Subgroups — C₇×#33704 #13210920

Order 1792.

682 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#493	C₂	1	1	yes	maximal, characteristic
3	C₇×#2244	C₂	1	4	yes	maximal, complemented
4	C₇×#2240	C₂	1	4	yes	maximal, complemented
5	C₇×#2140	C₂	1	4	yes	maximal, complemented
6	C₄×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₇×D₈	—	2	1	no	—
8	C₂²×C₇×D₈	C₂²	1	2	yes	—
9	C₇×#533	—	2	1	no	—
10	C₂×C₇×#127	—	2	1	no	—
11	C₂×C₇×M₃₂	C₂²	1	2	yes	—
12	C₇×#500	C₂²	1	2	yes	complemented
13	C₇×#499	—	2	1	no	—
14	C₇×#499	C₂²	1	2	yes	complemented
15	C₇×#498	—	2	1	no	—
16	C₇×#493	—	2	1	no	—
17	C₇×#493	—	2	1	no	—
18	C₇×#493	C₂²	1	2	yes	—
19	C₇×#493	C₂²	1	2	yes	—
20	C₇×#427	—	2	1	no	—
21	C₇×#427	C₂²	1	2	yes	—
22	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
23	C₇×#357	C₂²	1	16	yes	complemented
24	C₇×#346	C₂²	1	2	yes	complemented
25	C₄×C₇×C₁₆	C₂²	1	2	yes	complemented
26	#33704	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂²×C₇×D₄	—	2	2	no	—
28	C₂²×C₇×D₄	—	4	2	no	—
29	C₂×C₇×D₈	—	2	2	no	—
30	C₂×C₇×D₈	—	2	2	no	—
31	C₂×C₇×D₈	—	2	2	no	—
32	C₂×C₇×D₈	—	2	8	no	—
33	C₂×C₇×D₈	—	4	4	no	—
34	C₂×C₇×D₈	C₂³	1	4	yes	—
35	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
36	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
37	C₇×C₄⋊Q₈	—	2	2	no	—
38	C₇×#127	—	2	2	no	—
39	C₇×#127	—	2	2	no	—
40	C₇×#127	—	4	2	no	—
41	C₇×#125	—	2	2	no	—
42	C₇×C₂²⋊Q₈	—	4	2	no	—
43	C₄×C₇×Q₈	—	4	2	no	—
44	C₇×#117	—	2	2	no	—
45	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
46	C₇×M₃₂	—	2	4	no	—
47	C₂×C₇×C₁₆	C₂³	1	4	yes	—
48	C₇×#107	—	2	2	no	—
49	C₇×#107	C₂³	1	4	yes	—
50	C₇×#106	—	2	2	no	—
51	C₇×#106	—	2	2	no	—
52	C₄×C₇×C₈	D₄	1	1	yes	characteristic
53	C₄×C₇×C₈	D₄	1	1	yes	characteristic
54	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
55	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
56	#3160	C₂×C₇	1	1	yes	characteristic, complemented
57	C₂×#493	C₂×C₇	1	1	yes	characteristic
58	#2244	C₂×C₇	1	4	yes	complemented
59	#2240	C₂×C₇	1	4	yes	complemented
60	#2140	C₂×C₇	1	4	yes	complemented
61	C₄×M₃₂	C₂×C₇	1	1	yes	characteristic, complemented
62	C₂⁴×C₇	—	4	2	no	—
63	C₂×C₇×Q₈	—	4	2	no	—
64	C₂×C₇×D₄	—	2	4	no	—
65	C₂×C₇×D₄	—	4	4	no	—
66	C₂×C₇×D₄	—	4	4	no	—
67	C₂×C₇×D₄	—	4	4	no	—
68	C₂×C₇×D₄	—	4	2	no	—
69	C₂×C₇×D₄	—	4	4	no	—
70	C₂×C₇×D₄	—	8	8	no	—
71	C₂²×C₄×C₇	D₈	1	2	yes	—
72	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
73	C₇×D₈	—	2	8	no	—
74	C₇×D₈	—	4	4	no	—
75	C₇×D₈	—	4	4	no	—
76	C₇×D₈	—	4	4	no	—
77	C₂×C₇×C₈	SD₁₆	1	4	yes	—
78	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
81	C₂×C₇×C₈	#41	1	4	yes	—
82	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
83	C₇×C₄⋊C₄	—	2	4	no	—
84	C₇×C₄⋊C₄	—	4	4	no	—
85	C₇×C₄⋊C₄	—	4	2	no	—
86	C₇×C₂²⋊C₄	—	4	2	no	—
87	C₄²×C₇	—	4	2	no	—
88	C₄²×C₇	D₈	1	2	yes	—
