Subgroups — C₂×C₃×Dic₇⋊C₈ #13055532

Order 1344.

276 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×Dic₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×Dic₇⋊C₈	C₂	1	4	yes	maximal, direct product factor
5	C₂×Dic₇⋊C₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂²×C₃×Dic₇	C₄	1	1	yes	characteristic
8	C₂²×C₃×Dic₇	C₄	1	1	yes	characteristic
9	C₂×C₃×C₇×C₈	—	2	2	no	—
10	C₂×C₃×C₇×C₈	C₂²	1	2	yes	—
11	C₃×C₄×Dic₇	C₄	1	2	yes	—
12	C₃×C₄×Dic₇	C₂²	1	2	yes	—
13	C₂×C₃×C₇⋊C₈	—	2	2	no	—
14	C₂×C₃×C₇⋊C₈	C₂²	1	2	yes	—
15	C₂²×C₇×C₈	C₂×C₃	1	1	yes	characteristic
16	C₂×C₄×Dic₇	C₂×C₃	1	1	yes	characteristic
17	C₂²×C₇⋊C₈	C₂×C₃	1	1	yes	characteristic
18	Dic₇⋊C₈	C₂×C₃	1	4	yes	direct product factor
19	C₂×C₃×C₄⋊C₈	—	7	1	no	maximal, Hall
20	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
23	C₂×C₃×C₄×C₇	Q₈	1	2	yes	—
24	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₂×C₃×Dic₇	—	2	2	no	—
26	C₂×C₃×Dic₇	C₈	1	4	yes	complemented
27	C₂×C₃×Dic₇	C₂×C₄	1	2	yes	—
28	C₂×C₃×Dic₇	C₂×C₄	1	2	yes	—
29	C₃×C₇×C₈	—	2	2	no	—
30	C₃×C₇⋊C₈	—	2	2	no	—
31	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
32	C₂²×Dic₇	C₃×C₄	1	1	yes	characteristic
33	C₂²×Dic₇	C₃×C₄	1	1	yes	characteristic
34	C₂×C₇×C₈	—	2	2	no	—
35	C₂×C₇×C₈	C₂²×C₃	1	2	yes	—
36	C₄×Dic₇	C₃×C₄	1	2	yes	—
37	C₄×Dic₇	C₂²×C₃	1	2	yes	—
38	C₂×C₇⋊C₈	—	2	2	no	—
39	C₂×C₇⋊C₈	C₂²×C₃	1	2	yes	—
40	C₂²×C₃×C₈	—	7	1	no	—
41	C₂²×C₃×C₈	D₇	1	1	yes	characteristic
42	C₂×C₃×C₄²	—	7	1	no	—
43	C₃×C₄⋊C₈	—	7	4	no	—
44	C₂²×C₃×C₇	C₄⋊C₄	1	2	yes	—
45	C₂²×C₃×C₇	C₂×C₈	1	2	yes	—
46	C₂²×C₃×C₇	M₁₆	1	2	yes	—
47	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
48	C₃×C₄×C₇	C₄⋊C₄	1	2	yes	—
49	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
50	C₃×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
51	C₃×Dic₇	—	2	2	no	—
52	C₃×Dic₇	C₂×C₈	1	4	yes	complemented
53	C₂×C₄⋊C₈	—	7	1	no	Sylow
54	C₂³×C₇	C₂×C₃×C₄	1	1	yes	characteristic
55	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
56	C₂×C₄×C₇	C₃×D₄	1	2	yes	—
57	C₂×C₄×C₇	C₃×Q₈	1	2	yes	—
58	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
59	C₂×Dic₇	—	2	2	no	—
60	C₂×Dic₇	C₃×C₈	1	4	yes	complemented
61	C₂×Dic₇	C₂×C₃×C₄	1	2	yes	—
62	C₂×Dic₇	C₂×C₃×C₄	1	2	yes	—
63	C₇×C₈	—	2	2	no	—
64	C₇⋊C₈	—	2	2	no	—
65	C₂²×C₃×C₄	—	7	1	no	—
66	C₂²×C₃×C₄	—	7	1	no	—
67	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
68	C₂×C₃×C₈	—	2	2	no	—
69	C₂×C₃×C₈	—	7	2	no	—
70	C₂×C₃×C₈	—	14	2	no	—
71	C₂×C₃×C₈	C₂×D₇	1	2	yes	—
72	C₃×C₄²	—	7	2	no	—
73	C₃×C₄²	—	7	2	no	—
74	C₂×C₃×C₇	C₄⋊C₈	1	4	yes	complemented
75	C₂×C₃×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₇	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₃×C₇	C₂×M₁₆	1	1	yes	characteristic
78	C₂²×C₈	—	7	1	no	—
79	C₂²×C₈	C₃×D₇	1	1	yes	characteristic
80	C₂×C₄²	—	7	1	no	—
81	C₄⋊C₈	—	7	4	no	—
82	C₂²×C₇	C₃×C₄⋊C₄	1	2	yes	—
83	C₂²×C₇	C₂×C₃×C₈	1	2	yes	—
84	C₂²×C₇	C₃×M₁₆	1	2	yes	—
85	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
86	C₄×C₇	C₃×C₄⋊C₄	1	2	yes	—
87	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
88	C₄×C₇	C₂×C₃×Q₈	1	1	yes	characteristic
89	Dic₇	—	2	2	no	—
90	Dic₇	C₂×C₃×C₈	1	4	yes	complemented
91	C₂³×C₃	C₄×D₇	1	1	yes	characteristic
