Subgroups — C₇×#13622 #13190838

Order 1792.

294 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#508	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2158	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#1418	C₂	1	4	yes	maximal, complemented
5	C₂×C₇×#135	—	2	1	no	—
6	C₇×#508	—	2	1	no	—
7	C₇×#508	—	2	1	no	—
8	C₇×#508	C₄	1	2	yes	complemented
9	C₇×#508	C₂²	1	2	yes	—
10	C₂×C₇×Q₃₂	C₄	1	1	yes	characteristic, complemented
11	C₂×C₇×SD₃₂	—	2	1	no	—
12	C₂×C₇×D₁₆	C₄	1	1	yes	characteristic, complemented
13	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
14	C₇×#427	—	2	1	no	—
15	C₇×M₆₄	—	2	1	no	—
16	C₇×M₆₄	C₂²	1	2	yes	complemented
17	C₂×C₇×C₃₂	—	2	1	no	—
18	C₇×#367	C₂²	1	2	yes	complemented
19	C₇×#366	—	2	1	no	—
20	#13622	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂×C₇×#41	—	4	1	no	—
22	C₇×#135	—	2	2	no	—
23	C₇×#135	—	2	2	no	—
24	C₇×#135	—	4	1	no	—
25	C₇×#135	—	4	1	no	—
26	C₂×C₇×Q₁₆	—	2	1	no	—
27	C₂×C₇×SD₁₆	—	4	1	no	—
28	C₂×C₇×D₈	—	2	1	no	—
29	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
30	C₇×#117	—	4	1	no	—
31	C₇×Q₃₂	—	2	1	no	—
32	C₇×Q₃₂	C₂×C₄	1	2	yes	—
33	C₇×SD₃₂	—	2	2	no	—
34	C₇×SD₃₂	—	2	2	no	—
35	C₇×D₁₆	—	2	1	no	—
36	C₇×D₁₆	C₂×C₄	1	2	yes	—
37	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
40	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
41	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
42	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
43	C₇×#107	—	2	2	no	—
44	C₇×#106	—	4	1	no	—
45	C₇×C₃₂	—	2	2	no	—
46	C₂×#508	C₂×C₇	1	1	yes	characteristic
47	C₂×M₆₄	C₂×C₇	1	1	yes	characteristic, complemented
48	#2158	C₂×C₇	1	1	yes	characteristic, complemented
49	#1418	C₂×C₇	1	4	yes	complemented
50	C₇×#41	—	4	2	no	—
51	C₇×#41	—	4	2	no	—
52	C₇×#41	—	8	1	no	—
53	C₇×#41	—	8	1	no	—
54	C₂×C₇×Q₈	—	4	1	no	—
55	C₂×C₇×D₄	—	4	1	no	—
56	C₂×C₇×D₄	—	8	1	no	—
57	C₂²×C₄×C₇	—	8	1	no	—
58	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
59	C₇×Q₁₆	—	2	2	no	—
60	C₇×Q₁₆	—	4	1	no	—
61	C₇×SD₁₆	—	4	2	no	—
62	C₇×SD₁₆	—	4	2	no	—
63	C₇×D₈	—	2	2	no	—
64	C₇×D₈	—	4	1	no	—
65	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₇×C₈	D₈	1	1	yes	characteristic
68	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
69	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
70	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
71	C₇×C₄⋊C₄	—	4	2	no	—
72	C₇×C₂²⋊C₄	—	8	1	no	—
73	C₄²×C₇	—	8	1	no	—
74	C₇×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
75	C₇×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
76	C₇×C₁₆	C₂²×C₄	1	1	yes	characteristic
77	C₇×C₁₆	C₂×D₄	1	1	yes	characteristic
78	C₂×#135	—	2	1	no	—
79	#508	—	2	1	no	—
80	#508	—	2	1	no	—
81	#508	C₄×C₇	1	2	yes	complemented
82	#508	C₂²×C₇	1	2	yes	—
