Subgroups — C₇×#9702 #13186918

Order 1792.

538 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2868	C₂	1	1	yes	maximal, characteristic
2	C₇×#1831	C₂	1	1	yes	maximal, characteristic
3	C₇×#1767	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×#333	C₂	1	4	yes	maximal
5	C₂×C₇×C₄⋊Q₈	C₄	1	4	yes	—
6	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₇×#97	—	2	1	no	—
10	C₇×#384	—	2	1	no	—
11	C₇×#384	—	2	2	no	—
12	C₇×#384	—	2	1	no	—
13	C₄³×C₇	C₂²	1	1	yes	characteristic
14	C₇×#336	—	2	1	no	—
15	C₇×#336	—	2	2	no	—
16	C₇×#333	—	2	8	no	—
17	#9702	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₇×Q₈	C₂×C₄	1	4	yes	—
19	C₂²×C₇×C₈	—	2	2	no	—
20	C₂²×C₇×C₈	—	2	2	no	—
21	C₇×C₄⋊Q₈	—	2	4	no	—
22	C₇×C₄⋊Q₈	—	2	4	no	—
23	C₇×C₄⋊Q₈	—	4	8	no	—
24	C₂×C₇×C₄⋊C₄	—	2	4	no	—
25	C₂×C₇×C₄⋊C₄	—	2	4	no	—
26	C₂×C₇×C₄⋊C₄	—	2	2	no	—
27	C₂×C₇×C₄⋊C₄	—	2	2	no	—
28	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₇	D₄	1	2	yes	—
31	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
33	C₇×C₄⋊C₈	—	2	4	no	—
34	C₇×C₄⋊C₈	—	2	2	no	—
35	C₇×C₄⋊C₈	D₄	1	4	yes	—
36	C₇×#97	—	4	2	no	—
37	#2868	C₂×C₇	1	1	yes	characteristic
38	#1831	C₂×C₇	1	1	yes	characteristic
39	#1767	C₂×C₇	1	1	yes	characteristic
40	C₂×#333	C₂×C₇	1	4	yes	—
41	C₂×C₇×Q₈	—	2	8	no	—
42	C₂×C₇×Q₈	—	4	8	no	—
43	C₂²×C₄×C₇	—	2	4	no	—
44	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₇×C₈	—	2	4	no	—
50	C₂×C₇×C₈	—	4	2	no	—
51	C₂×C₇×C₈	—	4	2	no	—
52	C₂×C₇×C₈	—	4	2	no	—
53	C₂×C₇×C₈	—	4	4	no	—
54	C₇×C₄⋊C₄	—	4	8	no	—
55	C₇×C₄⋊C₄	—	4	8	no	—
56	C₇×C₄⋊C₄	—	4	4	no	—
57	C₇×C₄⋊C₄	—	4	2	no	—
58	C₇×C₄⋊C₄	—	4	2	no	—
59	C₄²×C₇	—	2	4	no	—
60	C₄²×C₇	—	2	2	no	—
61	C₄²×C₇	—	2	4	no	—
62	C₄²×C₇	SD₁₆	1	2	yes	—
63	C₄²×C₇	Q₁₆	1	2	yes	—
64	C₄²×C₇	C₂×D₄	1	2	yes	—
65	C₄²×C₇	#41	1	2	yes	—
66	C₂×C₄⋊Q₈	C₄×C₇	1	4	yes	—
67	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
68	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
69	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
70	C₂×#97	—	2	1	no	—
71	#384	—	2	1	no	—
72	#384	—	2	2	no	—
73	#384	—	2	1	no	—
74	C₄³	C₂²×C₇	1	1	yes	characteristic
75	#336	—	2	2	no	—
76	#336	—	2	1	no	—
77	#333	—	2	8	no	—
78	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₇×Q₈	—	8	8	no	—
80	C₂×C₄×C₇	—	2	2	no	—
81	C₂×C₄×C₇	—	2	2	no	—
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	—	2	2	no	—
84	C₂×C₄×C₇	—	2	2	no	—
85	C₂×C₄×C₇	—	4	4	no	—
86	C₂×C₄×C₇	—	4	8	no	—
87	C₂×C₄×C₇	#100	1	4	yes	—
88	C₂×C₄×C₇	#103	1	4	yes	—
89	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₇	#121	1	2	yes	—
91	C₂×C₄×C₇	#124	1	1	yes	characteristic
92	C₂×C₄×C₇	#127	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×C₄×C₇	#137	1	2	yes	—
96	C₂×C₄×C₇	#137	1	4	yes	—
97	C₇×C₈	—	4	4	no	—
98	C₇×C₈	—	8	2	no	—
99	C₂²×Q₈	C₂×C₄×C₇	1	4	yes	—
100	C₂²×C₈	—	2	2	no	—
101	C₂²×C₈	—	2	2	no	—
102	C₄⋊Q₈	—	2	4	no	—
103	C₄⋊Q₈	—	2	4	no	—
104	C₄⋊Q₈	—	4	8	no	—
105	C₂×C₄⋊C₄	—	2	4	no	—
106	C₂×C₄⋊C₄	—	2	4	no	—
