Subgroups — C₇×#34638 #13211854

Order 1792.

1598 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2647	C₂	1	8	yes	maximal, complemented
2	C₇×#2646	C₂	1	8	yes	maximal
3	C₇×#2402	C₂	1	8	yes	maximal, complemented
4	C₇×#2296	C₂	1	4	yes	maximal
5	C₇×#2285	C₂	1	2	yes	maximal
6	C₂×C₇×#376	C₂	1	1	yes	maximal, characteristic
7	C₂×C₇×#117	C₂²	1	4	yes	—
8	C₂²×C₄²×C₇	C₂²	1	2	yes	—
9	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₇×#400	C₂²	1	32	yes	—
11	C₇×#398	C₂²	1	32	yes	—
12	C₇×#389	C₂²	1	16	yes	—
13	C₇×#385	C₂²	1	16	yes	—
14	C₇×#384	C₂²	1	4	yes	—
15	C₇×#382	C₂²	1	4	yes	—
16	C₇×#380	C₂²	1	8	yes	—
17	C₄×C₇×C₄⋊C₄	C₂²	1	8	yes	—
18	C₄×C₇×C₄⋊C₄	C₂²	1	16	yes	—
19	C₇×#376	C₂²	1	4	yes	—
20	C₇×#376	C₂²	1	4	yes	—
21	C₂×C₇×#95	C₂²	1	4	yes	—
22	#34638	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂³×C₄×C₇	C₂³	1	4	yes	—
24	C₂³×C₄×C₇	C₂³	1	2	yes	—
25	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
26	C₇×#117	—	2	16	no	—
27	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
28	C₂×C₇×C₄⋊C₄	C₂³	1	16	yes	—
29	C₂×C₇×C₄⋊C₄	C₂³	1	32	yes	—
30	C₂×C₇×C₂²⋊C₄	C₂³	1	8	yes	—
31	C₂×C₄²×C₇	—	2	8	no	—
32	C₂×C₄²×C₇	Q₈	1	8	yes	—
33	C₂×C₄²×C₇	C₂³	1	4	yes	—
34	C₂×C₄²×C₇	C₂³	1	4	yes	—
35	C₂×C₄²×C₇	C₂³	1	8	yes	—
36	C₂×C₄²×C₇	C₂³	1	2	yes	—
37	C₂×C₄²×C₇	C₂³	1	2	yes	—
38	C₇×#95	C₂³	1	16	yes	—
39	C₇×#95	C₂³	1	16	yes	—
40	C₇×#95	C₂³	1	32	yes	—
41	#2647	C₂×C₇	1	8	yes	complemented
42	#2646	C₂×C₇	1	8	yes	—
43	#2402	C₂×C₇	1	8	yes	complemented
44	#2296	C₂×C₇	1	4	yes	—
45	#2285	C₂×C₇	1	2	yes	—
46	C₂×#376	C₂×C₇	1	1	yes	characteristic
47	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
48	C₂²×C₄×C₇	—	2	8	no	—
49	C₂²×C₄×C₇	—	4	4	no	—
50	C₂²×C₄×C₇	—	4	8	no	—
51	C₂²×C₄×C₇	C₂×Q₈	1	8	yes	—
52	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
53	C₂²×C₄×C₇	#41	1	8	yes	—
54	C₂²×C₄×C₇	C₂⁴	1	8	yes	—
55	C₂²×C₄×C₇	C₂⁴	1	16	yes	—
56	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
57	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
58	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
59	C₇×C₄⋊C₄	—	2	16	no	—
60	C₇×C₄⋊C₄	—	4	16	no	—
61	C₇×C₄⋊C₄	—	4	32	no	—
62	C₇×C₂²⋊C₄	—	2	16	no	—
63	C₄²×C₇	—	2	8	no	—
64	C₄²×C₇	—	2	8	no	—
65	C₄²×C₇	—	2	16	no	—
66	C₄²×C₇	C₂×Q₈	1	8	yes	—
67	C₄²×C₇	C₂×Q₈	1	8	yes	—
68	C₂×#117	C₂²×C₇	1	4	yes	—
69	C₂²×C₄²	C₂²×C₇	1	2	yes	—
70	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
71	#400	C₂²×C₇	1	32	yes	—
72	#398	C₂²×C₇	1	32	yes	—
73	#389	C₂²×C₇	1	16	yes	—
74	#385	C₂²×C₇	1	16	yes	—
75	#384	C₂²×C₇	1	4	yes	—
76	#382	C₂²×C₇	1	4	yes	—
77	#380	C₂²×C₇	1	8	yes	—
78	C₄×C₄⋊C₄	C₂²×C₇	1	8	yes	—
79	C₄×C₄⋊C₄	C₂²×C₇	1	16	yes	—
80	#376	C₂²×C₇	1	4	yes	—
81	#376	C₂²×C₇	1	4	yes	—
82	C₂×#95	C₂²×C₇	1	4	yes	—
83	C₂³×C₇	—	2	4	no	—
84	C₂³×C₇	C₂²×Q₈	1	2	yes	—
85	C₂³×C₇	C₂×#41	1	4	yes	—
86	C₂³×C₇	C₂⁵	1	1	yes	characteristic
87	C₂×C₄×C₇	—	2	8	no	—
88	C₂×C₄×C₇	—	2	8	no	—
89	C₂×C₄×C₇	—	2	16	no	—
90	C₂×C₄×C₇	—	2	32	no	—
91	C₂×C₄×C₇	—	2	16	no	—
92	C₂×C₄×C₇	—	4	8	no	—
93	C₂×C₄×C₇	—	4	16	no	—
94	C₂×C₄×C₇	#125	1	16	yes	—
95	C₂×C₄×C₇	C₂²×Q₈	1	8	yes	—
96	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
