Finite Group Databasebeta

Subgroups — #20291623

Order 6784.

322 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 148 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811674	C₂	1	4	yes	maximal, complemented
2	#16811673	C₂	1	2	yes	maximal, complemented
3	#16811672	C₂	1	2	yes	maximal, complemented
4	#16811648	C₂	1	4	yes	maximal
5	#16811645	C₂	1	1	yes	maximal, characteristic
6	#16811641	C₂	1	1	yes	maximal, characteristic
7	#16811178	C₂	1	1	yes	maximal, characteristic
8	#13133284	C₂²	1	2	yes	—
9	#13133284	C₂²	1	4	yes	—
10	C₄⋊#88324	C₂²	1	2	yes	—
11	C₄⋊#88324	C₂²	1	4	yes	—
12	#13133274	C₂²	1	2	yes	—
13	#13133274	C₂²	1	2	yes	—
14	#13133272	C₂²	1	2	yes	—
15	#13133272	C₂²	1	4	yes	—
16	C₄×#88324	C₂²	1	1	yes	characteristic
17	C₄×#88324	C₂²	1	1	yes	characteristic
18	C₄×#88324	C₂²	1	4	yes	—
19	#13133217	C₄	1	1	yes	characteristic
20	#13133208	C₄	1	2	yes	—
21	#13133197	C₄	1	2	yes	—
22	#13133193	C₄	1	2	yes	—
23	Q₈×Dic₅₃	C₂²	1	2	yes	—
24	#13133182	C₂²	1	2	yes	—
25	#13133175	C₂²	1	2	yes	—
26	C₄²×D₅₃	C₂²	1	1	yes	characteristic
27	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
28	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
33	C₂×#88324	C₂³	1	2	yes	—
34	C₂×#88324	C₂³	1	2	yes	—
35	C₂×#88324	C₂³	1	4	yes	—
36	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
37	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
38	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	#11719921	C₂×C₄	1	4	yes	—
40	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
41	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
42	C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
43	#11719919	C₂×C₄	1	4	yes	—
44	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
46	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
47	C₄×Dic₅₃	C₂³	1	4	yes	—
48	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
49	Q₈×C₅₃	—	4	2	no	—
50	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
53	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
55	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
58	C₄×D₅₃	—	2	2	no	—
59	C₄×D₅₃	—	2	2	no	—
60	C₄×D₅₃	—	4	1	no	—
61	Dic₁₀₆	—	2	4	no	—
62	#88324	—	2	2	no	—
63	#88324	—	2	2	no	—
64	#88324	—	2	4	no	—
65	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
66	C₂×D₅₃	—	2	1	no	—
67	C₂×D₅₃	—	2	1	no	—
68	C₂×D₅₃	—	2	1	no	—
69	C₄×C₅₃	—	2	1	no	—
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	—	2	4	no	—
72	C₄×C₅₃	#131	1	2	yes	—
73	Dic₅₃	—	2	4	no	—
74	Dic₅₃	—	2	1	no	—
75	Dic₅₃	—	2	1	no	—
76	Dic₅₃	2₋¹⁺⁴	1	2	yes	—
77	#2985	—	53	1	no	maximal, Sylow
78	C₂×C₅₃	#519	1	1	yes	characteristic
79	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
80	C₂×C₅₃	#568	1	1	yes	characteristic
81	D₅₃	—	4	1	no	—
82	#533	—	53	1	no	—
83	#446	—	53	2	no	—
84	#446	—	53	4	no	—
85	C₈×Q₈	—	53	2	no	—
86	#433	—	53	4	no	—
87	#432	—	53	1	no	—
88	#431	—	53	1	no	—
89	C₅₃	#2985	1	1	yes	characteristic, complemented, Sylow
90	C₄⋊Q₈	—	53	1	no	—
91	C₄⋊Q₈	#6157	1	1	yes	characteristic
92	#125	—	53	2	no	—
93	C₂²⋊Q₈	—	53	2	no	—
94	C₂²⋊Q₈	—	53	2	no	—
95	C₄×Q₈	—	53	2	no	—
96	C₄×Q₈	—	53	2	no	—
97	#117	—	53	2	no	—
98	C₂×C₄²	—	53	1	no	—
99	C₄⋊C₈	—	53	2	no	—
100	C₄⋊C₈	—	53	2	no	—
101	C₄⋊C₈	—	53	4	no	—
102	C₄⋊C₈	—	53	4	no	—
103	C₂²⋊C₈	—	53	2	no	—
104	C₂²⋊C₈	—	53	2	no	—
105	#97	—	53	2	no	—
106	#97	—	53	4	no	—
107	C₄×C₈	—	53	1	no	—
108	C₄×C₈	—	53	1	no	—
109	C₄×C₈	—	53	4	no	—
110	C₂×Q₈	—	53	2	no	—
111	C₂×Q₈	C₂×#6157	1	2	yes	—
112	C₂²×C₄	—	53	1	no	—
113	C₂²×C₄	—	53	1	no	—
114	C₂²×C₄	—	53	1	no	—
115	C₂×C₈	—	53	2	no	—
116	C₂×C₈	—	53	2	no	—
117	C₂×C₈	—	53	4	no	—
118	C₄⋊C₄	—	53	2	no	—
119	C₄⋊C₄	—	53	4	no	—
120	C₄⋊C₄	—	53	4	no	—
121	C₄⋊C₄	—	53	2	no	—
122	C₄⋊C₄	C₂×#6157	1	2	yes	—
123	C₄⋊C₄	C₂×#6157	1	2	yes	—
124	C₂²⋊C₄	—	53	4	no	—
125	C₄²	—	53	1	no	—
126	C₄²	—	53	4	no	—
127	C₄²	—	53	1	no	—
128	C₄²	—	53	1	no	—
129	C₄²	C₂×#6157	1	1	yes	characteristic
130	C₂³	—	53	1	no	—
131	Q₈	—	4	2	no	—
132	Q₈	—	106	4	no	—
133	C₂×C₄	—	53	1	no	—
134	C₂×C₄	—	53	1	no	—
135	C₂×C₄	—	53	1	no	—
136	C₂×C₄	—	53	4	no	—
137	C₂×C₄	—	106	2	no	—
138	C₂×C₄	—	106	2	no	—
139	C₂×C₄	—	212	1	no	—
140	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
141	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
142	C₂×C₄	C₂²×#6157	1	4	yes	—
143	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
144	C₈	—	106	2	no	—
145	C₈	—	106	2	no	—
146	C₈	—	106	4	no	—
147	C₂²	—	106	1	no	—
148	C₂²	—	106	1	no	—
149	C₂²	—	106	1	no	—
150	C₂²	C₂³×#6157	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	4	no	—
154	C₄	—	53	2	no	—
155	C₄	—	106	1	no	—
156	C₄	—	106	4	no	—
157	C₄	—	106	1	no	—
158	C₄	#13133305	1	2	yes	—
159	C₂	—	212	1	no	—
160	C₂	#16811691	1	1	yes	characteristic
161	C₂	C₂×#13133305	1	1	yes	characteristic
162	C₂	#16811701	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₄⋊Q₈×C₅₃	C₄
hypercenter	C₄⋊Q₈	#6157
hypocenter	C₅₃	#2985
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #16811674	C₂	semidirect
2. #16811673	C₂	semidirect
3. #16811672	C₂	semidirect
89. C₅₃	#2985	semidirect