Subgroups — C₇×#2584 #11741893

Order 896.

468 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#398	C₂	1	2	yes	maximal
4	C₇×#395	C₂	1	2	yes	maximal
5	C₇×#389	C₂	1	2	yes	maximal
6	C₇×#389	C₂	1	4	yes	maximal
7	C₇×#389	C₂	1	2	yes	maximal
8	C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
10	C₇×#125	—	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	1	yes	characteristic
16	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₇×C₄⋊C₄	C₂²	1	4	yes	—
18	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₇×C₄⋊C₄	C₂²	1	4	yes	—
20	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₄²×C₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₇	C₂²	1	2	yes	—
23	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
24	C₇×#95	C₂²	1	2	yes	—
25	C₇×#95	C₂²	1	4	yes	—
26	C₇×#95	C₂²	1	4	yes	—
27	#2584	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×C₇×Q₈	—	2	2	no	—
29	C₂×C₇×Q₈	—	2	4	no	—
30	C₂²×C₄×C₇	C₂³	1	2	yes	—
31	C₂²×C₄×C₇	C₂³	1	2	yes	—
32	C₂²×C₄×C₇	C₂³	1	2	yes	—
33	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₇	C₂³	1	4	yes	—
36	C₂²×C₄×C₇	C₂³	1	2	yes	—
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	4	no	—
41	C₇×C₄⋊C₄	—	2	4	no	—
42	C₇×C₄⋊C₄	—	2	2	no	—
43	C₇×C₄⋊C₄	—	2	2	no	—
44	C₇×C₄⋊C₄	—	2	8	no	—
45	C₇×C₄⋊C₄	—	4	4	no	—
46	C₇×C₄⋊C₄	Q₈	1	8	yes	complemented
47	C₄²×C₇	—	2	4	no	—
48	C₄²×C₇	—	4	2	no	—
49	C₄²×C₇	D₄	1	4	yes	—
50	C₂×#125	C₂×C₇	1	1	yes	characteristic
51	C₂×C₄×Q₈	C₂×C₇	1	1	yes	characteristic
52	#398	C₂×C₇	1	2	yes	—
53	#395	C₂×C₇	1	2	yes	—
54	#389	C₂×C₇	1	2	yes	—
55	#389	C₂×C₇	1	4	yes	—
56	#389	C₂×C₇	1	2	yes	—
57	C₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
58	C₂³×C₇	C₂⁴	1	1	yes	characteristic
59	C₇×Q₈	—	2	8	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	2	no	—
64	C₂×C₄×C₇	—	2	1	no	—
65	C₂×C₄×C₇	—	2	4	no	—
66	C₂×C₄×C₇	—	2	1	no	—
67	C₂×C₄×C₇	—	2	2	no	—
68	C₂×C₄×C₇	—	2	2	no	—
69	C₂×C₄×C₇	—	2	2	no	—
70	C₂×C₄×C₇	—	2	4	no	—
71	C₂×C₄×C₇	—	2	2	no	—
72	C₂×C₄×C₇	—	2	2	no	—
73	C₂×C₄×C₇	—	2	2	no	—
74	C₂×C₄×C₇	—	2	2	no	—
75	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₇	C₂×Q₈	1	8	yes	—
79	C₂×C₄×C₇	C₂×Q₈	1	4	yes	—
80	C₂×C₄×C₇	#41	1	4	yes	—
81	C₂²×Q₈	C₂²×C₇	1	1	yes	characteristic
82	#125	—	2	4	no	—
83	C₄×Q₈	—	2	4	no	—
84	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
85	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
86	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
87	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
88	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
89	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
90	C₂×C₄⋊C₄	C₂²×C₇	1	4	yes	—
91	C₂×C₄⋊C₄	C₂²×C₇	1	4	yes	—
92	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
93	C₂×C₄²	C₂²×C₇	1	2	yes	—
94	C₂×C₄²	C₂²×C₇	1	2	yes	—
95	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
96	#95	C₂²×C₇	1	2	yes	—
97	#95	C₂²×C₇	1	4	yes	—
98	#95	C₂²×C₇	1	4	yes	—
99	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
100	C₂²×C₇	C₂²×Q₈	1	2	yes	—
101	C₂²×C₇	C₂×#41	1	2	yes	—
102	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
103	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
104	C₄×C₇	—	2	2	no	—
105	C₄×C₇	—	2	4	no	—
106	C₄×C₇	—	2	2	no	—
107	C₄×C₇	—	2	8	no	—
108	C₄×C₇	—	4	2	no	—
109	C₄×C₇	—	4	2	no	—
110	C₄×C₇	—	4	2	no	—
111	C₄×C₇	C₄⋊Q₈	1	4	yes	—
112	C₂×Q₈	—	2	2	no	—
113	C₂×Q₈	—	2	4	no	—
114	C₂²×C₄	C₂³×C₇	1	2	yes	—
115	C₂²×C₄	C₂³×C₇	1	2	yes	—
116	C₂²×C₄	C₂³×C₇	1	2	yes	—
117	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
118	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
119	C₂²×C₄	C₂³×C₇	1	4	yes	—
120	C₂²×C₄	C₂³×C₇	1	2	yes	—
121	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
122	C₄⋊C₄	—	2	4	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	2	4	no	—
125	C₄⋊C₄	—	2	4	no	—
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	—	2	8	no	—
129	C₄⋊C₄	—	4	4	no	—
130	C₄⋊C₄	C₇×Q₈	1	8	yes	complemented
131	C₄²	—	2	4	no	—
132	C₄²	—	4	2	no	—
133	C₄²	C₇×D₄	1	4	yes	—
134	C₂×C₇	#525	1	1	yes	characteristic
135	C₂×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
136	C₂×C₇	#537	1	1	yes	characteristic
137	C₂×C₇	#554	1	2	yes	—
138	C₂×C₇	#557	1	2	yes	—
139	C₂³	C₂⁴×C₇	1	1	yes	characteristic
140	Q₈	—	2	8	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
157	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
158	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
159	C₂×C₄	C₂×C₇×Q₈	1	8	yes	—
160	C₂×C₄	C₂×C₇×Q₈	1	4	yes	—
161	C₂×C₄	C₇×#41	1	4	yes	—
162	C₇	#2584	1	1	yes	characteristic, direct product factor, Sylow
163	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
164	C₂²	C₂²×C₇×Q₈	1	2	yes	—
165	C₂²	C₂×C₇×#41	1	2	yes	—
166	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
167	C₂²	C₇×2₋¹⁺⁴	1	1	yes	characteristic
168	C₄	—	2	2	no	—
169	C₄	—	2	4	no	—
170	C₄	—	2	2	no	—
171	C₄	—	2	8	no	—
172	C₄	—	4	2	no	—
173	C₄	—	4	2	no	—
174	C₄	—	4	2	no	—
175	C₄	C₇×C₄⋊Q₈	1	4	yes	—
176	C₂	C₇×#525	1	1	yes	characteristic
177	C₂	C₂×C₇×C₄⋊Q₈	1	1	yes	characteristic
178	C₂	C₇×#537	1	1	yes	characteristic
179	C₂	C₇×#554	1	2	yes	—
180	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
27. #2584	C₇	direct (N × H)
46. C₇×C₄⋊C₄	Q₈	semidirect
130. C₄⋊C₄	C₇×Q₈	semidirect
162. C₇	#2584	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×#125	2
C₂×C₄×C₇×Q₈	2
C₇×#398	2
C₇×#395	2
C₇×#389	2
C₄×C₇×C₄⋊C₄	2
#2584	7

Subgroups — C7×#2584 #11741893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2584 #11741893