Subgroups — C₇×#20825 #13198041

Order 1792.

842 conjugacy classes of subgroups.

The subgroup lattice has 232 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×#2283	C₂	1	2	yes	maximal
3	C₇×#1846	C₂	1	2	yes	maximal, complemented
4	C₇×#1835	C₂	1	2	yes	maximal, complemented
5	C₇×#1809	C₂	1	4	yes	maximal
6	C₇×#1776	C₂	1	4	yes	maximal, complemented
7	C₂²×C₇×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₇×#107	C₂²	1	4	yes	—
10	C₂×C₇×#106	C₂²	1	4	yes	—
11	C₄×C₇×M₁₆	C₄	1	8	yes	complemented
12	C₂×C₇×#97	C₂²	1	2	yes	—
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₇×#383	—	2	2	no	—
16	C₇×#382	C₂²	1	4	yes	—
17	C₇×#381	—	2	2	no	—
18	C₇×#380	C₂²	1	4	yes	—
19	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
20	C₄×C₇×C₂²⋊C₄	—	2	2	no	—
21	C₇×#376	—	2	2	no	—
22	C₇×#340	C₂²	1	8	yes	—
23	#20825	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂³×C₄×C₇	D₄	1	2	yes	—
25	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
26	C₂×C₇×M₁₆	—	2	4	no	—
27	C₂×C₇×M₁₆	—	2	4	no	—
28	C₂×C₇×M₁₆	C₂×C₄	1	8	yes	—
29	C₂²×C₇×C₈	C₂³	1	2	yes	—
30	C₂²×C₇×C₈	C₂³	1	2	yes	—
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	8	no	—
40	C₇×#106	—	2	8	no	—
41	C₇×#97	C₂×C₄	1	8	yes	—
42	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
43	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
44	C₇×#95	—	2	4	no	—
45	C₇×#95	—	2	4	no	—
46	C₇×#95	—	2	4	no	—
47	C₇×#95	—	2	4	no	—
48	C₂×C₄×M₁₆	C₂×C₇	1	1	yes	characteristic
49	#2283	C₂×C₇	1	2	yes	—
50	#1846	C₂×C₇	1	2	yes	complemented
51	#1835	C₂×C₇	1	2	yes	complemented
52	#1809	C₂×C₇	1	4	yes	—
53	#1776	C₂×C₇	1	4	yes	complemented
54	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇	—	2	2	no	—
56	C₂²×C₄×C₇	—	2	2	no	—
57	C₂²×C₄×C₇	—	2	8	no	—
58	C₂²×C₄×C₇	—	4	4	no	—
59	C₂²×C₄×C₇	C₄⋊C₄	1	8	yes	—
60	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
66	C₇×M₁₆	—	2	16	no	—
67	C₂×C₇×C₈	—	2	2	no	—
68	C₂×C₇×C₈	—	2	2	no	—
69	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
70	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
71	C₂×C₇×C₈	#41	1	4	yes	—
72	C₂×C₇×C₈	#41	1	4	yes	—
73	C₇×C₄⋊C₄	—	2	8	no	—
74	C₇×C₄⋊C₄	—	2	8	no	—
75	C₇×C₂²⋊C₄	—	4	8	no	—
76	C₄²×C₇	—	4	8	no	—
77	C₄²×C₇	C₄⋊C₄	1	8	yes	—
78	C₄²×C₇	C₂²×C₄	1	2	yes	—
79	C₄²×C₇	C₂²×C₄	1	2	yes	—
80	C₄²×C₇	C₂×Q₈	1	2	yes	—
81	C₄²×C₇	C₂×Q₈	1	2	yes	—
82	C₂²×M₁₆	C₂²×C₇	1	2	yes	—
83	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
84	C₂×#107	C₂²×C₇	1	4	yes	—
85	C₂×#106	C₂²×C₇	1	4	yes	—
86	C₄×M₁₆	C₄×C₇	1	8	yes	complemented
87	C₂×#97	C₂²×C₇	1	2	yes	—
88	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
90	#383	—	2	2	no	—
91	#382	C₂²×C₇	1	4	yes	—
92	#381	—	2	2	no	—
93	#380	C₂²×C₇	1	4	yes	—
94	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
95	C₄×C₂²⋊C₄	—	2	2	no	—
96	#376	—	2	2	no	—
97	#340	C₂²×C₇	1	8	yes	—
98	C₂³×C₇	—	2	2	no	—
99	C₂³×C₇	—	2	4	no	—
100	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
101	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₇	—	2	2	no	—
103	C₂×C₄×C₇	—	2	4	no	—
104	C₂×C₄×C₇	—	2	4	no	—
105	C₂×C₄×C₇	—	2	4	no	—
106	C₂×C₄×C₇	—	2	2	no	—
107	C₂×C₄×C₇	—	2	2	no	—
108	C₂×C₄×C₇	—	4	8	no	—
109	C₂×C₄×C₇	—	4	8	no	—
110	C₂×C₄×C₇	—	4	8	no	—
111	C₂×C₄×C₇	—	4	8	no	—
112	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
114	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
115	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₇	C₂×#41	1	2	yes	—
118	C₂×C₄×C₇	C₂×#41	1	2	yes	—
119	C₇×C₈	—	2	4	no	—
120	C₇×C₈	#117	1	8	yes	—
121	C₂³×C₄	C₇×D₄	1	2	yes	—
122	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
123	C₂×M₁₆	—	2	4	no	—
124	C₂×M₁₆	—	2	4	no	—
125	C₂×M₁₆	C₂×C₄×C₇	1	8	yes	—
126	C₂²×C₈	C₂³×C₇	1	2	yes	—
127	C₂²×C₈	C₂³×C₇	1	2	yes	—
