Subgroups — C₇×#34642 #13211858

Order 1792.

2110 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#533	C₂	1	2	yes	maximal
2	C₇×#2660	C₂	1	8	yes	maximal, complemented
3	C₇×#2659	C₂	1	8	yes	maximal
4	C₇×#2399	C₂	1	8	yes	maximal, complemented
5	C₇×#2285	C₂	1	4	yes	maximal
6	C₂×C₇×#376	C₂	1	1	yes	maximal, characteristic
7	C₇×#533	—	2	16	no	—
8	C₂×C₇×C₄⋊Q₈	C₂²	1	4	yes	—
9	C₂×C₇×#125	C₂²	1	4	yes	—
10	C₂×C₇×C₂²⋊Q₈	C₂²	1	8	yes	—
11	C₂×C₄×C₇×Q₈	C₂²	1	8	yes	—
12	C₂×C₇×#117	C₂²	1	4	yes	—
13	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₇	C₂²	1	2	yes	—
15	C₇×#398	C₂²	1	32	yes	—
16	C₇×#395	C₂²	1	32	yes	—
17	C₇×#384	C₂²	1	8	yes	—
18	C₇×#382	C₂²	1	8	yes	—
19	C₇×#380	C₂²	1	16	yes	—
20	C₄×C₇×C₄⋊C₄	C₂²	1	16	yes	—
21	C₇×#376	C₂²	1	4	yes	—
22	C₇×#376	C₂²	1	4	yes	—
23	C₂×C₇×#95	C₂²	1	4	yes	—
24	#34642	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂²×C₇×Q₈	C₂³	1	8	yes	—
26	C₂³×C₄×C₇	C₂³	1	2	yes	—
27	C₂³×C₄×C₇	C₂³	1	4	yes	—
28	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
29	C₇×C₄⋊Q₈	—	2	16	no	—
30	C₇×#125	—	2	16	no	—
31	C₇×C₂²⋊Q₈	—	2	32	no	—
32	C₄×C₇×Q₈	—	2	32	no	—
33	C₇×#117	—	2	16	no	—
34	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
35	C₂×C₇×C₄⋊C₄	C₂³	1	16	yes	—
36	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
37	C₂×C₇×C₄⋊C₄	C₂³	1	32	yes	—
38	C₂×C₇×C₂²⋊C₄	C₂³	1	8	yes	—
39	C₂×C₄²×C₇	D₄	1	8	yes	—
40	C₂×C₄²×C₇	Q₈	1	16	yes	complemented
41	C₂×C₄²×C₇	C₂³	1	8	yes	—
42	C₂×C₄²×C₇	C₂³	1	2	yes	—
43	C₂×C₄²×C₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₇	C₂³	1	4	yes	—
45	C₂×C₄²×C₇	C₂³	1	4	yes	—
46	C₇×#95	C₂³	1	16	yes	—
47	C₇×#95	C₂³	1	32	yes	—
48	C₂×#533	C₂×C₇	1	2	yes	—
49	#2660	C₂×C₇	1	8	yes	complemented
50	#2659	C₂×C₇	1	8	yes	—
51	#2399	C₂×C₇	1	8	yes	complemented
52	#2285	C₂×C₇	1	4	yes	—
53	C₂×#376	C₂×C₇	1	1	yes	characteristic
54	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
55	C₂×C₇×Q₈	—	2	16	no	—
56	C₂×C₇×Q₈	—	4	16	no	—
57	C₂²×C₄×C₇	—	4	8	no	—
58	C₂²×C₄×C₇	—	4	4	no	—
59	C₂²×C₄×C₇	C₂×D₄	1	8	yes	—
60	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
61	C₂²×C₄×C₇	C₂×Q₈	1	16	yes	—
62	C₂²×C₄×C₇	C₂×Q₈	1	8	yes	—
63	C₂²×C₄×C₇	C₂⁴	1	16	yes	—
64	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
65	C₂²×C₄×C₇	C₂⁴	1	8	yes	—
66	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
67	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
68	C₇×C₄⋊C₄	—	2	16	no	—
69	C₇×C₄⋊C₄	—	2	32	no	—
