Subgroups — #95×#88322 #27289249

Order 13568.

1174 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×#88322	C₂	1	3	yes	maximal
2	C₂×#16810378	C₂	1	3	yes	maximal
3	C₄×#95×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#88322	C₂²	1	3	yes	—
5	C₂³×#88322	C₂²	1	1	yes	characteristic
6	C₂×C₄×#88322	—	2	12	no	—
7	C₂×C₄×#88322	C₄	1	12	yes	complemented
8	#16810378	C₄	1	12	yes	complemented
9	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
10	C₂×#95×C₅₃	C₄	1	4	yes	—
11	C₂²×#88322	—	2	6	no	—
12	C₂²×#88322	—	2	3	no	—
13	C₂²×#88322	—	2	3	no	—
14	C₂²×#88322	C₂×C₄	1	6	yes	—
15	C₂²×#88322	C₂×C₄	1	12	yes	—
16	C₂²×#88322	C₂×C₄	1	6	yes	—
17	C₂²×#88322	D₄	1	6	yes	—
18	C₂²×#88322	Q₈	1	2	yes	—
19	C₄×#88322	—	2	24	no	—
20	C₂³×C₄×C₅₃	C₂×C₄	1	6	yes	—
21	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₅₃	—	2	6	no	—
23	C₂×C₄²×C₅₃	—	2	6	no	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
26	#95×C₅₃	C₈	1	16	yes	complemented
27	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	—	2	12	no	—
29	C₂²×C₄×C₅₃	—	2	12	no	—
30	C₂²×C₄×C₅₃	C₄²	1	12	yes	—
31	C₂²×C₄×C₅₃	C₂×C₈	1	24	yes	—
32	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
33	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
34	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
35	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	#41	1	3	yes	—
38	C₄²×C₅₃	—	2	6	no	—
39	C₄²×C₅₃	—	2	6	no	—
40	C₄²×C₅₃	—	2	12	no	—
41	C₂×#88322	—	2	6	no	—
42	C₂×#88322	—	2	6	no	—
43	C₂×#88322	—	2	12	no	—
44	C₂×#88322	—	2	12	no	—
45	C₂×#88322	C₄²	1	12	yes	—
46	C₂×#88322	C₄²	1	4	yes	—
47	C₂×#88322	C₂²⋊C₄	1	12	yes	—
48	C₂×#88322	C₄⋊C₄	1	12	yes	—
49	C₂³×C₅₃	C₂×C₄²	1	3	yes	—
50	C₂³×C₅₃	C₄×D₄	1	3	yes	—
51	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
52	C₂³×C₅₃	C₂²×C₈	1	4	yes	—
53	C₂³×C₅₃	#131	1	4	yes	—
54	C₂×C₄×C₅₃	—	2	12	no	—
55	C₂×C₄×C₅₃	—	2	12	no	—
56	C₂×C₄×C₅₃	—	2	48	no	—
57	C₂×C₄×C₅₃	C₄×C₈	1	24	yes	complemented
58	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×C₄²	1	3	yes	—
60	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
61	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
62	C₂×C₄×C₅₃	#117	1	3	yes	—
63	C₂×C₄×C₅₃	#117	1	3	yes	—
64	C₂×C₄×C₅₃	C₄×D₄	1	6	yes	—
65	C₂×C₄×C₅₃	C₄×D₄	1	3	yes	—
66	C₂×C₄×C₅₃	C₄×Q₈	1	3	yes	—
67	#88322	—	2	12	no	—
68	#88322	#95	1	8	yes	direct product factor
69	C₈×#95	—	53	1	no	maximal, Sylow
70	C₂²×C₅₃	C₄³	1	1	yes	characteristic
71	C₂²×C₅₃	C₄×C₂²⋊C₄	1	3	yes	—
72	C₂²×C₅₃	C₄×C₄⋊C₄	1	3	yes	—
73	C₂²×C₅₃	C₂×C₄×C₈	1	6	yes	—
74	C₂²×C₅₃	#405	1	6	yes	—
75	C₂²×C₅₃	C₈×D₄	1	12	yes	—
76	C₂²×C₅₃	C₈×Q₈	1	4	yes	—
77	C₄×C₅₃	—	2	24	no	—
78	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
79	C₄×C₅₃	#376	1	1	yes	characteristic
80	C₄×C₅₃	C₄×C₂²⋊C₄	1	3	yes	—
81	C₄×C₅₃	C₄×C₄⋊C₄	1	3	yes	—
82	C₂²×C₄×C₈	—	53	3	no	—
83	C₂×#336	—	53	3	no	—
84	C₄×#95	D₅₃	1	1	yes	characteristic
85	C₂×C₅₃	C₄×#95	1	1	yes	characteristic
86	C₂×C₅₃	C₄²×C₈	1	1	yes	characteristic
87	C₂×C₅₃	#1726	1	1	yes	characteristic
88	C₂×C₅₃	C₈×C₂²⋊C₄	1	6	yes	—
89	C₂×C₅₃	C₈×C₄⋊C₄	1	6	yes	—
90	C₂³×C₈	—	53	1	no	—
91	C₂³×C₈	—	53	3	no	—
92	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
