Subgroups — #14219496

Order 1792.

1964 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3483	C₂	1	1	yes	maximal, characteristic
2	#11740723	C₂	1	9	yes	maximal, complemented
3	#11740720	C₂	1	9	yes	maximal, complemented
4	#11739968	C₂	1	6	yes	maximal, complemented
5	#41×C₇⋊C₈	C₂	1	6	yes	maximal, complemented
6	C₇×Q₈²	C₄	1	1	yes	characteristic
7	C₇×#555	C₄	1	9	yes	—
8	C₇×#550	C₄	1	6	yes	—
9	C₇×#532	C₂²	1	9	yes	—
10	C₄×C₇×#41	C₂²	1	6	yes	—
11	C₂×#6511	C₂²	1	6	yes	—
12	#90006	C₂²	1	18	yes	—
13	Q₈×C₇⋊C₈	C₂²	1	2	yes	—
14	Q₈×C₇⋊C₈	C₂²	1	6	yes	—
15	#89784	C₂²	1	36	yes	—
16	D₄×C₇⋊C₈	C₂²	1	18	yes	—
17	D₄×C₇⋊C₈	C₂²	1	18	yes	—
18	#89697	C₂²	1	18	yes	—
19	#89696	C₂²	1	9	yes	—
20	C₂×C₄×C₇⋊C₈	C₂²	1	9	yes	—
21	#60308	—	7	1	no	maximal, Sylow
22	C₂×C₇×#41	C₂×C₄	1	6	yes	—
23	C₂×C₇×#41	C₂³	1	6	yes	—
24	C₇×C₄⋊Q₈	C₂×C₄	1	9	yes	—
25	C₇×#127	C₂×C₄	1	9	yes	—
26	C₇×#124	C₂×C₄	1	18	yes	—
27	C₇×#121	C₂×C₄	1	36	yes	—
28	C₄×C₇×Q₈	C₂×C₄	1	6	yes	—
29	C₄×C₇×Q₈	C₂³	1	2	yes	—
30	C₄×C₇×D₄	C₂×C₄	1	18	yes	—
31	C₄×C₇×D₄	C₂³	1	18	yes	—
32	C₇×#117	C₂×C₄	1	18	yes	—
33	C₂×C₄²×C₇	C₂³	1	9	yes	—
34	#6511	—	4	12	no	—
35	C₂×#1136	C₂³	1	18	yes	—
36	C₂²×C₇⋊C₈	C₂³	1	9	yes	—
37	C₂²×C₇⋊C₈	C₂³	1	9	yes	—
38	C₂²⋊(C₇⋊C₈)	C₂³	1	36	yes	—
39	C₄⋊(C₇⋊C₈)	C₂³	1	18	yes	—
40	C₄⋊(C₇⋊C₈)	C₂³	1	6	yes	—
41	C₄×C₇⋊C₈	C₂³	1	6	yes	—
42	C₄×C₇⋊C₈	C₂³	1	18	yes	—
43	#3483	D₇	1	1	yes	characteristic
44	#2991	—	7	6	no	—
45	#2979	—	7	9	no	—
46	C₈×#41	—	7	6	no	—
47	#2931	—	7	9	no	—
48	C₇×#41	—	2	24	no	—
49	C₇×#41	—	4	6	no	—
50	C₇×#41	—	4	6	no	—
51	C₂×C₇×Q₈	C₂²×C₄	1	6	yes	—
52	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
53	C₂×C₇×D₄	C₂²×C₄	1	18	yes	—
54	C₂×C₇×D₄	C₂²×C₄	1	18	yes	—
55	C₂²×C₄×C₇	C₂²×C₄	1	18	yes	—
56	C₂²×C₄×C₇	C₂⁴	1	9	yes	—
57	C₇×C₄⋊C₄	C₂²×C₄	1	6	yes	—
58	C₇×C₄⋊C₄	C₂²×C₄	1	18	yes	—
59	C₇×C₂²⋊C₄	C₂²×C₄	1	36	yes	—
60	C₄²×C₇	C₂²×C₄	1	9	yes	—
61	C₄²×C₇	C₂²×C₄	1	9	yes	—
62	C₄²×C₇	C₂⁴	1	6	yes	—
