Subgroups — H(5,𝔽₂)×#88322 #28249751

Order 13568.

2422 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283826	C₂	1	6	yes	maximal, complemented
2	#20283739	C₂	1	9	yes	maximal, complemented
3	#41×#88322	C₂	1	6	yes	maximal, complemented
4	C₂×D₄×#88322	C₂	1	9	yes	maximal, complemented
5	C₄×H(5,𝔽₂)×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×#88322	C₂²	1	6	yes	complemented
7	Q₈×#88322	C₂²	1	2	yes	complemented
8	Q₈×#88322	C₂²	1	6	yes	complemented
9	D₄×#88322	C₂²	1	18	yes	complemented
10	D₄×#88322	C₂²	1	18	yes	complemented
11	D₄×#88322	C₂²	1	36	yes	complemented
12	#16810676	C₂²	1	18	yes	complemented
13	C₂×C₄⋊#88322	C₂²	1	9	yes	complemented
14	C₂×C₂²⋊#88322	C₂²	1	18	yes	complemented
15	C₂×C₄×#88322	C₂²	1	9	yes	complemented
16	C₂×H(5,𝔽₂)×C₅₃	C₄	1	1	yes	characteristic
17	C₅₃×#520	C₄	1	6	yes	—
18	C₅₃×#518	C₄	1	9	yes	—
19	C₄×#41×C₅₃	C₂²	1	6	yes	—
20	C₂×C₄×D₄×C₅₃	C₂²	1	9	yes	—
21	C₂²×#88322	—	4	12	no	—
22	C₂²×#88322	C₂³	1	9	yes	—
23	C₂²×#88322	C₂³	1	9	yes	—
24	C₂²×#88322	C₂³	1	18	yes	—
25	C₄⋊#88322	C₂³	1	18	yes	—
26	C₄⋊#88322	C₂³	1	6	yes	—
27	C₂²⋊#88322	C₂³	1	36	yes	—
28	C₄×#88322	C₂³	1	6	yes	—
29	C₄×#88322	C₂³	1	18	yes	—
30	H(5,𝔽₂)×C₅₃	C₈	1	16	yes	complemented
31	C₂×#41×C₅₃	C₂×C₄	1	6	yes	—
32	C₂²×D₄×C₅₃	C₂×C₄	1	9	yes	—
33	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
34	C₄×Q₈×C₅₃	C₂×C₄	1	6	yes	—
35	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
36	C₄×D₄×C₅₃	C₂×C₄	1	36	yes	—
37	C₄×D₄×C₅₃	C₂×C₄	1	18	yes	—
38	C₄×D₄×C₅₃	C₂³	1	18	yes	—
39	#117×C₅₃	C₂×C₄	1	18	yes	—
40	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	9	yes	—
41	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	18	yes	—
42	C₂×C₄²×C₅₃	C₂³	1	9	yes	—
43	C₂⁴×C₅₃	C₂²×C₄	1	6	yes	—
44	#41×C₅₃	C₂×C₈	1	48	yes	complemented
45	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
46	C₂×D₄×C₅₃	C₂×C₈	1	72	yes	complemented
47	C₂×D₄×C₅₃	C₂²×C₄	1	18	yes	—
48	C₂²×C₄×C₅₃	—	4	6	no	—
49	C₂²×C₄×C₅₃	—	4	6	no	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	9	yes	—
51	C₂²×C₄×C₅₃	C₂²×C₄	1	18	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	9	yes	—
53	C₂²×C₄×C₅₃	C₂⁴	1	9	yes	—
54	C₄⋊C₄×C₅₃	C₂²×C₄	1	18	yes	—
55	C₄⋊C₄×C₅₃	C₂²×C₄	1	6	yes	—
56	C₂²⋊C₄×C₅₃	C₂²×C₄	1	36	yes	—
57	C₄²×C₅₃	C₂²×C₄	1	18	yes	—
58	C₄²×C₅₃	C₂⁴	1	6	yes	—
59	C₂×#88322	—	2	18	no	—
60	C₂×#88322	—	4	18	no	—
61	C₂×#88322	—	4	9	no	—
62	C₂×#88322	C₂⁴	1	6	yes	—
63	C₂×#88322	C₂⁴	1	9	yes	—
64	C₂×#88322	C₂⁴	1	1	yes	characteristic
65	C₂³×C₅₃	—	4	6	no	—
66	C₂³×C₅₃	—	4	6	no	—
67	C₂³×C₅₃	C₂²×C₈	1	24	yes	complemented
68	C₂³×C₅₃	C₂³×C₄	1	9	yes	—
69	Q₈×C₅₃	C₂²×C₈	1	8	yes	complemented
70	D₄×C₅₃	C₂²×C₈	1	72	yes	complemented
71	C₂×C₄×C₅₃	—	2	9	no	—