93	C₂×C₄×C₈	—	53	12	no	—
94	C₂×C₄×C₈	—	106	12	no	—
95	C₂×#95	Dic₅₃	1	4	yes	—
96	#336	—	53	12	no	—
97	C₅₃	C₈×#95	1	1	yes	characteristic, complemented, Sylow
98	C₂³×C₄	C₂×Dic₅₃	1	6	yes	—
99	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
100	C₂²×C₈	—	53	6	no	—
101	C₂²×C₈	—	53	6	no	—
102	C₂²×C₈	—	53	2	no	—
103	C₂²×C₈	—	53	12	no	—
104	C₂²×C₈	—	53	6	no	—
105	C₂²×C₈	—	106	6	no	—
106	C₂²×C₈	—	106	3	no	—
107	C₂²×C₈	—	106	3	no	—
108	C₂×C₄²	—	2	6	no	—
109	C₂×C₄²	—	2	6	no	—
110	C₂×C₄²	C₄×D₅₃	1	6	yes	—
111	C₂×C₄²	C₄×D₅₃	1	6	yes	—
112	C₄×C₈	—	106	24	no	—
113	#95	#88322	1	16	yes	direct product factor
114	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
115	C₂²×C₄	—	2	12	no	—
116	C₂²×C₄	—	2	12	no	—
117	C₂²×C₄	C₂×#88322	1	24	yes	—
118	C₂²×C₄	C₄×Dic₅₃	1	12	yes	—
119	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
120	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
121	C₂²×C₄	D₄×D₅₃	1	3	yes	—
122	C₂²×C₄	#11719947	1	3	yes	—
123	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
124	C₂²×C₄	#11719949	1	1	yes	characteristic
125	C₂×C₈	—	53	4	no	—
126	C₂×C₈	—	53	12	no	—
127	C₂×C₈	—	53	12	no	—
128	C₂×C₈	—	53	12	no	—
129	C₂×C₈	—	106	6	no	—
130	C₂×C₈	—	106	6	no	—
131	C₂×C₈	—	106	12	no	—
132	C₂×C₈	—	106	12	no	—
133	C₄²	—	2	6	no	—
134	C₄²	—	2	6	no	—
135	C₄²	—	2	12	no	—
136	C₂³	C₂×C₄×Dic₅₃	1	3	yes	—
137	C₂³	D₄×Dic₅₃	1	3	yes	—
138	C₂³	Q₈×Dic₅₃	1	1	yes	characteristic
139	C₂³	C₂²×#88322	1	4	yes	—
140	C₂³	#13133224	1	4	yes	—
141	C₂×C₄	—	2	12	no	—
142	C₂×C₄	—	2	48	no	—
143	C₂×C₄	—	2	12	no	—
144	C₂×C₄	C₄×#88322	1	24	yes	complemented
145	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
146	C₂×C₄	C₄²×D₅₃	1	3	yes	—
147	C₂×C₄	C₂²⋊C₄×D₅₃	1	3	yes	—
148	C₂×C₄	C₄⋊C₄×D₅₃	1	3	yes	—
149	C₂×C₄	#13133173	1	3	yes	—
150	C₂×C₄	#13133175	1	3	yes	—
151	C₂×C₄	#13133177	1	6	yes	—
152	C₂×C₄	#13133179	1	3	yes	—
153	C₂×C₄	#13133182	1	3	yes	—
154	C₈	—	53	8	no	—
155	C₈	—	106	12	no	—
156	C₂²	C₄²×Dic₅₃	1	1	yes	characteristic
157	C₂²	C₂²⋊C₄×Dic₅₃	1	3	yes	—
158	C₂²	C₄⋊C₄×Dic₅₃	1	3	yes	—
159	C₂²	C₂×C₄×#88322	1	6	yes	—
160	C₂²	#16810579	1	6	yes	—
161	C₂²	D₄×#88322	1	12	yes	—
162	C₂²	Q₈×#88322	1	4	yes	—
163	C₄	—	2	24	no	—
164	C₄	#95×D₅₃	1	1	yes	characteristic
165	C₄	#16810468	1	1	yes	characteristic
166	C₄	#16810470	1	3	yes	—
167	C₄	#16810474	1	3	yes	—
168	C₂	#95×Dic₅₃	1	1	yes	characteristic
169	C₂	C₄²×#88322	1	1	yes	characteristic
170	C₂	#20274402	1	1	yes	characteristic
171	C₂	C₂²⋊C₄×#88322	1	6	yes	—
172	C₂	C₄⋊C₄×#88322	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₄²×C₈
Fitting	C₄×#95×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₄×#95	D₅₃
hypocenter	C₅₃	C₈×#95
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₄×#95

Complements

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

N	H ≅ G/N	type
7. C₂×C₄×#88322	C₄	semidirect
8. #16810378	C₄	semidirect
26. #95×C₅₃	C₈	semidirect
57. C₂×C₄×C₅₃	C₄×C₈	semidirect
68. #88322	#95	direct (N × H)
97. C₅₃	C₈×#95	semidirect
113. #95	#88322	direct (N × H)
144. C₂×C₄	C₄×#88322	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×#88322	2
C₂×#16810378	2
C₄×#95×C₅₃	2
C₈×#95	53