Subgroups — 2₋¹⁺⁴×C₇⋊C₈ #14219833

Order 1792.

2190 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
2	#11739965	C₂	1	10	yes	maximal, complemented
3	#41×C₇⋊C₈	C₂	1	10	yes	maximal, complemented
4	#11739880	C₂	1	5	yes	maximal, complemented
5	C₂×Q₈×C₇⋊C₈	C₂	1	5	yes	maximal, complemented
6	C₂×C₇×2₋¹⁺⁴	C₄	1	1	yes	characteristic
7	C₇×#520	C₄	1	10	yes	—
8	C₇×#519	C₄	1	5	yes	—
9	C₄×C₇×#41	C₂²	1	10	yes	—
10	C₂×C₄×C₇×Q₈	C₂²	1	5	yes	—
11	Q₈×C₇⋊C₈	C₂²	1	10	yes	—
12	Q₈×C₇⋊C₈	C₂²	1	10	yes	—
13	Q₈×C₇⋊C₈	C₂²	1	20	yes	—
14	D₄×C₇⋊C₈	C₂²	1	10	yes	—
15	D₄×C₇⋊C₈	C₂²	1	30	yes	—
16	#89697	C₂²	1	30	yes	—
17	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	15	yes	—
18	C₂×C₄×C₇⋊C₈	C₂²	1	15	yes	—
19	C₈×2₋¹⁺⁴	—	7	1	no	maximal, Sylow
20	C₇×2₋¹⁺⁴	C₈	1	16	yes	complemented
21	C₂×C₇×#41	C₂×C₄	1	10	yes	—
22	C₂²×C₇×Q₈	C₂×C₄	1	5	yes	—
23	C₄×C₇×Q₈	C₂×C₄	1	20	yes	—
24	C₄×C₇×Q₈	C₂×C₄	1	10	yes	—
25	C₄×C₇×Q₈	C₂³	1	10	yes	—
26	C₄×C₇×D₄	C₂×C₄	1	30	yes	—
27	C₄×C₇×D₄	C₂³	1	10	yes	—
28	C₇×#117	C₂×C₄	1	30	yes	—
29	C₂×C₇×C₄⋊C₄	C₂×C₄	1	15	yes	—
30	C₂×C₄²×C₇	C₂³	1	15	yes	—
31	C₂²×C₇⋊C₈	C₂³	1	5	yes	—
32	C₂²×C₇⋊C₈	C₂³	1	15	yes	—
33	C₂²⋊(C₇⋊C₈)	C₂³	1	20	yes	—
34	C₄⋊(C₇⋊C₈)	C₂³	1	30	yes	—
35	C₄⋊(C₇⋊C₈)	C₂³	1	10	yes	—
36	C₄×C₇⋊C₈	C₂³	1	10	yes	—
37	C₄×C₇⋊C₈	C₂³	1	30	yes	—
38	C₄×2₋¹⁺⁴	D₇	1	1	yes	characteristic
39	#2988	—	7	10	no	—
40	#2982	—	7	5	no	—
41	C₈×#41	—	7	10	no	—
42	C₂×C₈×Q₈	—	7	5	no	—
43	C₇×#41	C₂×C₈	1	80	yes	complemented
44	C₂×C₇×Q₈	C₂×C₈	1	40	yes	complemented
45	C₂×C₇×Q₈	C₂²×C₄	1	10	yes	—
46	C₂×C₇×D₄	C₂²×C₄	1	10	yes	—
47	C₂²×C₄×C₇	C₂²×C₄	1	15	yes	—
48	C₂²×C₄×C₇	C₂²×C₄	1	15	yes	—
49	C₂²×C₄×C₇	C₂⁴	1	5	yes	—
50	C₇×C₄⋊C₄	C₂²×C₄	1	30	yes	—
51	C₇×C₄⋊C₄	C₂²×C₄	1	10	yes	—
52	C₇×C₂²⋊C₄	C₂²×C₄	1	20	yes	—
53	C₄²×C₇	C₂²×C₄	1	30	yes	—
54	C₄²×C₇	C₂⁴	1	10	yes	—
55	C₂×C₇⋊C₈	—	2	10	no	—
56	C₂×C₇⋊C₈	—	4	15	no	—
57	C₂×C₇⋊C₈	C₂⁴	1	5	yes	—
58	C₂×C₇⋊C₈	C₂⁴	1	10	yes	—
59	C₂×C₇⋊C₈	C₂⁴	1	1	yes	characteristic