63	#1136	—	4	18	no	—
64	C₂×C₇⋊C₈	—	2	18	no	—
65	C₂×C₇⋊C₈	—	4	9	no	—
66	C₂×C₇⋊C₈	C₂⁴	1	6	yes	—
67	C₂×C₇⋊C₈	C₂⁴	1	9	yes	—
68	C₂×C₇⋊C₈	C₂⁴	1	1	yes	characteristic
69	C₂×#131	—	7	6	no	—
70	Q₈²	Dic₇	1	1	yes	characteristic
71	#555	Dic₇	1	9	yes	—
72	#550	Dic₇	1	6	yes	—
73	#532	C₂×D₇	1	9	yes	—
74	C₄×#41	C₂×D₇	1	6	yes	—
75	C₈×Q₈	—	7	2	no	—
76	C₈×Q₈	—	7	6	no	—
77	#436	—	7	36	no	—
78	C₈×D₄	—	7	18	no	—
79	C₈×D₄	—	7	18	no	—
80	#431	—	7	18	no	—
81	#423	—	7	9	no	—
82	#408	—	7	18	no	—
83	C₂×C₄×C₈	—	7	9	no	—
84	C₂³×C₇	C₂³×C₄	1	9	yes	—
85	C₇×Q₈	—	4	6	no	—
86	C₇×Q₈	#131	1	8	yes	complemented
87	C₇×D₄	—	2	36	no	—
88	C₇×D₄	—	4	18	no	—
89	C₂×C₄×C₇	—	2	36	no	—
90	C₂×C₄×C₇	—	2	9	no	—
91	C₂×C₄×C₇	—	2	9	no	—
92	C₂×C₄×C₇	C₂³×C₄	1	6	yes	—
93	C₂×C₄×C₇	C₂³×C₄	1	9	yes	—
94	C₂×C₄×C₇	C₂³×C₄	1	6	yes	—
95	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
96	C₇⋊C₈	—	2	9	no	—
97	C₇⋊C₈	—	2	6	no	—
98	C₇⋊C₈	H(5,𝔽₂)	1	2	yes	—
99	C₂×#41	C₂×Dic₇	1	6	yes	—
100	C₂×#41	C₂²×D₇	1	6	yes	—
101	#131	—	28	12	no	—
102	C₂×M₁₆	—	7	18	no	—
103	C₂²×C₈	—	7	9	no	—
104	C₂²×C₈	—	7	9	no	—
105	C₄⋊Q₈	C₂×Dic₇	1	9	yes	—
106	#127	C₂×Dic₇	1	9	yes	—
107	#124	C₂×Dic₇	1	18	yes	—
108	#121	C₂×Dic₇	1	36	yes	—
109	C₄×Q₈	C₂×Dic₇	1	6	yes	—
110	C₄×Q₈	C₂²×D₇	1	2	yes	—
111	C₄×D₄	C₂×Dic₇	1	18	yes	—
112	C₄×D₄	C₂²×D₇	1	18	yes	—
113	#117	C₂×Dic₇	1	18	yes	—
114	C₂×C₄²	C₂²×D₇	1	9	yes	—
115	C₄⋊C₈	—	7	18	no	—
116	C₄⋊C₈	—	7	6	no	—
117	C₂²⋊C₈	—	7	36	no	—
118	C₄×C₈	—	7	6	no	—
119	C₄×C₈	—	7	18	no	—
120	C₂²×C₇	—	2	9	no	—
121	C₂²×C₇	—	2	18	no	—
122	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
123	C₄×C₇	—	2	9	no	—
124	C₄×C₇	—	2	6	no	—
125	C₄×C₇	C₂×#131	1	12	yes	—
126	C₄×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
127	C₄×C₇	#585	1	1	yes	characteristic
128	#41	—	2	24	no	—
129	#41	—	4	6	no	—
130	#41	—	4	6	no	—
131	C₂×Q₈	C₂²×Dic₇	1	6	yes	—
132	C₂×Q₈	C₂²×Dic₇	1	2	yes	—
133	C₂×D₄	C₂²×Dic₇	1	18	yes	—
