Subgroups — C₇×#2568 #11741877

Order 896.

468 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#400	C₂	1	4	yes	maximal
4	C₇×#391	C₂	1	4	yes	maximal
5	C₇×#391	C₂	1	2	yes	maximal
6	C₇×#382	C₂	1	2	yes	maximal
7	C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₇×Q₈	C₂²	1	2	yes	—
9	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
10	C₇×C₄⋊Q₈	—	2	4	no	—
11	C₄×C₇×Q₈	—	2	4	no	—
12	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₄²×C₇	C₂²	1	2	yes	—
18	C₂×C₄²×C₇	C₂²	1	2	yes	—
19	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
20	C₇×#95	C₂²	1	2	yes	—
21	C₇×#95	C₂²	1	8	yes	—
22	C₇×#95	C₂²	1	8	yes	—
23	#2568	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂×C₇×Q₈	—	2	2	no	—
25	C₂×C₇×Q₈	—	2	4	no	—
26	C₂×C₇×Q₈	—	2	2	no	—
27	C₂×C₇×Q₈	—	2	8	no	—
28	C₂×C₇×Q₈	Q₈	1	4	yes	complemented
29	C₂²×C₄×C₇	C₂³	1	4	yes	—
30	C₂²×C₄×C₇	C₂³	1	2	yes	—
31	C₂²×C₄×C₇	C₂³	1	4	yes	—
32	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₇	C₂³	1	2	yes	—
34	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
36	C₇×C₄⋊C₄	—	2	4	no	—
37	C₇×C₄⋊C₄	—	2	2	no	—
38	C₇×C₄⋊C₄	—	2	4	no	—
39	C₇×C₄⋊C₄	—	2	4	no	—
40	C₇×C₄⋊C₄	—	2	4	no	—
41	C₄²×C₇	—	2	4	no	—
42	C₄²×C₇	—	2	4	no	—
43	C₄²×C₇	D₄	1	4	yes	—
44	C₂×C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic
45	C₂×C₄×Q₈	C₂×C₇	1	1	yes	characteristic
46	#400	C₂×C₇	1	4	yes	—
47	#391	C₂×C₇	1	4	yes	—
48	#391	C₂×C₇	1	2	yes	—
49	#382	C₂×C₇	1	2	yes	—
50	C₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
51	C₂³×C₇	C₂⁴	1	1	yes	characteristic
52	C₇×Q₈	—	2	8	no	—
53	C₇×Q₈	—	4	8	no	—
54	C₂×C₄×C₇	—	2	2	no	—
55	C₂×C₄×C₇	—	2	2	no	—
56	C₂×C₄×C₇	—	2	2	no	—
57	C₂×C₄×C₇	—	2	2	no	—
58	C₂×C₄×C₇	—	2	4	no	—
59	C₂×C₄×C₇	—	2	4	no	—
60	C₂×C₄×C₇	—	2	4	no	—
61	C₂×C₄×C₇	—	2	8	no	—
62	C₂×C₄×C₇	—	2	4	no	—
63	C₂×C₄×C₇	—	2	4	no	—
64	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
67	C₂×C₄×C₇	C₂×Q₈	1	4	yes	—
68	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
69	C₂×C₄×C₇	#41	1	2	yes	—
70	C₂×C₄×C₇	#41	1	4	yes	—
71	C₂²×Q₈	C₂²×C₇	1	2	yes	—
72	C₂²×Q₈	C₂²×C₇	1	1	yes	characteristic
73	C₄⋊Q₈	—	2	4	no	—
74	C₄×Q₈	—	2	4	no	—
75	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
76	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
77	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
78	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
79	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
80	C₂×C₄²	C₂²×C₇	1	2	yes	—
81	C₂×C₄²	C₂²×C₇	1	2	yes	—
82	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
83	#95	C₂²×C₇	1	2	yes	—
84	#95	C₂²×C₇	1	8	yes	—
85	#95	C₂²×C₇	1	8	yes	—
86	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
87	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
88	C₂²×C₇	C₂×#41	1	1	yes	characteristic
89	C₂²×C₇	C₂×#41	1	2	yes	—
90	C₂²×C₇	2₋¹⁺⁴	1	2	yes	—
91	C₄×C₇	—	2	4	no	—
92	C₄×C₇	—	2	2	no	—
93	C₄×C₇	—	4	4	no	—
94	C₄×C₇	—	4	2	no	—
95	C₄×C₇	—	4	4	no	—
96	C₄×C₇	C₂²⋊Q₈	1	4	yes	—
97	C₄×C₇	#124	1	4	yes	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	2	4	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	—	2	8	no	—
102	C₂×Q₈	C₇×Q₈	1	4	yes	complemented
103	C₂²×C₄	C₂³×C₇	1	4	yes	—
104	C₂²×C₄	C₂³×C₇	1	2	yes	—
105	C₂²×C₄	C₂³×C₇	1	4	yes	—
106	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
107	C₂²×C₄	C₂³×C₇	1	2	yes	—
108	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
109	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
110	C₄⋊C₄	—	2	4	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	4	no	—
113	C₄⋊C₄	—	2	4	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄²	—	2	4	no	—
116	C₄²	—	2	4	no	—
117	C₄²	C₇×D₄	1	4	yes	—
118	C₂×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₇	C₂×#124	1	1	yes	characteristic
120	C₂×C₇	#536	1	1	yes	characteristic
121	C₂×C₇	#556	1	2	yes	—
122	C₂×C₇	#557	1	2	yes	—
123	C₂³	C₂⁴×C₇	1	1	yes	characteristic
124	Q₈	—	2	8	no	—
125	Q₈	—	4	8	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	8	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
137	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
138	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
139	C₂×C₄	C₂×C₇×Q₈	1	4	yes	—
140	C₂×C₄	C₂×C₇×Q₈	1	2	yes	—
141	C₂×C₄	C₇×#41	1	2	yes	—
142	C₂×C₄	C₇×#41	1	4	yes	—
143	C₇	#2568	1	1	yes	characteristic, direct product factor, Sylow
144	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
145	C₂²	C₂²×C₇×Q₈	1	1	yes	characteristic
146	C₂²	C₂×C₇×#41	1	1	yes	characteristic
147	C₂²	C₂×C₇×#41	1	2	yes	—
148	C₂²	C₇×2₋¹⁺⁴	1	2	yes	—
149	C₄	—	2	4	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	4	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	4	no	—
154	C₄	C₇×C₂²⋊Q₈	1	4	yes	—
155	C₄	C₇×#124	1	4	yes	—
156	C₂	C₂×C₇×C₂²⋊Q₈	1	1	yes	characteristic
157	C₂	C₂×C₇×#124	1	1	yes	characteristic
158	C₂	C₇×#536	1	1	yes	characteristic
159	C₂	C₇×#556	1	2	yes	—
160	C₂	C₇×#557	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
23. #2568	C₇	direct (N × H)
28. C₂×C₇×Q₈	Q₈	semidirect
102. C₂×Q₈	C₇×Q₈	semidirect
143. C₇	#2568	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×C₄⋊Q₈	2
C₂×C₄×C₇×Q₈	2
C₇×#400	2
C₇×#391	2
C₇×#382	2
C₄×C₇×C₄⋊C₄	2
#2568	7

Subgroups — C7×#2568 #11741877

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2568 #11741877