Subgroups — C₇×#5766 #13061482

Order 1344.

496 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1007	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#942	C₂	1	2	yes	maximal, complemented
4	C₇×#919	C₂	1	2	yes	maximal, complemented
5	C₇×#918	C₂	1	4	yes	maximal, complemented
6	D₃×C₇×Q₁₆	C₂	1	2	yes	maximal, complemented
7	C₇×#904	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#901	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#578	—	3	1	no	maximal, Hall
10	C₂×C₃×C₇×Q₈	C₂²	1	2	yes	—
11	C₇×#239	—	2	2	no	—
12	C₇×#239	C₂²	1	4	yes	—
13	D₃×C₇×Q₈	—	2	2	no	—
14	D₃×C₇×Q₈	C₂²	1	4	yes	—
15	C₇×#235	—	2	4	no	—
16	C₇×#235	C₂²	1	1	yes	characteristic
17	C₇×#235	C₂²	1	2	yes	—
18	C₃×C₇×Q₁₆	C₂²	1	2	yes	—
19	C₃×C₇×Q₁₆	C₂²	1	2	yes	—
20	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
21	C₇×#216	C₂²	1	4	yes	—
22	C₇×#215	C₂²	1	4	yes	—
23	C₇×#208	C₂²	1	1	yes	characteristic
24	C₇×Dic₁₂	C₂²	1	1	yes	characteristic
25	C₇×D₂₄	C₂²	1	1	yes	characteristic
26	C₇×#204	C₂²	1	2	yes	—
27	C₇×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	—
28	D₃×C₇×C₈	C₂²	1	2	yes	—
29	C₇×2₋¹⁺⁴	—	3	2	no	—
30	C₇×#137	—	3	2	no	—
31	C₇×#137	—	3	4	no	—
32	C₇×#135	—	3	2	no	—
33	C₇×#135	—	3	1	no	—
34	C₂×C₇×Q₁₆	—	3	2	no	—
35	C₂×C₇×Q₁₆	D₃	1	1	yes	characteristic, complemented
36	C₇×#131	—	3	1	no	—
37	#5766	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
38	C₃×C₇×Q₈	—	2	2	no	—
39	C₃×C₇×Q₈	C₂³	1	4	yes	—
40	C₂×C₃×C₄×C₇	—	2	2	no	—
41	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₇×#67	—	2	2	no	—
43	C₇×#67	D₄	1	2	yes	—
44	C₇×D₁₂	—	2	4	no	—
45	C₇×D₁₂	D₄	1	1	yes	characteristic
46	C₇×D₁₂	C₂³	1	2	yes	—
47	C₄×D₃×C₇	—	2	4	no	—
48	C₄×D₃×C₇	—	2	2	no	—
49	C₄×D₃×C₇	—	2	4	no	—
50	C₄×D₃×C₇	C₂³	1	2	yes	—
51	C₇×Dic₆	—	2	4	no	—
52	C₇×Dic₆	D₄	1	1	yes	characteristic
53	C₇×Dic₆	C₂³	1	2	yes	—
54	C₃×C₇×C₈	C₂³	1	1	yes	characteristic
55	C₃×C₇×C₈	C₂³	1	1	yes	characteristic
56	C₇×C₃⋊C₈	C₂³	1	2	yes	—
57	C₇×#41	—	3	1	no	—
58	C₇×#41	—	3	4	no	—
59	C₇×#41	—	3	2	no	—
60	C₇×#41	—	6	4	no	—
61	C₇×#41	—	6	2	no	—
62	C₂×C₇×Q₈	—	3	4	no	—
63	C₂×C₇×Q₈	—	6	2	no	—
64	C₂×C₇×Q₈	C₂×D₃	1	2	yes	—
65	C₇×Q₁₆	—	3	4	no	—
66	C₇×Q₁₆	—	3	1	no	—
67	C₇×Q₁₆	C₂×D₃	1	2	yes	—
68	C₇×Q₁₆	C₂×D₃	1	2	yes	—
69	C₇×SD₁₆	—	3	4	no	—
70	C₇×SD₁₆	—	3	2	no	—
71	C₇×D₈	—	3	1	no	—
72	C₇×M₁₆	—	3	1	no	—
73	C₇×M₁₆	—	3	2	no	—
74	C₂×C₇×C₈	—	3	2	no	—
75	C₂×C₇×C₈	C₂×D₃	1	1	yes	characteristic
76	#1007	C₂×C₇	1	2	yes	complemented
77	C₂×C₃×Q₁₆	C₂×C₇	1	1	yes	characteristic, complemented
78	#942	C₂×C₇	1	2	yes	complemented
79	#919	C₂×C₇	1	2	yes	complemented
80	#918	C₂×C₇	1	4	yes	complemented
81	D₃×Q₁₆	C₂×C₇	1	2	yes	complemented
82	#904	C₂×C₇	1	1	yes	characteristic, complemented
83	#901	C₂×C₇	1	1	yes	characteristic, complemented
