Subgroups — C₇×#20826 #13198042

Order 1792.

810 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2920	C₂	1	2	yes	maximal
2	C₂²×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#1848	C₂	1	2	yes	maximal, complemented
4	C₇×#1837	C₂	1	2	yes	maximal, complemented
5	C₇×#1810	C₂	1	4	yes	maximal
6	C₄×C₇×#108	C₂	1	4	yes	maximal, complemented
7	C₂³×C₇×C₈	C₂²	1	2	yes	—
8	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₇×#432	—	2	4	no	—
10	C₇×#431	—	2	4	no	—
11	C₂×C₇×#108	C₂²	1	8	yes	—
12	C₇×#423	C₂²	1	4	yes	—
13	C₇×#407	C₂²	1	4	yes	—
14	C₄×C₇×M₁₆	C₂²	1	4	yes	—
15	C₂×C₄×C₇×C₈	C₄	1	4	yes	—
16	C₂×C₄×C₇×C₈	C₄	1	4	yes	—
17	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₇×#341	C₂²	1	8	yes	—
21	#20826	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂³×C₄×C₇	Q₈	1	2	yes	—
23	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₂×C₇×M₁₆	C₂³	1	8	yes	—
25	C₂²×C₇×C₈	—	2	8	no	—
26	C₂²×C₇×C₈	C₂×C₄	1	8	yes	—
27	C₂²×C₇×C₈	C₂³	1	2	yes	—
28	C₂²×C₇×C₈	C₂³	1	2	yes	—
29	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₇	D₄	1	2	yes	—
32	C₂×C₄²×C₇	D₄	1	2	yes	—
33	C₂×C₄²×C₇	Q₈	1	2	yes	—
34	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
36	C₇×#108	—	2	16	no	—
37	C₇×C₄⋊C₈	—	2	8	no	—
38	C₇×C₂²⋊C₈	—	2	8	no	—
39	C₇×#97	—	2	4	no	—
40	C₄×C₇×C₈	—	2	4	no	—
41	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
42	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
43	C₄×C₇×C₈	C₂×C₄	1	8	yes	—
44	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
45	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
46	#2920	C₂×C₇	1	2	yes	—
47	C₂²×C₄×C₈	C₂×C₇	1	1	yes	characteristic
48	#1848	C₂×C₇	1	2	yes	complemented
49	#1837	C₂×C₇	1	2	yes	complemented
50	#1810	C₂×C₇	1	4	yes	—
51	C₄×#108	C₂×C₇	1	4	yes	complemented
52	C₂⁴×C₇	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₇	—	2	8	no	—
54	C₂²×C₄×C₇	—	2	4	no	—
55	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
63	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
65	C₇×M₁₆	—	2	16	no	—
66	C₂×C₇×C₈	—	2	4	no	—
67	C₂×C₇×C₈	—	2	16	no	—
68	C₂×C₇×C₈	—	2	8	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	8	yes	—
72	C₄²×C₇	C₄⋊C₄	1	2	yes	—
73	C₄²×C₇	C₄⋊C₄	1	2	yes	—
74	C₄²×C₇	C₄⋊C₄	1	2	yes	—
75	C₄²×C₇	C₄⋊C₄	1	2	yes	—
76	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₇	C₂²×C₄	1	2	yes	—
78	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₇	C₂×D₄	1	2	yes	—
81	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
82	C₂³×C₈	C₂²×C₇	1	2	yes	—
83	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
84	#432	—	2	4	no	—
85	#431	—	2	4	no	—
86	C₂×#108	C₂²×C₇	1	8	yes	—
87	#423	C₂²×C₇	1	4	yes	—
88	#407	C₂²×C₇	1	4	yes	—
89	C₄×M₁₆	C₂²×C₇	1	4	yes	—
90	C₂×C₄×C₈	C₄×C₇	1	4	yes	—
91	C₂×C₄×C₈	C₄×C₇	1	4	yes	—
92	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
94	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
95	#341	C₂²×C₇	1	8	yes	—
96	C₂³×C₇	—	2	2	no	—
97	C₂³×C₇	—	2	4	no	—
98	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₇	—	2	2	no	—
101	C₂×C₄×C₇	—	2	4	no	—
102	C₂×C₄×C₇	—	2	4	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	4	no	—
107	C₂×C₄×C₇	—	2	4	no	—
108	C₂×C₄×C₇	#108	1	8	yes	complemented
109	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇	C₂×#41	1	4	yes	—
117	C₇×C₈	—	2	4	no	—
118	C₇×C₈	—	4	8	no	—
119	C₇×C₈	#117	1	8	yes	—
120	C₂³×C₄	C₇×Q₈	1	2	yes	—
121	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
122	C₂×M₁₆	C₂³×C₇	1	8	yes	—
123	C₂²×C₈	—	2	8	no	—
124	C₂²×C₈	C₂×C₄×C₇	1	8	yes	—
125	C₂²×C₈	C₂³×C₇	1	2	yes	—
126	C₂²×C₈	C₂³×C₇	1	2	yes	—
127	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
128	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
129	C₂×C₄²	C₇×D₄	1	2	yes	—
130	C₂×C₄²	C₇×D₄	1	2	yes	—
131	C₂×C₄²	C₇×Q₈	1	2	yes	—
132	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
133	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
134	#108	—	2	16	no	—
135	C₄⋊C₈	—	2	8	no	—
136	C₂²⋊C₈	—	2	8	no	—
137	#97	—	2	4	no	—
