Subgroups — C₇×#20765 #13197981

Order 1792.

926 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₇×#2285	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#382	C₂	1	1	yes	maximal, characteristic
4	C₇×#1836	C₂	1	4	yes	maximal, complemented
5	C₇×#1835	C₂	1	4	yes	maximal, complemented
6	C₇×#1736	C₂	1	4	yes	maximal
7	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
8	C₂×C₇×#117	C₂²	1	2	yes	—
9	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₄×C₇×M₁₆	C₄	1	8	yes	complemented
12	C₂×C₇×#97	C₂²	1	2	yes	—
13	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
14	C₇×#384	C₂²	1	2	yes	—
15	C₇×#382	—	2	2	no	—
16	C₇×#382	C₂²	1	2	yes	—
17	C₇×#382	C₂²	1	2	yes	—
18	C₇×#382	C₂²	1	2	yes	—
19	C₇×#380	—	2	2	no	—
20	C₄×C₇×C₄⋊C₄	—	2	2	no	—
21	C₂×C₇×#95	—	2	2	no	—
22	C₇×#340	C₂²	1	16	yes	—
23	#20765	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₇	—	2	2	no	—
25	C₂³×C₄×C₇	D₄	1	2	yes	—
26	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
27	C₂×C₇×M₁₆	—	2	4	no	—
28	C₂×C₇×M₁₆	—	2	4	no	—
29	C₂×C₇×M₁₆	C₂×C₄	1	8	yes	—
30	C₂²×C₇×C₈	C₂³	1	4	yes	—
31	C₇×#117	—	2	8	no	—
32	C₂×C₇×C₄⋊C₄	—	2	8	no	—
33	C₂×C₇×C₄⋊C₄	—	2	2	no	—
34	C₂×C₇×C₄⋊C₄	—	2	4	no	—
35	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
36	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
37	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
38	C₂×C₄²×C₇	—	2	2	no	—
39	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₇	D₄	1	2	yes	—
41	C₂×C₄²×C₇	Q₈	1	4	yes	—
42	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
44	C₇×#97	C₂×C₄	1	8	yes	complemented
45	C₄×C₇×C₈	C₂×C₄	1	8	yes	complemented
46	C₇×#95	—	2	4	no	—
47	C₇×#95	—	2	4	no	—
48	C₇×#95	—	2	4	no	—
49	C₂×C₄×M₁₆	C₂×C₇	1	1	yes	characteristic
50	#2285	C₂×C₇	1	1	yes	characteristic
51	C₂×#382	C₂×C₇	1	1	yes	characteristic
52	#1836	C₂×C₇	1	4	yes	complemented
53	#1835	C₂×C₇	1	4	yes	complemented
54	#1736	C₂×C₇	1	4	yes	—
55	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₇	—	2	4	no	—
57	C₂²×C₄×C₇	—	2	4	no	—
58	C₂²×C₄×C₇	—	4	4	no	—
59	C₂²×C₄×C₇	—	4	8	no	—
60	C₂²×C₄×C₇	—	4	4	no	—
61	C₂²×C₄×C₇	C₄⋊C₄	1	8	yes	—
62	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₇	#41	1	4	yes	—
68	C₂²×C₄×C₇	#41	1	4	yes	—
69	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
70	C₇×M₁₆	—	2	16	no	—
71	C₂×C₇×C₈	—	2	4	no	—
72	C₂×C₇×C₈	—	2	4	no	—
73	C₂×C₇×C₈	C₂²×C₄	1	8	yes	—
74	C₇×C₄⋊C₄	—	2	8	no	—
75	C₇×C₄⋊C₄	—	2	8	no	—
76	C₇×C₄⋊C₄	—	4	8	no	—
77	C₇×C₄⋊C₄	—	4	4	no	—
78	C₇×C₂²⋊C₄	—	4	4	no	—
79	C₄²×C₇	—	4	4	no	—
80	C₄²×C₇	C₄⋊C₄	1	8	yes	—
81	C₄²×C₇	C₂²×C₄	1	2	yes	—
82	C₄²×C₇	C₂²×C₄	1	2	yes	—
83	C₄²×C₇	C₂×Q₈	1	2	yes	—
84	C₄²×C₇	C₂×Q₈	1	2	yes	—
85	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
86	C₂×#117	C₂²×C₇	1	2	yes	—
87	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