97	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
98	C₂×C₄×C₇	C₂×#41	1	4	yes	—
99	C₂×C₄×C₇	C₂×#41	1	4	yes	—
100	C₂³×C₄	C₂³×C₇	1	4	yes	—
101	C₂³×C₄	C₂³×C₇	1	2	yes	—
102	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
103	#117	—	2	16	no	—
104	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
105	C₂×C₄⋊C₄	C₂³×C₇	1	16	yes	—
106	C₂×C₄⋊C₄	C₂³×C₇	1	32	yes	—
107	C₂×C₂²⋊C₄	C₂³×C₇	1	8	yes	—
108	C₂×C₄²	—	2	8	no	—
109	C₂×C₄²	C₇×Q₈	1	8	yes	—
110	C₂×C₄²	C₂³×C₇	1	8	yes	—
111	C₂×C₄²	C₂³×C₇	1	4	yes	—
112	C₂×C₄²	C₂³×C₇	1	4	yes	—
113	C₂×C₄²	C₂³×C₇	1	2	yes	—
114	C₂×C₄²	C₂³×C₇	1	2	yes	—
115	#95	C₂³×C₇	1	16	yes	—
116	#95	C₂³×C₇	1	16	yes	—
117	#95	C₂³×C₇	1	32	yes	—
118	C₂²×C₇	—	2	8	no	—
119	C₂²×C₇	—	2	4	no	—
120	C₂²×C₇	C₂×#125	1	4	yes	—
121	C₂²×C₇	C₂³×Q₈	1	1	yes	characteristic
122	C₂²×C₇	C₂²×#41	1	2	yes	—
123	C₂²×C₇	#585	1	4	yes	—
124	C₄×C₇	—	2	8	no	—
125	C₄×C₇	—	4	16	no	—
126	C₄×C₇	—	4	8	no	—
127	C₄×C₇	C₂×#125	1	4	yes	—
128	C₄×C₇	C₂×#125	1	4	yes	—
129	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
130	C₂²×C₄	—	2	8	no	—
131	C₂²×C₄	—	4	4	no	—
132	C₂²×C₄	—	4	8	no	—
133	C₂²×C₄	C₂×C₇×Q₈	1	8	yes	—
134	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
135	C₂²×C₄	C₇×#41	1	8	yes	—
136	C₂²×C₄	C₂⁴×C₇	1	8	yes	—
137	C₂²×C₄	C₂⁴×C₇	1	16	yes	—
138	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
139	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
140	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
141	C₄⋊C₄	—	2	16	no	—
142	C₄⋊C₄	—	4	16	no	—
143	C₄⋊C₄	—	4	32	no	—
144	C₂²⋊C₄	—	2	16	no	—
145	C₄²	—	2	8	no	—
146	C₄²	—	2	8	no	—
147	C₄²	—	2	16	no	—
148	C₄²	C₂×C₇×Q₈	1	8	yes	—
149	C₄²	C₂×C₇×Q₈	1	8	yes	—
150	C₂×C₇	—	2	4	no	—
151	C₂×C₇	C₂²×#125	1	1	yes	characteristic
152	C₂×C₇	#3455	1	4	yes	—
153	C₂×C₇	#3458	1	2	yes	—
154	C₂³	—	2	4	no	—
155	C₂³	C₂²×C₇×Q₈	1	2	yes	—
156	C₂³	C₂×C₇×#41	1	4	yes	—
157	C₂³	C₂⁵×C₇	1	1	yes	characteristic
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	2	16	no	—
161	C₂×C₄	—	2	32	no	—
162	C₂×C₄	—	2	16	no	—
163	C₂×C₄	—	4	8	no	—
164	C₂×C₄	—	4	16	no	—
165	C₂×C₄	C₇×#125	1	16	yes	—
166	C₂×C₄	C₂²×C₇×Q₈	1	8	yes	—
167	C₂×C₄	C₂²×C₇×Q₈	1	2	yes	—
168	C₂×C₄	C₂²×C₇×Q₈	1	2	yes	—
169	C₂×C₄	C₂×C₇×#41	1	4	yes	—
170	C₂×C₄	C₂×C₇×#41	1	4	yes	—
171	C₇	#34638	1	1	yes	characteristic, direct product factor, Sylow
172	C₂²	—	2	8	no	—
173	C₂²	—	2	4	no	—
174	C₂²	C₂×C₇×#125	1	4	yes	—
175	C₂²	C₂³×C₇×Q₈	1	1	yes	characteristic
176	C₂²	C₂²×C₇×#41	1	2	yes	—
177	C₂²	C₇×#585	1	4	yes	—
178	C₄	—	2	8	no	—
179	C₄	—	4	8	no	—
180	C₄	—	4	16	no	—
181	C₄	C₂×C₇×#125	1	4	yes	—
182	C₄	C₂×C₇×#125	1	4	yes	—
183	C₂	—	2	4	no	—
184	C₂	C₂²×C₇×#125	1	1	yes	characteristic
185	C₂	C₇×#3455	1	4	yes	—
186	C₂	C₇×#3458	1	2	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
1. C₇×#2647	C₂	semidirect
3. C₇×#2402	C₂	semidirect
22. #34638	C₇	direct (N × H)
41. #2647	C₂×C₇	semidirect
43. #2402	C₂×C₇	semidirect
171. C₇	#34638	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×#2647	2
C₇×#2646	2
C₇×#2402	2
C₇×#2296	2
C₇×#2285	2
C₂×C₇×#376	2
#34638	7

Subgroups — C7×#34638 #13211854

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#34638 #13211854