70	C₇×C₄⋊C₄	—	2	16	no	—
71	C₇×C₄⋊C₄	—	4	32	no	—
72	C₇×C₂²⋊C₄	—	2	16	no	—
73	C₄²×C₇	—	2	16	no	—
74	C₄²×C₇	C₂×D₄	1	8	yes	—
75	C₄²×C₇	C₂×D₄	1	8	yes	—
76	C₄²×C₇	C₂×Q₈	1	16	yes	—
77	C₄²×C₇	C₂×Q₈	1	16	yes	—
78	#533	—	2	16	no	—
79	C₂×C₄⋊Q₈	C₂²×C₇	1	4	yes	—
80	C₂×#125	C₂²×C₇	1	4	yes	—
81	C₂×C₂²⋊Q₈	C₂²×C₇	1	8	yes	—
82	C₂×C₄×Q₈	C₂²×C₇	1	8	yes	—
83	C₂×#117	C₂²×C₇	1	4	yes	—
84	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
85	C₂²×C₄²	C₂²×C₇	1	2	yes	—
86	#398	C₂²×C₇	1	32	yes	—
87	#395	C₂²×C₇	1	32	yes	—
88	#384	C₂²×C₇	1	8	yes	—
89	#382	C₂²×C₇	1	8	yes	—
90	#380	C₂²×C₇	1	16	yes	—
91	C₄×C₄⋊C₄	C₂²×C₇	1	16	yes	—
92	#376	C₂²×C₇	1	4	yes	—
93	#376	C₂²×C₇	1	4	yes	—
94	C₂×#95	C₂²×C₇	1	4	yes	—
95	C₂³×C₇	—	2	4	no	—
96	C₂³×C₇	C₂²×D₄	1	2	yes	—
97	C₂³×C₇	C₂²×Q₈	1	4	yes	—
98	C₂³×C₇	C₂⁵	1	1	yes	characteristic
99	C₇×Q₈	—	4	32	no	—
100	C₂×C₄×C₇	—	2	16	no	—
101	C₂×C₄×C₇	—	2	8	no	—
102	C₂×C₄×C₇	—	2	16	no	—
103	C₂×C₄×C₇	—	2	32	no	—
104	C₂×C₄×C₇	—	4	8	no	—
105	C₂×C₄×C₇	—	4	16	no	—
106	C₂×C₄×C₇	C₄⋊Q₈	1	16	yes	—
107	C₂×C₄×C₇	C₂²×D₄	1	8	yes	—
108	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
109	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
110	C₂×C₄×C₇	C₂²×Q₈	1	16	yes	—
111	C₂×C₄×C₇	C₂²×Q₈	1	4	yes	—
112	C₂×C₄×C₇	C₂²×Q₈	1	4	yes	—
113	C₂²×Q₈	C₂³×C₇	1	8	yes	—
114	C₂³×C₄	C₂³×C₇	1	2	yes	—
115	C₂³×C₄	C₂³×C₇	1	4	yes	—
116	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
117	C₄⋊Q₈	—	2	16	no	—
118	#125	—	2	16	no	—
119	C₂²⋊Q₈	—	2	32	no	—
120	C₄×Q₈	—	2	32	no	—
121	#117	—	2	16	no	—
122	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
123	C₂×C₄⋊C₄	C₂³×C₇	1	16	yes	—
124	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
125	C₂×C₄⋊C₄	C₂³×C₇	1	32	yes	—
126	C₂×C₂²⋊C₄	C₂³×C₇	1	8	yes	—
127	C₂×C₄²	C₇×D₄	1	8	yes	—
128	C₂×C₄²	C₇×Q₈	1	16	yes	complemented
129	C₂×C₄²	C₂³×C₇	1	8	yes	—
130	C₂×C₄²	C₂³×C₇	1	2	yes	—
131	C₂×C₄²	C₂³×C₇	1	2	yes	—
132	C₂×C₄²	C₂³×C₇	1	4	yes	—
133	C₂×C₄²	C₂³×C₇	1	4	yes	—
134	#95	C₂³×C₇	1	16	yes	—
135	#95	C₂³×C₇	1	32	yes	—
136	C₂²×C₇	—	2	4	no	—
137	C₂²×C₇	—	2	8	no	—
138	C₂²×C₇	C₂×C₄⋊Q₈	1	4	yes	—
139	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
140	C₂²×C₇	C₂³×Q₈	1	2	yes	—
141	C₂²×C₇	#585	1	4	yes	—
142	C₄×C₇	—	2	8	no	—
143	C₄×C₇	—	2	16	no	—
144	C₄×C₇	—	4	16	no	—
145	C₄×C₇	C₂×C₄⋊Q₈	1	4	yes	—
146	C₄×C₇	C₂×C₄⋊Q₈	1	4	yes	—
147	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
148	C₂×Q₈	—	2	16	no	—
149	C₂×Q₈	—	4	16	no	—
