Subgroups — #28328429

Order 13568.

490 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291130	C₂	1	8	yes	maximal, complemented
2	#20291050	C₂	1	2	yes	maximal
3	#20291047	C₂	1	1	yes	maximal, characteristic
4	#20290924	C₂	1	2	yes	maximal
5	C₂×#16811511	C₂	1	1	yes	maximal, characteristic
6	#20283650	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133291	C₂²	1	2	yes	—
8	C₂²×#11719910	C₂²	1	1	yes	characteristic
9	C₂²×#11719910	C₂²	1	1	yes	characteristic
10	C₄⋊#11719910	C₂²	1	4	yes	—
11	C₄⋊#11719910	C₂²	1	8	yes	—
12	#16811511	C₂²	1	4	yes	—
13	#16811511	C₂²	1	4	yes	—
14	C₄×#11719910	C₂²	1	4	yes	—
15	#16810754	C₄	1	4	yes	—
16	Q₈×#88322	C₂²	1	4	yes	—
17	#16810650	C₄	1	2	yes	—
18	#16810647	C₄	1	1	yes	characteristic
19	#16810579	C₂²	1	2	yes	—
20	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
21	C₅₃×#533	C₄	1	1	yes	characteristic
22	#13133291	—	2	4	no	—
23	C₂×#11719910	—	2	2	no	—
24	C₂×#11719910	Q₈	1	4	yes	complemented
25	C₂×#11719910	C₂³	1	2	yes	—
26	C₂×#11719910	C₂³	1	4	yes	—
27	C₂×#11719910	C₂³	1	2	yes	—
28	C₂×#11719917	C₂×C₄	1	2	yes	—
29	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
30	C₂²×#88322	C₂³	1	1	yes	characteristic
31	C₄⋊#88322	C₂×C₄	1	4	yes	—
32	C₄⋊#88322	C₂×C₄	1	8	yes	—
33	#13133128	C₂×C₄	1	4	yes	—
34	C₄×#88322	C₂×C₄	1	2	yes	—
35	C₄×#88322	C₂³	1	2	yes	—
36	C₄×#88322	C₂³	1	4	yes	—
37	C₄⋊Q₈×C₅₃	C₈	1	2	yes	—
38	#125×C₅₃	C₈	1	2	yes	—
39	C₂²⋊Q₈×C₅₃	C₈	1	4	yes	—
40	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
41	#117×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
47	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
48	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
49	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
50	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
51	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
52	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
53	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
54	#11719917	—	2	4	no	—
55	C₂×#88322	—	2	2	no	—
56	C₂×#88322	C₂²×C₄	1	2	yes	—
57	C₂×#88322	C₂²×C₄	1	4	yes	—
58	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
59	C₂×#88322	C₂×Q₈	1	2	yes	—
60	C₂×#88322	#41	1	2	yes	—
61	C₂×#88322	C₂⁴	1	1	yes	characteristic
62	#11719910	—	2	2	no	—
63	#11719910	—	2	4	no	—
64	#11719910	C₂×Q₈	1	4	yes	—
65	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
66	Q₈×C₅₃	—	4	4	no	—
67	C₂×C₄×C₅₃	—	4	2	no	—
68	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
69	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
71	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
72	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#131	1	2	yes	—
74	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
75	#88322	—	2	2	no	—
76	#88322	—	2	4	no	—
77	#88322	C₄×Q₈	1	2	yes	—
78	#88322	C₂²×Q₈	1	1	yes	characteristic
79	#88322	C₂×#41	1	1	yes	characteristic
80	#33475	—	53	1	no	maximal, Sylow
81	C₂²×C₅₃	—	2	2	no	—
82	C₂²×C₅₃	C₈×Q₈	1	2	yes	—
83	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
84	C₄×C₅₃	—	2	2	no	—
85	C₄×C₅₃	—	2	2	no	—
86	C₄×C₅₃	—	2	8	no	—
87	C₄×C₅₃	C₈×Q₈	1	2	yes	—
