Subgroups — H(5,𝔽₂)×C₇⋊C₈ #14219455

Order 1792.

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	C₄×C₇×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	#11739965	C₂	1	6	yes	maximal, complemented
3	#41×C₇⋊C₈	C₂	1	6	yes	maximal, complemented
4	#11739746	C₂	1	9	yes	maximal, complemented
5	C₂×D₄×C₇⋊C₈	C₂	1	9	yes	maximal, complemented
6	C₂×C₇×H(5,𝔽₂)	C₄	1	1	yes	characteristic
7	C₇×#520	C₄	1	6	yes	—
8	C₇×#518	C₄	1	9	yes	—
9	C₄×C₇×#41	C₂²	1	6	yes	—
10	C₂×C₄×C₇×D₄	C₂²	1	9	yes	—
11	C₂³×C₇⋊C₈	C₂²	1	6	yes	complemented
12	C₂×C₂²⋊(C₇⋊C₈)	C₂²	1	18	yes	complemented
13	Q₈×C₇⋊C₈	C₂²	1	2	yes	complemented
14	Q₈×C₇⋊C₈	C₂²	1	6	yes	complemented
15	D₄×C₇⋊C₈	C₂²	1	18	yes	complemented
16	D₄×C₇⋊C₈	C₂²	1	18	yes	complemented
17	D₄×C₇⋊C₈	C₂²	1	36	yes	complemented
18	#89697	C₂²	1	18	yes	complemented
19	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	9	yes	complemented
20	C₂×C₄×C₇⋊C₈	C₂²	1	9	yes	complemented
21	C₈×H(5,𝔽₂)	—	7	1	no	maximal, Sylow
22	C₇×H(5,𝔽₂)	C₈	1	16	yes	complemented
23	C₂×C₇×#41	C₂×C₄	1	6	yes	—
24	C₂²×C₇×D₄	C₂×C₄	1	9	yes	—
25	C₂³×C₄×C₇	C₂³	1	6	yes	—
26	C₄×C₇×Q₈	C₂×C₄	1	6	yes	—
27	C₄×C₇×Q₈	C₂³	1	2	yes	—
28	C₄×C₇×D₄	C₂×C₄	1	36	yes	—
29	C₄×C₇×D₄	C₂×C₄	1	18	yes	—
30	C₄×C₇×D₄	C₂³	1	18	yes	—
31	C₇×#117	C₂×C₄	1	18	yes	—
32	C₂×C₇×C₄⋊C₄	C₂×C₄	1	9	yes	—
33	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	18	yes	—
34	C₂×C₄²×C₇	C₂³	1	9	yes	—
35	C₂²×C₇⋊C₈	—	4	12	no	—
36	C₂²×C₇⋊C₈	C₂³	1	9	yes	—
37	C₂²×C₇⋊C₈	C₂³	1	9	yes	—
38	C₂²×C₇⋊C₈	C₂³	1	18	yes	—
39	C₂²⋊(C₇⋊C₈)	C₂³	1	36	yes	—
40	C₄⋊(C₇⋊C₈)	C₂³	1	18	yes	—
41	C₄⋊(C₇⋊C₈)	C₂³	1	6	yes	—
42	C₄×C₇⋊C₈	C₂³	1	6	yes	—
43	C₄×C₇⋊C₈	C₂³	1	18	yes	—
44	C₄×H(5,𝔽₂)	D₇	1	1	yes	characteristic
45	#2988	—	7	6	no	—
46	#2972	—	7	9	no	—
47	C₈×#41	—	7	6	no	—
48	C₂×C₈×D₄	—	7	9	no	—
49	C₂⁴×C₇	C₂²×C₄	1	6	yes	—
50	C₇×#41	C₂×C₈	1	48	yes	complemented
