Subgroups — C₇×#2722 #11742031

Order 896.

448 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#126	C₂	1	2	yes	maximal
2	C₇×#398	C₂	1	2	yes	maximal, complemented
3	C₇×#398	C₂	1	2	yes	maximal, complemented
4	C₇×#397	C₂	1	1	yes	maximal, characteristic
5	C₇×#397	C₂	1	1	yes	maximal, characteristic
6	C₇×#389	C₂	1	2	yes	maximal, complemented
7	C₇×#387	C₂	1	2	yes	maximal, complemented
8	C₇×#385	C₂	1	1	yes	maximal, characteristic
9	C₇×#380	C₂	1	2	yes	maximal
10	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₇×#126	—	2	8	no	—
12	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₇	C₂²	1	2	yes	—
25	C₇×#95	C₂²	1	2	yes	—
26	C₇×#95	C₂²	1	2	yes	—
27	C₇×#95	C₂²	1	2	yes	—
28	C₇×#95	C₂²	1	1	yes	characteristic
29	C₇×#95	C₂²	1	2	yes	—
30	C₇×#95	C₂²	1	2	yes	—
31	C₇×#95	C₂²	1	1	yes	characteristic
32	#2722	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂⁴×C₇	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₇	—	2	4	no	—
35	C₂²×C₄×C₇	—	4	1	no	—
36	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₇	C₂³	1	2	yes	—
38	C₂²×C₄×C₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₇	C₂³	1	2	yes	—
44	C₇×C₄⋊C₄	—	2	4	no	—
45	C₇×C₄⋊C₄	—	2	4	no	—
46	C₇×C₄⋊C₄	—	2	4	no	—
47	C₇×C₄⋊C₄	—	2	4	no	—
48	C₇×C₄⋊C₄	—	2	4	no	—
49	C₇×C₄⋊C₄	—	4	2	no	—
50	C₇×C₄⋊C₄	—	4	1	no	—
51	C₇×C₄⋊C₄	—	4	1	no	—
52	C₇×C₂²⋊C₄	—	2	4	no	—
53	C₇×C₂²⋊C₄	—	2	4	no	—
54	C₇×C₂²⋊C₄	—	4	1	no	—
55	C₇×C₂²⋊C₄	D₄	1	4	yes	—
56	C₄²×C₇	—	2	4	no	—
57	C₂×#126	C₂×C₇	1	2	yes	—
58	#398	C₂×C₇	1	2	yes	complemented
59	#398	C₂×C₇	1	2	yes	complemented
60	#397	C₂×C₇	1	1	yes	characteristic
61	#397	C₂×C₇	1	1	yes	characteristic
62	#389	C₂×C₇	1	2	yes	complemented
63	#387	C₂×C₇	1	2	yes	complemented
64	#385	C₂×C₇	1	1	yes	characteristic
65	#380	C₂×C₇	1	2	yes	—
66	C₂³×C₇	—	2	2	no	—
67	C₂³×C₇	—	2	1	no	—
68	C₂³×C₇	—	2	1	no	—
69	C₂³×C₇	—	2	2	no	—
70	C₂³×C₇	C₂×D₄	1	2	yes	—
71	C₂³×C₇	C₂⁴	1	1	yes	characteristic
72	C₂×C₄×C₇	—	2	2	no	—
73	C₂×C₄×C₇	—	2	2	no	—
74	C₂×C₄×C₇	—	2	2	no	—
75	C₂×C₄×C₇	—	2	2	no	—
76	C₂×C₄×C₇	—	2	2	no	—
77	C₂×C₄×C₇	—	2	2	no	—
78	C₂×C₄×C₇	—	2	2	no	—
79	C₂×C₄×C₇	—	2	2	no	—
80	C₂×C₄×C₇	—	2	2	no	—
81	C₂×C₄×C₇	—	2	2	no	—
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	—	2	1	no	—
84	C₂×C₄×C₇	—	2	1	no	—
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	—	2	2	no	—
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	2	2	no	—
90	C₂×C₄×C₇	—	2	2	no	—
91	C₂×C₄×C₇	—	2	2	no	—
92	C₂×C₄×C₇	—	4	4	no	—
93	C₂×C₄×C₇	C₂×D₄	1	4	yes	—
94	C₂×C₄×C₇	#41	1	4	yes	—
95	C₂³×C₄	C₂²×C₇	1	1	yes	characteristic
96	#126	—	2	8	no	—
97	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
98	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
99	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
100	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
101	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
102	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
103	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
104	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
105	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
106	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
