Subgroups — C₇×#20797 #13198013

Order 1792.

1126 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#427	C₂	1	4	yes	maximal
2	C₂²×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#2285	C₂	1	2	yes	maximal
4	C₇×#1846	C₂	1	4	yes	maximal, complemented
5	C₇×#1845	C₂	1	2	yes	maximal, complemented
6	C₇×#1844	C₂	1	2	yes	maximal, complemented
7	C₂³×C₇×C₈	C₂²	1	2	yes	—
8	C₂×C₇×#117	C₂²	1	4	yes	—
9	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₇×#427	—	2	16	no	—
11	C₂×C₇×#107	C₂²	1	8	yes	—
12	C₂×C₇×#106	C₂²	1	8	yes	—
13	C₂×C₄×C₇×C₈	C₄	1	8	yes	complemented
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
17	C₇×#384	C₂²	1	4	yes	—
18	C₇×#382	C₂²	1	4	yes	—
19	C₇×#380	—	2	4	no	—
20	C₄×C₇×C₄⋊C₄	—	2	4	no	—
21	#20797	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂³×C₄×C₇	D₄	1	2	yes	—
23	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₂²×C₇×C₈	C₂×C₄	1	8	yes	—
25	C₂²×C₇×C₈	D₄	1	8	yes	—
26	C₂²×C₇×C₈	Q₈	1	8	yes	—
27	C₂²×C₇×C₈	C₂³	1	2	yes	—
28	C₂²×C₇×C₈	C₂³	1	2	yes	—
29	C₇×#117	—	2	16	no	—
30	C₂×C₇×C₄⋊C₄	—	2	8	no	—
31	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
32	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
33	C₂×C₄²×C₇	—	2	4	no	—
34	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₇	D₄	1	2	yes	—
36	C₂×C₄²×C₇	Q₈	1	4	yes	—
37	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
39	C₇×#107	—	2	16	no	—
40	C₇×#106	—	2	16	no	—
41	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
42	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
43	C₄×C₇×C₈	C₂×C₄	1	8	yes	—
44	C₇×#95	—	2	8	no	—
45	C₂×#427	C₂×C₇	1	4	yes	—
46	C₂²×C₄×C₈	C₂×C₇	1	1	yes	characteristic
47	#2285	C₂×C₇	1	2	yes	—
48	#1846	C₂×C₇	1	4	yes	complemented
49	#1845	C₂×C₇	1	2	yes	complemented
50	#1844	C₂×C₇	1	2	yes	complemented
51	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₇	—	2	8	no	—
53	C₂²×C₄×C₇	—	2	8	no	—
54	C₂²×C₄×C₇	C₄⋊C₄	1	8	yes	—
55	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
58	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
61	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
63	C₂×C₇×C₈	C₄⋊C₄	1	32	yes	complemented
64	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
65	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
66	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
67	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
68	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
69	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
70	C₇×C₄⋊C₄	—	2	16	no	—
71	C₇×C₄⋊C₄	—	4	16	no	—
72	C₇×C₂²⋊C₄	—	4	8	no	—
73	C₄²×C₇	—	4	8	no	—
74	C₄²×C₇	C₄⋊C₄	1	8	yes	—
75	C₄²×C₇	C₂²×C₄	1	2	yes	—
76	C₄²×C₇	C₂²×C₄	1	2	yes	—
77	C₄²×C₇	C₂×Q₈	1	2	yes	—
78	C₄²×C₇	C₂×Q₈	1	2	yes	—
79	C₂³×C₈	C₂²×C₇	1	2	yes	—
80	C₂×#117	C₂²×C₇	1	4	yes	—
81	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
82	#427	—	2	16	no	—
83	C₂×#107	C₂²×C₇	1	8	yes	—
84	C₂×#106	C₂²×C₇	1	8	yes	—
85	C₂×C₄×C₈	C₄×C₇	1	8	yes	complemented
86	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
89	#384	C₂²×C₇	1	4	yes	—
90	#382	C₂²×C₇	1	4	yes	—
91	#380	—	2	4	no	—
92	C₄×C₄⋊C₄	—	2	4	no	—
93	C₂³×C₇	—	2	4	no	—
94	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³×C₇	#127	1	2	yes	—
96	C₂³×C₇	C₄⋊Q₈	1	2	yes	—
97	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₇	—	2	4	no	—
99	C₂×C₄×C₇	—	2	16	no	—
100	C₂×C₄×C₇	—	4	8	no	—
101	C₂×C₄×C₇	—	4	16	no	—
102	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
103	C₂×C₄×C₇	C₂×C₄⋊C₄	1	8	yes	—
104	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
106	C₂×C₄×C₇	#127	1	2	yes	—
107	C₂×C₄×C₇	C₄⋊Q₈	1	8	yes	—
108	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
109	C₂×C₄×C₇	#135	1	8	yes	—
110	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
112	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
113	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
114	C₇×C₈	C₂×C₄⋊C₄	1	8	yes	—
115	C₇×C₈	C₂×C₄⋊C₄	1	8	yes	—
116	C₂³×C₄	C₇×D₄	1	2	yes	—
117	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
118	C₂²×C₈	C₂×C₄×C₇	1	8	yes	—
119	C₂²×C₈	C₇×D₄	1	8	yes	—
120	C₂²×C₈	C₇×Q₈	1	8	yes	—
121	C₂²×C₈	C₂³×C₇	1	2	yes	—
122	C₂²×C₈	C₂³×C₇	1	2	yes	—
123	#117	—	2	16	no	—
124	C₂×C₄⋊C₄	—	2	8	no	—
125	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
126	C₂×C₂²⋊C₄	—	2	4	no	—
127	C₂×C₄²	—	2	4	no	—
128	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
129	C₂×C₄²	C₇×D₄	1	2	yes	—
130	C₂×C₄²	C₇×Q₈	1	4	yes	—