83	C₂×Q₃₂	C₄×C₇	1	1	yes	characteristic, complemented
84	C₂×SD₃₂	—	2	1	no	—
85	C₂×D₁₆	C₄×C₇	1	1	yes	characteristic, complemented
86	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
87	#427	—	2	1	no	—
88	M₆₄	—	2	1	no	—
89	M₆₄	C₂²×C₇	1	2	yes	complemented
90	C₂×C₃₂	—	2	1	no	—
91	#367	C₂²×C₇	1	2	yes	complemented
92	#366	—	2	1	no	—
93	C₂³×C₇	—	8	1	no	—
94	C₂³×C₇	D₁₆	1	1	yes	characteristic
95	C₇×Q₈	—	4	2	no	—
96	C₇×Q₈	—	8	1	no	—
97	C₇×D₄	—	4	2	no	—
98	C₇×D₄	—	8	2	no	—
99	C₇×D₄	—	8	2	no	—
100	C₇×D₄	—	8	1	no	—
101	C₂×C₄×C₇	—	8	1	no	—
102	C₂×C₄×C₇	—	8	2	no	—
103	C₂×C₄×C₇	—	8	2	no	—
104	C₂×C₄×C₇	—	8	2	no	—
105	C₂×C₄×C₇	#102	1	1	yes	characteristic
106	C₂×C₄×C₇	#102	1	1	yes	characteristic
107	C₂×C₄×C₇	D₁₆	1	1	yes	characteristic
108	C₂×C₄×C₇	SD₃₂	1	1	yes	characteristic
109	C₂×C₄×C₇	SD₃₂	1	1	yes	characteristic
110	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
111	C₇×C₈	#102	1	1	yes	characteristic
112	C₇×C₈	#102	1	1	yes	characteristic
113	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₇×C₈	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×#41	—	4	1	no	—
116	#135	—	2	2	no	—
117	#135	—	2	2	no	—
118	#135	—	4	1	no	—
119	#135	—	4	1	no	—
120	C₂×Q₁₆	—	2	1	no	—
121	C₂×SD₁₆	—	4	1	no	—
122	C₂×D₈	—	2	1	no	—
123	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
124	#117	—	4	1	no	—
125	Q₃₂	—	2	1	no	—
126	Q₃₂	C₂×C₄×C₇	1	2	yes	—
127	SD₃₂	—	2	2	no	—
128	SD₃₂	—	2	2	no	—
129	D₁₆	—	2	1	no	—
130	D₁₆	C₂×C₄×C₇	1	2	yes	—
131	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
132	C₂×C₁₆	C₂×C₄×C₇	1	1	yes	characteristic
133	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
134	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
135	C₂×C₁₆	C₇×D₄	1	1	yes	characteristic
136	C₂×C₁₆	C₂³×C₇	1	1	yes	characteristic
137	#107	—	2	2	no	—
138	#106	—	4	1	no	—
139	C₃₂	—	2	2	no	—
140	C₂²×C₇	—	4	1	no	—
141	C₂²×C₇	—	8	2	no	—
142	C₂²×C₇	—	16	2	no	—
143	C₂²×C₇	#357	1	1	yes	characteristic
144	C₂²×C₇	#357	1	1	yes	characteristic
145	C₂²×C₇	C₂×D₁₆	1	1	yes	characteristic
146	C₄×C₇	—	8	2	no	—
147	C₄×C₇	—	16	2	no	—
148	C₄×C₇	#357	1	1	yes	characteristic
149	C₄×C₇	#357	1	1	yes	characteristic
150	C₄×C₇	C₂×#102	1	1	yes	characteristic
151	C₄×C₇	C₂×SD₃₂	1	1	yes	characteristic
152	#41	—	4	2	no	—
153	#41	—	4	2	no	—
154	#41	—	8	1	no	—
155	#41	—	8	1	no	—
156	C₂×Q₈	—	4	1	no	—
157	C₂×D₄	—	4	1	no	—
158	C₂×D₄	—	8	1	no	—
159	C₂²×C₄	—	8	1	no	—
160	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
161	Q₁₆	—	2	2	no	—
162	Q₁₆	—	4	1	no	—
163	SD₁₆	—	4	2	no	—
164	SD₁₆	—	4	2	no	—
165	D₈	—	2	2	no	—
166	D₈	—	4	1	no	—
167	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₈	C₇×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₈	C₇×D₈	1	1	yes	characteristic
