Subgroups — C₇×#2731 #11742040

Order 896.

492 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#124	C₂	1	1	yes	maximal, characteristic
3	C₇×#396	C₂	1	2	yes	maximal, complemented
4	C₇×#394	C₂	1	2	yes	maximal
5	C₇×#393	C₂	1	2	yes	maximal, complemented
6	C₇×#391	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#388	C₂	1	2	yes	maximal, complemented
8	C₇×#388	C₂	1	2	yes	maximal, complemented
9	C₇×#387	C₂	1	2	yes	maximal, complemented
10	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
11	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×C₇×D₄	C₂²	1	1	yes	characteristic
13	C₇×C₄⋊Q₈	—	2	4	no	—
14	C₇×#124	—	2	4	no	—
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×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×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
28	C₇×#95	C₂²	1	2	yes	—
29	C₇×#95	C₂²	1	2	yes	—
30	C₇×#95	C₂²	1	2	yes	—
31	C₇×#95	C₂²	1	2	yes	—
32	C₇×#95	C₂²	1	2	yes	—
33	#2731	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂⁴×C₇	C₂³	1	2	yes	—
35	C₂×C₇×Q₈	—	2	2	no	—
36	C₂×C₇×Q₈	—	4	1	no	—
37	C₂×C₇×Q₈	—	4	1	no	—
38	C₂×C₇×Q₈	—	4	1	no	—
39	C₂×C₇×Q₈	—	4	1	no	—
40	C₂×C₇×Q₈	D₄	1	4	yes	—
41	C₂×C₇×D₄	—	2	2	no	—
42	C₂×C₇×D₄	—	4	1	no	—
43	C₂×C₇×D₄	—	4	1	no	—
44	C₂²×C₄×C₇	C₂³	1	2	yes	—
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	1	yes	characteristic
49	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₇	C₂³	1	2	yes	—
52	C₇×C₄⋊C₄	—	2	4	no	—
53	C₇×C₄⋊C₄	—	2	4	no	—
54	C₇×C₄⋊C₄	—	2	4	no	—
55	C₇×C₄⋊C₄	—	4	1	no	—
56	C₇×C₂²⋊C₄	—	2	4	no	—
57	C₇×C₂²⋊C₄	—	2	4	no	—
58	C₇×C₂²⋊C₄	—	2	4	no	—
59	C₇×C₂²⋊C₄	—	2	4	no	—
60	C₇×C₂²⋊C₄	—	4	2	no	—
61	C₇×C₂²⋊C₄	—	4	2	no	—
62	C₄²×C₇	—	2	2	no	—
63	C₄²×C₇	—	2	2	no	—
64	C₄²×C₇	—	4	1	no	—
65	C₂×C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
66	C₂×#124	C₂×C₇	1	1	yes	characteristic
67	#396	C₂×C₇	1	2	yes	complemented
68	#394	C₂×C₇	1	2	yes	—
69	#393	C₂×C₇	1	2	yes	complemented
70	#391	C₂×C₇	1	1	yes	characteristic, complemented
71	#388	C₂×C₇	1	2	yes	complemented
72	#388	C₂×C₇	1	2	yes	complemented
73	#387	C₂×C₇	1	2	yes	complemented
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	C₂⁴	1	1	yes	characteristic
82	C₇×Q₈	—	4	4	no	—
83	C₇×Q₈	—	4	4	no	—
84	C₇×D₄	—	4	4	no	—
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	—	2	1	no	—
87	C₂×C₄×C₇	—	2	1	no	—
88	C₂×C₄×C₇	—	2	1	no	—
89	C₂×C₄×C₇	—	2	1	no	—
90	C₂×C₄×C₇	—	2	1	no	—
91	C₂×C₄×C₇	—	2	1	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	2	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	2	no	—
102	C₂×C₄×C₇	—	2	2	no	—
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	2	2	no	—
105	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₇	#41	1	4	yes	—
109	C₂²×Q₈	C₂²×C₇	1	1	yes	characteristic
110	C₂²×Q₈	C₂²×C₇	1	1	yes	characteristic
111	C₂²×D₄	C₂²×C₇	1	1	yes	characteristic
112	C₄⋊Q₈	—	2	4	no	—
113	#124	—	2	4	no	—
114	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
115	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
116	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
117	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
118	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
119	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
120	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
121	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
122	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
123	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
124	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
