Subgroups — C₇×#2546 #11741855

Order 896.

540 conjugacy classes of subgroups.

The subgroup lattice has 170 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	2	yes	maximal
2	C₂×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
4	C₇×#398	C₂	1	4	yes	maximal
5	C₇×#391	C₂	1	2	yes	maximal
6	C₇×#389	C₂	1	4	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	8	no	—
11	C₇×C₄⋊Q₈	—	2	4	no	—
12	C₄×C₇×Q₈	—	2	4	no	—
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	8	yes	—
17	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₄²×C₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
22	C₇×#95	C₂²	1	2	yes	—
23	C₇×#95	C₂²	1	8	yes	—
24	#2546	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂×C₇×Q₈	—	2	2	no	—
26	C₂×C₇×Q₈	—	2	4	no	—
27	C₂×C₇×Q₈	—	2	4	no	—
28	C₂×C₇×Q₈	—	2	2	no	—
29	C₂×C₇×Q₈	—	2	4	no	—
30	C₂×C₇×Q₈	D₄	1	4	yes	—
31	C₂²×C₄×C₇	C₂³	1	2	yes	—
32	C₂²×C₄×C₇	C₂³	1	4	yes	—
33	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₇	C₂³	1	2	yes	—
35	C₂²×C₄×C₇	C₂³	1	4	yes	—
36	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
38	C₇×C₄⋊C₄	—	2	4	no	—
39	C₇×C₄⋊C₄	—	2	2	no	—
40	C₇×C₄⋊C₄	—	2	16	no	—
41	C₇×C₄⋊C₄	—	2	4	no	—
42	C₇×C₄⋊C₄	—	2	4	no	—
43	C₇×C₄⋊C₄	Q₈	1	8	yes	complemented
44	C₄²×C₇	—	2	4	no	—
45	C₄²×C₇	—	2	4	no	—
46	C₄²×C₇	D₄	1	4	yes	—
47	C₂×C₄⋊Q₈	C₂×C₇	1	2	yes	—
48	C₂×C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic
49	C₂×C₄×Q₈	C₂×C₇	1	1	yes	characteristic
50	#398	C₂×C₇	1	4	yes	—
51	#391	C₂×C₇	1	2	yes	—
52	#389	C₂×C₇	1	4	yes	—
53	C₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
54	C₂³×C₇	C₂⁴	1	1	yes	characteristic
55	C₇×Q₈	—	2	8	no	—
56	C₇×Q₈	—	4	8	no	—
57	C₂×C₄×C₇	—	2	4	no	—
58	C₂×C₄×C₇	—	2	2	no	—
59	C₂×C₄×C₇	—	2	2	no	—
60	C₂×C₄×C₇	—	2	2	no	—
61	C₂×C₄×C₇	—	2	4	no	—
62	C₂×C₄×C₇	—	2	2	no	—
63	C₂×C₄×C₇	—	2	4	no	—
64	C₂×C₄×C₇	—	2	4	no	—
65	C₂×C₄×C₇	—	2	4	no	—
66	C₂×C₄×C₇	—	2	4	no	—
67	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₇	C₂×D₄	1	4	yes	—
69	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₇	C₂×Q₈	1	8	yes	—
73	C₂×C₄×C₇	C₂×Q₈	1	4	yes	—
74	C₂×C₄×C₇	#41	1	2	yes	—
75	C₂²×Q₈	C₂²×C₇	1	2	yes	—
76	C₂²×Q₈	C₂²×C₇	1	1	yes	characteristic
77	C₄⋊Q₈	—	2	8	no	—
78	C₄⋊Q₈	—	2	4	no	—
79	C₄×Q₈	—	2	4	no	—
80	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
81	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
82	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
83	C₂×C₄⋊C₄	C₂²×C₇	1	8	yes	—
84	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
85	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
86	C₂×C₄²	C₂²×C₇	1	2	yes	—
87	C₂×C₄²	C₂²×C₇	1	2	yes	—
88	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
89	#95	C₂²×C₇	1	2	yes	—