138	C₄×C₈	—	2	4	no	—
139	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
140	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
141	C₄×C₈	C₂×C₄×C₇	1	8	yes	—
142	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
143	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
144	C₂²×C₇	—	2	8	no	—
145	C₂²×C₇	—	2	2	no	—
146	C₂²×C₇	—	2	2	no	—
147	C₂²×C₇	—	2	2	no	—
148	C₂²×C₇	C₂×#108	1	4	yes	—
149	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₂²×C₇	#533	1	2	yes	—
151	C₄×C₇	—	2	2	no	—
152	C₄×C₇	—	2	2	no	—
153	C₄×C₇	—	2	2	no	—
154	C₄×C₇	C₂×#108	1	4	yes	—
155	C₄×C₇	C₂×#108	1	4	yes	—
156	C₄×C₇	C₂×#117	1	2	yes	—
157	C₄×C₇	#532	1	2	yes	—
158	C₂⁴	C₂×C₇×Q₈	1	1	yes	characteristic
159	C₂²×C₄	—	2	8	no	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
162	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
163	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
164	C₂²×C₄	C₇×C₄⋊C₄	1	2	yes	—
165	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
166	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
167	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
169	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
170	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
171	M₁₆	—	2	16	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	16	no	—
174	C₂×C₈	—	2	8	no	—
175	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
176	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
177	C₂×C₈	C₇×#41	1	8	yes	—
178	C₄²	C₇×C₄⋊C₄	1	2	yes	—
179	C₄²	C₇×C₄⋊C₄	1	2	yes	—
180	C₄²	C₇×C₄⋊C₄	1	2	yes	—
181	C₄²	C₇×C₄⋊C₄	1	2	yes	—
182	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
183	C₄²	C₂²×C₄×C₇	1	2	yes	—
184	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
185	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
186	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
187	C₄²	C₂×C₇×D₄	1	2	yes	—
188	C₂×C₇	—	2	2	no	—
189	C₂×C₇	—	2	4	no	—
190	C₂×C₇	#2285	1	1	yes	characteristic
191	C₂×C₇	C₂²×#108	1	2	yes	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	4	no	—
194	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
195	C₂³	C₂²×C₇×Q₈	1	1	yes	characteristic
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	C₇×#108	1	8	yes	complemented
205	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
206	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
207	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
208	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
209	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
210	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
211	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
212	C₂×C₄	C₂×C₇×#41	1	4	yes	—
213	C₈	—	2	4	no	—
214	C₈	—	4	8	no	—
215	C₈	C₇×#117	1	8	yes	—
216	C₇	#20826	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²	—	2	8	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	C₂×C₇×#108	1	4	yes	—
222	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
223	C₂²	C₇×#533	1	2	yes	—
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	C₂×C₇×#108	1	4	yes	—
228	C₄	C₂×C₇×#108	1	4	yes	—
229	C₄	C₂×C₇×#117	1	2	yes	—
230	C₄	C₇×#532	1	2	yes	—
231	C₂	—	2	2	no	—
232	C₂	—	2	4	no	—
233	C₂	C₇×#2285	1	1	yes	characteristic
234	C₂	C₂²×C₇×#108	1	2	yes	—

	subgroup	G/H
center	C₄²×C₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂²×C₇	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₇	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1848	C₂	semidirect
4. C₇×#1837	C₂	semidirect
6. C₄×C₇×#108	C₂	semidirect
21. #20826	C₇	direct (N × H)
48. #1848	C₂×C₇	semidirect
49. #1837	C₂×C₇	semidirect
51. C₄×#108	C₂×C₇	semidirect
108. C₂×C₄×C₇	#108	semidirect
204. C₂×C₄	C₇×#108	semidirect
216. C₇	#20826	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×#2920	2
C₂²×C₄×C₇×C₈	2
C₇×#1848	2
C₇×#1837	2
C₇×#1810	2
C₄×C₇×#108	2
#20826	7

Subgroups — C7×#20826 #13198042

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#20826 #13198042