	C₄×Q₈	1	2	yes	—
75	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
76	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
77	Dic₆₁	—	2	4	no	—
78	Dic₆₁	—	2	2	no	—
79	Dic₆₁	#131	1	4	yes	—
80	#2960	—	61	1	no	maximal, Sylow
81	C₂×C₆₁	—	2	2	no	—
82	C₂×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
83	C₂×C₆₁	C₂×#131	1	2	yes	—
84	#533	—	61	1	no	—
85	#446	—	61	4	no	—
86	C₈×Q₈	—	61	4	no	—
87	#424	—	61	1	no	—
88	#424	—	61	1	no	—
89	#423	—	61	1	no	—
90	#405	—	61	1	no	—
91	#405	D₆₁	1	1	yes	characteristic
92	C₂×C₄×C₈	—	61	1	no	—
93	C₆₁	#2960	1	1	yes	characteristic, complemented, Sylow
94	C₂×M₁₆	—	61	1	no	—
95	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
96	C₂²×C₈	—	61	1	no	—
97	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
98	C₄⋊Q₈	—	61	2	no	—
99	#125	—	61	2	no	—
100	C₂²⋊Q₈	—	61	4	no	—
101	C₄×Q₈	—	61	4	no	—
102	#117	—	61	1	no	—
103	#117	C₂×D₆₁	1	1	yes	characteristic
104	C₂×C₄²	—	61	1	no	—
105	C₄⋊C₈	—	61	4	no	—
106	C₄⋊C₈	—	61	4	no	—
107	C₄⋊C₈	—	61	4	no	—
108	#97	—	61	2	no	—
109	#97	C₂×D₆₁	1	2	yes	—
110	C₄×C₈	—	61	2	no	—
111	C₄×C₈	—	61	2	no	—
112	C₄×C₈	—	61	2	no	—
113	C₄×C₈	C₂×D₆₁	1	2	yes	—
114	C₂×Q₈	—	61	4	no	—
115	C₂²×C₄	—	61	2	no	—
116	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
117	M₁₆	—	2	2	no	—
118	M₁₆	—	122	2	no	—
119	C₂×C₈	—	61	2	no	—
120	C₂×C₈	—	61	2	no	—
121	C₂×C₈	—	122	2	no	—
122	C₂×C₈	Dic₁₂₂	1	4	yes	complemented
123	C₂×C₈	C₂²×D₆₁	1	2	yes	—
124	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
125	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
126	C₄⋊C₄	—	61	4	no	—
127	C₄⋊C₄	—	61	4	no	—
128	C₄⋊C₄	—	61	2	no	—
129	C₄⋊C₄	—	61	4	no	—
130	C₄⋊C₄	C₄×D₆₁	1	2	yes	—
131	C₂²⋊C₄	—	61	2	no	—
132	C₂²⋊C₄	C₄×D₆₁	1	2	yes	—
133	C₄²	—	61	1	no	—
134	C₄²	—	61	1	no	—
135	C₄²	—	61	2	no	—
136	C₄²	—	61	2	no	—
137	C₄²	C₂²×D₆₁	1	2	yes	—
138	C₂³	C₂×C₄×D₆₁	1	1	yes	characteristic
139	Q₈	—	122	8	no	—
140	C₂×C₄	—	61	2	no	—
141	C₂×C₄	—	61	2	no	—
142	C₂×C₄	—	61	4	no	—
143	C₂×C₄	—	122	4	no	—
144	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
145	C₂×C₄	C₂×C₄×D₆₁	1	4	yes	—
146	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
147	C₂×C₄	#11758222	1	2	yes	—
148	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	122	2	no	—
151	C₈	—	122	2	no	—
152	C₈	C₂×Dic₁₂₂	1	2	yes	—
153	C₈	C₂×Dic₁₂₂	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	C₄×Dic₁₂₂	1	2	yes	—
156	C₂²	C₂²×C₄×D₆₁	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	61	4	no	—
159	C₄	—	122	2	no	—
160	C₄	—	122	4	no	—
161	C₄	C₄×Dic₁₂₂	1	2	yes	—
162	C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
163	C₄	C₂×#11758222	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	C₂×C₄×Dic₁₂₂	1	1	yes	characteristic
166	C₂	#131×D₆₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#405	C₂
hypercenter	#405	D₆₁
hypocenter	C₆₁	#2960
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #16839890	C₂	semidirect
3. C₈×Dic₁₂₂	C₂	semidirect
34. C₂×C₈×C₆₁	Q₈	semidirect
93. C₆₁	#2960	semidirect
122. C₂×C₈	Dic₁₂₂	semidirect