84	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
85	C₂×D₃×C₇	—	2	2	no	—
86	C₂×D₃×C₇	C₂×D₄	1	2	yes	—
87	C₃×C₄×C₇	—	2	4	no	—
88	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
89	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
90	C₇×Dic₃	—	2	2	no	—
91	C₇×Dic₃	C₂×D₄	1	2	yes	—
92	#578	—	3	1	no	Sylow
93	C₇×Q₈	—	2	2	no	—
94	C₇×Q₈	—	3	1	no	—
95	C₇×Q₈	—	3	2	no	—
96	C₇×Q₈	—	6	4	no	—
97	C₇×Q₈	C₂²×D₃	1	4	yes	—
98	C₇×D₄	—	3	2	no	—
99	C₇×D₄	—	3	1	no	—
100	C₇×D₄	—	3	2	no	—
101	C₇×D₄	—	6	4	no	—
102	C₇×D₄	—	6	2	no	—
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	3	2	no	—
105	C₂×C₄×C₇	—	6	4	no	—
106	C₂×C₄×C₇	—	6	4	no	—
107	C₂×C₄×C₇	—	6	2	no	—
108	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
109	C₇×C₈	—	3	2	no	—
110	C₇×C₈	C₂²×D₃	1	1	yes	characteristic
111	C₇×C₈	C₂²×D₃	1	1	yes	characteristic
112	C₂×C₃×Q₈	C₂²×C₇	1	2	yes	—
113	#239	—	2	2	no	—
114	#239	C₂²×C₇	1	4	yes	—
115	D₃×Q₈	—	2	2	no	—
116	D₃×Q₈	C₂²×C₇	1	4	yes	—
117	#235	—	2	4	no	—
118	#235	C₂²×C₇	1	1	yes	characteristic
119	#235	C₂²×C₇	1	2	yes	—
120	C₃×Q₁₆	C₂²×C₇	1	2	yes	—
121	C₃×Q₁₆	C₂²×C₇	1	2	yes	—
122	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
123	#216	C₂²×C₇	1	4	yes	—
124	#215	C₂²×C₇	1	4	yes	—
125	#208	C₂²×C₇	1	1	yes	characteristic
126	Dic₁₂	C₂²×C₇	1	1	yes	characteristic
127	D₂₄	C₂²×C₇	1	1	yes	characteristic
128	#204	C₂²×C₇	1	2	yes	—
129	ΓL(1,𝔽₂₅)	C₂²×C₇	1	2	yes	—
130	D₃×C₈	C₂²×C₇	1	2	yes	—
131	C₂×C₃×C₇	—	2	1	no	—
132	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
133	D₃×C₇	—	2	2	no	—
134	D₃×C₇	—	4	2	no	—
135	2₋¹⁺⁴	—	3	2	no	—
136	#137	—	3	2	no	—
137	#137	—	3	4	no	—
138	#135	—	3	2	no	—
139	#135	—	3	1	no	—
140	C₂×Q₁₆	—	3	2	no	—
141	C₂×Q₁₆	D₃×C₇	1	1	yes	characteristic, complemented
142	#131	—	3	1	no	—
143	C₂²×C₇	—	3	2	no	—
144	C₂²×C₇	—	6	2	no	—
145	C₂²×C₇	D₃×D₄	1	1	yes	characteristic
146	C₄×C₇	—	2	4	no	—
147	C₄×C₇	—	3	2	no	—
148	C₄×C₇	—	6	2	no	—
149	C₄×C₇	D₃×D₄	1	1	yes	characteristic
150	C₄×C₇	C₂³×D₃	1	1	yes	characteristic
151	C₃×Q₈	—	2	2	no	—
152	C₃×Q₈	C₂³×C₇	1	4	yes	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
155	#67	—	2	2	no	—
156	#67	C₇×D₄	1	2	yes	—
157	D₁₂	—	2	4	no	—
158	D₁₂	C₇×D₄	1	1	yes	characteristic
159	D₁₂	C₂³×C₇	1	2	yes	—
160	C₄×D₃	—	2	4	no	—
161	C₄×D₃	—	2	2	no	—
162	C₄×D₃	—	2	4	no	—
163	C₄×D₃	C₂³×C₇	1	2	yes	—
164	Dic₆	—	2	4	no	—
165	Dic₆	C₇×D₄	1	1	yes	characteristic
166	Dic₆	C₂³×C₇	1	2	yes	—
167	C₃×C₈	C₂³×C₇	1	1	yes	characteristic
168	C₃×C₈	C₂³×C₇	1	1	yes	characteristic
169	C₃⋊C₈	C₂³×C₇	1	2	yes	—
170	C₃×C₇	#578	1	1	yes	characteristic, complemented, Hall
171	#41	—	3	1	no	—
172	#41	—	3	4	no	—
173	#41	—	3	2	no	—
174	#41	—	6	4	no	—
175	#41	—	6	2	no	—
176	C₂×Q₈	—	3	4	no	—
177	C₂×Q₈	—	6	2	no	—
178	C₂×Q₈	C₂×D₃×C₇	1	2	yes	—
179	Q₁₆	—	3	4	no	—
