Subgroups — #20283664

Order 6784.

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	#16810754	C₂	1	4	yes	maximal, complemented
2	#16810754	C₂	1	2	yes	maximal, complemented
3	Q₈×#88322	C₂	1	2	yes	maximal, complemented
4	#16810650	C₂	1	2	yes	maximal
5	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
6	#16810579	C₂	1	2	yes	maximal
7	C₄×#11719917	C₂	1	1	yes	maximal, characteristic
8	C₅₃×#533	C₂	1	1	yes	maximal, characteristic
9	C₂×#11719917	C₂²	1	1	yes	characteristic
10	C₂×#11719917	C₂²	1	1	yes	characteristic
11	C₂²×#88322	C₂²	1	2	yes	—
12	C₄⋊#88322	C₂²	1	2	yes	—
13	C₄⋊#88322	C₂²	1	1	yes	characteristic
14	C₄⋊#88322	C₂²	1	4	yes	—
15	C₄⋊#88322	C₂²	1	4	yes	—
16	C₄⋊#88322	C₂²	1	1	yes	characteristic
17	#13133128	C₂²	1	2	yes	—
18	#13133128	C₂²	1	4	yes	—
19	C₄×#88322	C₂²	1	2	yes	—
20	C₄×#88322	C₂²	1	4	yes	—
21	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
22	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
23	#125×C₅₃	C₄	1	2	yes	—
24	C₂²⋊Q₈×C₅₃	C₄	1	2	yes	—
25	C₂²⋊Q₈×C₅₃	C₄	1	2	yes	—
26	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
27	C₄×Q₈×C₅₃	C₂²	1	2	yes	—
28	#117×C₅₃	C₂²	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
30	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
36	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
37	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
38	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
39	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
40	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
41	C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
42	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	C₄²×C₅₃	C₂³	1	1	yes	characteristic
46	C₄²×C₅₃	C₂³	1	4	yes	—
47	#11719917	—	2	2	no	—
48	#11719917	Q₈	1	4	yes	complemented
49	C₂×#88322	—	2	4	no	—
50	C₂×#88322	C₂³	1	2	yes	—
51	C₂×#88322	C₂³	1	2	yes	—
52	C₂×#88322	C₂³	1	2	yes	—
53	C₂×#88322	C₂³	1	2	yes	—
54	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	Q₈×C₅₃	—	2	4	no	—
56	Q₈×C₅₃	—	4	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	4	1	no	—
59	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
63	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
64	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
67	C₂×C₄×C₅₃	#41	1	2	yes	—
68	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
69	#88322	—	2	2	no	—
70	#88322	—	2	2	no	—
71	#88322	—	2	2	no	—
72	#88322	C₂×Q₈	1	4	yes	—
73	C₂²×C₅₃	—	2	1	no	—
74	C₂²×C₅₃	—	2	1	no	—
75	C₂²×C₅₃	C₄×Q₈	1	2	yes	—
76	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
77	C₄×C₅₃	—	2	1	no	—
78	C₄×C₅₃	—	2	4	no	—
79	C₄×C₅₃	—	2	1	no	—
80	C₄×C₅₃	—	2	1	no	—
81	C₄×C₅₃	—	2	4	no	—
82	C₄×C₅₃	C₄×Q₈	1	2	yes	—
83	C₄×C₅₃	#131	1	2	yes	—
84	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
85	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
86	#2962	—	53	1	no	maximal, Sylow
87	C₂×C₅₃	—	2	2	no	—
88	C₂×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
89	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
90	C₂×C₅₃	#568	1	1	yes	characteristic
91	#533	D₅₃	1	1	yes	characteristic
92	#446	—	53	2	no	—
93	#446	—	53	4	no	—
94	C₈×Q₈	—	53	2	no	—
95	#424	—	53	2	no	—
96	C₂×C₄⋊C₈	—	53	1	no	—
97	#405	—	53	2	no	—
98	C₄×M₁₆	—	53	1	no	—
99	C₅₃	#2962	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₁₆	—	53	1	no	—
101	C₂×M₁₆	—	53	1	no	—
102	C₂²×C₈	—	53	2	no	—
103	C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
104	C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
105	#125	Dic₅₃	1	2	yes	—
106	C₂²⋊Q₈	Dic₅₃	1	2	yes	—
107	C₂²⋊Q₈	Dic₅₃	1	2	yes	—
108	C₄×Q₈	C₂×D₅₃	1	2	yes	—
109	C₄×Q₈	C₂×D₅₃	1	2	yes	—
110	#117	C₂×D₅₃	1	2	yes	—
111	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
112	C₄⋊C₈	—	53	1	no	—
113	C₄⋊C₈	—	53	1	no	—
114	C₄⋊C₈	—	53	2	no	—
115	C₄⋊C₈	—	53	4	no	—
116	C₄⋊C₈	—	53	4	no	—
117	#97	—	53	2	no	—
118	#97	—	53	4	no	—
119	C₄×C₈	—	53	2	no	—
120	C₄×C₈	—	53	4	no	—
121	C₂×Q₈	C₂×Dic₅₃	1	2	yes	—
122	C₂×Q₈	C₂×Dic₅₃	1	2	yes	—
123	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
124	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
126	M₁₆	—	53	4	no	—
127	M₁₆	—	106	2	no	—
128	C₂×C₈	—	53	2	no	—
129	C₂×C₈	—	53	2	no	—
130	C₂×C₈	—	53	2	no	—
131	C₂×C₈	—	53	2	no	—
132	C₂×C₈	—	106	4	no	—
133	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
134	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
135	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
136	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
137	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
138	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
139	C₂²⋊C₄	C₂×Dic₅₃	1	4	yes	—
140	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
141	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
142	C₄²	C₂²×D₅₃	1	1	yes	characteristic
143	C₄²	C₂²×D₅₃	1	4	yes	—
144	C₄²	C₂²×D₅₃	1	1	yes	characteristic
145	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
146	Q₈	—	2	4	no	—
147	Q₈	—	4	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	Q₈×D₅₃	1	2	yes	—
151	C₂×C₄	#11719949	1	2	yes	—
152	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
153	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
154	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
155	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
156	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
157	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
158	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
159	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
160	C₈	—	53	4	no	—
161	C₈	—	106	2	no	—
162	C₈	—	106	2	no	—
163	C₈	—	106	2	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	Q₈×Dic₅₃	1	2	yes	—
167	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	4	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	4	no	—
173	C₄	Q₈×Dic₅₃	1	2	yes	—
174	C₄	#13133224	1	2	yes	—
175	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
176	C₄	C₂×#11719949	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
179	C₂	C₂×#13133224	1	1	yes	characteristic
180	C₂	#16811390	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#533	C₂
hypercenter	#533	D₅₃
hypocenter	C₅₃	#2962
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #16810754	C₂	semidirect
2. #16810754	C₂	semidirect
3. Q₈×#88322	C₂	semidirect
48. #11719917	Q₈	semidirect
99. C₅₃	#2962	semidirect