Subgroups — #21454766

Order 7808.

356 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16840319	C₂	1	1	yes	maximal, characteristic
2	#16839892	C₂	1	2	yes	maximal, complemented
3	#16839890	C₂	1	4	yes	maximal, complemented
4	C₈×Dic₁₂₂	C₂	1	2	yes	maximal, complemented
5	#16839790	C₂	1	2	yes	maximal
6	C₂×C₄⋊#92418	C₂	1	1	yes	maximal, characteristic
7	#16839720	C₂	1	2	yes	maximal
8	C₄×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
9	C₂²×#92418	C₂²	1	2	yes	—
10	#14519777	C₄	1	1	yes	characteristic
11	#14519777	C₄	1	1	yes	characteristic
12	#14519769	C₄	1	2	yes	—
13	#14519756	C₄	1	2	yes	—
14	#14519756	C₄	1	2	yes	—
15	C₄×Dic₁₂₂	C₂²	1	2	yes	—
16	C₄×Dic₁₂₂	C₂²	1	2	yes	—
17	#14519736	C₂²	1	2	yes	—
18	C₄⋊#92418	C₂²	1	1	yes	characteristic
19	C₄⋊#92418	C₂²	1	2	yes	—
20	C₄⋊#92418	C₂²	1	1	yes	characteristic
21	#14519706	C₂²	1	4	yes	—
22	#14519706	C₂²	1	4	yes	—
23	#14519687	C₂²	1	4	yes	—
24	C₈×Dic₆₁	C₂²	1	4	yes	—
25	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
26	C₂×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
27	C₂×C₄²×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	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
34	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
35	M₁₆×C₆₁	—	2	2	no	—
36	M₁₆×C₆₁	Q₈	1	4	yes	complemented
37	C₂×C₈×C₆₁	C₂³	1	2	yes	—
38	C₂×C₈×C₆₁	C₂³	1	2	yes	—
39	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₆₁	C₂³	1	1	yes	characteristic
42	C₄²×C₆₁	C₂³	1	1	yes	characteristic
43	C₂²⋊Dic₆₁	C₂×C₄	1	4	yes	—
44	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
45	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
46	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
47	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
48	#11758196	C₂×C₄	1	4	yes	—
49	#11758196	C₂×C₄	1	4	yes	—
50	C₄×Dic₆₁	C₂³	1	4	yes	—
51	C₂×#92418	—	2	4	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	—	2	2	no	—
56	C₂×C₄×C₆₁	—	4	1	no	—
57	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₆₁	#41	1	2	yes	—
64	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
65	C₂×Dic₆₁	C₂²×C₄	1	4	yes	—
66	C₂×Dic₆₁	C₂²×C₄	1	4	yes	—
67	Dic₁₂₂	—	2	4	no	—
68	Dic₁₂₂	—	4	2	no	—
69	C₈×C₆₁	—	2	2	no	—
70	C₈×C₆₁	C₂×Q₈	1	4	yes	—
71	#92418	—	2	2	no	—
72	#92418	—	2	2	no	—
73	C₂²×C₆₁	—	2	1	no	—
74	C₂²×C₆₁	—	2	1	no	—
75	C₂²×C₆₁	C₄×Q₈	1	2	yes	—
76	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
77	C₄×C₆₁	—	2	1	no	—
78	C₄×C₆₁	—	2	1	no	—
79	C₄×C₆₁	—	2	1	no	—
80	C₄×C₆₁	C₄×Q₈	1	2	yes	—
81	C₄×C₆₁	#131	1	2	yes	—
82	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
83	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
84	Dic₆₁	—	2	4	no	—
85	Dic₆₁	—	2	4	no	—
86	#2962	—	61	1	no	maximal, Sylow
87	C₂×C₆₁	—	2	2	no	—
88	C₂×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
89	C₂×C₆₁	C₂×#131	1	1	yes	characteristic