88	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
89	C₄×M₁₆	C₄×C₇	1	8	yes	complemented
90	C₂×#97	C₂²×C₇	1	2	yes	—
91	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
92	#384	C₂²×C₇	1	2	yes	—
93	#382	—	2	2	no	—
94	#382	C₂²×C₇	1	2	yes	—
95	#382	C₂²×C₇	1	2	yes	—
96	#382	C₂²×C₇	1	2	yes	—
97	#380	—	2	2	no	—
98	C₄×C₄⋊C₄	—	2	2	no	—
99	C₂×#95	—	2	2	no	—
100	#340	C₂²×C₇	1	16	yes	—
101	C₂³×C₇	—	2	4	no	—
102	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂³×C₇	#124	1	2	yes	—
104	C₂³×C₇	#124	1	2	yes	—
105	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₇	—	2	4	no	—
107	C₂×C₄×C₇	—	2	4	no	—
108	C₂×C₄×C₇	—	4	8	no	—
109	C₂×C₄×C₇	—	4	4	no	—
110	C₂×C₄×C₇	—	4	4	no	—
111	C₂×C₄×C₇	—	4	8	no	—
112	C₂×C₄×C₇	—	4	8	no	—
113	C₂×C₄×C₇	—	8	8	no	—
114	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
115	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
116	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₇	#117	1	8	yes	—
118	C₂×C₄×C₇	#124	1	2	yes	—
119	C₂×C₄×C₇	#124	1	2	yes	—
120	C₂×C₄×C₇	#125	1	8	yes	—
121	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂×#41	1	2	yes	—
124	C₂×C₄×C₇	C₂×#41	1	2	yes	—
125	C₇×C₈	—	2	8	no	—
126	C₂³×C₄	—	2	2	no	—
127	C₂³×C₄	C₇×D₄	1	2	yes	—
128	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
129	C₂×M₁₆	—	2	4	no	—
130	C₂×M₁₆	—	2	4	no	—
131	C₂×M₁₆	C₂×C₄×C₇	1	8	yes	—
132	C₂²×C₈	C₂³×C₇	1	4	yes	—
133	#117	—	2	8	no	—
134	C₂×C₄⋊C₄	—	2	8	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	2	4	no	—
137	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
138	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
139	C₂×C₂²⋊C₄	—	2	2	no	—
140	C₂×C₄²	—	2	2	no	—
141	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
142	C₂×C₄²	C₇×D₄	1	2	yes	—
143	C₂×C₄²	C₇×Q₈	1	4	yes	—
144	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
145	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
146	#97	C₂×C₄×C₇	1	8	yes	complemented
147	C₄×C₈	C₂×C₄×C₇	1	8	yes	complemented
148	#95	—	2	4	no	—
149	#95	—	2	4	no	—
150	#95	—	2	4	no	—
151	C₂²×C₇	—	2	4	no	—
152	C₂²×C₇	—	2	4	no	—
153	C₂²×C₇	—	2	4	no	—
154	C₂²×C₇	#382	1	4	yes	—
155	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
156	C₂²×C₇	C₂×#124	1	1	yes	characteristic
157	C₂²×C₇	C₂×#124	1	1	yes	characteristic
158	C₂²×C₇	#575	1	4	yes	—
159	C₄×C₇	—	2	4	no	—
160	C₄×C₇	—	2	4	no	—
161	C₄×C₇	—	8	4	no	—
162	C₄×C₇	—	8	4	no	—
163	C₄×C₇	#382	1	4	yes	—
164	C₄×C₇	C₂×#117	1	2	yes	—
165	C₄×C₇	C₂×#125	1	2	yes	—
166	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	4	4	no	—
170	C₂²×C₄	—	4	8	no	—
171	C₂²×C₄	—	4	4	no	—
172	C₂²×C₄	C₇×C₄⋊C₄	1	8	yes	—
173	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
174	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
