Subgroups — H(5,𝔽₂)×#10596842 #34035047

Order 18688.

2422 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627175	C₂	1	6	yes	maximal, complemented
2	#22627173	C₂	1	9	yes	maximal, complemented
3	#41×#10596842	C₂	1	6	yes	maximal, complemented
4	C₂×D₄×#10596842	C₂	1	9	yes	maximal, complemented
5	H(5,𝔽₂)×#64408	C₂	1	1	yes	maximal, characteristic
6	C₂³×#10596842	C₂²	1	6	yes	complemented
7	Q₈×#10596842	C₂²	1	2	yes	complemented
8	Q₈×#10596842	C₂²	1	6	yes	complemented
9	D₄×#10596842	C₂²	1	18	yes	complemented
10	D₄×#10596842	C₂²	1	36	yes	complemented
11	D₄×#10596842	C₂²	1	18	yes	complemented
12	#17989700	C₂²	1	18	yes	complemented
13	C₂×C₄⋊#10596842	C₂²	1	9	yes	complemented
14	C₂×C₂²⋊#10596842	C₂²	1	18	yes	complemented
15	C₂×C₄×#10596842	C₂²	1	9	yes	complemented
16	#17989656	C₄	1	6	yes	—
17	#17989654	C₄	1	9	yes	—
18	#41×#64408	C₂²	1	6	yes	—
19	C₂×D₄×#64408	C₂²	1	9	yes	—
20	H(5,𝔽₂)×D₇₃	C₄	1	1	yes	characteristic
21	C₂²×#10596842	—	4	12	no	—
22	C₂²×#10596842	C₂³	1	9	yes	—
23	C₂²×#10596842	C₂³	1	9	yes	—
24	C₂²×#10596842	C₂³	1	18	yes	—
25	C₄⋊#10596842	C₂³	1	18	yes	—
26	C₄⋊#10596842	C₂³	1	6	yes	—
27	C₂²⋊#10596842	C₂³	1	36	yes	—
28	C₄×#10596842	C₂³	1	6	yes	—
29	C₄×#10596842	C₂³	1	18	yes	—
30	C₂³×#64408	C₂³	1	6	yes	—
31	Q₈×#64408	C₂×C₄	1	6	yes	—
32	Q₈×#64408	C₂³	1	2	yes	—
33	D₄×#64408	C₂×C₄	1	36	yes	—
34	D₄×#64408	C₂×C₄	1	18	yes	—
35	D₄×#64408	C₂³	1	18	yes	—
36	#14556670	C₂×C₄	1	18	yes	—
37	C₂×C₄⋊#64408	C₂×C₄	1	9	yes	—
38	C₂×C₂²⋊#64408	C₂×C₄	1	18	yes	—
39	C₂×C₄×#64408	C₂³	1	9	yes	—
40	#14556637	C₈	1	6	yes	complemented
41	#14556636	C₈	1	9	yes	complemented
42	#41×D₇₃	C₂×C₄	1	6	yes	—
43	C₂×D₄×D₇₃	C₂×C₄	1	9	yes	—
44	H(5,𝔽₂)×C₇₃	C₈	1	1	yes	characteristic, complemented
45	C₂×#10596842	—	2	18	no	—
46	C₂×#10596842	—	4	18	no	—
47	C₂×#10596842	—	4	9	no	—
48	C₂×#10596842	C₂⁴	1	6	yes	—
49	C₂×#10596842	C₂⁴	1	9	yes	—
50	C₂×#10596842	C₂⁴	1	1	yes	characteristic
51	C₂²×#64408	—	4	6	no	—
52	C₂²×#64408	—	4	6	no	—
53	C₂²×#64408	C₂²×C₄	1	18	yes	—
54	C₂²×#64408	C₂²×C₄	1	9	yes	—
55	C₂²×#64408	C₂⁴	1	9	yes	—
56	C₄⋊#64408	C₂²×C₄	1	18	yes	—
57	C₄⋊#64408	C₂²×C₄	1	6	yes	—
58	C₂²⋊#64408	C₂²×C₄	1	36	yes	—
59	C₄×#64408	C₂²×C₄	1	18	yes	—
60	C₄×#64408	C₂⁴	1	6	yes	—
