Subgroups — #11743338

Order 896.

890 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#533	C₂	1	1	yes	maximal, characteristic
2	#90412	C₂	1	4	yes	maximal, complemented
3	C₇⋊#564	C₂	1	4	yes	maximal, complemented
4	#90316	C₂	1	8	yes	maximal
5	#90261	C₂	1	4	yes	maximal, complemented
6	#90258	C₂	1	4	yes	maximal, complemented
7	#90209	C₂	1	4	yes	maximal
8	#90161	C₂	1	2	yes	maximal
9	C₇×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₇×#125	C₂²	1	2	yes	—
11	C₇×C₂²⋊Q₈	C₂²	1	4	yes	—
12	C₄×C₇×Q₈	C₂²	1	4	yes	—
13	C₇×#117	C₂²	1	2	yes	—
14	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
15	Q₈×Dic₇	C₂²	1	8	yes	—
16	#6507	C₂²	1	8	yes	—
17	#6489	C₂²	1	4	yes	—
18	#6487	C₂²	1	4	yes	—
19	#6487	C₂²	1	4	yes	—
20	#6453	C₂²	1	16	yes	—
21	#6453	C₂²	1	8	yes	—
22	#6452	C₂²	1	16	yes	—
23	#6452	C₂²	1	16	yes	—
24	#6450	C₂²	1	8	yes	—
25	#6442	C₂²	1	16	yes	—
26	#6433	C₂²	1	4	yes	—
27	#6433	C₂²	1	4	yes	—
28	#6432	C₂²	1	2	yes	—
29	#6432	C₂²	1	2	yes	—
30	#6432	C₂²	1	4	yes	—
31	C₄×Dic₁₄	C₂²	1	4	yes	—
32	C₄×Dic₁₄	C₂²	1	4	yes	—
33	C₄×Dic₁₄	C₂²	1	8	yes	—
34	#3556	—	7	1	no	maximal, Sylow
35	C₂×C₇×Q₈	C₂³	1	4	yes	—
36	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₇	C₂³	1	2	yes	—
38	C₂×Dic₁₄	C₂³	1	4	yes	—
39	C₂×Dic₁₄	C₂³	1	4	yes	—
40	C₇×C₄⋊C₄	C₂³	1	4	yes	—
41	C₇×C₄⋊C₄	C₂³	1	4	yes	—
42	C₇×C₄⋊C₄	C₂³	1	8	yes	—
43	C₇×C₂²⋊C₄	C₂³	1	4	yes	—
44	C₄²×C₇	C₂³	1	1	yes	characteristic
45	C₄²×C₇	C₂³	1	2	yes	—
46	C₄²×C₇	C₂³	1	4	yes	—
47	C₄²×C₇	C₂³	1	1	yes	characteristic
48	C₂²⋊Dic₇	C₂³	1	8	yes	—
49	C₄⋊Dic₇	C₂³	1	4	yes	—
50	C₄⋊Dic₇	C₂³	1	4	yes	—
51	C₄⋊Dic₇	C₂³	1	4	yes	—
52	C₄⋊Dic₇	C₂³	1	4	yes	—
53	C₄⋊Dic₇	C₂³	1	16	yes	—
54	Dic₇⋊C₄	C₂³	1	8	yes	—
55	Dic₇⋊C₄	C₂³	1	8	yes	—
56	Dic₇⋊C₄	C₂³	1	16	yes	—
57	Dic₇⋊C₄	C₂³	1	16	yes	—
58	C₄×Dic₇	C₂³	1	8	yes	—
59	C₄×Dic₇	C₂³	1	16	yes	—
60	#564	—	7	4	no	—
61	#557	—	7	4	no	—
62	#557	—	7	4	no	—
63	#557	—	7	4	no	—
64	#541	—	7	8	no	—
65	#541	—	7	4	no	—
66	#533	—	7	2	no	—
67	#533	D₇	1	1	yes	characteristic
68	C₂³×C₇	C₂⁴	1	1	yes	characteristic
69	C₇×Q₈	—	2	8	no	—
70	C₂×C₄×C₇	—	2	2	no	—
71	C₂×C₄×C₇	—	2	4	no	—
72	C₂×C₄×C₇	C₂⁴	1	8	yes	—
73	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
74	C₂×C₄×C₇	C₂⁴	1	2	yes	—
75	C₂×C₄×C₇	C₂⁴	1	2	yes	—
76	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
77	C₂×Dic₇	C₂⁴	1	8	yes	—
78	C₂×Dic₇	C₂⁴	1	8	yes	—
79	Dic₁₄	—	2	8	no	—
80	Dic₁₄	—	2	8	no	—
81	C₄⋊Q₈	—	7	8	no	—
82	C₄⋊Q₈	—	7	2	no	—
83	C₄⋊Q₈	—	7	2	no	—
84	C₄⋊Q₈	—	7	4	no	—
85	C₄⋊Q₈	C₂×D₇	1	2	yes	—
86	#126	—	7	16	no	—
87	#125	—	7	8	no	—
88	#125	—	7	16	no	—
89	#125	—	7	16	no	—
90	#125	—	7	16	no	—
91	#125	—	7	4	no	—
92	#125	—	7	4	no	—
