Subgroups — C₇×#2552 #11741861

Order 896.

452 conjugacy classes of subgroups.

The subgroup lattice has 190 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	2	yes	maximal
2	C₇×#398	C₂	1	4	yes	maximal
3	C₇×#389	C₂	1	2	yes	maximal
4	C₇×#389	C₂	1	2	yes	maximal
5	C₇×#384	C₂	1	2	yes	maximal
6	C₇×#384	C₂	1	1	yes	maximal, characteristic
7	C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
8	C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₇×#125	—	2	8	no	—
10	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₇×C₄⋊C₄	C₂²	1	4	yes	—
15	C₂×C₇×C₄⋊C₄	C₂²	1	4	yes	—
16	C₂×C₇×C₄⋊C₄	C₂²	1	4	yes	—
17	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
23	C₇×#95	C₂²	1	2	yes	—
24	C₇×#95	C₂²	1	2	yes	—
25	C₇×#95	C₂²	1	4	yes	—
26	#2552	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂²×C₄×C₇	C₂³	1	2	yes	—
28	C₂²×C₄×C₇	C₂³	1	2	yes	—
29	C₂²×C₄×C₇	C₂³	1	4	yes	—
30	C₂²×C₄×C₇	C₂³	1	2	yes	—
31	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
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	C₇×C₄⋊C₄	—	2	8	no	—
36	C₇×C₄⋊C₄	—	2	8	no	—
37	C₇×C₄⋊C₄	—	2	4	no	—
38	C₇×C₄⋊C₄	—	2	2	no	—
39	C₇×C₄⋊C₄	—	2	2	no	—
40	C₇×C₄⋊C₄	—	2	4	no	—
41	C₇×C₄⋊C₄	—	4	4	no	—
42	C₇×C₄⋊C₄	—	4	2	no	—
43	C₇×C₄⋊C₄	D₄	1	4	yes	—
44	C₇×C₄⋊C₄	Q₈	1	4	yes	—
45	C₄²×C₇	—	2	4	no	—
46	C₄²×C₇	—	2	4	no	—
47	C₄²×C₇	Q₈	1	4	yes	—
48	C₂×#125	C₂×C₇	1	2	yes	—
49	#398	C₂×C₇	1	4	yes	—
50	#389	C₂×C₇	1	2	yes	—
51	#389	C₂×C₇	1	2	yes	—
52	#384	C₂×C₇	1	2	yes	—
53	#384	C₂×C₇	1	1	yes	characteristic
54	C₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
55	C₄×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
56	C₂³×C₇	C₂⁴	1	1	yes	characteristic
57	C₂×C₄×C₇	—	2	2	no	—
58	C₂×C₄×C₇	—	2	2	no	—
59	C₂×C₄×C₇	—	2	1	no	—
60	C₂×C₄×C₇	—	2	2	no	—
61	C₂×C₄×C₇	—	2	1	no	—
62	C₂×C₄×C₇	—	2	2	no	—
63	C₂×C₄×C₇	—	2	2	no	—
64	C₂×C₄×C₇	—	2	2	no	—
65	C₂×C₄×C₇	—	2	4	no	—
66	C₂×C₄×C₇	—	2	2	no	—
67	C₂×C₄×C₇	—	2	2	no	—
68	C₂×C₄×C₇	—	2	4	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	C₂×D₄	1	4	yes	—
73	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
75	C₂×C₄×C₇	C₂×Q₈	1	4	yes	—
76	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
77	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
78	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
79	C₂×C₄×C₇	#41	1	2	yes	—
80	C₂×C₄×C₇	#41	1	2	yes	—
81	#125	—	2	8	no	—
82	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
83	C₂×C₄⋊C₄	C₂²×C₇	1	4	yes	—
84	C₂×C₄⋊C₄	C₂²×C₇	1	4	yes	—
85	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
86	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
87	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
88	C₂×C₄⋊C₄	C₂²×C₇	1	4	yes	—
89	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
90	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
91	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
92	C₂×C₄²	C₂²×C₇	1	2	yes	—
93	C₂×C₄²	C₂²×C₇	1	2	yes	—
94	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
95	#95	C₂²×C₇	1	2	yes	—
96	#95	C₂²×C₇	1	2	yes	—
97	#95	C₂²×C₇	1	4	yes	—
98	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