60	C₂×2₋¹⁺⁴	Dic₇	1	1	yes	characteristic
61	#520	Dic₇	1	10	yes	—
62	#519	Dic₇	1	5	yes	—
63	C₄×#41	C₂×D₇	1	10	yes	—
64	C₂×C₄×Q₈	C₂×D₇	1	5	yes	—
65	C₈×Q₈	—	7	10	no	—
66	C₈×Q₈	—	7	20	no	—
67	C₈×Q₈	—	7	10	no	—
68	C₈×D₄	—	7	10	no	—
69	C₈×D₄	—	7	30	no	—
70	#431	—	7	30	no	—
71	C₂×C₄⋊C₈	—	7	15	no	—
72	C₂×C₄×C₈	—	7	15	no	—
73	C₂³×C₇	C₂³×C₄	1	5	yes	—
74	C₇×Q₈	C₂²×C₈	1	40	yes	—
75	C₇×D₄	C₂²×C₈	1	40	yes	—
76	C₂×C₄×C₇	—	2	5	no	—
77	C₂×C₄×C₇	—	2	5	no	—
78	C₂×C₄×C₇	—	4	15	no	—
79	C₂×C₄×C₇	C₂²×C₈	1	60	yes	—
80	C₂×C₄×C₇	C₂³×C₄	1	5	yes	—
81	C₂×C₄×C₇	C₂³×C₄	1	10	yes	—
82	C₂×C₄×C₇	C₂³×C₄	1	10	yes	—
83	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
84	C₇⋊C₈	—	2	5	no	—
85	C₇⋊C₈	—	2	10	no	—
86	C₇⋊C₈	2₋¹⁺⁴	1	2	yes	direct product factor
87	2₋¹⁺⁴	C₇⋊C₈	1	16	yes	direct product factor
88	C₂×#41	C₂×Dic₇	1	10	yes	—
89	C₂²×Q₈	C₂×Dic₇	1	5	yes	—
90	C₂²×C₈	—	7	5	no	—
91	C₂²×C₈	—	7	15	no	—
92	C₄×Q₈	C₂×Dic₇	1	20	yes	—
93	C₄×Q₈	C₂×Dic₇	1	10	yes	—
94	C₄×Q₈	C₂²×D₇	1	10	yes	—
95	C₄×D₄	C₂×Dic₇	1	30	yes	—
96	C₄×D₄	C₂²×D₇	1	10	yes	—
97	#117	C₂×Dic₇	1	30	yes	—
98	C₂×C₄⋊C₄	C₂×Dic₇	1	15	yes	—
99	C₂×C₄²	C₂²×D₇	1	15	yes	—
100	C₄⋊C₈	—	7	30	no	—
101	C₄⋊C₈	—	7	10	no	—
102	C₂²⋊C₈	—	7	20	no	—
103	C₄×C₈	—	7	10	no	—
104	C₄×C₈	—	7	30	no	—
105	C₂²×C₇	—	2	5	no	—
106	C₂²×C₇	—	2	5	no	—
107	C₂²×C₇	C₂³×C₈	1	10	yes	—
108	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
109	C₄×C₇	—	2	5	no	—
110	C₄×C₇	—	2	10	no	—
111	C₄×C₇	C₂³×C₈	1	20	yes	—
112	C₄×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
113	C₄×C₇	#585	1	1	yes	characteristic
114	#41	C₂×C₇⋊C₈	1	80	yes	complemented
115	C₂×Q₈	C₂×C₇⋊C₈	1	40	yes	complemented
116	C₂×Q₈	C₂²×Dic₇	1	10	yes	—
117	C₂×D₄	C₂²×Dic₇	1	10	yes	—
118	C₂²×C₄	C₂²×Dic₇	1	15	yes	—
119	C₂²×C₄	C₂²×Dic₇	1	15	yes	—
120	C₂²×C₄	C₂³×D₇	1	5	yes	—
121	C₂×C₈	—	7	1	no	—
122	C₂×C₈	—	7	5	no	—
123	C₂×C₈	—	7	10	no	—
124	C₂×C₈	—	14	10	no	—
125	C₂×C₈	—	28	15	no	—
126	C₄⋊C₄	C₂²×Dic₇	1	30	yes	—
