Subgroups — C₇×#2748 #11742057

Order 896.

464 conjugacy classes of subgroups.

The subgroup lattice has 250 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	2	yes	maximal, complemented
2	C₇×#398	C₂	1	2	yes	maximal, complemented
3	C₇×#396	C₂	1	2	yes	maximal, complemented
4	C₇×#390	C₂	1	1	yes	maximal, characteristic
5	C₇×#387	C₂	1	2	yes	maximal, complemented
6	C₇×#386	C₂	1	1	yes	maximal, characteristic
7	C₇×#386	C₂	1	1	yes	maximal, characteristic
8	C₇×#385	C₂	1	2	yes	maximal, complemented
9	C₇×#380	C₂	1	2	yes	maximal, complemented
10	C₂²×C₇×D₄	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₇	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	2	yes	—
16	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
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	1	yes	characteristic
24	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
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	2	yes	—
30	C₇×#95	C₂²	1	2	yes	—
31	#2748	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂⁴×C₇	C₂³	1	1	yes	characteristic
33	C₂⁴×C₇	C₂³	1	1	yes	characteristic
34	C₂×C₇×D₄	—	4	1	no	—
35	C₂×C₇×D₄	—	4	1	no	—
36	C₂×C₇×D₄	Q₈	1	4	yes	—
37	C₂²×C₄×C₇	—	2	4	no	—
38	C₂²×C₄×C₇	—	4	1	no	—
39	C₂²×C₄×C₇	—	4	2	no	—
40	C₂²×C₄×C₇	—	4	1	no	—
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	C₂²×C₄×C₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₇	C₂³	1	2	yes	—
46	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂³	1	2	yes	—
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	2	no	—
52	C₇×C₄⋊C₄	—	4	2	no	—
53	C₇×C₄⋊C₄	—	4	2	no	—
54	C₇×C₂²⋊C₄	—	2	4	no	—
55	C₇×C₂²⋊C₄	—	4	2	no	—
56	C₇×C₂²⋊C₄	—	4	2	no	—
57	C₇×C₂²⋊C₄	—	4	1	no	—
58	C₇×C₂²⋊C₄	—	4	1	no	—
59	C₇×C₂²⋊C₄	—	4	2	no	—
60	C₄²×C₇	—	4	1	no	—
61	#399	C₂×C₇	1	2	yes	complemented
62	#398	C₂×C₇	1	2	yes	complemented
63	#396	C₂×C₇	1	2	yes	complemented
64	#390	C₂×C₇	1	1	yes	characteristic
65	#387	C₂×C₇	1	2	yes	complemented
66	#386	C₂×C₇	1	1	yes	characteristic
67	#386	C₂×C₇	1	1	yes	characteristic
68	#385	C₂×C₇	1	2	yes	complemented
69	#380	C₂×C₇	1	2	yes	complemented
70	C₂³×C₇	—	2	2	no	—
71	C₂³×C₇	—	2	2	no	—
72	C₂³×C₇	—	2	2	no	—
73	C₂³×C₇	—	2	2	no	—
74	C₂³×C₇	—	2	1	no	—
75	C₂³×C₇	—	2	1	no	—
76	C₂³×C₇	—	2	1	no	—
77	C₂³×C₇	—	2	1	no	—
78	C₂³×C₇	C₂×Q₈	1	2	yes	—
79	C₂³×C₇	C₂×Q₈	1	2	yes	—
80	C₂³×C₇	C₂⁴	1	1	yes	characteristic
81	C₇×D₄	—	4	4	no	—
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	—	2	1	no	—
84	C₂×C₄×C₇	—	2	2	no	—
85	C₂×C₄×C₇	—	2	1	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	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	—	4	4	no	—
102	C₂×C₄×C₇	—	4	4	no	—
103	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
104	C₂×C₄×C₇	#41	1	4	yes	—
105	C₂²×D₄	C₂²×C₇	1	1	yes	characteristic
106	C₂³×C₄	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	2	yes	—
111	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
112	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
113	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
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	1	yes	characteristic
119	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
120	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
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	2	yes	—
125	#95	C₂²×C₇	1	2	yes	—
126	C₂²×C₇	—	2	2	no	—
127	C₂²×C₇	—	2	2	no	—
128	C₂²×C₇	—	2	4	no	—