61	C₂³×D₇₃	C₂²×C₄	1	6	yes	—
62	#11924029	C₂×C₈	1	6	yes	complemented
63	#11924028	C₂×C₈	1	18	yes	complemented
64	#11924027	C₂×C₈	1	18	yes	complemented
65	Q₈×D₇₃	C₂²×C₄	1	2	yes	—
66	C₂×D₂₉₂	C₂×C₈	1	9	yes	complemented
67	C₂×C₂²⋊D₇₃	C₂×C₈	1	18	yes	complemented
68	D₄×D₇₃	C₂×C₈	1	36	yes	complemented
69	D₄×D₇₃	C₂²×C₄	1	18	yes	—
70	C₂×C₄×D₇₃	C₂²×C₄	1	9	yes	—
71	#41×C₇₃	C₂×C₈	1	6	yes	complemented
72	C₂×D₄×C₇₃	C₂×C₈	1	9	yes	complemented
73	C₂³×C₇₃	C₂²×C₈	1	6	yes	complemented
74	C₂²×D₇₃	—	4	6	no	—
75	C₂²×D₇₃	—	4	6	no	—
76	C₂²×D₇₃	C₂²×C₈	1	18	yes	complemented
77	C₂²×D₇₃	C₂³×C₄	1	9	yes	—
78	C₂×#64408	—	2	9	no	—
79	C₂×#64408	—	2	9	no	—
80	C₂×#64408	—	4	18	no	—
81	C₂×#64408	—	4	9	no	—
82	C₂×#64408	C₂³×C₄	1	6	yes	—
83	C₂×#64408	C₂³×C₄	1	9	yes	—
84	C₂×#64408	C₂⁵	1	1	yes	characteristic
85	#10596842	—	2	9	no	—
86	#10596842	—	2	6	no	—
87	#10596842	H(5,𝔽₂)	1	2	yes	direct product factor
88	Q₈×C₇₃	C₂²×C₈	1	2	yes	complemented
89	D₄×C₇₃	C₂²×C₈	1	18	yes	complemented
90	C₂×C₄×C₇₃	C₂²×C₈	1	9	yes	complemented
91	C₂²⋊D₇₃	C₂²×C₈	1	36	yes	complemented
92	C₂×Dic₇₃	C₂²×C₈	1	9	yes	complemented
93	D₂₉₂	C₂²×C₈	1	18	yes	complemented
94	C₄×D₇₃	C₂²×C₈	1	18	yes	complemented
95	C₄×D₇₃	C₂³×C₄	1	6	yes	—
96	Dic₁₄₆	C₂²×C₈	1	6	yes	complemented
97	C₂²×C₇₃	—	4	6	no	—
98	C₂²×C₇₃	C₂³×C₈	1	9	yes	—
99	C₂×D₇₃	—	2	9	no	—
100	C₂×D₇₃	—	2	9	no	—
101	C₂×D₇₃	—	4	18	no	—
102	C₂×D₇₃	C₂³×C₈	1	9	yes	—
103	C₂×D₇₃	C₂⁴×C₄	1	1	yes	characteristic
104	#64408	—	2	9	no	—
105	#64408	—	2	6	no	—
106	#64408	C₂×H(5,𝔽₂)	1	1	yes	characteristic
107	#64408	#585	1	1	yes	characteristic
108	C₄×C₇₃	C₂³×C₈	1	6	yes	—
109	Dic₇₃	C₂³×C₈	1	6	yes	—
110	C₈×H(5,𝔽₂)	—	73	1	no	maximal, Sylow
111	C₂×C₇₃	—	2	9	no	—
112	C₂×C₇₃	C₂⁴×C₈	1	1	yes	characteristic
113	D₇₃	—	2	9	no	—
114	D₇₃	C₄×H(5,𝔽₂)	1	1	yes	characteristic
115	D₇₃	#3573	1	1	yes	characteristic
116	C₄×H(5,𝔽₂)	—	73	1	no	—
117	#2988	—	73	6	no	—
118	#2972	—	73	9	no	—
119	C₈×#41	—	73	6	no	—
120	C₂×C₈×D₄	—	73	9	no	—
121	C₇₃	C₈×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
122	C₂×H(5,𝔽₂)	—	73	1	no	—
123	C₂³×C₈	—	73	6	no	—
124	#520	—	73	6	no	—
125	#518	—	73	9	no	—
126	C₄×#41	—	73	6	no	—
127	C₂×C₄×D₄	—	73	9	no	—
128	C₈×Q₈	—	73	2	no	—
129	C₈×Q₈	—	73	6	no	—
130	C₈×D₄	—	73	18	no	—