89	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
91	C₇×C₁₆	—	2	4	no	—
92	C₂²×D₈	—	2	1	no	—
93	C₂²×D₈	C₂²×C₇	1	2	yes	—
94	#533	—	2	1	no	—
95	C₂×#127	—	2	1	no	—
96	C₂×M₃₂	C₂²×C₇	1	2	yes	—
97	#500	C₂²×C₇	1	2	yes	complemented
98	#499	—	2	1	no	—
99	#499	C₂²×C₇	1	2	yes	complemented
100	#498	—	2	1	no	—
101	#493	—	2	1	no	—
102	#493	—	2	1	no	—
103	#493	C₂²×C₇	1	2	yes	—
104	#493	C₂²×C₇	1	2	yes	—
105	#427	—	2	1	no	—
106	#427	C₂²×C₇	1	2	yes	—
107	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
108	#357	C₂²×C₇	1	16	yes	complemented
109	#346	C₂²×C₇	1	2	yes	complemented
110	C₄×C₁₆	C₂²×C₇	1	2	yes	complemented
111	C₂³×C₇	—	4	4	no	—
112	C₂³×C₇	—	8	8	no	—
113	C₂³×C₇	—	8	4	no	—
114	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
115	C₇×Q₈	—	8	4	no	—
116	C₇×D₄	—	4	4	no	—
117	C₇×D₄	—	4	4	no	—
118	C₇×D₄	—	8	8	no	—
119	C₇×D₄	—	8	4	no	—
120	C₇×D₄	—	8	8	no	—
121	C₂×C₄×C₇	—	2	4	no	—
122	C₂×C₄×C₇	—	4	4	no	—
123	C₂×C₄×C₇	#124	1	2	yes	—
124	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
127	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
128	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
129	C₇×C₈	#124	1	2	yes	—
130	C₇×C₈	C₂×SD₁₆	1	2	yes	—
131	C₇×C₈	C₂×SD₁₆	1	2	yes	—
132	C₇×C₈	C₂×#41	1	2	yes	—
133	C₂²×D₄	—	2	2	no	—
134	C₂²×D₄	—	4	2	no	—
135	C₂×D₈	—	2	2	no	—
136	C₂×D₈	—	2	2	no	—
137	C₂×D₈	—	2	2	no	—
138	C₂×D₈	—	2	8	no	—
139	C₂×D₈	—	4	4	no	—
140	C₂×D₈	C₂³×C₇	1	4	yes	—
141	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
142	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
143	C₄⋊Q₈	—	2	2	no	—
144	#127	—	2	2	no	—
145	#127	—	2	2	no	—
146	#127	—	4	2	no	—
147	#125	—	2	2	no	—
148	C₂²⋊Q₈	—	4	2	no	—
149	C₄×Q₈	—	4	2	no	—
150	#117	—	2	2	no	—
151	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
152	M₃₂	—	2	4	no	—
153	C₂×C₁₆	C₂³×C₇	1	4	yes	—
154	#107	—	2	2	no	—
155	#107	C₂³×C₇	1	4	yes	—
156	#106	—	2	2	no	—
157	#106	—	2	2	no	—
158	C₄×C₈	C₇×D₄	1	1	yes	characteristic
159	C₄×C₈	C₇×D₄	1	1	yes	characteristic
160	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
161	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
162	C₂²×C₇	—	2	2	no	—
163	C₂²×C₇	—	8	4	no	—
164	C₂²×C₇	—	8	8	no	—
165	C₂²×C₇	—	16	8	no	—
166	C₂²×C₇	#486	1	2	yes	—
167	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
168	C₄×C₇	—	2	2	no	—
169	C₄×C₇	—	8	4	no	—
170	C₄×C₇	#486	1	2	yes	—
171	C₄×C₇	#509	1	4	yes	—
172	C₄×C₇	C₂×#124	1	1	yes	characteristic
173	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
174	C₂⁴	—	4	2	no	—
175	C₂×Q₈	—	4	2	no	—
176	C₂×D₄	—	2	4	no	—
177	C₂×D₄	—	4	4	no	—
178	C₂×D₄	—	4	4	no	—
179	C₂×D₄	—	4	4	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂×D₄	—	4	4	no	—
182	C₂×D₄	—	8	8	no	—
183	C₂²×C₄	C₇×D₈	1	2	yes	—
184	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
185	D₈	—	2	8	no	—
186	D₈	—	4	4	no	—