170	C₂×C₈	C₇×SD₁₆	1	1	yes	characteristic
171	C₂×C₈	C₇×SD₁₆	1	1	yes	characteristic
172	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
173	C₄⋊C₄	—	4	2	no	—
174	C₂²⋊C₄	—	8	1	no	—
175	C₄²	—	8	1	no	—
176	C₁₆	C₇×C₂²⋊C₄	1	1	yes	characteristic
177	C₁₆	C₇×C₂²⋊C₄	1	1	yes	characteristic
178	C₁₆	C₂²×C₄×C₇	1	1	yes	characteristic
179	C₁₆	C₂×C₇×D₄	1	1	yes	characteristic
180	C₂×C₇	—	2	1	no	—
181	C₂×C₇	—	2	1	no	—
182	C₂×C₇	—	2	1	no	—
183	C₂×C₇	—	16	2	no	—
184	C₂×C₇	C₂×#357	1	1	yes	characteristic
185	C₂³	—	8	1	no	—
186	C₂³	C₇×D₁₆	1	1	yes	characteristic
187	Q₈	—	4	2	no	—
188	Q₈	—	8	1	no	—
189	D₄	—	4	2	no	—
190	D₄	—	8	2	no	—
191	D₄	—	8	2	no	—
192	D₄	—	8	1	no	—
193	C₂×C₄	—	8	1	no	—
194	C₂×C₄	—	8	2	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	C₇×#102	1	1	yes	characteristic
198	C₂×C₄	C₇×#102	1	1	yes	characteristic
199	C₂×C₄	C₇×D₁₆	1	1	yes	characteristic
200	C₂×C₄	C₇×SD₃₂	1	1	yes	characteristic
201	C₂×C₄	C₇×SD₃₂	1	1	yes	characteristic
202	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
203	C₈	C₇×#102	1	1	yes	characteristic
204	C₈	C₇×#102	1	1	yes	characteristic
205	C₈	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
206	C₈	C₂×C₇×SD₁₆	1	1	yes	characteristic
207	C₇	#13622	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²	—	4	1	no	—
209	C₂²	—	8	2	no	—
210	C₂²	—	16	2	no	—
211	C₂²	C₇×#357	1	1	yes	characteristic
212	C₂²	C₇×#357	1	1	yes	characteristic
213	C₂²	C₂×C₇×D₁₆	1	1	yes	characteristic
214	C₄	—	8	2	no	—
215	C₄	—	16	2	no	—
216	C₄	C₇×#357	1	1	yes	characteristic
217	C₄	C₇×#357	1	1	yes	characteristic
218	C₄	C₂×C₇×#102	1	1	yes	characteristic
219	C₄	C₂×C₇×SD₃₂	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	—	2	1	no	—
223	C₂	—	16	2	no	—
224	C₂	C₂×C₇×#357	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇	#357
commutator	C₁₆	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
socle	C₂×C₇	C₂×#357
prime radical	C₂×C₇×#508	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ C₂²×C₇×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×M₆₄	C₂	semidirect
3. C₇×#2158	C₂	semidirect
4. C₇×#1418	C₂	semidirect
8. C₇×#508	C₄	semidirect
10. C₂×C₇×Q₃₂	C₄	semidirect
12. C₂×C₇×D₁₆	C₄	semidirect
16. C₇×M₆₄	C₂²	semidirect
18. C₇×#367	C₂²	semidirect
20. #13622	C₇	direct (N × H)
47. C₂×M₆₄	C₂×C₇	semidirect
48. #2158	C₂×C₇	semidirect
49. #1418	C₂×C₇	semidirect
81. #508	C₄×C₇	semidirect
83. C₂×Q₃₂	C₄×C₇	semidirect
85. C₂×D₁₆	C₄×C₇	semidirect
89. M₆₄	C₂²×C₇	semidirect
91. #367	C₂²×C₇	semidirect
207. C₇	#13622	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×#508	2
C₂×C₇×M₆₄	2
C₇×#2158	2
C₇×#1418	2
#13622	7

Subgroups — C7×#13622 #13190838

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#13622 #13190838