128	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
129	C₂×C₂²⋊C₄	—	2	4	no	—
130	C₂×C₄²	—	2	4	no	—
131	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
132	C₂×C₄²	C₇×D₄	1	2	yes	—
133	C₂×C₄²	C₇×Q₈	1	4	yes	—
134	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
135	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
136	#107	—	2	8	no	—
137	#106	—	2	8	no	—
138	#97	C₂×C₄×C₇	1	8	yes	—
139	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
140	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
141	#95	—	2	4	no	—
142	#95	—	2	4	no	—
143	#95	—	2	4	no	—
144	#95	—	2	4	no	—
145	C₂²×C₇	—	2	4	no	—
146	C₂²×C₇	—	2	2	no	—
147	C₂²×C₇	—	4	4	no	—
148	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₇	#532	1	2	yes	—
150	C₂²×C₇	#575	1	4	yes	—
151	C₄×C₇	—	2	4	no	—
152	C₄×C₇	—	2	4	no	—
153	C₄×C₇	—	2	2	no	—
154	C₄×C₇	—	8	8	no	—
155	C₄×C₇	C₂×#117	1	2	yes	—
156	C₄×C₇	#533	1	1	yes	characteristic
157	C₄×C₇	#533	1	1	yes	characteristic
158	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	8	no	—
162	C₂²×C₄	—	4	4	no	—
163	C₂²×C₄	C₇×C₄⋊C₄	1	8	yes	—
164	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
165	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
166	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
167	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
168	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
169	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
170	M₁₆	—	2	16	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
174	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
175	C₂×C₈	C₇×#41	1	4	yes	—
176	C₂×C₈	C₇×#41	1	4	yes	—
177	C₄⋊C₄	—	2	8	no	—
178	C₄⋊C₄	—	2	8	no	—
179	C₂²⋊C₄	—	4	8	no	—
180	C₄²	—	4	8	no	—
181	C₄²	C₇×C₄⋊C₄	1	8	yes	—
182	C₄²	C₂²×C₄×C₇	1	2	yes	—
183	C₄²	C₂²×C₄×C₇	1	2	yes	—
184	C₄²	C₂×C₇×Q₈	1	2	yes	—
185	C₄²	C₂×C₇×Q₈	1	2	yes	—
186	C₂×C₇	—	4	2	no	—
187	C₂×C₇	#2285	1	1	yes	characteristic
188	C₂×C₇	#2911	1	2	yes	—
189	C₂×C₇	#3183	1	4	yes	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	4	no	—
192	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
193	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	8	no	—
201	C₂×C₄	—	4	8	no	—
202	C₂×C₄	—	4	8	no	—
203	C₂×C₄	—	4	8	no	—
204	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
205	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
206	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
207	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
208	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
209	C₂×C₄	C₂×C₇×#41	1	2	yes	—
210	C₂×C₄	C₂×C₇×#41	1	2	yes	—
211	C₈	—	2	4	no	—
212	C₈	C₇×#117	1	8	yes	—
213	C₇	#20825	1	1	yes	characteristic, direct product factor, Sylow
214	C₂²	—	2	4	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	4	4	no	—
217	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
218	C₂²	C₇×#532	1	2	yes	—
219	C₂²	C₇×#575	1	4	yes	—
220	C₄	—	2	4	no	—
221	C₄	—	2	4	no	—
222	C₄	—	2	2	no	—
223	C₄	—	8	8	no	—
224	C₄	C₂×C₇×#117	1	2	yes	—
225	C₄	C₇×#533	1	1	yes	characteristic
226	C₄	C₇×#533	1	1	yes	characteristic
227	C₂	—	4	2	no	—
228	C₂	C₇×#2285	1	1	yes	characteristic
229	C₂	C₇×#2911	1	2	yes	—
230	C₂	C₇×#3183	1	4	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
3. C₇×#1846	C₂	semidirect
4. C₇×#1835	C₂	semidirect
6. C₇×#1776	C₂	semidirect
11. C₄×C₇×M₁₆	C₄	semidirect
23. #20825	C₇	direct (N × H)
50. #1846	C₂×C₇	semidirect
51. #1835	C₂×C₇	semidirect
53. #1776	C₂×C₇	semidirect
86. C₄×M₁₆	C₄×C₇	semidirect
213. C₇	#20825	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×#2283	2
C₇×#1846	2
C₇×#1835	2
C₇×#1809	2
C₇×#1776	2
#20825	7

Subgroups — C7×#20825 #13198041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#20825 #13198041