107	C₂×C₄⋊C₄	—	2	2	no	—
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
110	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
111	C₂×C₄²	C₇×D₄	1	2	yes	—
112	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
113	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
114	C₄⋊C₈	—	2	4	no	—
115	C₄⋊C₈	—	2	2	no	—
116	C₄⋊C₈	C₇×D₄	1	4	yes	—
117	#97	—	4	2	no	—
118	C₂²×C₇	#390	1	1	yes	characteristic
119	C₂²×C₇	C₂×#100	1	2	yes	—
120	C₂²×C₇	C₂×#103	1	1	yes	characteristic
121	C₂²×C₇	#419	1	2	yes	—
122	C₂²×C₇	#419	1	1	yes	characteristic
123	C₄×C₇	—	2	4	no	—
124	C₄×C₇	—	2	4	no	—
125	C₄×C₇	—	2	2	no	—
126	C₄×C₇	—	8	4	no	—
127	C₄×C₇	#461	1	2	yes	—
128	C₄×C₇	#462	1	2	yes	—
129	C₄×C₇	#491	1	2	yes	—
130	C₄×C₇	#497	1	2	yes	—
131	C₂×Q₈	—	2	8	no	—
132	C₂×Q₈	—	4	8	no	—
133	C₂²×C₄	—	2	4	no	—
134	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
135	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
136	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	4	4	no	—
144	C₄⋊C₄	—	4	8	no	—
145	C₄⋊C₄	—	4	8	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	2	no	—
149	C₄²	—	2	4	no	—
150	C₄²	—	2	4	no	—
151	C₄²	—	2	2	no	—
152	C₄²	C₇×SD₁₆	1	2	yes	—
153	C₄²	C₇×Q₁₆	1	2	yes	—
154	C₄²	C₂×C₇×D₄	1	2	yes	—
155	C₄²	C₇×#41	1	2	yes	—
156	C₂×C₇	#1548	1	2	yes	—
157	C₂×C₇	#1558	1	2	yes	—
158	C₂×C₇	#1960	1	1	yes	characteristic
159	C₂×C₇	#1965	1	1	yes	characteristic
160	C₂×C₇	#1981	1	1	yes	characteristic
161	C₂³	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
162	Q₈	—	8	8	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	4	8	no	—
170	C₂×C₄	C₇×#100	1	4	yes	—
171	C₂×C₄	C₇×#103	1	4	yes	—
172	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
173	C₂×C₄	C₇×#121	1	2	yes	—
174	C₂×C₄	C₇×#124	1	1	yes	characteristic
175	C₂×C₄	C₇×#127	1	1	yes	characteristic
176	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
177	C₂×C₄	C₂×C₇×Q₁₆	1	1	yes	characteristic
178	C₂×C₄	C₇×#137	1	4	yes	—
179	C₂×C₄	C₇×#137	1	2	yes	—
180	C₈	—	4	4	no	—
181	C₈	—	8	2	no	—
182	C₇	#9702	1	1	yes	characteristic, direct product factor, Sylow
183	C₂²	C₇×#390	1	1	yes	characteristic
184	C₂²	C₂×C₇×#100	1	2	yes	—
185	C₂²	C₂×C₇×#103	1	1	yes	characteristic
186	C₂²	C₇×#419	1	1	yes	characteristic
187	C₂²	C₇×#419	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	4	no	—
191	C₄	—	8	4	no	—
192	C₄	C₇×#461	1	2	yes	—
193	C₄	C₇×#462	1	2	yes	—
194	C₄	C₇×#491	1	2	yes	—
195	C₄	C₇×#497	1	2	yes	—
196	C₂	C₇×#1548	1	2	yes	—
197	C₂	C₇×#1558	1	2	yes	—
198	C₂	C₇×#1960	1	1	yes	characteristic
199	C₂	C₇×#1965	1	1	yes	characteristic
200	C₂	C₇×#1981	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₄³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
17. #9702	C₇	direct (N × H)
182. C₇	#9702	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×#2868	2
C₇×#1831	2
C₇×#1767	2
C₂×C₇×#333	2
#9702	7

Subgroups — C7×#9702 #13186918

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#9702 #13186918