Subgroups — C₇×#2752 #11742061

Order 896.

484 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#399	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#397	C₂	1	2	yes	maximal, complemented
3	C₇×#394	C₂	1	2	yes	maximal, complemented
4	C₇×#392	C₂	1	1	yes	maximal, characteristic
5	C₇×#390	C₂	1	2	yes	maximal, complemented
6	C₇×#388	C₂	1	2	yes	maximal, complemented
7	C₇×#387	C₂	1	2	yes	maximal, complemented
8	C₇×#386	C₂	1	2	yes	maximal, complemented
9	C₇×#381	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₇×D₄	C₂²	1	2	yes	—
11	C₂²×C₇×D₄	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
16	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
17	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
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	2	yes	—
23	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
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	2	yes	—
29	C₇×#95	C₂²	1	1	yes	characteristic
30	C₇×#95	C₂²	1	1	yes	characteristic
31	#2752	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂⁴×C₇	C₂³	1	2	yes	—
33	C₂⁴×C₇	C₂³	1	1	yes	characteristic
34	C₂×C₇×D₄	—	2	4	no	—
35	C₂×C₇×D₄	—	2	4	no	—
36	C₂×C₇×D₄	—	4	2	no	—
37	C₂×C₇×D₄	—	4	1	no	—
38	C₂×C₇×D₄	—	4	2	no	—
39	C₂²×C₄×C₇	—	2	2	no	—
40	C₂²×C₄×C₇	—	4	1	no	—
41	C₂²×C₄×C₇	—	4	1	no	—
42	C₂²×C₄×C₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₇	C₂³	1	2	yes	—
47	C₂²×C₄×C₇	C₂³	1	2	yes	—
48	C₂²×C₄×C₇	C₂³	1	2	yes	—
49	C₇×C₄⋊C₄	—	2	4	no	—
50	C₇×C₄⋊C₄	—	4	2	no	—
51	C₇×C₄⋊C₄	—	4	1	no	—
52	C₇×C₂²⋊C₄	—	2	4	no	—
53	C₇×C₂²⋊C₄	—	4	2	no	—
54	C₇×C₂²⋊C₄	—	4	1	no	—
55	C₇×C₂²⋊C₄	—	4	2	no	—
56	C₇×C₂²⋊C₄	—	4	2	no	—
57	C₇×C₂²⋊C₄	—	4	2	no	—
58	C₇×C₂²⋊C₄	—	4	2	no	—
59	C₇×C₂²⋊C₄	—	4	1	no	—
60	C₄²×C₇	—	4	2	no	—
61	#399	C₂×C₇	1	1	yes	characteristic, complemented
62	#397	C₂×C₇	1	2	yes	complemented
63	#394	C₂×C₇	1	2	yes	complemented
64	#392	C₂×C₇	1	1	yes	characteristic
65	#390	C₂×C₇	1	2	yes	complemented
66	#388	C₂×C₇	1	2	yes	complemented
67	#387	C₂×C₇	1	2	yes	complemented
68	#386	C₂×C₇	1	2	yes	complemented
69	#381	C₂×C₇	1	1	yes	characteristic, complemented
70	C₂³×C₇	—	2	2	no	—
71	C₂³×C₇	—	2	1	no	—
72	C₂³×C₇	—	2	1	no	—
73	C₂³×C₇	—	2	2	no	—
74	C₂³×C₇	—	2	2	no	—
75	C₂³×C₇	—	2	2	no	—
76	C₂³×C₇	—	2	2	no	—
77	C₂³×C₇	—	2	2	no	—
78	C₂³×C₇	—	2	2	no	—
79	C₂³×C₇	—	2	2	no	—
80	C₂³×C₇	—	2	2	no	—
81	C₂³×C₇	#41	1	2	yes	—
82	C₂³×C₇	C₂⁴	1	1	yes	characteristic
83	C₇×D₄	—	4	8	no	—
84	C₇×D₄	—	4	4	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	—	2	2	no	—
93	C₂×C₄×C₇	—	2	2	no	—
94	C₂×C₄×C₇	—	2	2	no	—
95	C₂×C₄×C₇	—	2	2	no	—
96	C₂×C₄×C₇	—	2	2	no	—
97	C₂×C₄×C₇	—	2	1	no	—
98	C₂×C₄×C₇	—	2	2	no	—
99	C₂×C₄×C₇	—	2	2	no	—
100	C₂×C₄×C₇	—	2	2	no	—
101	C₂×C₄×C₇	—	2	1	no	—
102	C₂×C₄×C₇	—	2	2	no	—
103	C₂×C₄×C₇	—	4	4	no	—
104	C₂×C₄×C₇	#41	1	4	yes	—
105	C₂²×D₄	C₂²×C₇	1	2	yes	—
106	C₂²×D₄	C₂²×C₇	1	1	yes	characteristic
107	C₂³×C₄	C₂²×C₇	1	1	yes	characteristic
108	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
109	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
110	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
111	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
112	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
113	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
114	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
115	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
116	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
117	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
118	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
119	C₂×C₄²	C₂²×C₇	1	2	yes	—
120	#95	C₂²×C₇	1	2	yes	—
121	#95	C₂²×C₇	1	2	yes	—
122	#95	C₂²×C₇	1	2	yes	—