93	#125	C₂×D₇	1	2	yes	—
94	C₂²⋊Q₈	—	7	4	no	—
95	C₂²⋊Q₈	—	7	4	no	—
96	C₂²⋊Q₈	C₂×D₇	1	4	yes	—
97	C₄×Q₈	—	7	8	no	—
98	C₄×Q₈	—	7	8	no	—
99	C₄×Q₈	—	7	4	no	—
100	C₄×Q₈	—	7	4	no	—
101	C₄×Q₈	—	7	8	no	—
102	C₄×Q₈	C₂×D₇	1	4	yes	—
103	#117	—	7	4	no	—
104	#117	C₂×D₇	1	2	yes	—
105	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
106	C₂²×C₇	—	2	1	no	—
107	C₂²×C₇	—	2	2	no	—
108	C₂²×C₇	C₂⁵	1	1	yes	characteristic
109	C₄×C₇	—	2	1	no	—
110	C₄×C₇	—	2	2	no	—
111	C₄×C₇	—	2	8	no	—
112	C₄×C₇	2₋¹⁺⁴	1	2	yes	—
113	C₄×C₇	2₋¹⁺⁴	1	4	yes	—
114	Dic₇	—	2	8	no	—
115	Dic₇	—	2	8	no	—
116	C₂×Q₈	—	7	4	no	—
117	C₂×Q₈	—	7	4	no	—
118	C₂×Q₈	C₂²×D₇	1	4	yes	—
119	C₂²×C₄	C₂²×D₇	1	2	yes	—
120	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
121	C₄⋊C₄	—	7	8	no	—
122	C₄⋊C₄	—	7	8	no	—
123	C₄⋊C₄	—	7	16	no	—
124	C₄⋊C₄	—	7	16	no	—
125	C₄⋊C₄	—	7	4	no	—
126	C₄⋊C₄	—	7	4	no	—
127	C₄⋊C₄	—	7	4	no	—
128	C₄⋊C₄	—	7	4	no	—
129	C₄⋊C₄	—	7	16	no	—
130	C₄⋊C₄	C₂²×D₇	1	4	yes	—
131	C₄⋊C₄	C₂²×D₇	1	4	yes	—
132	C₄⋊C₄	C₂²×D₇	1	8	yes	—
133	C₂²⋊C₄	—	7	8	no	—
134	C₂²⋊C₄	C₂²×D₇	1	4	yes	—
135	C₄²	—	7	8	no	—
136	C₄²	—	7	16	no	—
137	C₄²	C₂²×D₇	1	1	yes	characteristic
138	C₄²	C₂²×D₇	1	2	yes	—
139	C₄²	C₂²×D₇	1	1	yes	characteristic
140	C₄²	C₂²×D₇	1	4	yes	—
141	C₂×C₇	—	4	1	no	—
142	C₂×C₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
143	C₂×C₇	C₂×2₋¹⁺⁴	1	2	yes	—
144	C₂³	C₂³×D₇	1	1	yes	characteristic
145	Q₈	—	2	8	no	—
146	Q₈	—	14	8	no	—
147	Q₈	—	14	8	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	7	8	no	—
151	C₂×C₄	—	7	8	no	—
152	C₂×C₄	C₂³×D₇	1	8	yes	—
153	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
154	C₂×C₄	C₂³×D₇	1	2	yes	—
155	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
156	C₂×C₄	C₂³×D₇	1	2	yes	—
157	C₇	#3556	1	1	yes	characteristic, complemented, Sylow
158	C₂²	—	2	1	no	—
159	C₂²	—	2	2	no	—
160	C₂²	C₂⁴×D₇	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	2	2	no	—
163	C₄	—	2	8	no	—
164	C₄	—	14	8	no	—
165	C₄	—	14	8	no	—
166	C₄	#6551	1	2	yes	—
167	C₄	#6554	1	4	yes	—
168	C₂	—	4	1	no	—
169	C₂	C₂×#6551	1	1	yes	characteristic
170	C₂	C₂×#6554	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₇
commutator, socle, cosocle	C₂²×C₇	C₂⁵
Fitting	C₇×#533	C₂
prime radical	C₂³×C₇	C₂⁴
hypercenter	#533	D₇
hypocenter	C₇	#3556
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
2. #90412	C₂	semidirect
3. C₇⋊#564	C₂	semidirect
5. #90261	C₂	semidirect
6. #90258	C₂	semidirect
157. C₇	#3556	semidirect

Maximal subgroups

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

subgroup	index
C₇×#533	2
#90412	2
C₇⋊#564	2
#90316	2
#90261	2
#90258	2
#90209	2
#90161	2
#3556	7