92	C₂×C₃×C₄	—	7	4	no	—
93	C₂×C₃×C₄	—	7	2	no	—
94	C₂×C₃×C₄	—	7	2	no	—
95	C₂×C₃×C₄	—	14	2	no	—
96	C₂×C₃×C₄	Dic₁₄	1	2	yes	—
97	C₂×C₃×C₄	C₄×D₇	1	1	yes	characteristic
98	C₂×C₃×C₄	C₂²⋊D₇	1	2	yes	—
99	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
100	C₃×C₈	—	2	2	no	—
101	C₃×C₈	—	14	2	no	—
102	C₃×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Hall
103	C₂²×C₄	—	7	1	no	—
104	C₂²×C₄	—	7	1	no	—
105	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	7	2	no	—
108	C₂×C₈	—	14	2	no	—
109	C₂×C₈	C₂×C₃×D₇	1	2	yes	—
110	C₄²	—	7	2	no	—
111	C₄²	—	7	2	no	—
112	C₂×C₇	C₃×C₄⋊C₈	1	4	yes	complemented
113	C₂×C₇	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₇	C₂²×C₃×C₈	1	1	yes	characteristic
115	C₂×C₇	C₂×C₃×M₁₆	1	1	yes	characteristic
116	C₂²×C₃	C₈×D₇	1	2	yes	—
117	C₂²×C₃	#1131	1	2	yes	—
118	C₂²×C₃	Dic₇⋊C₄	1	2	yes	—
119	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
120	C₃×C₄	—	7	4	no	—
121	C₃×C₄	—	14	2	no	—
122	C₃×C₄	Dic₇⋊C₄	1	2	yes	—
123	C₃×C₄	C₂×Dic₁₄	1	1	yes	characteristic
124	C₃×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
125	C₂³	C₃×C₄×D₇	1	1	yes	characteristic
126	C₂×C₄	—	7	4	no	—
127	C₂×C₄	—	7	2	no	—
128	C₂×C₄	—	7	2	no	—
129	C₂×C₄	—	14	2	no	—
130	C₂×C₄	C₃×Dic₁₄	1	2	yes	—
131	C₂×C₄	C₃×C₄×D₇	1	1	yes	characteristic
132	C₂×C₄	C₃×C₂²⋊D₇	1	2	yes	—
133	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	—	14	2	no	—
136	C₇	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
137	C₂×C₃	Dic₇⋊C₈	1	4	yes	direct product factor
138	C₂×C₃	C₂×C₈×D₇	1	1	yes	characteristic
139	C₂×C₃	C₂×#1131	1	1	yes	characteristic
140	C₂×C₃	C₂×Dic₇⋊C₄	1	1	yes	characteristic
141	C₂²	C₃×C₈×D₇	1	2	yes	—
142	C₂²	C₃×#1131	1	2	yes	—
143	C₂²	C₃×Dic₇⋊C₄	1	2	yes	—
144	C₂²	C₂×C₃×C₄×D₇	1	1	yes	characteristic
145	C₄	—	7	4	no	—
146	C₄	—	14	2	no	—
147	C₄	C₃×Dic₇⋊C₄	1	2	yes	—
148	C₄	C₂×C₃×Dic₁₄	1	1	yes	characteristic
149	C₄	C₂×C₃×C₂²⋊D₇	1	1	yes	characteristic
150	C₃	C₂×Dic₇⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
151	C₂	C₃×Dic₇⋊C₈	1	4	yes	direct product factor
152	C₂	C₂×C₃×C₈×D₇	1	1	yes	characteristic
153	C₂	C₂×C₃×#1131	1	1	yes	characteristic
154	C₂	C₂×C₃×Dic₇⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂×D₇
commutator	C₂×C₇	C₂²×C₃×C₈
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₂²×C₃×C₇×C₈	C₂
socle, prime radical	C₂³×C₃×C₇	C₂×C₄
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	C₂²×C₃×C₈	D₇
hypocenter	C₇	C₂×C₃×C₄⋊C₈
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
4. C₃×Dic₇⋊C₈	C₂	direct (N × H)
5. C₂×Dic₇⋊C₈	C₃	direct (N × H)
18. Dic₇⋊C₈	C₂×C₃	direct (N × H)
26. C₂×C₃×Dic₇	C₈	semidirect
52. C₃×Dic₇	C₂×C₈	semidirect
60. C₂×Dic₇	C₃×C₈	semidirect
74. C₂×C₃×C₇	C₄⋊C₈	semidirect
90. Dic₇	C₂×C₃×C₈	semidirect
102. C₃×C₇	C₂×C₄⋊C₈	semidirect
112. C₂×C₇	C₃×C₄⋊C₈	semidirect
136. C₇	C₂×C₃×C₄⋊C₈	semidirect
137. C₂×C₃	Dic₇⋊C₈	direct (N × H)
150. C₃	C₂×Dic₇⋊C₈	direct (N × H)
151. C₂	C₃×Dic₇⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₇×C₈	2
C₂×C₃×C₄×Dic₇	2
C₂²×C₃×C₇⋊C₈	2
C₃×Dic₇⋊C₈	2
C₂×Dic₇⋊C₈	3
C₂×C₃×C₄⋊C₈	7

Subgroups — C2×C3×Dic7⋊C8 #13055532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×Dic₇⋊C₈ #13055532