Subgroups — C₇×#2604 #11741913

Order 896.

470 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	C₂×C₇×#126	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#400	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#399	C₂	1	1	yes	maximal, characteristic
5	C₇×#395	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#393	C₂	1	1	yes	maximal, characteristic
7	C₇×#389	C₂	1	2	yes	maximal, complemented
8	C₇×#388	C₂	1	2	yes	maximal
9	C₇×#388	C₂	1	2	yes	maximal
10	C₇×#387	C₂	1	2	yes	maximal, complemented
11	C₇×#376	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
13	C₇×#126	—	2	4	no	—
14	C₄×C₇×Q₈	—	2	4	no	—
15	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₇	C₂²	1	2	yes	—
28	C₂×C₄²×C₇	C₂²	1	2	yes	—
29	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
30	C₇×#95	C₂²	1	2	yes	—
31	C₇×#95	C₂²	1	1	yes	characteristic
32	C₇×#95	C₂²	1	1	yes	characteristic
33	C₇×#95	C₂²	1	2	yes	—
34	C₇×#95	C₂²	1	2	yes	—
35	C₇×#95	C₂²	1	2	yes	—
36	C₇×#95	C₂²	1	2	yes	—
37	#2604	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂⁴×C₇	C₂³	1	1	yes	characteristic
39	C₂×C₇×Q₈	—	2	2	no	—
40	C₂×C₇×Q₈	—	2	4	no	—
41	C₂²×C₄×C₇	C₂³	1	2	yes	—
42	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₇	C₂³	1	2	yes	—
44	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₂³	1	2	yes	—
50	C₂²×C₄×C₇	C₂³	1	2	yes	—
51	C₇×C₄⋊C₄	—	2	4	no	—
52	C₇×C₄⋊C₄	—	2	2	no	—
53	C₇×C₄⋊C₄	—	2	2	no	—
54	C₇×C₄⋊C₄	—	2	2	no	—
55	C₇×C₄⋊C₄	—	2	2	no	—
56	C₇×C₄⋊C₄	—	2	4	no	—
57	C₇×C₄⋊C₄	—	2	4	no	—
58	C₇×C₂²⋊C₄	—	2	2	no	—
59	C₇×C₂²⋊C₄	—	2	2	no	—
60	C₇×C₂²⋊C₄	—	2	2	no	—
61	C₇×C₂²⋊C₄	—	2	4	no	—
62	C₇×C₂²⋊C₄	—	2	4	no	—
63	C₄²×C₇	—	2	4	no	—
64	C₄²×C₇	—	2	4	no	—
65	C₄²×C₇	D₄	1	4	yes	—
66	C₂×#126	C₂×C₇	1	1	yes	characteristic
67	C₂×C₄×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
68	#400	C₂×C₇	1	1	yes	characteristic, complemented
69	#399	C₂×C₇	1	1	yes	characteristic
70	#395	C₂×C₇	1	1	yes	characteristic, complemented
71	#393	C₂×C₇	1	1	yes	characteristic
72	#389	C₂×C₇	1	2	yes	complemented
73	#388	C₂×C₇	1	2	yes	—
74	#388	C₂×C₇	1	2	yes	—
75	#387	C₂×C₇	1	2	yes	complemented
76	#376	C₂×C₇	1	1	yes	characteristic, complemented
77	C₂³×C₇	—	2	1	no	—
78	C₂³×C₇	—	2	2	no	—
79	C₂³×C₇	—	2	2	no	—
80	C₂³×C₇	—	2	1	no	—
81	C₂³×C₇	—	2	1	no	—
82	C₂³×C₇	C₂⁴	1	1	yes	characteristic
83	C₇×Q₈	—	2	8	no	—
84	C₂×C₄×C₇	—	2	2	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	1	no	—
90	C₂×C₄×C₇	—	2	1	no	—
91	C₂×C₄×C₇	—	2	2	no	—
92	C₂×C₄×C₇	—	2	2	no	—
93	C₂×C₄×C₇	—	2	1	no	—
94	C₂×C₄×C₇	—	2	1	no	—
95	C₂×C₄×C₇	—	2	1	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	1	no	—
103	C₂×C₄×C₇	—	2	1	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	C₂×C₄×C₇	#41	1	4	yes	—
110	C₂²×Q₈	C₂²×C₇	1	1	yes	characteristic
111	#126	—	2	4	no	—
112	C₄×Q₈	—	2	4	no	—
113	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
114	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
115	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
116	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
117	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
118	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
119	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
120	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
121	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
122	C₂×C₂²⋊C₄	C₂²×C₇	1	1	yes	characteristic
123	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
124	C₂×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
125	C₂×C₄²	C₂²×C₇	1	2	yes	—
126	C₂×C₄²	C₂²×C₇	1	2	yes	—
