Subgroups — #27275285

Order 13568.

438 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	#20274568	C₂	1	2	yes	maximal
2	C₂×#16810376	C₂	1	2	yes	maximal
3	#20272550	C₂	1	2	yes	maximal
4	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	C₂²	1	4	yes	—
6	C₂×#13133128	—	2	2	no	—
7	#16810378	—	2	2	no	—
8	#16810376	C₄	1	8	yes	complemented
9	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₄³×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×#88322	—	2	4	no	—
15	C₄⋊#88322	—	2	4	no	—
16	C₄⋊#88322	C₂×C₄	1	8	yes	—
17	#13133128	—	4	4	no	—
18	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
25	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
26	C₄×C₈×C₅₃	—	2	4	no	—
27	C₄×C₈×C₅₃	—	2	4	no	—
28	C₄×C₈×C₅₃	—	2	2	no	—
29	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
31	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈×C₅₃	—	2	2	no	—
36	C₂×C₈×C₅₃	—	2	2	no	—
37	C₂×C₈×C₅₃	—	2	2	no	—
38	C₂×C₈×C₅₃	C₄²	1	2	yes	—
39	C₂×C₈×C₅₃	C₄²	1	2	yes	—
40	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
41	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
42	C₄²×C₅₃	—	2	4	no	—
43	C₄²×C₅₃	—	2	2	no	—
44	C₄²×C₅₃	—	2	2	no	—
45	C₄²×C₅₃	—	2	2	no	—
46	C₄²×C₅₃	D₈	1	2	yes	—
47	C₄²×C₅₃	Q₁₆	1	2	yes	—
48	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
49	C₄²×C₅₃	#41	1	2	yes	—
50	C₂×#88322	—	2	8	no	—
51	C₂×#88322	—	4	4	no	—
52	C₂×#88322	—	4	4	no	—
53	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	—	2	2	no	—
55	C₂×C₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	#107	1	4	yes	—
60	C₂×C₄×C₅₃	#108	1	4	yes	—
61	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
62	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
64	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
65	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
66	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
67	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#135	1	2	yes	—
70	C₈×C₅₃	—	2	2	no	—
71	C₈×C₅₃	—	2	4	no	—
72	C₈×C₅₃	#97	1	4	yes	complemented
73	#88322	—	4	8	no	—
74	#8274	—	53	1	no	maximal, Sylow
75	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₅₃	#423	1	2	yes	—
77	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
78	C₂²×C₅₃	#427	1	1	yes	characteristic
79	C₂²×C₅₃	C₂×#108	1	2	yes	—
80	C₄×C₅₃	—	2	4	no	—
81	C₄×C₅₃	—	2	4	no	—
82	C₄×C₅₃	—	2	2	no	—
83	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
84	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
85	C₄×C₅₃	#432	1	2	yes	—
86	C₄×C₅₃	C₄×D₈	1	2	yes	—
87	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
88	#1767	—	53	2	no	—
89	C₄²×C₈	D₅₃	1	1	yes	characteristic
90	C₂×#335	—	53	2	no	—
91	#1277	—	53	2	no	—
92	C₂×C₅₃	#1567	1	2	yes	—
93	C₂×C₅₃	#1568	1	2	yes	—
94	C₂×C₅₃	#1767	1	1	yes	characteristic
95	C₂×C₅₃	C₄×#107	1	1	yes	characteristic
96	C₂×C₅₃	C₄×#108	1	1	yes	characteristic
97	C₂×C₄⋊C₈	—	53	4	no	—
98	C₂×#97	—	106	2	no	—
99	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
100	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
101	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
103	C₄³	C₂×D₅₃	1	1	yes	characteristic
104	#336	—	106	2	no	—
105	#335	—	53	8	no	—
106	C₅₃	#8274	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₈	—	106	4	no	—
108	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
109	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
110	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
111	C₂×C₄²	Dic₁₀₆	1	2	yes	—
112	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
113	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
114	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
115	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
116	C₄⋊C₈	—	53	8	no	—
117	C₄⋊C₈	—	106	4	no	—
118	#97	—	212	4	no	—
119	C₄×C₈	—	2	4	no	—
120	C₄×C₈	—	2	4	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	C₄×D₅₃	1	4	yes	—
123	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
124	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
125	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
126	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
127	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	106	8	no	—
132	C₂×C₈	—	212	4	no	—
133	C₂×C₈	—	212	4	no	—
134	C₂×C₈	#11719917	1	4	yes	—
135	C₂×C₈	#11719917	1	4	yes	—
136	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
137	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
138	C₄²	—	2	4	no	—
139	C₄²	—	2	2	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	2	no	—
142	C₄²	D₄₂₄	1	2	yes	—
143	C₄²	Dic₂₁₂	1	2	yes	—
144	C₄²	C₂×C₄×D₅₃	1	2	yes	—
145	C₄²	#11719945	1	2	yes	—
146	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	#13133151	1	4	yes	—
153	C₂×C₄	#13133153	1	4	yes	—
154	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
155	C₂×C₄	#13133176	1	1	yes	characteristic
156	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
157	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
158	C₂×C₄	C₂×#11719917	1	2	yes	—
159	C₂×C₄	C₂×#11719917	1	2	yes	—
160	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
161	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
162	C₂×C₄	#13133239	1	2	yes	—
163	C₈	—	2	2	no	—
164	C₈	—	2	4	no	—
165	C₈	—	212	8	no	—
166	C₈	#13133128	1	4	yes	complemented
167	C₂²	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
168	C₂²	#16810647	1	2	yes	—
169	C₂²	C₂×#13133151	1	1	yes	characteristic
170	C₂²	#16810661	1	1	yes	characteristic
171	C₂²	C₂×#13133153	1	2	yes	—
172	C₄	—	2	4	no	—
173	C₄	—	2	4	no	—
174	C₄	—	2	2	no	—
175	C₄	C₂×#13133128	1	1	yes	characteristic
176	C₄	C₄×#11719917	1	1	yes	characteristic
177	C₄	#16810680	1	2	yes	—
178	C₄	C₄×D₄₂₄	1	2	yes	—
179	C₄	C₄×Dic₂₁₂	1	2	yes	—
180	C₂	#20273474	1	2	yes	—
181	C₂	#20273478	1	2	yes	—
182	C₂	#20274568	1	1	yes	characteristic
183	C₂	C₄×#13133151	1	1	yes	characteristic
184	C₂	C₄×#13133153	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄²×C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	#8274
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
8. #16810376	C₄	semidirect
72. C₈×C₅₃	#97	semidirect
106. C₅₃	#8274	semidirect
166. C₈	#13133128	semidirect

Maximal subgroups

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

subgroup	index
#20274568	2
C₂×#16810376	2
#20272550	2
C₄²×C₈×C₅₃	2
#8274	53