90	C₂×C₆₁	#568	1	1	yes	characteristic
91	#533	—	61	1	no	—
92	#446	—	61	4	no	—
93	#446	—	61	2	no	—
94	C₈×Q₈	—	61	2	no	—
95	#424	—	61	2	no	—
96	C₂×C₄⋊C₈	—	61	1	no	—
97	#405	—	61	2	no	—
98	C₄×M₁₆	D₆₁	1	1	yes	characteristic
99	C₆₁	#2962	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
101	C₂×M₁₆	C₂×D₆₁	1	1	yes	characteristic
102	C₂²×C₈	—	61	2	no	—
103	C₄⋊Q₈	—	61	1	no	—
104	C₄⋊Q₈	—	61	1	no	—
105	#125	—	61	2	no	—
106	C₂²⋊Q₈	—	61	2	no	—
107	C₂²⋊Q₈	—	61	2	no	—
108	C₄×Q₈	—	61	2	no	—
109	C₄×Q₈	—	61	2	no	—
110	#117	—	61	2	no	—
111	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
112	C₄⋊C₈	—	61	4	no	—
113	C₄⋊C₈	—	61	4	no	—
114	C₄⋊C₈	—	61	1	no	—
115	C₄⋊C₈	—	61	1	no	—
116	C₄⋊C₈	—	61	2	no	—
117	#97	—	61	4	no	—
118	#97	C₂×D₆₁	1	2	yes	—
119	C₄×C₈	—	61	4	no	—
120	C₄×C₈	C₂×D₆₁	1	2	yes	—
121	C₂×Q₈	—	61	2	no	—
122	C₂×Q₈	—	61	2	no	—
123	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
124	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
126	M₁₆	—	2	2	no	—
127	M₁₆	Dic₁₂₂	1	4	yes	complemented
128	C₂×C₈	—	61	2	no	—
129	C₂×C₈	—	61	2	no	—
130	C₂×C₈	—	122	4	no	—
131	C₂×C₈	C₂²×D₆₁	1	2	yes	—
132	C₂×C₈	C₂²×D₆₁	1	2	yes	—
133	C₄⋊C₄	—	61	4	no	—
134	C₄⋊C₄	—	61	4	no	—
135	C₄⋊C₄	—	61	2	no	—
136	C₄⋊C₄	—	61	2	no	—
137	C₄⋊C₄	—	61	2	no	—
138	C₄⋊C₄	—	61	2	no	—
139	C₂²⋊C₄	—	61	4	no	—
140	C₄²	—	61	4	no	—
141	C₄²	C₄×D₆₁	1	1	yes	characteristic
142	C₄²	C₄×D₆₁	1	1	yes	characteristic
143	C₄²	C₂²×D₆₁	1	1	yes	characteristic
144	C₄²	C₂²×D₆₁	1	1	yes	characteristic
145	C₂³	C₂×C₄×D₆₁	1	1	yes	characteristic
146	Q₈	—	122	4	no	—
147	Q₈	—	244	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	61	4	no	—
151	C₂×C₄	—	61	4	no	—
152	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
153	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
155	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
157	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
158	C₂×C₄	#11758222	1	2	yes	—
159	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	122	2	no	—
162	C₈	—	122	2	no	—
163	C₈	C₂×Dic₁₂₂	1	4	yes	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₄×Dic₁₂₂	1	2	yes	—
167	C₂²	C₂²×C₄×D₆₁	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	122	4	no	—
172	C₄	—	122	4	no	—
173	C₄	C₄×Dic₁₂₂	1	2	yes	—
174	C₄	#14519785	1	2	yes	—
175	C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
176	C₄	C₂×#11758222	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	C₂×C₄×Dic₁₂₂	1	1	yes	characteristic
179	C₂	C₂×#14519785	1	1	yes	characteristic
180	C₂	#16840528	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₄×M₁₆×C₆₁	C₂
hypercenter	C₄×M₁₆	D₆₁
hypocenter	C₆₁	#2962
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #16839892	C₂	semidirect
3. #16839890	C₂	semidirect
4. C₈×Dic₁₂₂	C₂	semidirect
36. M₁₆×C₆₁	Q₈	semidirect
99. C₆₁	#2962	semidirect
127. M₁₆	Dic₁₂₂	semidirect