131	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
132	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
133	#107	—	2	16	no	—
134	#106	—	2	16	no	—
135	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
136	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
137	C₄×C₈	C₂×C₄×C₇	1	8	yes	—
138	#95	—	2	8	no	—
139	C₂²×C₇	—	2	4	no	—
140	C₂²×C₇	—	2	4	no	—
141	C₂²×C₇	—	2	4	no	—
142	C₂²×C₇	#384	1	4	yes	—
143	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×C₇	C₂×#127	1	1	yes	characteristic
145	C₂²×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
146	C₂²×C₇	C₂×#135	1	4	yes	—
147	C₄×C₇	—	2	4	no	—
148	C₄×C₇	—	8	8	no	—
149	C₄×C₇	#384	1	4	yes	—
150	C₄×C₇	#427	1	8	yes	complemented
151	C₄×C₇	C₂²×C₄⋊C₄	1	2	yes	—
152	C₄×C₇	C₂×C₄⋊Q₈	1	2	yes	—
153	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
154	C₂²×C₄	—	2	8	no	—
155	C₂²×C₄	—	2	8	no	—
156	C₂²×C₄	C₇×C₄⋊C₄	1	8	yes	—
157	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
158	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
159	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
160	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
161	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
162	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
163	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
164	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
165	C₂×C₈	C₇×C₄⋊C₄	1	32	yes	complemented
166	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
167	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
168	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
169	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
170	C₂×C₈	C₂×C₇×Q₈	1	4	yes	—
171	C₂×C₈	C₂×C₇×Q₈	1	4	yes	—
172	C₄⋊C₄	—	2	16	no	—
173	C₄⋊C₄	—	4	16	no	—
174	C₂²⋊C₄	—	4	8	no	—
175	C₄²	—	4	8	no	—
176	C₄²	C₇×C₄⋊C₄	1	8	yes	—
177	C₄²	C₂²×C₄×C₇	1	2	yes	—
178	C₄²	C₂²×C₄×C₇	1	2	yes	—
179	C₄²	C₂×C₇×Q₈	1	2	yes	—
180	C₄²	C₂×C₇×Q₈	1	2	yes	—
181	C₂×C₇	—	2	4	no	—
182	C₂×C₇	C₂×#384	1	1	yes	characteristic
183	C₂×C₇	C₂×#427	1	2	yes	—
184	C₂×C₇	#3147	1	2	yes	—
185	C₂×C₇	#3160	1	2	yes	—
186	C₂³	—	2	4	no	—
187	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
188	C₂³	C₇×#127	1	2	yes	—
189	C₂³	C₇×C₄⋊Q₈	1	2	yes	—
190	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	16	no	—
193	C₂×C₄	—	4	8	no	—
194	C₂×C₄	—	4	16	no	—
195	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
196	C₂×C₄	C₂×C₇×C₄⋊C₄	1	8	yes	—
197	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
198	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
199	C₂×C₄	C₇×#127	1	2	yes	—
200	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
201	C₂×C₄	C₇×C₄⋊Q₈	1	8	yes	—
202	C₂×C₄	C₇×#135	1	8	yes	—
203	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
204	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
205	C₂×C₄	C₂²×C₇×Q₈	1	2	yes	—
206	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
207	C₈	C₂×C₇×C₄⋊C₄	1	8	yes	—
208	C₈	C₂×C₇×C₄⋊C₄	1	8	yes	—
209	C₇	#20797	1	1	yes	characteristic, direct product factor, Sylow
210	C₂²	—	2	4	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	4	no	—
213	C₂²	C₇×#384	1	4	yes	—
214	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
215	C₂²	C₂×C₇×#127	1	1	yes	characteristic
216	C₂²	C₂×C₇×C₄⋊Q₈	1	1	yes	characteristic
217	C₂²	C₂×C₇×#135	1	4	yes	—
218	C₄	—	2	4	no	—
219	C₄	—	8	8	no	—
220	C₄	C₇×#384	1	4	yes	—
221	C₄	C₇×#427	1	8	yes	complemented
222	C₄	C₂²×C₇×C₄⋊C₄	1	2	yes	—
223	C₄	C₂×C₇×C₄⋊Q₈	1	2	yes	—
224	C₂	—	2	4	no	—
225	C₂	C₂×C₇×#384	1	1	yes	characteristic
226	C₂	C₂×C₇×#427	1	2	yes	—
227	C₂	C₇×#3147	1	2	yes	—
228	C₂	C₇×#3160	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂³×C₇	C₂²×D₄
prime radical	C₂⁴×C₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#1846	C₂	semidirect
5. C₇×#1845	C₂	semidirect
6. C₇×#1844	C₂	semidirect
13. C₂×C₄×C₇×C₈	C₄	semidirect
21. #20797	C₇	direct (N × H)
48. #1846	C₂×C₇	semidirect
49. #1845	C₂×C₇	semidirect
50. #1844	C₂×C₇	semidirect
63. C₂×C₇×C₈	C₄⋊C₄	semidirect
85. C₂×C₄×C₈	C₄×C₇	semidirect
150. C₄×C₇	#427	semidirect
165. C₂×C₈	C₇×C₄⋊C₄	semidirect
209. C₇	#20797	direct (N × H)
221. C₄	C₇×#427	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#427	2
C₂²×C₄×C₇×C₈	2
C₇×#2285	2
C₇×#1846	2
C₇×#1845	2
C₇×#1844	2
#20797	7

Subgroups — C7×#20797 #13198013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#20797 #13198013