107	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
108	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
109	C₂×C₄²	C₂²×C₇	1	2	yes	—
110	#95	C₂²×C₇	1	2	yes	—
111	#95	C₂²×C₇	1	2	yes	—
112	#95	C₂²×C₇	1	2	yes	—
113	#95	C₂²×C₇	1	1	yes	characteristic
114	#95	C₂²×C₇	1	2	yes	—
115	#95	C₂²×C₇	1	2	yes	—
116	#95	C₂²×C₇	1	1	yes	characteristic
117	C₂²×C₇	—	2	2	no	—
118	C₂²×C₇	—	2	4	no	—
119	C₂²×C₇	—	4	2	no	—
120	C₂²×C₇	—	4	1	no	—
121	C₂²×C₇	—	4	1	no	—
122	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
123	C₂²×C₇	C₂×#41	1	2	yes	—
124	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
126	C₂²×C₇	2₋¹⁺⁴	1	2	yes	—
127	C₄×C₇	—	2	4	no	—
128	C₄×C₇	—	2	4	no	—
129	C₄×C₇	—	4	2	no	—
130	C₄×C₇	—	4	2	no	—
131	C₄×C₇	—	4	2	no	—
132	C₄×C₇	—	4	2	no	—
133	C₄×C₇	—	4	1	no	—
134	C₄×C₇	—	4	1	no	—
135	C₂⁴	C₂³×C₇	1	1	yes	characteristic
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	4	1	no	—
138	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
139	C₂²×C₄	C₂³×C₇	1	2	yes	—
140	C₂²×C₄	C₂³×C₇	1	2	yes	—
141	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
142	C₂²×C₄	C₂³×C₇	1	2	yes	—
143	C₂²×C₄	C₂³×C₇	1	2	yes	—
144	C₂²×C₄	C₂³×C₇	1	2	yes	—
145	C₂²×C₄	C₂³×C₇	1	2	yes	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	4	2	no	—
152	C₄⋊C₄	—	4	1	no	—
153	C₄⋊C₄	—	4	1	no	—
154	C₂²⋊C₄	—	2	4	no	—
155	C₂²⋊C₄	—	2	4	no	—
156	C₂²⋊C₄	—	4	1	no	—
157	C₂²⋊C₄	C₇×D₄	1	4	yes	—
158	C₄²	—	2	4	no	—
159	C₂×C₇	—	4	2	no	—
160	C₂×C₇	#537	1	1	yes	characteristic
161	C₂×C₇	#541	1	2	yes	—
162	C₂×C₇	#547	1	2	yes	—
163	C₂×C₇	#551	1	1	yes	characteristic
164	C₂×C₇	#563	1	1	yes	characteristic
165	C₂³	—	2	2	no	—
166	C₂³	—	2	1	no	—
167	C₂³	—	2	1	no	—
168	C₂³	—	2	2	no	—
169	C₂³	C₂×C₇×D₄	1	2	yes	—
170	C₂³	C₂⁴×C₇	1	1	yes	characteristic
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	C₂×C₇×D₄	1	4	yes	—
193	C₂×C₄	C₇×#41	1	4	yes	—
194	C₇	#2722	1	1	yes	characteristic, direct product factor, Sylow
195	C₂²	—	2	2	no	—
196	C₂²	—	2	4	no	—
197	C₂²	—	4	2	no	—
198	C₂²	—	4	1	no	—
199	C₂²	—	4	1	no	—
200	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
201	C₂²	C₂×C₇×#41	1	2	yes	—
202	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
203	C₂²	C₇×2₋¹⁺⁴	1	1	yes	characteristic
204	C₂²	C₇×2₋¹⁺⁴	1	2	yes	—
205	C₄	—	2	4	no	—
206	C₄	—	2	4	no	—
207	C₄	—	4	2	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	2	no	—
211	C₄	—	4	1	no	—
212	C₄	—	4	1	no	—
213	C₂	—	4	2	no	—
214	C₂	C₇×#537	1	1	yes	characteristic
215	C₂	C₇×#541	1	2	yes	—
216	C₂	C₇×#547	1	2	yes	—
217	C₂	C₇×#551	1	1	yes	characteristic
218	C₂	C₇×#563	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₇×#398	C₂	semidirect
3. C₇×#398	C₂	semidirect
6. C₇×#389	C₂	semidirect
7. C₇×#387	C₂	semidirect
32. #2722	C₇	direct (N × H)
58. #398	C₂×C₇	semidirect
59. #398	C₂×C₇	semidirect
62. #389	C₂×C₇	semidirect
63. #387	C₂×C₇	semidirect
194. C₇	#2722	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×#126	2
C₇×#398	2
C₇×#397	2
C₇×#389	2
C₇×#387	2
C₇×#385	2
C₇×#380	2
#2722	7

Subgroups — C7×#2722 #11742031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2722 #11742031