175	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
176	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
178	C₂²×C₄	C₇×#41	1	4	yes	—
179	C₂²×C₄	C₇×#41	1	4	yes	—
180	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
181	M₁₆	—	2	16	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	C₂²×C₄×C₇	1	8	yes	—
185	C₄⋊C₄	—	2	8	no	—
186	C₄⋊C₄	—	2	8	no	—
187	C₄⋊C₄	—	4	8	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₂²⋊C₄	—	4	4	no	—
190	C₄²	—	4	4	no	—
191	C₄²	C₇×C₄⋊C₄	1	8	yes	—
192	C₄²	C₂²×C₄×C₇	1	2	yes	—
193	C₄²	C₂²×C₄×C₇	1	2	yes	—
194	C₄²	C₂×C₇×Q₈	1	2	yes	—
195	C₄²	C₂×C₇×Q₈	1	2	yes	—
196	C₂×C₇	—	2	4	no	—
197	C₂×C₇	C₂×#382	1	1	yes	characteristic
198	C₂×C₇	#2911	1	2	yes	—
199	C₂×C₇	#3140	1	2	yes	—
200	C₂×C₇	#3142	1	2	yes	—
201	C₂³	—	2	4	no	—
202	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
203	C₂³	C₇×#124	1	2	yes	—
204	C₂³	C₇×#124	1	2	yes	—
205	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	4	8	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	8	no	—
212	C₂×C₄	—	4	8	no	—
213	C₂×C₄	—	8	8	no	—
214	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
215	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
216	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
217	C₂×C₄	C₇×#117	1	8	yes	—
218	C₂×C₄	C₇×#124	1	2	yes	—
219	C₂×C₄	C₇×#124	1	2	yes	—
220	C₂×C₄	C₇×#125	1	8	yes	—
221	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
222	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
223	C₂×C₄	C₂×C₇×#41	1	2	yes	—
224	C₂×C₄	C₂×C₇×#41	1	2	yes	—
225	C₈	—	2	8	no	—
226	C₇	#20765	1	1	yes	characteristic, direct product factor, Sylow
227	C₂²	—	2	4	no	—
228	C₂²	—	2	4	no	—
229	C₂²	—	2	4	no	—
230	C₂²	C₇×#382	1	4	yes	—
231	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
232	C₂²	C₂×C₇×#124	1	1	yes	characteristic
233	C₂²	C₂×C₇×#124	1	1	yes	characteristic
234	C₂²	C₇×#575	1	4	yes	—
235	C₄	—	2	4	no	—
236	C₄	—	2	4	no	—
237	C₄	—	8	4	no	—
238	C₄	—	8	4	no	—
239	C₄	C₇×#382	1	4	yes	—
240	C₄	C₂×C₇×#117	1	2	yes	—
241	C₄	C₂×C₇×#125	1	2	yes	—
242	C₂	—	2	4	no	—
243	C₂	C₂×C₇×#382	1	1	yes	characteristic
244	C₂	C₇×#2911	1	2	yes	—
245	C₂	C₇×#3140	1	2	yes	—
246	C₂	C₇×#3142	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₇×#1836	C₂	semidirect
5. C₇×#1835	C₂	semidirect
11. C₄×C₇×M₁₆	C₄	semidirect
23. #20765	C₇	direct (N × H)
44. C₇×#97	C₂×C₄	semidirect
45. C₄×C₇×C₈	C₂×C₄	semidirect
52. #1836	C₂×C₇	semidirect
53. #1835	C₂×C₇	semidirect
89. C₄×M₁₆	C₄×C₇	semidirect
146. #97	C₂×C₄×C₇	semidirect
147. C₄×C₈	C₂×C₄×C₇	semidirect
226. C₇	#20765	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×C₄×C₇×M₁₆	2
C₇×#2285	2
C₂×C₇×#382	2
C₇×#1836	2
C₇×#1835	2
C₇×#1736	2
#20765	7

Subgroups — C7×#20765 #13197981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#20765 #13197981