131	C₈×D₄	—	73	36	no	—
132	C₈×D₄	—	73	18	no	—
133	#431	—	73	18	no	—
134	C₂×C₄⋊C₈	—	73	9	no	—
135	C₂×C₂²⋊C₈	—	73	18	no	—
136	C₂×C₄×C₈	—	73	9	no	—
137	H(5,𝔽₂)	—	73	6	no	—
138	H(5,𝔽₂)	—	73	9	no	—
139	H(5,𝔽₂)	#10596842	1	1	yes	characteristic, direct product factor
140	C₂×#41	—	73	6	no	—
141	C₂²×D₄	—	73	9	no	—
142	C₂³×C₄	—	73	6	no	—
143	C₂²×C₈	—	292	12	no	—
144	C₂²×C₈	—	73	18	no	—
145	C₂²×C₈	—	73	9	no	—
146	C₂²×C₈	—	73	9	no	—
147	C₄×Q₈	—	73	6	no	—
148	C₄×Q₈	—	73	2	no	—
149	C₄×D₄	—	73	18	no	—
150	C₄×D₄	—	73	36	no	—
151	C₄×D₄	—	73	18	no	—
152	#117	—	73	18	no	—
153	C₂×C₄⋊C₄	—	73	9	no	—
154	C₂×C₂²⋊C₄	—	73	18	no	—
155	C₂×C₄²	—	73	9	no	—
156	C₄⋊C₈	—	73	18	no	—
157	C₄⋊C₈	—	73	6	no	—
158	C₂²⋊C₈	—	73	36	no	—
159	C₄×C₈	—	73	6	no	—
160	C₄×C₈	—	73	18	no	—
161	C₂⁴	—	73	6	no	—
162	#41	—	73	6	no	—
163	#41	—	73	18	no	—
164	#41	—	73	18	no	—
165	#41	C₂×#10596842	1	6	yes	complemented
166	C₂×Q₈	—	73	2	no	—
167	C₂×D₄	—	73	18	no	—
168	C₂×D₄	—	73	36	no	—
169	C₂×D₄	—	73	18	no	—
170	C₂×D₄	—	73	9	no	—
171	C₂×D₄	C₂×#10596842	1	9	yes	complemented
172	C₂²×C₄	—	292	6	no	—
173	C₂²×C₄	—	292	6	no	—
174	C₂²×C₄	—	73	9	no	—
175	C₂²×C₄	—	73	18	no	—
176	C₂²×C₄	—	73	9	no	—
177	C₂²×C₄	—	73	9	no	—
178	C₂×C₈	—	292	18	no	—
179	C₂×C₈	—	292	9	no	—
180	C₂×C₈	—	73	1	no	—
181	C₂×C₈	—	73	6	no	—
182	C₂×C₈	—	73	9	no	—
183	C₂×C₈	—	146	18	no	—
184	C₄⋊C₄	—	73	18	no	—
185	C₄⋊C₄	—	73	6	no	—
186	C₂²⋊C₄	—	73	36	no	—
187	C₄²	—	73	18	no	—
188	C₄²	—	73	6	no	—
189	C₂³	—	292	6	no	—
190	C₂³	—	292	6	no	—
191	C₂³	—	73	18	no	—
192	C₂³	—	73	9	no	—
193	C₂³	C₂²×#10596842	1	6	yes	complemented
194	Q₈	—	73	6	no	—
195	Q₈	C₂²×#10596842	1	2	yes	complemented
196	D₄	—	73	36	no	—
197	D₄	—	73	18	no	—
198	D₄	C₂²×#10596842	1	18	yes	complemented
199	C₂×C₄	—	292	18	no	—
200	C₂×C₄	—	292	9	no	—
201	C₂×C₄	—	73	1	no	—
202	C₂×C₄	—	73	9	no	—
203	C₂×C₄	—	73	6	no	—
204	C₂×C₄	—	73	18	no	—
205	C₂×C₄	—	73	6	no	—
206	C₂×C₄	—	73	9	no	—
207	C₂×C₄	—	146	9	no	—
208	C₂×C₄	—	146	9	no	—
209	C₂×C₄	C₂²×#10596842	1	9	yes	complemented
210	C₈	—	73	2	no	—
211	C₈	—	146	9	no	—
212	C₈	—	146	6	no	—
213	C₂²	—	4	6	no	—
214	C₂²	—	292	18	no	—
215	C₂²	—	73	1	no	—
216	C₂²	—	73	9	no	—
217	C₂²	—	146	9	no	—
218	C₂²	—	146	9	no	—
219	C₂²	C₂³×#10596842	1	9	yes	—