51	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
52	C₂×C₇×D₄	C₂×C₈	1	72	yes	complemented
53	C₂×C₇×D₄	C₂²×C₄	1	18	yes	—
54	C₂²×C₄×C₇	—	4	6	no	—
55	C₂²×C₄×C₇	—	4	6	no	—
56	C₂²×C₄×C₇	C₂²×C₄	1	9	yes	—
57	C₂²×C₄×C₇	C₂²×C₄	1	18	yes	—
58	C₂²×C₄×C₇	C₂²×C₄	1	9	yes	—
59	C₂²×C₄×C₇	C₂⁴	1	9	yes	—
60	C₇×C₄⋊C₄	C₂²×C₄	1	18	yes	—
61	C₇×C₄⋊C₄	C₂²×C₄	1	6	yes	—
62	C₇×C₂²⋊C₄	C₂²×C₄	1	36	yes	—
63	C₄²×C₇	C₂²×C₄	1	18	yes	—
64	C₄²×C₇	C₂⁴	1	6	yes	—
65	C₂×C₇⋊C₈	—	2	18	no	—
66	C₂×C₇⋊C₈	—	4	18	no	—
67	C₂×C₇⋊C₈	—	4	9	no	—
68	C₂×C₇⋊C₈	C₂⁴	1	6	yes	—
69	C₂×C₇⋊C₈	C₂⁴	1	9	yes	—
70	C₂×C₇⋊C₈	C₂⁴	1	1	yes	characteristic
71	C₂×H(5,𝔽₂)	Dic₇	1	1	yes	characteristic
72	C₂³×C₈	—	7	6	no	—
73	#520	Dic₇	1	6	yes	—
74	#518	Dic₇	1	9	yes	—
75	C₄×#41	C₂×D₇	1	6	yes	—
76	C₂×C₄×D₄	C₂×D₇	1	9	yes	—
77	C₈×Q₈	—	7	2	no	—
78	C₈×Q₈	—	7	6	no	—
79	C₈×D₄	—	7	18	no	—
80	C₈×D₄	—	7	36	no	—
81	C₈×D₄	—	7	18	no	—
82	#431	—	7	18	no	—
83	C₂×C₄⋊C₈	—	7	9	no	—
84	C₂×C₂²⋊C₈	—	7	18	no	—
85	C₂×C₄×C₈	—	7	9	no	—
86	C₂³×C₇	—	4	6	no	—
87	C₂³×C₇	—	4	6	no	—
88	C₂³×C₇	C₂²×C₈	1	24	yes	complemented
89	C₂³×C₇	C₂³×C₄	1	9	yes	—
90	C₇×Q₈	C₂²×C₈	1	8	yes	complemented
91	C₇×D₄	C₂²×C₈	1	72	yes	complemented
92	C₂×C₄×C₇	—	2	9	no	—
93	C₂×C₄×C₇	—	2	9	no	—
94	C₂×C₄×C₇	—	4	18	no	—
95	C₂×C₄×C₇	—	4	9	no	—
96	C₂×C₄×C₇	C₂²×C₈	1	36	yes	complemented
97	C₂×C₄×C₇	C₂³×C₄	1	6	yes	—
98	C₂×C₄×C₇	C₂³×C₄	1	9	yes	—
99	C₂×C₄×C₇	C₂³×C₄	1	6	yes	—
100	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
101	C₇⋊C₈	—	2	9	no	—
102	C₇⋊C₈	—	2	6	no	—
103	C₇⋊C₈	H(5,𝔽₂)	1	2	yes	direct product factor
104	H(5,𝔽₂)	C₇⋊C₈	1	16	yes	direct product factor
105	C₂×#41	C₂×Dic₇	1	6	yes	—
106	C₂²×D₄	C₂×Dic₇	1	9	yes	—
107	C₂³×C₄	C₂²×D₇	1	6	yes	—
108	C₂²×C₈	—	7	9	no	—
109	C₂²×C₈	—	7	18	no	—
110	C₂²×C₈	—	7	9	no	—
111	C₂²×C₈	—	28	12	no	—
112	C₄×Q₈	C₂×Dic₇	1	6	yes	—
113	C₄×Q₈	C₂²×D₇	1	2	yes	—
114	C₄×D₄	C₂×Dic₇	1	36	yes	—
115	C₄×D₄	C₂×Dic₇	1	18	yes	—
