Finite Group Databasebeta

← C₇×#13657 #13190873

Subgroups — C₇×#13657 #13190873

Order 1792.

182 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2264	C₂	1	1	yes	maximal, characteristic
2	C₇×#2179	C₂	1	1	yes	maximal, characteristic
3	C₇×#2176	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1426	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1420	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1418	C₂	1	1	yes	maximal, characteristic
7	C₇×#1398	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#508	C₂²	1	1	yes	characteristic
9	C₂×C₇×Q₃₂	C₂²	1	1	yes	characteristic
10	C₇×#443	—	2	1	no	—
11	C₄×C₇×Q₁₆	—	2	1	no	—
12	C₇×Q₆₄	C₄	1	2	yes	complemented
13	C₇×SD₆₄	C₄	1	2	yes	complemented
14	C₇×M₆₄	C₂²	1	1	yes	characteristic
15	C₇×M₆₄	C₂²	1	1	yes	characteristic
16	C₇×#368	C₂²	1	1	yes	characteristic
17	C₇×#367	C₂²	1	1	yes	characteristic
18	C₇×#359	—	2	1	no	—
19	C₇×#358	—	2	1	no	—
20	C₇×#346	C₂²	1	1	yes	characteristic
21	#13657	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₇×#135	—	2	1	no	—
23	C₂×C₇×Q₁₆	—	2	1	no	—
24	C₇×#131	—	4	1	no	—
25	C₄×C₇×Q₈	—	4	1	no	—
26	C₇×Q₃₂	—	2	1	no	—
27	C₇×Q₃₂	C₂×C₄	1	2	yes	—
28	C₇×Q₃₂	C₂×C₄	1	1	yes	characteristic
29	C₇×SD₃₂	—	2	1	no	—
30	C₇×D₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
32	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
33	C₇×#108	—	2	1	no	—
34	C₇×#107	—	2	1	no	—
35	C₇×#104	—	4	1	no	—
36	C₇×#103	—	4	1	no	—
37	C₄×C₇×C₈	D₄	1	1	yes	characteristic
38	C₇×C₃₂	—	2	1	no	—
39	C₇×C₃₂	C₂×C₄	1	2	yes	—
40	#2264	C₂×C₇	1	1	yes	characteristic
41	#2179	C₂×C₇	1	1	yes	characteristic
42	#2176	C₂×C₇	1	1	yes	characteristic, complemented
43	#1426	C₂×C₇	1	1	yes	characteristic, complemented
44	#1420	C₂×C₇	1	1	yes	characteristic, complemented
45	#1418	C₂×C₇	1	1	yes	characteristic
46	#1398	C₂×C₇	1	1	yes	characteristic, complemented
47	C₇×#41	—	4	1	no	—
48	C₂×C₇×Q₈	—	4	1	no	—
49	C₇×Q₁₆	—	2	1	no	—
50	C₇×Q₁₆	—	2	2	no	—
51	C₇×Q₁₆	—	4	1	no	—
52	C₇×SD₁₆	—	4	1	no	—
53	C₇×D₈	—	2	1	no	—
54	C₇×M₁₆	—	4	1	no	—
55	C₇×M₁₆	—	8	1	no	—
56	C₂×C₇×C₈	—	8	1	no	—
57	C₂×C₇×C₈	D₈	1	1	yes	characteristic
58	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
59	C₇×C₄⋊C₄	—	4	1	no	—
60	C₇×C₄⋊C₄	—	8	1	no	—
61	C₄²×C₇	—	8	1	no	—
62	C₄²×C₇	D₈	1	1	yes	characteristic
63	C₇×C₁₆	—	2	1	no	—
64	C₇×C₁₆	C₂²×C₄	1	1	yes	characteristic
65	C₇×C₁₆	#41	1	1	yes	characteristic
66	#508	C₂²×C₇	1	1	yes	characteristic
67	C₂×Q₃₂	C₂²×C₇	1	1	yes	characteristic
68	#443	—	2	1	no	—
69	C₄×Q₁₆	—	2	1	no	—
70	Q₆₄	C₄×C₇	1	2	yes	complemented
71	SD₆₄	C₄×C₇	1	2	yes	complemented
72	M₆₄	C₂²×C₇	1	1	yes	characteristic
73	M₆₄	C₂²×C₇	1	1	yes	characteristic
74	#368	C₂²×C₇	1	1	yes	characteristic
75	#367	C₂²×C₇	1	1	yes	characteristic
76	#359	—	2	1	no	—
77	#358	—	2	1	no	—
78	#346	C₂²×C₇	1	1	yes	characteristic
79	C₇×Q₈	—	4	1	no	—
80	C₇×Q₈	—	4	2	no	—
81	C₇×Q₈	—	8	1	no	—
82	C₇×D₄	—	4	1	no	—
83	C₇×D₄	—	8	1	no	—
84	C₂×C₄×C₇	—	2	1	no	—
85	C₂×C₄×C₇	—	8	1	no	—
86	C₂×C₄×C₇	—	8	1	no	—