150	C₂²×C₄	—	4	8	no	—
151	C₂²×C₄	—	4	4	no	—
152	C₂²×C₄	C₂×C₇×D₄	1	8	yes	—
153	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
154	C₂²×C₄	C₂×C₇×Q₈	1	16	yes	—
155	C₂²×C₄	C₂×C₇×Q₈	1	8	yes	—
156	C₂²×C₄	C₂⁴×C₇	1	16	yes	—
157	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
158	C₂²×C₄	C₂⁴×C₇	1	8	yes	—
159	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
160	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
161	C₄⋊C₄	—	2	16	no	—
162	C₄⋊C₄	—	2	32	no	—
163	C₄⋊C₄	—	2	16	no	—
164	C₄⋊C₄	—	4	32	no	—
165	C₂²⋊C₄	—	2	16	no	—
166	C₄²	—	2	16	no	—
167	C₄²	C₂×C₇×D₄	1	8	yes	—
168	C₄²	C₂×C₇×D₄	1	8	yes	—
169	C₄²	C₂×C₇×Q₈	1	16	yes	—
170	C₄²	C₂×C₇×Q₈	1	16	yes	—
171	C₂×C₇	—	2	4	no	—
172	C₂×C₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
173	C₂×C₇	#3449	1	2	yes	—
174	C₂×C₇	#3458	1	4	yes	—
175	C₂³	—	2	4	no	—
176	C₂³	C₂²×C₇×D₄	1	2	yes	—
177	C₂³	C₂²×C₇×Q₈	1	4	yes	—
178	C₂³	C₂⁵×C₇	1	1	yes	characteristic
179	Q₈	—	4	32	no	—
180	C₂×C₄	—	2	16	no	—
181	C₂×C₄	—	2	8	no	—
182	C₂×C₄	—	2	16	no	—
183	C₂×C₄	—	2	32	no	—
184	C₂×C₄	—	4	8	no	—
185	C₂×C₄	—	4	16	no	—
186	C₂×C₄	C₇×C₄⋊Q₈	1	16	yes	—
187	C₂×C₄	C₂²×C₇×D₄	1	8	yes	—
188	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
189	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
190	C₂×C₄	C₂²×C₇×Q₈	1	16	yes	—
191	C₂×C₄	C₂²×C₇×Q₈	1	4	yes	—
192	C₂×C₄	C₂²×C₇×Q₈	1	4	yes	—
193	C₇	#34642	1	1	yes	characteristic, direct product factor, Sylow
194	C₂²	—	2	4	no	—
195	C₂²	—	2	8	no	—
196	C₂²	C₂×C₇×C₄⋊Q₈	1	4	yes	—
197	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
198	C₂²	C₂³×C₇×Q₈	1	2	yes	—
199	C₂²	C₇×#585	1	4	yes	—
200	C₄	—	2	8	no	—
201	C₄	—	2	16	no	—
202	C₄	—	4	16	no	—
203	C₄	C₂×C₇×C₄⋊Q₈	1	4	yes	—
204	C₄	C₂×C₇×C₄⋊Q₈	1	4	yes	—
205	C₂	—	2	4	no	—
206	C₂	C₂²×C₇×C₄⋊Q₈	1	1	yes	characteristic
207	C₂	C₇×#3449	1	2	yes	—
208	C₂	C₇×#3458	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₇
socle	C₂³×C₇	C₂⁵
prime radical	C₂⁴×C₇	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2660	C₂	semidirect
4. C₇×#2399	C₂	semidirect
24. #34642	C₇	direct (N × H)
40. C₂×C₄²×C₇	Q₈	semidirect
49. #2660	C₂×C₇	semidirect
51. #2399	C₂×C₇	semidirect
128. C₂×C₄²	C₇×Q₈	semidirect
193. C₇	#34642	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#533	2
C₇×#2660	2
C₇×#2659	2
C₇×#2399	2
C₇×#2285	2
C₂×C₇×#376	2
#34642	7

Subgroups — C7×#34642 #13211858

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#34642 #13211858