187	D₈	—	4	4	no	—
188	D₈	—	4	4	no	—
189	C₂×C₈	C₇×SD₁₆	1	4	yes	—
190	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
191	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
192	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
193	C₂×C₈	C₇×#41	1	4	yes	—
194	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₂²⋊C₄	—	4	2	no	—
199	C₄²	—	4	2	no	—
200	C₄²	C₇×D₈	1	2	yes	—
201	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
202	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
203	C₁₆	—	2	4	no	—
204	C₂×C₇	—	2	2	no	—
205	C₂×C₇	—	16	4	no	—
206	C₂×C₇	C₂×#486	1	1	yes	characteristic
207	C₂×C₇	C₂×#509	1	2	yes	—
208	C₂³	—	4	4	no	—
209	C₂³	—	8	8	no	—
210	C₂³	—	8	4	no	—
211	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
212	Q₈	—	8	4	no	—
213	D₄	—	4	4	no	—
214	D₄	—	4	4	no	—
215	D₄	—	8	8	no	—
216	D₄	—	8	4	no	—
217	D₄	—	8	8	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	C₇×#124	1	2	yes	—
221	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
222	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
223	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
224	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
225	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
226	C₈	C₇×#124	1	2	yes	—
227	C₈	C₂×C₇×SD₁₆	1	2	yes	—
228	C₈	C₂×C₇×SD₁₆	1	2	yes	—
229	C₈	C₂×C₇×#41	1	2	yes	—
230	C₇	#33704	1	1	yes	characteristic, direct product factor, Sylow
231	C₂²	—	2	2	no	—
232	C₂²	—	8	8	no	—
233	C₂²	—	8	4	no	—
234	C₂²	—	16	8	no	—
235	C₂²	C₇×#486	1	2	yes	—
236	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
237	C₄	—	2	2	no	—
238	C₄	—	8	4	no	—
239	C₄	C₇×#486	1	2	yes	—
240	C₄	C₇×#509	1	4	yes	—
241	C₄	C₂×C₇×#124	1	1	yes	characteristic
242	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	—	16	4	no	—
245	C₂	C₂×C₇×#486	1	1	yes	characteristic
246	C₂	C₂×C₇×#509	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇
prime radical	C₂×C₇×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3160	C₂	semidirect
3. C₇×#2244	C₂	semidirect
4. C₇×#2240	C₂	semidirect
5. C₇×#2140	C₂	semidirect
6. C₄×C₇×M₃₂	C₂	semidirect
12. C₇×#500	C₂²	semidirect
14. C₇×#499	C₂²	semidirect
23. C₇×#357	C₂²	semidirect
24. C₇×#346	C₂²	semidirect
25. C₄×C₇×C₁₆	C₂²	semidirect
26. #33704	C₇	direct (N × H)
56. #3160	C₂×C₇	semidirect
58. #2244	C₂×C₇	semidirect
59. #2240	C₂×C₇	semidirect
60. #2140	C₂×C₇	semidirect
61. C₄×M₃₂	C₂×C₇	semidirect
97. #500	C₂²×C₇	semidirect
99. #499	C₂²×C₇	semidirect
108. #357	C₂²×C₇	semidirect
109. #346	C₂²×C₇	semidirect
110. C₄×C₁₆	C₂²×C₇	semidirect
230. C₇	#33704	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×#3160	2
C₂×C₇×#493	2
C₇×#2244	2
C₇×#2240	2
C₇×#2140	2
C₄×C₇×M₃₂	2
#33704	7

Subgroups — C7×#33704 #13210920

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#33704 #13210920