180	Q₁₆	—	3	1	no	—
181	Q₁₆	C₂×D₃×C₇	1	2	yes	—
182	Q₁₆	C₂×D₃×C₇	1	2	yes	—
183	SD₁₆	—	3	4	no	—
184	SD₁₆	—	3	2	no	—
185	D₈	—	3	1	no	—
186	M₁₆	—	3	1	no	—
187	M₁₆	—	3	2	no	—
188	C₂×C₈	—	3	2	no	—
189	C₂×C₈	C₂×D₃×C₇	1	1	yes	characteristic
190	C₂×C₇	—	2	1	no	—
191	C₂×C₇	—	6	2	no	—
192	C₂×C₇	—	12	2	no	—
193	C₂×C₇	C₂×D₃×D₄	1	1	yes	characteristic
194	C₂²×C₃	C₂×C₇×D₄	1	1	yes	characteristic
195	C₂×D₃	—	2	2	no	—
196	C₂×D₃	C₂×C₇×D₄	1	2	yes	—
197	C₃×C₄	—	2	4	no	—
198	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
199	C₃×C₄	C₂⁴×C₇	1	1	yes	characteristic
200	Dic₃	—	2	2	no	—
201	Dic₃	C₂×C₇×D₄	1	2	yes	—
202	Q₈	—	2	2	no	—
203	Q₈	—	3	1	no	—
204	Q₈	—	3	2	no	—
205	Q₈	—	6	4	no	—
206	Q₈	C₂²×D₃×C₇	1	4	yes	—
207	D₄	—	3	2	no	—
208	D₄	—	3	1	no	—
209	D₄	—	3	2	no	—
210	D₄	—	6	4	no	—
211	D₄	—	6	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	3	2	no	—
214	C₂×C₄	—	6	4	no	—
215	C₂×C₄	—	6	4	no	—
216	C₂×C₄	—	6	2	no	—
217	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
218	C₈	—	3	2	no	—
219	C₈	C₂²×D₃×C₇	1	1	yes	characteristic
220	C₈	C₂²×D₃×C₇	1	1	yes	characteristic
221	C₇	#5766	1	1	yes	characteristic, direct product factor, Sylow
222	C₂×C₃	—	2	1	no	—
223	C₂×C₃	C₂²×C₇×D₄	1	1	yes	characteristic
224	D₃	—	2	2	no	—
225	D₃	—	4	2	no	—
226	C₂²	—	3	2	no	—
227	C₂²	—	6	2	no	—
228	C₂²	D₃×C₇×D₄	1	1	yes	characteristic
229	C₄	—	2	4	no	—
230	C₄	—	3	2	no	—
231	C₄	—	6	2	no	—
232	C₄	D₃×C₇×D₄	1	1	yes	characteristic
233	C₄	C₂³×D₃×C₇	1	1	yes	characteristic
234	C₃	C₇×#578	1	1	yes	characteristic, complemented, Sylow
235	C₂	—	2	1	no	—
236	C₂	—	6	2	no	—
237	C₂	—	12	2	no	—
238	C₂	C₂×D₃×C₇×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇	C₂×D₃×D₄
commutator, cosocle	C₃×C₄	C₂⁴×C₇
Frattini	C₄	C₂³×D₃×C₇
Fitting	C₂×C₃×C₇×Q₁₆	C₂
socle	C₂×C₃×C₇	C₂²×D₄
hypercenter	C₂×C₇×Q₁₆	D₃
hypocenter	C₃	C₇×#578
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₇×#1007	C₂	semidirect
2. C₂×C₃×C₇×Q₁₆	C₂	semidirect
3. C₇×#942	C₂	semidirect
4. C₇×#919	C₂	semidirect
5. C₇×#918	C₂	semidirect
6. D₃×C₇×Q₁₆	C₂	semidirect
7. C₇×#904	C₂	semidirect
8. C₇×#901	C₂	semidirect
35. C₂×C₇×Q₁₆	D₃	semidirect
37. #5766	C₇	direct (N × H)
76. #1007	C₂×C₇	semidirect
77. C₂×C₃×Q₁₆	C₂×C₇	semidirect
78. #942	C₂×C₇	semidirect
79. #919	C₂×C₇	semidirect
80. #918	C₂×C₇	semidirect
81. D₃×Q₁₆	C₂×C₇	semidirect
82. #904	C₂×C₇	semidirect
83. #901	C₂×C₇	semidirect
141. C₂×Q₁₆	D₃×C₇	semidirect
170. C₃×C₇	#578	semidirect
221. C₇	#5766	direct (N × H)
234. C₃	C₇×#578	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇×#1007	2
C₂×C₃×C₇×Q₁₆	2
C₇×#942	2
C₇×#919	2
C₇×#918	2
D₃×C₇×Q₁₆	2
C₇×#904	2
C₇×#901	2
C₇×#578	3
#5766	7

Subgroups — C7×#5766 #13061482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#5766 #13061482