127	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
128	#95	C₂²×C₇	1	2	yes	—
129	#95	C₂²×C₇	1	1	yes	characteristic
130	#95	C₂²×C₇	1	1	yes	characteristic
131	#95	C₂²×C₇	1	2	yes	—
132	#95	C₂²×C₇	1	2	yes	—
133	#95	C₂²×C₇	1	2	yes	—
134	#95	C₂²×C₇	1	2	yes	—
135	C₂²×C₇	—	4	1	no	—
136	C₂²×C₇	—	4	1	no	—
137	C₂²×C₇	—	4	1	no	—
138	C₂²×C₇	—	4	2	no	—
139	C₂²×C₇	—	4	2	no	—
140	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
141	C₂²×C₇	C₂×#41	1	2	yes	—
142	C₂²×C₇	C₂×#41	1	2	yes	—
143	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
144	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
145	C₄×C₇	—	2	2	no	—
146	C₄×C₇	—	2	4	no	—
147	C₄×C₇	—	2	2	no	—
148	C₄×C₇	—	2	2	no	—
149	C₄×C₇	—	2	4	no	—
150	C₄×C₇	—	4	1	no	—
151	C₄×C₇	—	4	2	no	—
152	C₄×C₇	—	4	1	no	—
153	C₄×C₇	—	4	1	no	—
154	C₄×C₇	—	4	2	no	—
155	C₂⁴	C₂³×C₇	1	1	yes	characteristic
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	—	2	4	no	—
158	C₂²×C₄	C₂³×C₇	1	2	yes	—
159	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
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	1	yes	characteristic
164	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
166	C₂²×C₄	C₂³×C₇	1	2	yes	—
167	C₂²×C₄	C₂³×C₇	1	2	yes	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	—	2	4	no	—
175	C₂²⋊C₄	—	2	2	no	—
176	C₂²⋊C₄	—	2	2	no	—
177	C₂²⋊C₄	—	2	2	no	—
178	C₂²⋊C₄	—	2	4	no	—
179	C₂²⋊C₄	—	2	4	no	—
180	C₄²	—	2	4	no	—
181	C₄²	—	2	4	no	—
182	C₄²	C₇×D₄	1	4	yes	—
183	C₂×C₇	—	8	1	no	—
184	C₂×C₇	#525	1	1	yes	characteristic
185	C₂×C₇	#532	1	1	yes	characteristic
186	C₂×C₇	#536	1	1	yes	characteristic
187	C₂×C₇	#552	1	2	yes	—
188	C₂×C₇	#556	1	2	yes	—
189	C₂³	—	2	1	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	1	no	—
193	C₂³	—	2	1	no	—
194	C₂³	C₂⁴×C₇	1	1	yes	characteristic
195	Q₈	—	2	8	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	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	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
218	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
219	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
220	C₂×C₄	C₇×#41	1	4	yes	—
221	C₂×C₄	C₇×#41	1	4	yes	—
222	C₇	#2604	1	1	yes	characteristic, direct product factor, Sylow
223	C₂²	—	4	1	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	C₂²×C₇×D₄	1	1	yes	characteristic
229	C₂²	C₂×C₇×#41	1	2	yes	—
230	C₂²	C₂×C₇×#41	1	2	yes	—
231	C₂²	C₇×2₋¹⁺⁴	1	1	yes	characteristic
232	C₂²	C₇×2₋¹⁺⁴	1	1	yes	characteristic
233	C₄	—	2	2	no	—
234	C₄	—	2	4	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	2	no	—
237	C₄	—	2	4	no	—
238	C₄	—	4	1	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	1	no	—
241	C₄	—	4	1	no	—
242	C₄	—	4	2	no	—
243	C₂	—	8	1	no	—
244	C₂	C₇×#525	1	1	yes	characteristic
245	C₂	C₇×#532	1	1	yes	characteristic
246	C₂	C₇×#536	1	1	yes	characteristic
247	C₂	C₇×#552	1	2	yes	—
248	C₂	C₇×#556	1	2	yes	—

	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. C₂×C₄×C₇×Q₈	C₂	semidirect
3. C₇×#400	C₂	semidirect
5. C₇×#395	C₂	semidirect
7. C₇×#389	C₂	semidirect
10. C₇×#387	C₂	semidirect
11. C₇×#376	C₂	semidirect
37. #2604	C₇	direct (N × H)
67. C₂×C₄×Q₈	C₂×C₇	semidirect
68. #400	C₂×C₇	semidirect
70. #395	C₂×C₇	semidirect
72. #389	C₂×C₇	semidirect
75. #387	C₂×C₇	semidirect
76. #376	C₂×C₇	semidirect
222. C₇	#2604	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
C₂×C₄×C₇×Q₈	2
C₇×#400	2
C₇×#399	2
C₇×#395	2
C₇×#393	2
C₇×#389	2
C₇×#388	2
C₇×#387	2
C₇×#376	2
#2604	7

Subgroups — C7×#2604 #11741913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2604 #11741913