220	C₄	—	73	1	no	—
221	C₄	—	73	1	no	—
222	C₄	—	73	6	no	—
223	C₄	—	146	9	no	—
224	C₄	—	146	6	no	—
225	C₄	C₂³×#10596842	1	6	yes	—
226	C₂	—	2	9	no	—
227	C₂	—	73	1	no	—
228	C₂	—	73	1	no	—
229	C₂	—	146	9	no	—
230	C₂	C₂⁴×#10596842	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×#10596842
commutator, socle	C₂×C₇₃	C₂⁴×C₈
Fitting	H(5,𝔽₂)×C₇₃	C₈
prime radical	H(5,𝔽₂)×D₇₃	C₄
cosocle	C₂×#64408	C₂⁵
hypercenter	H(5,𝔽₂)	#10596842
hypocenter	C₇₃	C₈×H(5,𝔽₂)
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. #22627175	C₂	semidirect
2. #22627173	C₂	semidirect
3. #41×#10596842	C₂	semidirect
4. C₂×D₄×#10596842	C₂	semidirect
6. C₂³×#10596842	C₂²	semidirect
7. Q₈×#10596842	C₂²	semidirect
8. Q₈×#10596842	C₂²	semidirect
9. D₄×#10596842	C₂²	semidirect
10. D₄×#10596842	C₂²	semidirect
11. D₄×#10596842	C₂²	semidirect
12. #17989700	C₂²	semidirect
13. C₂×C₄⋊#10596842	C₂²	semidirect
14. C₂×C₂²⋊#10596842	C₂²	semidirect
15. C₂×C₄×#10596842	C₂²	semidirect
40. #14556637	C₈	semidirect
41. #14556636	C₈	semidirect
44. H(5,𝔽₂)×C₇₃	C₈	semidirect
62. #11924029	C₂×C₈	semidirect
63. #11924028	C₂×C₈	semidirect
64. #11924027	C₂×C₈	semidirect
66. C₂×D₂₉₂	C₂×C₈	semidirect
67. C₂×C₂²⋊D₇₃	C₂×C₈	semidirect
68. D₄×D₇₃	C₂×C₈	semidirect
71. #41×C₇₃	C₂×C₈	semidirect
72. C₂×D₄×C₇₃	C₂×C₈	semidirect
73. C₂³×C₇₃	C₂²×C₈	semidirect
76. C₂²×D₇₃	C₂²×C₈	semidirect
87. #10596842	H(5,𝔽₂)	direct (N × H)
88. Q₈×C₇₃	C₂²×C₈	semidirect
89. D₄×C₇₃	C₂²×C₈	semidirect
90. C₂×C₄×C₇₃	C₂²×C₈	semidirect
91. C₂²⋊D₇₃	C₂²×C₈	semidirect
92. C₂×Dic₇₃	C₂²×C₈	semidirect
93. D₂₉₂	C₂²×C₈	semidirect
94. C₄×D₇₃	C₂²×C₈	semidirect
96. Dic₁₄₆	C₂²×C₈	semidirect
121. C₇₃	C₈×H(5,𝔽₂)	semidirect
139. H(5,𝔽₂)	#10596842	direct (N × H)
165. #41	C₂×#10596842	semidirect
171. C₂×D₄	C₂×#10596842	semidirect
193. C₂³	C₂²×#10596842	semidirect
195. Q₈	C₂²×#10596842	semidirect
198. D₄	C₂²×#10596842	semidirect
209. C₂×C₄	C₂²×#10596842	semidirect

Maximal subgroups

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

subgroup	index
#22627175	2
#22627173	2
#41×#10596842	2
C₂×D₄×#10596842	2
H(5,𝔽₂)×#64408	2
C₈×H(5,𝔽₂)	73

Subgroups — H(5,𝔽2)×#10596842 #34035047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(5,𝔽₂)×#10596842 #34035047