127	C₄⋊C₄	C₂²×Dic₇	1	10	yes	—
128	C₂²⋊C₄	C₂²×Dic₇	1	20	yes	—
129	C₄²	C₂²×Dic₇	1	30	yes	—
130	C₄²	C₂³×D₇	1	10	yes	—
131	C₂×C₇	—	2	10	no	—
132	C₂×C₇	C₄×2₋¹⁺⁴	1	1	yes	characteristic
133	C₂×C₇	C₂⁴×C₈	1	1	yes	characteristic
134	C₂×C₇	#3573	1	1	yes	characteristic
135	C₂³	C₂³×Dic₇	1	5	yes	—
136	Q₈	C₂²×C₇⋊C₈	1	40	yes	—
137	D₄	C₂²×C₇⋊C₈	1	40	yes	—
138	C₂×C₄	—	2	5	no	—
139	C₂×C₄	—	2	5	no	—
140	C₂×C₄	—	4	15	no	—
141	C₂×C₄	C₂²×C₇⋊C₈	1	60	yes	—
142	C₂×C₄	C₂³×Dic₇	1	5	yes	—
143	C₂×C₄	C₂³×Dic₇	1	10	yes	—
144	C₂×C₄	C₂³×Dic₇	1	10	yes	—
145	C₂×C₄	C₂⁴×D₇	1	1	yes	characteristic
146	C₈	—	7	2	no	—
147	C₈	—	14	5	no	—
148	C₈	—	14	10	no	—
149	C₇	C₈×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
150	C₂²	—	2	5	no	—
151	C₂²	—	2	5	no	—
152	C₂²	C₂³×C₇⋊C₈	1	10	yes	—
153	C₂²	C₂⁴×Dic₇	1	1	yes	characteristic
154	C₄	—	2	5	no	—
155	C₄	—	2	10	no	—
156	C₄	C₂³×C₇⋊C₈	1	20	yes	—
157	C₄	D₇×2₋¹⁺⁴	1	1	yes	characteristic
158	C₄	#90620	1	1	yes	characteristic
159	C₂	—	2	10	no	—
160	C₂	C₂⁴×C₇⋊C₈	1	1	yes	characteristic
161	C₂	Dic₇×2₋¹⁺⁴	1	1	yes	characteristic
162	C₂	#11744364	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₇
commutator	C₂×C₇	C₂⁴×C₈
Fitting	C₄×C₇×2₋¹⁺⁴	C₂
socle	C₂²×C₇	C₂⁴×C₄
prime radical	C₂×C₇×2₋¹⁺⁴	C₄
cosocle	C₂×C₄×C₇	C₂⁵
hypercenter	C₄×2₋¹⁺⁴	D₇
hypocenter	C₇	C₈×2₋¹⁺⁴
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×2₋¹⁺⁴

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. #11739880	C₂	semidirect
5. C₂×Q₈×C₇⋊C₈	C₂	semidirect
20. C₇×2₋¹⁺⁴	C₈	semidirect
43. C₇×#41	C₂×C₈	semidirect
44. C₂×C₇×Q₈	C₂×C₈	semidirect
86. C₇⋊C₈	2₋¹⁺⁴	direct (N × H)
87. 2₋¹⁺⁴	C₇⋊C₈	direct (N × H)
114. #41	C₂×C₇⋊C₈	semidirect
115. C₂×Q₈	C₂×C₇⋊C₈	semidirect
149. C₇	C₈×2₋¹⁺⁴	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇×2₋¹⁺⁴	2
#11739965	2
#41×C₇⋊C₈	2
#11739880	2
C₂×Q₈×C₇⋊C₈	2
C₈×2₋¹⁺⁴	7

Subgroups — 2−1+4×C7⋊C8 #14219833

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 2₋¹⁺⁴×C₇⋊C₈ #14219833