116	C₄×D₄	C₂²×D₇	1	18	yes	—
117	#117	C₂×Dic₇	1	18	yes	—
118	C₂×C₄⋊C₄	C₂×Dic₇	1	9	yes	—
119	C₂×C₂²⋊C₄	C₂×Dic₇	1	18	yes	—
120	C₂×C₄²	C₂²×D₇	1	9	yes	—
121	C₄⋊C₈	—	7	18	no	—
122	C₄⋊C₈	—	7	6	no	—
123	C₂²⋊C₈	—	7	36	no	—
124	C₄×C₈	—	7	6	no	—
125	C₄×C₈	—	7	18	no	—
126	C₂²×C₇	—	2	9	no	—
127	C₂²×C₇	—	2	9	no	—
128	C₂²×C₇	—	4	24	no	—
129	C₂²×C₇	C₂³×C₈	1	18	yes	—
130	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
131	C₄×C₇	—	2	9	no	—
132	C₄×C₇	—	2	6	no	—
133	C₄×C₇	C₂³×C₈	1	12	yes	—
134	C₄×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
135	C₄×C₇	#585	1	1	yes	characteristic
136	C₂⁴	C₂²×Dic₇	1	6	yes	—
137	#41	C₂×C₇⋊C₈	1	48	yes	complemented
138	C₂×Q₈	C₂²×Dic₇	1	2	yes	—
139	C₂×D₄	C₂×C₇⋊C₈	1	72	yes	complemented
140	C₂×D₄	C₂²×Dic₇	1	18	yes	—
141	C₂²×C₄	—	4	6	no	—
142	C₂²×C₄	—	4	6	no	—
143	C₂²×C₄	C₂²×Dic₇	1	9	yes	—
144	C₂²×C₄	C₂²×Dic₇	1	18	yes	—
145	C₂²×C₄	C₂²×Dic₇	1	9	yes	—
146	C₂²×C₄	C₂³×D₇	1	9	yes	—
147	C₂×C₈	—	7	1	no	—
148	C₂×C₈	—	7	6	no	—
149	C₂×C₈	—	7	9	no	—
150	C₂×C₈	—	14	18	no	—
151	C₂×C₈	—	28	9	no	—
152	C₂×C₈	—	28	18	no	—
153	C₄⋊C₄	C₂²×Dic₇	1	18	yes	—
154	C₄⋊C₄	C₂²×Dic₇	1	6	yes	—
155	C₂²⋊C₄	C₂²×Dic₇	1	36	yes	—
156	C₄²	C₂²×Dic₇	1	18	yes	—
157	C₄²	C₂³×D₇	1	6	yes	—
158	C₂×C₇	—	2	18	no	—
159	C₂×C₇	C₄×H(5,𝔽₂)	1	1	yes	characteristic
160	C₂×C₇	C₂⁴×C₈	1	1	yes	characteristic
161	C₂×C₇	#3573	1	1	yes	characteristic
162	C₂³	—	4	6	no	—
163	C₂³	—	4	6	no	—
164	C₂³	C₂²×C₇⋊C₈	1	24	yes	complemented
165	C₂³	C₂³×Dic₇	1	9	yes	—
166	Q₈	C₂²×C₇⋊C₈	1	8	yes	complemented
167	D₄	C₂²×C₇⋊C₈	1	72	yes	complemented
168	C₂×C₄	—	2	9	no	—
169	C₂×C₄	—	2	9	no	—
170	C₂×C₄	—	4	18	no	—
171	C₂×C₄	—	4	9	no	—
172	C₂×C₄	C₂²×C₇⋊C₈	1	36	yes	complemented
173	C₂×C₄	C₂³×Dic₇	1	6	yes	—
174	C₂×C₄	C₂³×Dic₇	1	9	yes	—
175	C₂×C₄	C₂³×Dic₇	1	6	yes	—
176	C₂×C₄	C₂⁴×D₇	1	1	yes	characteristic
177	C₈	—	7	2	no	—
178	C₈	—	14	9	no	—
179	C₈	—	14	6	no	—