72	C₂×C₄×C₅₃	—	2	9	no	—
73	C₂×C₄×C₅₃	—	4	18	no	—
74	C₂×C₄×C₅₃	—	4	9	no	—
75	C₂×C₄×C₅₃	C₂²×C₈	1	36	yes	complemented
76	C₂×C₄×C₅₃	C₂³×C₄	1	9	yes	—
77	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
78	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
79	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
80	#88322	—	2	9	no	—
81	#88322	—	2	6	no	—
82	#88322	H(5,𝔽₂)	1	2	yes	direct product factor
83	C₈×H(5,𝔽₂)	—	53	1	no	maximal, Sylow
84	C₂²×C₅₃	—	2	9	no	—
85	C₂²×C₅₃	—	2	9	no	—
86	C₂²×C₅₃	—	4	24	no	—
87	C₂²×C₅₃	C₂³×C₈	1	18	yes	—
88	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
89	C₄×C₅₃	—	2	9	no	—
90	C₄×C₅₃	—	2	6	no	—
91	C₄×C₅₃	C₂³×C₈	1	12	yes	—
92	C₄×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
93	C₄×C₅₃	#585	1	1	yes	characteristic
94	C₄×H(5,𝔽₂)	D₅₃	1	1	yes	characteristic
95	#2988	—	53	6	no	—
96	#2972	—	53	9	no	—
97	C₈×#41	—	53	6	no	—
98	C₂×C₈×D₄	—	53	9	no	—
99	C₂×C₅₃	—	2	18	no	—
100	C₂×C₅₃	C₄×H(5,𝔽₂)	1	1	yes	characteristic
101	C₂×C₅₃	C₂⁴×C₈	1	1	yes	characteristic
102	C₂×C₅₃	#3573	1	1	yes	characteristic
103	C₂×H(5,𝔽₂)	Dic₅₃	1	1	yes	characteristic
104	C₂³×C₈	—	53	6	no	—
105	#520	Dic₅₃	1	6	yes	—
106	#518	Dic₅₃	1	9	yes	—
107	C₄×#41	C₂×D₅₃	1	6	yes	—
108	C₂×C₄×D₄	C₂×D₅₃	1	9	yes	—
109	C₈×Q₈	—	53	2	no	—
110	C₈×Q₈	—	53	6	no	—
111	C₈×D₄	—	53	18	no	—
112	C₈×D₄	—	53	36	no	—
113	C₈×D₄	—	53	18	no	—
114	#431	—	53	18	no	—
115	C₂×C₄⋊C₈	—	53	9	no	—
116	C₂×C₂²⋊C₈	—	53	18	no	—
117	C₂×C₄×C₈	—	53	9	no	—
118	C₅₃	C₈×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
119	H(5,𝔽₂)	#88322	1	16	yes	direct product factor
120	C₂×#41	C₂×Dic₅₃	1	6	yes	—
121	C₂²×D₄	C₂×Dic₅₃	1	9	yes	—
122	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
123	C₂²×C₈	—	53	9	no	—
124	C₂²×C₈	—	53	18	no	—
125	C₂²×C₈	—	53	9	no	—
126	C₂²×C₈	—	212	12	no	—
127	C₄×Q₈	C₂×Dic₅₃	1	6	yes	—
128	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
129	C₄×D₄	C₂×Dic₅₃	1	36	yes	—
130	C₄×D₄	C₂×Dic₅₃	1	18	yes	—
131	C₄×D₄	C₂²×D₅₃	1	18	yes	—
132	#117	C₂×Dic₅₃	1	18	yes	—
133	C₂×C₄⋊C₄	C₂×Dic₅₃	1	9	yes	—
134	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	18	yes	—
135	C₂×C₄²	C₂²×D₅₃	1	9	yes	—
136	C₄⋊C₈	—	53	18	no	—
137	C₄⋊C₈	—	53	6	no	—
138	C₂²⋊C₈	—	53	36	no	—
139	C₄×C₈	—	53	6	no	—
140	C₄×C₈	—	53	18	no	—
141	C₂⁴	C₂²×Dic₅₃	1	6	yes	—
142	#41	C₂×#88322	1	48	yes	complemented
143	C₂×Q₈	C₂²×Dic₅₃	1	2	yes	—
144	C₂×D₄	C₂×#88322	1	72	yes	complemented
145	C₂×D₄	C₂²×Dic₅₃	1	18	yes	—
146	C₂²×C₄	—	4	6	no	—
147	C₂²×C₄	—	4	6	no	—
148	C₂²×C₄	C₂²×Dic₅₃	1	9	yes	—
149	C₂²×C₄	C₂²×Dic₅₃	1	18	yes	—
150	C₂²×C₄	C₂²×Dic₅₃	1	9	yes	—
151	C₂²×C₄	C₂³×D₅₃	1	9	yes	—
152	C₂×C₈	—	53	1	no	—
153	C₂×C₈	—	53	6	no	—
154	C₂×C₈	—	53	9	no	—
155	C₂×C₈	—	106	18	no	—