123	#95	C₂²×C₇	1	2	yes	—
124	#95	C₂²×C₇	1	1	yes	characteristic
125	#95	C₂²×C₇	1	1	yes	characteristic
126	C₂²×C₇	—	2	2	no	—
127	C₂²×C₇	—	2	4	no	—
128	C₂²×C₇	—	4	2	no	—
129	C₂²×C₇	—	4	1	no	—
130	C₂²×C₇	—	4	1	no	—
131	C₂²×C₇	—	4	2	no	—
132	C₂²×C₇	—	4	2	no	—
133	C₂²×C₇	—	4	2	no	—
134	C₂²×C₇	—	4	2	no	—
135	C₂²×C₇	—	4	2	no	—
136	C₂²×C₇	—	4	2	no	—
137	C₂²×C₇	—	4	2	no	—
138	C₂²×C₇	C₂×#41	1	1	yes	characteristic
139	C₂²×C₇	C₂×#41	1	2	yes	—
140	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂²×C₇	H(5,𝔽₂)	1	2	yes	—
143	C₄×C₇	—	2	4	no	—
144	C₄×C₇	—	4	2	no	—
145	C₄×C₇	—	4	2	no	—
146	C₄×C₇	—	4	2	no	—
147	C₄×C₇	—	4	1	no	—
148	C₄×C₇	—	4	2	no	—
149	C₄×C₇	—	4	1	no	—
150	C₂⁴	C₂³×C₇	1	2	yes	—
151	C₂⁴	C₂³×C₇	1	1	yes	characteristic
152	C₂×D₄	—	2	4	no	—
153	C₂×D₄	—	2	4	no	—
154	C₂×D₄	—	4	2	no	—
155	C₂×D₄	—	4	1	no	—
156	C₂×D₄	—	4	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	4	1	no	—
159	C₂²×C₄	—	4	1	no	—
160	C₂²×C₄	C₂³×C₇	1	2	yes	—
161	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
162	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₇	1	2	yes	—
164	C₂²×C₄	C₂³×C₇	1	2	yes	—
165	C₂²×C₄	C₂³×C₇	1	2	yes	—
166	C₂²×C₄	C₂³×C₇	1	2	yes	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	1	no	—
170	C₂²⋊C₄	—	2	4	no	—
171	C₂²⋊C₄	—	4	2	no	—
172	C₂²⋊C₄	—	4	1	no	—
173	C₂²⋊C₄	—	4	2	no	—
174	C₂²⋊C₄	—	4	2	no	—
175	C₂²⋊C₄	—	4	2	no	—
176	C₂²⋊C₄	—	4	2	no	—
177	C₂²⋊C₄	—	4	1	no	—
178	C₄²	—	4	2	no	—
179	C₂×C₇	—	4	2	no	—
180	C₂×C₇	—	8	2	no	—
181	C₂×C₇	#538	1	1	yes	characteristic
182	C₂×C₇	#540	1	2	yes	—
183	C₂×C₇	#551	1	2	yes	—
184	C₂×C₇	#559	1	1	yes	characteristic
185	C₂×C₇	#560	1	1	yes	characteristic
186	C₂³	—	2	2	no	—
187	C₂³	—	2	1	no	—
188	C₂³	—	2	1	no	—
189	C₂³	—	2	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	2	no	—
193	C₂³	—	2	2	no	—
194	C₂³	—	2	2	no	—
195	C₂³	—	2	2	no	—
196	C₂³	—	2	2	no	—
197	C₂³	C₇×#41	1	2	yes	—
198	C₂³	C₂⁴×C₇	1	1	yes	characteristic
199	D₄	—	4	8	no	—
200	D₄	—	4	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	C₇×#41	1	4	yes	—
221	C₇	#2752	1	1	yes	characteristic, direct product factor, Sylow
222	C₂²	—	2	2	no	—
223	C₂²	—	2	4	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	4	2	no	—
230	C₂²	—	4	2	no	—
231	C₂²	—	4	2	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	4	2	no	—
234	C₂²	C₂×C₇×#41	1	1	yes	characteristic
235	C₂²	C₂×C₇×#41	1	2	yes	—
236	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
237	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
238	C₂²	C₇×H(5,𝔽₂)	1	2	yes	—
239	C₄	—	2	4	no	—
240	C₄	—	4	2	no	—
241	C₄	—	4	2	no	—
242	C₄	—	4	2	no	—
243	C₄	—	4	1	no	—
244	C₄	—	4	1	no	—
245	C₄	—	4	2	no	—
246	C₂	—	4	2	no	—
247	C₂	—	8	2	no	—
248	C₂	C₇×#538	1	1	yes	characteristic
249	C₂	C₇×#540	1	2	yes	—
250	C₂	C₇×#551	1	2	yes	—
251	C₂	C₇×#559	1	1	yes	characteristic
252	C₂	C₇×#560	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
prime radical	C₇×#392	C₂
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
1. C₇×#399	C₂	semidirect
2. C₇×#397	C₂	semidirect
3. C₇×#394	C₂	semidirect
5. C₇×#390	C₂	semidirect
6. C₇×#388	C₂	semidirect
7. C₇×#387	C₂	semidirect
8. C₇×#386	C₂	semidirect
9. C₇×#381	C₂	semidirect
31. #2752	C₇	direct (N × H)
61. #399	C₂×C₇	semidirect
62. #397	C₂×C₇	semidirect
63. #394	C₂×C₇	semidirect
65. #390	C₂×C₇	semidirect
66. #388	C₂×C₇	semidirect
67. #387	C₂×C₇	semidirect
68. #386	C₂×C₇	semidirect
69. #381	C₂×C₇	semidirect
221. C₇	#2752	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×#399	2
C₇×#397	2
C₇×#394	2
C₇×#392	2
C₇×#390	2
C₇×#388	2
C₇×#387	2
C₇×#386	2
C₇×#381	2
#2752	7

Subgroups — C7×#2752 #11742061

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2752 #11742061