180	C₇	C₈×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
181	C₂²	—	2	9	no	—
182	C₂²	—	2	9	no	—
183	C₂²	—	4	24	no	—
184	C₂²	C₂³×C₇⋊C₈	1	18	yes	—
185	C₂²	C₂⁴×Dic₇	1	1	yes	characteristic
186	C₄	—	2	9	no	—
187	C₄	—	2	6	no	—
188	C₄	C₂³×C₇⋊C₈	1	12	yes	—
189	C₄	D₇×H(5,𝔽₂)	1	1	yes	characteristic
190	C₄	#90618	1	1	yes	characteristic
191	C₂	—	2	18	no	—
192	C₂	C₂⁴×C₇⋊C₈	1	1	yes	characteristic
193	C₂	Dic₇×H(5,𝔽₂)	1	1	yes	characteristic
194	C₂	#11744364	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₇
commutator	C₂×C₇	C₂⁴×C₈
Fitting	C₄×C₇×H(5,𝔽₂)	C₂
socle	C₂²×C₇	C₂⁴×C₄
prime radical	C₂×C₇×H(5,𝔽₂)	C₄
cosocle	C₂×C₄×C₇	C₂⁵
hypercenter	C₄×H(5,𝔽₂)	D₇
hypocenter	C₇	C₈×H(5,𝔽₂)
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
2. #11739965	C₂	semidirect
3. #41×C₇⋊C₈	C₂	semidirect
4. #11739746	C₂	semidirect
5. C₂×D₄×C₇⋊C₈	C₂	semidirect
11. C₂³×C₇⋊C₈	C₂²	semidirect
12. C₂×C₂²⋊(C₇⋊C₈)	C₂²	semidirect
13. Q₈×C₇⋊C₈	C₂²	semidirect
14. Q₈×C₇⋊C₈	C₂²	semidirect
15. D₄×C₇⋊C₈	C₂²	semidirect
16. D₄×C₇⋊C₈	C₂²	semidirect
17. D₄×C₇⋊C₈	C₂²	semidirect
18. #89697	C₂²	semidirect
19. C₂×C₄⋊(C₇⋊C₈)	C₂²	semidirect
20. C₂×C₄×C₇⋊C₈	C₂²	semidirect
22. C₇×H(5,𝔽₂)	C₈	semidirect
50. C₇×#41	C₂×C₈	semidirect
52. C₂×C₇×D₄	C₂×C₈	semidirect
88. C₂³×C₇	C₂²×C₈	semidirect
90. C₇×Q₈	C₂²×C₈	semidirect
91. C₇×D₄	C₂²×C₈	semidirect
96. C₂×C₄×C₇	C₂²×C₈	semidirect
103. C₇⋊C₈	H(5,𝔽₂)	direct (N × H)
104. H(5,𝔽₂)	C₇⋊C₈	direct (N × H)
137. #41	C₂×C₇⋊C₈	semidirect
139. C₂×D₄	C₂×C₇⋊C₈	semidirect
164. C₂³	C₂²×C₇⋊C₈	semidirect
166. Q₈	C₂²×C₇⋊C₈	semidirect
167. D₄	C₂²×C₇⋊C₈	semidirect
172. C₂×C₄	C₂²×C₇⋊C₈	semidirect
180. C₇	C₈×H(5,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇×H(5,𝔽₂)	2
#11739965	2
#41×C₇⋊C₈	2
#11739746	2
C₂×D₄×C₇⋊C₈	2
C₈×H(5,𝔽₂)	7

Subgroups — H(5,𝔽2)×C7⋊C8 #14219455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(5,𝔽₂)×C₇⋊C₈ #14219455