129	C₂²×C₇	—	2	4	no	—
130	C₂²×C₇	—	4	2	no	—
131	C₂²×C₇	—	4	2	no	—
132	C₂²×C₇	—	4	1	no	—
133	C₂²×C₇	—	4	1	no	—
134	C₂²×C₇	—	4	1	no	—
135	C₂²×C₇	—	4	1	no	—
136	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
137	C₂²×C₇	C₂×#41	1	2	yes	—
138	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂²×C₇	H(5,𝔽₂)	1	2	yes	—
140	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
141	C₄×C₇	—	2	2	no	—
142	C₄×C₇	—	2	4	no	—
143	C₄×C₇	—	4	2	no	—
144	C₄×C₇	—	4	2	no	—
145	C₄×C₇	—	4	2	no	—
146	C₄×C₇	—	4	2	no	—
147	C₄×C₇	—	4	2	no	—
148	C₂⁴	C₂³×C₇	1	1	yes	characteristic
149	C₂⁴	C₂³×C₇	1	1	yes	characteristic
150	C₂×D₄	—	4	1	no	—
151	C₂×D₄	—	4	1	no	—
152	C₂×D₄	C₇×Q₈	1	4	yes	—
153	C₂²×C₄	—	2	4	no	—
154	C₂²×C₄	—	4	1	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	—	4	1	no	—
157	C₂²×C₄	C₂³×C₇	1	2	yes	—
158	C₂²×C₄	C₂³×C₇	1	2	yes	—
159	C₂²×C₄	C₂³×C₇	1	2	yes	—
160	C₂²×C₄	C₂³×C₇	1	2	yes	—
161	C₂²×C₄	C₂³×C₇	1	2	yes	—
162	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₇	1	2	yes	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	1	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₂²⋊C₄	—	2	4	no	—
171	C₂²⋊C₄	—	4	2	no	—
172	C₂²⋊C₄	—	4	2	no	—
173	C₂²⋊C₄	—	4	1	no	—
174	C₂²⋊C₄	—	4	1	no	—
175	C₂²⋊C₄	—	4	2	no	—
176	C₄²	—	4	1	no	—
177	C₂×C₇	—	4	2	no	—
178	C₂×C₇	—	4	2	no	—
179	C₂×C₇	#540	1	2	yes	—
180	C₂×C₇	#543	1	1	yes	characteristic
181	C₂×C₇	#551	1	1	yes	characteristic
182	C₂×C₇	#554	1	2	yes	—
183	C₂×C₇	#560	1	1	yes	characteristic
184	C₂³	—	2	2	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	2	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	1	no	—
189	C₂³	—	2	1	no	—
190	C₂³	—	2	1	no	—
191	C₂³	—	2	1	no	—
192	C₂³	C₂×C₇×Q₈	1	2	yes	—
193	C₂³	C₂×C₇×Q₈	1	2	yes	—
194	C₂³	C₂⁴×C₇	1	1	yes	characteristic
195	D₄	—	4	4	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	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	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	C₂×C₇×Q₈	1	2	yes	—
218	C₂×C₄	C₇×#41	1	4	yes	—
219	C₇	#2748	1	1	yes	characteristic, direct product factor, Sylow
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	4	no	—
223	C₂²	—	2	4	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	1	no	—
230	C₂²	C₂²×C₇×Q₈	1	1	yes	characteristic
231	C₂²	C₂×C₇×#41	1	2	yes	—
232	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
233	C₂²	C₇×H(5,𝔽₂)	1	2	yes	—
234	C₂²	C₇×H(5,𝔽₂)	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	4	no	—
237	C₄	—	4	2	no	—
238	C₄	—	4	2	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	2	no	—
241	C₄	—	4	2	no	—
242	C₂	—	4	2	no	—
243	C₂	—	4	2	no	—
244	C₂	C₇×#540	1	2	yes	—
245	C₂	C₇×#543	1	1	yes	characteristic
246	C₂	C₇×#551	1	1	yes	characteristic
247	C₂	C₇×#554	1	2	yes	—
248	C₂	C₇×#560	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. C₇×#399	C₂	semidirect
2. C₇×#398	C₂	semidirect
3. C₇×#396	C₂	semidirect
5. C₇×#387	C₂	semidirect
8. C₇×#385	C₂	semidirect
9. C₇×#380	C₂	semidirect
31. #2748	C₇	direct (N × H)
61. #399	C₂×C₇	semidirect
62. #398	C₂×C₇	semidirect
63. #396	C₂×C₇	semidirect
65. #387	C₂×C₇	semidirect
68. #385	C₂×C₇	semidirect
69. #380	C₂×C₇	semidirect
219. C₇	#2748	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×#398	2
C₇×#396	2
C₇×#390	2
C₇×#387	2
C₇×#386	2
C₇×#385	2
C₇×#380	2
#2748	7

Subgroups — C7×#2748 #11742057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2748 #11742057