90	#95	C₂²×C₇	1	8	yes	—
91	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
92	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
93	C₂²×C₇	C₂²×Q₈	1	2	yes	—
94	C₂²×C₇	C₂×#41	1	1	yes	characteristic
95	C₂²×C₇	2₋¹⁺⁴	1	2	yes	—
96	C₄×C₇	—	2	4	no	—
97	C₄×C₇	—	2	2	no	—
98	C₄×C₇	—	2	8	no	—
99	C₄×C₇	—	4	4	no	—
100	C₄×C₇	—	4	2	no	—
101	C₄×C₇	#121	1	4	yes	—
102	C₄×C₇	C₄⋊Q₈	1	4	yes	complemented
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	4	no	—
105	C₂×Q₈	—	2	4	no	—
106	C₂×Q₈	—	2	2	no	—
107	C₂×Q₈	—	2	4	no	—
108	C₂×Q₈	C₇×D₄	1	4	yes	—
109	C₂²×C₄	C₂³×C₇	1	2	yes	—
110	C₂²×C₄	C₂³×C₇	1	4	yes	—
111	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
112	C₂²×C₄	C₂³×C₇	1	2	yes	—
113	C₂²×C₄	C₂³×C₇	1	4	yes	—
114	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
115	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	2	16	no	—
119	C₄⋊C₄	—	2	4	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	C₇×Q₈	1	8	yes	complemented
122	C₄²	—	2	4	no	—
123	C₄²	—	2	4	no	—
124	C₄²	C₇×D₄	1	4	yes	—
125	C₂×C₇	C₂×#121	1	1	yes	characteristic
126	C₂×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
127	C₂×C₇	#536	1	1	yes	characteristic
128	C₂×C₇	D₄×Q₈	1	2	yes	—
129	C₂×C₇	#557	1	2	yes	—
130	C₂³	C₂⁴×C₇	1	1	yes	characteristic
131	Q₈	—	2	8	no	—
132	Q₈	—	4	8	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
144	C₂×C₄	C₂×C₇×D₄	1	4	yes	—
145	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
146	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
147	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
148	C₂×C₄	C₂×C₇×Q₈	1	8	yes	—
149	C₂×C₄	C₂×C₇×Q₈	1	4	yes	—
150	C₂×C₄	C₇×#41	1	2	yes	—
151	C₇	#2546	1	1	yes	characteristic, direct product factor, Sylow
152	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
153	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
154	C₂²	C₂²×C₇×Q₈	1	2	yes	—
155	C₂²	C₂×C₇×#41	1	1	yes	characteristic
156	C₂²	C₇×2₋¹⁺⁴	1	2	yes	—
157	C₄	—	2	4	no	—
158	C₄	—	2	2	no	—
159	C₄	—	2	8	no	—
160	C₄	—	4	4	no	—
161	C₄	—	4	2	no	—
162	C₄	C₇×#121	1	4	yes	—
163	C₄	C₇×C₄⋊Q₈	1	4	yes	complemented
164	C₂	C₂×C₇×#121	1	1	yes	characteristic
165	C₂	C₂×C₇×C₄⋊Q₈	1	1	yes	characteristic
166	C₂	C₇×#536	1	1	yes	characteristic
167	C₂	C₇×D₄×Q₈	1	2	yes	—
168	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
24. #2546	C₇	direct (N × H)
43. C₇×C₄⋊C₄	Q₈	semidirect
102. C₄×C₇	C₄⋊Q₈	semidirect
121. C₄⋊C₄	C₇×Q₈	semidirect
151. C₇	#2546	direct (N × H)
163. C₄	C₇×C₄⋊Q₈	semidirect

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×#398	2
C₇×#391	2
C₇×#389	2
C₄×C₇×C₄⋊C₄	2
#2546	7

Subgroups — C7×#2546 #11741855

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2546 #11741855