88	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
89	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
90	#2960	—	53	1	no	—
91	#2183	—	53	8	no	—
92	#2151	—	53	2	no	—
93	#2150	—	53	1	no	—
94	#2108	—	53	2	no	—
95	C₂×#346	—	53	1	no	—
96	C₂×C₅₃	—	2	2	no	—
97	C₂×C₅₃	C₂×C₈×Q₈	1	1	yes	characteristic
98	C₂×C₅₃	#3407	1	2	yes	—
99	#533	#6157	1	1	yes	characteristic
100	C₂×M₃₂	—	53	2	no	—
101	C₂²×C₁₆	—	53	1	no	—
102	C₂²×C₁₆	—	53	1	no	—
103	#446	—	53	4	no	—
104	C₈×Q₈	—	53	4	no	—
105	#424	—	53	2	no	—
106	#423	—	53	1	no	—
107	#405	—	53	2	no	—
108	C₂×C₄×C₈	—	53	1	no	—
109	C₄⋊C₁₆	—	53	4	no	—
110	C₄⋊C₁₆	—	53	8	no	—
111	#346	—	53	4	no	—
112	#346	—	53	4	no	—
113	C₄×C₁₆	—	53	4	no	—
114	C₅₃	#33475	1	1	yes	characteristic, complemented, Sylow
115	C₂×M₁₆	—	53	2	no	—
116	C₂²×C₈	—	53	1	no	—
117	C₂²×C₈	—	53	1	no	—
118	C₄⋊Q₈	#88324	1	2	yes	—
119	#125	#88324	1	2	yes	—
120	C₂²⋊Q₈	#88324	1	4	yes	—
121	C₄×Q₈	C₂×#6157	1	4	yes	—
122	#117	C₂×#6157	1	2	yes	—
123	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
124	M₃₂	—	106	4	no	—
125	C₂×C₁₆	—	53	2	no	—
126	C₂×C₁₆	—	53	4	no	—
127	C₂×C₁₆	—	53	4	no	—
128	C₂×C₁₆	—	53	2	no	—
129	C₂×C₁₆	—	106	2	no	—
130	C₄⋊C₈	—	53	4	no	—
131	C₄⋊C₈	—	53	8	no	—
132	#97	—	53	4	no	—
133	C₄×C₈	—	53	2	no	—
134	C₄×C₈	—	53	4	no	—
135	C₄×C₈	—	53	2	no	—
136	C₂×Q₈	C₂×#88324	1	4	yes	—
137	C₂²×C₄	C₂×#88324	1	2	yes	—
138	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
139	M₁₆	—	106	4	no	—
140	C₂×C₈	—	53	2	no	—
141	C₂×C₈	—	53	2	no	—
142	C₂×C₈	—	53	1	no	—
143	C₂×C₈	—	53	1	no	—
144	C₂×C₈	—	53	2	no	—
145	C₂×C₈	—	53	4	no	—
146	C₂×C₈	—	106	2	no	—
147	C₄⋊C₄	C₂×#88324	1	4	yes	—
148	C₄⋊C₄	C₂×#88324	1	8	yes	—
149	C₄⋊C₄	C₂×#88324	1	4	yes	—
150	C₂²⋊C₄	C₂×#88324	1	4	yes	—
151	C₄²	C₂×#88324	1	1	yes	characteristic
152	C₄²	C₂×#88324	1	1	yes	characteristic
153	C₄²	C₂²×#6157	1	2	yes	—
154	C₄²	C₂²×#6157	1	4	yes	—
155	C₁₆	—	53	4	no	—
156	C₁₆	—	106	2	no	—
157	C₁₆	—	106	4	no	—
158	C₂³	C₂²×#88324	1	1	yes	characteristic
159	Q₈	—	4	4	no	—
160	C₂×C₄	—	4	2	no	—
161	C₂×C₄	Q₈×#6157	1	2	yes	—
162	C₂×C₄	C₂²×#88324	1	2	yes	—
163	C₂×C₄	C₂²×#88324	1	2	yes	—
164	C₂×C₄	C₂²×#88324	1	8	yes	—
165	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
166	C₂×C₄	#13133305	1	2	yes	—
167	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
168	C₈	—	53	2	no	—
169	C₈	—	53	1	no	—
170	C₈	—	53	1	no	—
171	C₈	—	106	2	no	—
172	C₈	—	106	4	no	—
173	C₂²	—	2	2	no	—
174	C₂²	Q₈×#88324	1	2	yes	—
175	C₂²	C₂³×#88324	1	1	yes	characteristic
176	C₄	—	2	2	no	—
177	C₄	—	2	2	no	—
178	C₄	—	2	8	no	—
179	C₄	Q₈×#88324	1	2	yes	—
180	C₄	C₂×Q₈×#6157	1	1	yes	characteristic
181	C₄	C₂×#13133305	1	1	yes	characteristic
182	C₂	—	2	2	no	—
183	C₂	C₂×Q₈×#88324	1	1	yes	characteristic
184	C₂	#20291682	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#533	C₄
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	#533	#6157
hypocenter	C₅₃	#33475
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. #20291130	C₂	semidirect
24. C₂×#11719910	Q₈	semidirect
114. C₅₃	#33475	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20291130	2
#20291050	2
#20291047	2
#20290924	2
C₂×#16811511	2
#20283650	2
#33475	53