156	C₂×C₈	—	212	18	no	—
157	C₂×C₈	—	212	9	no	—
158	C₄⋊C₄	C₂²×Dic₅₃	1	18	yes	—
159	C₄⋊C₄	C₂²×Dic₅₃	1	6	yes	—
160	C₂²⋊C₄	C₂²×Dic₅₃	1	36	yes	—
161	C₄²	C₂²×Dic₅₃	1	18	yes	—
162	C₄²	C₂³×D₅₃	1	6	yes	—
163	C₂³	—	4	6	no	—
164	C₂³	—	4	6	no	—
165	C₂³	C₂²×#88322	1	24	yes	complemented
166	C₂³	C₂³×Dic₅₃	1	9	yes	—
167	Q₈	C₂²×#88322	1	8	yes	complemented
168	D₄	C₂²×#88322	1	72	yes	complemented
169	C₂×C₄	—	2	9	no	—
170	C₂×C₄	—	2	9	no	—
171	C₂×C₄	—	4	18	no	—
172	C₂×C₄	—	4	9	no	—
173	C₂×C₄	C₂²×#88322	1	36	yes	complemented
174	C₂×C₄	C₂³×Dic₅₃	1	6	yes	—
175	C₂×C₄	C₂³×Dic₅₃	1	9	yes	—
176	C₂×C₄	C₂³×Dic₅₃	1	6	yes	—
177	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
178	C₈	—	53	2	no	—
179	C₈	—	106	9	no	—
180	C₈	—	106	6	no	—
181	C₂²	—	2	9	no	—
182	C₂²	—	2	9	no	—
183	C₂²	—	4	24	no	—
184	C₂²	C₂³×#88322	1	18	yes	—
185	C₂²	C₂⁴×Dic₅₃	1	1	yes	characteristic
186	C₄	—	2	6	no	—
187	C₄	—	2	9	no	—
188	C₄	C₂³×#88322	1	12	yes	—
189	C₄	H(5,𝔽₂)×D₅₃	1	1	yes	characteristic
190	C₄	#16811458	1	1	yes	characteristic
191	C₂	—	2	18	no	—
192	C₂	H(5,𝔽₂)×Dic₅₃	1	1	yes	characteristic
193	C₂	C₂⁴×#88322	1	1	yes	characteristic
194	C₂	#20289427	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₅₃
commutator	C₂×C₅₃	C₂⁴×C₈
Fitting	C₄×H(5,𝔽₂)×C₅₃	C₂
socle	C₂²×C₅₃	C₂⁴×C₄
prime radical	C₂×H(5,𝔽₂)×C₅₃	C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₄×H(5,𝔽₂)	D₅₃
hypocenter	C₅₃	C₈×H(5,𝔽₂)
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. #20283826	C₂	semidirect
2. #20283739	C₂	semidirect
3. #41×#88322	C₂	semidirect
4. C₂×D₄×#88322	C₂	semidirect
6. C₂³×#88322	C₂²	semidirect
7. Q₈×#88322	C₂²	semidirect
8. Q₈×#88322	C₂²	semidirect
9. D₄×#88322	C₂²	semidirect
10. D₄×#88322	C₂²	semidirect
11. D₄×#88322	C₂²	semidirect
12. #16810676	C₂²	semidirect
13. C₂×C₄⋊#88322	C₂²	semidirect
14. C₂×C₂²⋊#88322	C₂²	semidirect
15. C₂×C₄×#88322	C₂²	semidirect
30. H(5,𝔽₂)×C₅₃	C₈	semidirect
44. #41×C₅₃	C₂×C₈	semidirect
46. C₂×D₄×C₅₃	C₂×C₈	semidirect
67. C₂³×C₅₃	C₂²×C₈	semidirect
69. Q₈×C₅₃	C₂²×C₈	semidirect
70. D₄×C₅₃	C₂²×C₈	semidirect
75. C₂×C₄×C₅₃	C₂²×C₈	semidirect
82. #88322	H(5,𝔽₂)	direct (N × H)
118. C₅₃	C₈×H(5,𝔽₂)	semidirect
119. H(5,𝔽₂)	#88322	direct (N × H)
142. #41	C₂×#88322	semidirect
144. C₂×D₄	C₂×#88322	semidirect
165. C₂³	C₂²×#88322	semidirect
167. Q₈	C₂²×#88322	semidirect
168. D₄	C₂²×#88322	semidirect
173. C₂×C₄	C₂²×#88322	semidirect

Maximal subgroups

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

subgroup	index
#20283826	2
#20283739	2
#41×#88322	2
C₂×D₄×#88322	2
C₄×H(5,𝔽₂)×C₅₃	2
C₈×H(5,𝔽₂)	53

Subgroups — H(5,𝔽2)×#88322 #28249751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(5,𝔽₂)×#88322 #28249751