134	C₂×D₄	C₂²×Dic₇	1	18	yes	—
135	C₂²×C₄	C₂²×Dic₇	1	18	yes	—
136	C₂²×C₄	C₂³×D₇	1	9	yes	—
137	M₁₆	—	28	18	no	—
138	C₂×C₈	—	7	1	no	—
139	C₂×C₈	—	7	6	no	—
140	C₂×C₈	—	7	9	no	—
141	C₂×C₈	—	14	18	no	—
142	C₂×C₈	—	28	9	no	—
143	C₄⋊C₄	C₂²×Dic₇	1	6	yes	—
144	C₄⋊C₄	C₂²×Dic₇	1	18	yes	—
145	C₂²⋊C₄	C₂²×Dic₇	1	36	yes	—
146	C₄²	C₂²×Dic₇	1	9	yes	—
147	C₄²	C₂²×Dic₇	1	9	yes	—
148	C₄²	C₂³×D₇	1	6	yes	—
149	C₂×C₇	—	4	9	no	—
150	C₂×C₇	C₄×H(5,𝔽₂)	1	1	yes	characteristic
151	C₂×C₇	C₂²×#131	1	2	yes	—
152	C₂³	C₂³×Dic₇	1	9	yes	—
153	Q₈	—	4	6	no	—
154	Q₈	#6511	1	8	yes	complemented
155	D₄	—	2	36	no	—
156	D₄	—	4	18	no	—
157	C₂×C₄	—	2	36	no	—
158	C₂×C₄	—	2	9	no	—
159	C₂×C₄	—	2	9	no	—
160	C₂×C₄	C₂³×Dic₇	1	6	yes	—
161	C₂×C₄	C₂³×Dic₇	1	9	yes	—
162	C₂×C₄	C₂³×Dic₇	1	6	yes	—
163	C₂×C₄	C₂⁴×D₇	1	1	yes	characteristic
164	C₈	—	7	2	no	—
165	C₈	—	14	9	no	—
166	C₈	—	14	6	no	—
167	C₇	#60308	1	1	yes	characteristic, complemented, Sylow
168	C₂²	—	2	18	no	—
169	C₂²	—	2	9	no	—
170	C₂²	C₂⁴×Dic₇	1	1	yes	characteristic
171	C₄	—	2	6	no	—
172	C₄	—	2	9	no	—
173	C₄	C₂×#6511	1	12	yes	—
174	C₄	D₇×H(5,𝔽₂)	1	1	yes	characteristic
175	C₄	#90618	1	1	yes	characteristic
176	C₂	—	4	9	no	—
177	C₂	C₂²×#6511	1	2	yes	—
178	C₂	Dic₇×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₇
commutator, socle	C₂²×C₇	C₂⁴×C₄
Fitting, prime radical	C₇×#3483	C₂
cosocle	C₂×C₄×C₇	C₂⁵
hypercenter	#3483	D₇
hypocenter	C₇	#60308
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #3483

Complements

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

N	H ≅ G/N	type
2. #11740723	C₂	semidirect
3. #11740720	C₂	semidirect
4. #11739968	C₂	semidirect
5. #41×C₇⋊C₈	C₂	semidirect
86. C₇×Q₈	#131	semidirect
154. Q₈	#6511	semidirect
167. C₇	#60308	semidirect

Maximal subgroups

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

subgroup	index
C₇×#3483	2
#11740723	2
#11740720	2
#11739968	2
#41×C₇⋊C₈	2
#60308	7