125	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
126	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
127	#95	C₂²×C₇	1	2	yes	—
128	#95	C₂²×C₇	1	2	yes	—
129	#95	C₂²×C₇	1	2	yes	—
130	#95	C₂²×C₇	1	2	yes	—
131	#95	C₂²×C₇	1	2	yes	—
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	—	4	2	no	—
139	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
140	C₂²×C₇	C₂×#41	1	2	yes	—
141	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂²×C₇	2₋¹⁺⁴	1	2	yes	—
144	C₄×C₇	—	2	2	no	—
145	C₄×C₇	—	2	2	no	—
146	C₄×C₇	—	2	2	no	—
147	C₄×C₇	—	2	4	no	—
148	C₄×C₇	—	4	2	no	—
149	C₄×C₇	—	4	2	no	—
150	C₄×C₇	—	4	2	no	—
151	C₄×C₇	—	4	2	no	—
152	C₂⁴	C₂³×C₇	1	2	yes	—
153	C₂×Q₈	—	2	2	no	—
154	C₂×Q₈	—	4	1	no	—
155	C₂×Q₈	—	4	1	no	—
156	C₂×Q₈	—	4	1	no	—
157	C₂×Q₈	—	4	1	no	—
158	C₂×Q₈	C₇×D₄	1	4	yes	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	—	4	1	no	—
161	C₂×D₄	—	4	1	no	—
162	C₂²×C₄	C₂³×C₇	1	2	yes	—
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	1	yes	characteristic
167	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₇	1	2	yes	—
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	1	no	—
174	C₂²⋊C₄	—	2	4	no	—
175	C₂²⋊C₄	—	2	4	no	—
176	C₂²⋊C₄	—	2	4	no	—
177	C₂²⋊C₄	—	2	4	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₂²⋊C₄	—	4	2	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	4	1	no	—
183	C₂×C₇	—	8	2	no	—
184	C₂×C₇	#534	1	1	yes	characteristic
185	C₂×C₇	#542	1	2	yes	—
186	C₂×C₇	#548	1	2	yes	—
187	C₂×C₇	#555	1	1	yes	characteristic
188	C₂×C₇	#563	1	1	yes	characteristic
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	C₂⁴×C₇	1	1	yes	characteristic
197	Q₈	—	4	4	no	—
198	Q₈	—	4	4	no	—
199	D₄	—	4	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	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	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
221	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
222	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
223	C₂×C₄	C₇×#41	1	4	yes	—
224	C₇	#2731	1	1	yes	characteristic, direct product factor, Sylow
225	C₂²	—	4	2	no	—
226	C₂²	—	4	2	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	C₂²×C₇×D₄	1	1	yes	characteristic
233	C₂²	C₂×C₇×#41	1	2	yes	—
234	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
235	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
236	C₂²	C₇×2₋¹⁺⁴	1	2	yes	—
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	2	4	no	—
241	C₄	—	4	2	no	—
242	C₄	—	4	2	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₂	—	8	2	no	—
246	C₂	C₇×#534	1	1	yes	characteristic
247	C₂	C₇×#542	1	2	yes	—
248	C₂	C₇×#548	1	2	yes	—
249	C₂	C₇×#555	1	1	yes	characteristic
250	C₂	C₇×#563	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₇
prime radical	C₂²×C₇×D₄	C₂²
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. C₂×C₇×C₄⋊Q₈	C₂	semidirect
3. C₇×#396	C₂	semidirect
5. C₇×#393	C₂	semidirect
6. C₇×#391	C₂	semidirect
7. C₇×#388	C₂	semidirect
8. C₇×#388	C₂	semidirect
9. C₇×#387	C₂	semidirect
33. #2731	C₇	direct (N × H)
65. C₂×C₄⋊Q₈	C₂×C₇	semidirect
67. #396	C₂×C₇	semidirect
69. #393	C₂×C₇	semidirect
70. #391	C₂×C₇	semidirect
71. #388	C₂×C₇	semidirect
72. #388	C₂×C₇	semidirect
73. #387	C₂×C₇	semidirect
224. C₇	#2731	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×C₄⋊Q₈	2
C₂×C₇×#124	2
C₇×#396	2
C₇×#394	2
C₇×#393	2
C₇×#391	2
C₇×#388	2
C₇×#387	2
#2731	7

Subgroups — C7×#2731 #11742040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2731 #11742040