99	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
100	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
101	C₂²×C₇	C₂×#41	1	1	yes	characteristic
102	C₂²×C₇	C₂×#41	1	1	yes	characteristic
103	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
104	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
105	C₄×C₇	—	2	2	no	—
106	C₄×C₇	—	2	4	no	—
107	C₄×C₇	—	2	4	no	—
108	C₄×C₇	—	4	2	no	—
109	C₄×C₇	—	4	2	no	—
110	C₄×C₇	—	4	4	no	—
111	C₄×C₇	#125	1	4	yes	complemented
112	C₄×C₇	C₄⋊Q₈	1	4	yes	—
113	C₂²×C₄	C₂³×C₇	1	2	yes	—
114	C₂²×C₄	C₂³×C₇	1	2	yes	—
115	C₂²×C₄	C₂³×C₇	1	4	yes	—
116	C₂²×C₄	C₂³×C₇	1	2	yes	—
117	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
118	C₂²×C₄	C₂³×C₇	1	2	yes	—
119	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
120	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
121	C₄⋊C₄	—	2	8	no	—
122	C₄⋊C₄	—	2	8	no	—
123	C₄⋊C₄	—	2	4	no	—
124	C₄⋊C₄	—	2	2	no	—
125	C₄⋊C₄	—	2	2	no	—
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	—	4	4	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	C₇×D₄	1	4	yes	—
130	C₄⋊C₄	C₇×Q₈	1	4	yes	—
131	C₄²	—	2	4	no	—
132	C₄²	—	2	4	no	—
133	C₄²	C₇×Q₈	1	4	yes	—
134	C₂×C₇	C₂×#125	1	1	yes	characteristic
135	C₂×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
136	C₂×C₇	#544	1	1	yes	characteristic
137	C₂×C₇	#547	1	1	yes	characteristic
138	C₂×C₇	#550	1	1	yes	characteristic
139	C₂×C₇	#554	1	1	yes	characteristic
140	C₂×C₇	#557	1	1	yes	characteristic
141	C₂³	C₂⁴×C₇	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	C₂×C₇×D₄	1	4	yes	—
158	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
159	C₂×C₄	C₂×C₇×Q₈	1	2	yes	—
160	C₂×C₄	C₂×C₇×Q₈	1	4	yes	—
161	C₂×C₄	C₂×C₇×Q₈	1	2	yes	—
162	C₂×C₄	C₂×C₇×Q₈	1	2	yes	—
163	C₂×C₄	C₂×C₇×Q₈	1	2	yes	—
164	C₂×C₄	C₇×#41	1	2	yes	—
165	C₂×C₄	C₇×#41	1	2	yes	—
166	C₇	#2552	1	1	yes	characteristic, direct product factor, Sylow
167	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
168	C₂²	C₂²×C₇×Q₈	1	1	yes	characteristic
169	C₂²	C₂²×C₇×Q₈	1	1	yes	characteristic
170	C₂²	C₂×C₇×#41	1	1	yes	characteristic
171	C₂²	C₂×C₇×#41	1	1	yes	characteristic
172	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
173	C₂²	C₇×2₋¹⁺⁴	1	1	yes	characteristic
174	C₄	—	2	2	no	—
175	C₄	—	2	4	no	—
176	C₄	—	2	4	no	—
177	C₄	—	4	2	no	—
178	C₄	—	4	2	no	—
179	C₄	—	4	4	no	—
180	C₄	C₇×#125	1	4	yes	complemented
181	C₄	C₇×C₄⋊Q₈	1	4	yes	—
182	C₂	C₂×C₇×#125	1	1	yes	characteristic
183	C₂	C₂×C₇×C₄⋊Q₈	1	1	yes	characteristic
184	C₂	C₇×#544	1	1	yes	characteristic
185	C₂	C₇×#547	1	1	yes	characteristic
186	C₂	C₇×#550	1	1	yes	characteristic
187	C₂	C₇×#554	1	1	yes	characteristic
188	C₂	C₇×#557	1	1	yes	characteristic

	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
26. #2552	C₇	direct (N × H)
111. C₄×C₇	#125	semidirect
166. C₇	#2552	direct (N × H)
180. C₄	C₇×#125	semidirect

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
C₇×#398	2
C₇×#389	2
C₇×#384	2
C₄×C₇×C₄⋊C₄	2
#2552	7

Subgroups — C7×#2552 #11741861

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2552 #11741861