87	C₂×C₄×C₇	—	8	1	no	—
88	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
89	C₇×C₈	—	2	1	no	—
90	C₇×C₈	—	8	1	no	—
91	C₇×C₈	C₄×D₄	1	1	yes	characteristic
92	C₇×C₈	#135	1	1	yes	characteristic
93	#135	—	2	1	no	—
94	C₂×Q₁₆	—	2	1	no	—
95	#131	—	4	1	no	—
96	C₄×Q₈	—	4	1	no	—
97	Q₃₂	—	2	1	no	—
98	Q₃₂	C₂×C₄×C₇	1	2	yes	—
99	Q₃₂	C₂×C₄×C₇	1	1	yes	characteristic
100	SD₃₂	—	2	1	no	—
101	D₁₆	C₂×C₄×C₇	1	1	yes	characteristic
102	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
103	C₂×C₁₆	C₂³×C₇	1	1	yes	characteristic
104	#108	—	2	1	no	—
105	#107	—	2	1	no	—
106	#104	—	4	1	no	—
107	#103	—	4	1	no	—
108	C₄×C₈	C₇×D₄	1	1	yes	characteristic
109	C₃₂	—	2	1	no	—
110	C₃₂	C₂×C₄×C₇	1	2	yes	—
111	C₂²×C₇	—	8	1	no	—
112	C₂²×C₇	#509	1	1	yes	characteristic
113	C₄×C₇	—	4	1	no	—
114	C₄×C₇	—	8	1	no	—
115	C₄×C₇	—	8	2	no	—
116	C₄×C₇	—	16	1	no	—
117	C₄×C₇	C₄×D₈	1	1	yes	characteristic
118	C₄×C₇	#509	1	1	yes	characteristic
119	#41	—	4	1	no	—
120	C₂×Q₈	—	4	1	no	—
121	Q₁₆	—	2	1	no	—
122	Q₁₆	—	2	2	no	—
123	Q₁₆	—	4	1	no	—
124	SD₁₆	—	4	1	no	—
125	D₈	—	2	1	no	—
126	M₁₆	—	4	1	no	—
127	M₁₆	—	8	1	no	—
128	C₂×C₈	—	8	1	no	—
129	C₂×C₈	C₇×D₈	1	1	yes	characteristic
130	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
131	C₄⋊C₄	—	4	1	no	—
132	C₄⋊C₄	—	8	1	no	—
133	C₄²	—	8	1	no	—
134	C₄²	C₇×D₈	1	1	yes	characteristic
135	C₁₆	—	2	1	no	—
136	C₁₆	C₂²×C₄×C₇	1	1	yes	characteristic
137	C₁₆	C₇×#41	1	1	yes	characteristic
138	C₂×C₇	—	2	1	no	—
139	C₂×C₇	—	16	1	no	—
140	C₂×C₇	#2177	1	1	yes	characteristic
141	Q₈	—	4	1	no	—
142	Q₈	—	4	2	no	—
143	Q₈	—	8	1	no	—
144	D₄	—	4	1	no	—
145	D₄	—	8	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	8	1	no	—
148	C₂×C₄	—	8	1	no	—
149	C₂×C₄	—	8	1	no	—
150	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	8	1	no	—
153	C₈	C₄×C₇×D₄	1	1	yes	characteristic
154	C₈	C₇×#135	1	1	yes	characteristic
155	C₇	#13657	1	1	yes	characteristic, direct product factor, Sylow
156	C₂²	—	8	1	no	—
157	C₂²	C₇×#509	1	1	yes	characteristic
158	C₄	—	4	1	no	—
159	C₄	—	8	1	no	—
160	C₄	—	8	2	no	—
161	C₄	—	16	1	no	—
162	C₄	C₄×C₇×D₈	1	1	yes	characteristic
163	C₄	C₇×#509	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	16	1	no	—
166	C₂	C₇×#2177	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	#2177
commutator	C₁₆	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
prime radical	C₇×#508	C₂²
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₄×C₇×C₈ ◁ C₇×#346 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#2176	C₂	semidirect
4. C₇×#1426	C₂	semidirect
5. C₇×#1420	C₂	semidirect
7. C₇×#1398	C₂	semidirect
12. C₇×Q₆₄	C₄	semidirect
13. C₇×SD₆₄	C₄	semidirect
21. #13657	C₇	direct (N × H)
42. #2176	C₂×C₇	semidirect
43. #1426	C₂×C₇	semidirect
44. #1420	C₂×C₇	semidirect
46. #1398	C₂×C₇	semidirect
70. Q₆₄	C₄×C₇	semidirect
71. SD₆₄	C₄×C₇	semidirect
155. C₇	#13657	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#2264	2
C₇×#2179	2
C₇×#2176	2
C₇×#1426	2
C₇×#1420	2
C₇×#1418	2
C₇×#1398	2
#13657	7