Subgroups — C₇×#36644 #13213860

Order 1792.

4086 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#532	C₂	1	6	yes	maximal, complemented
2	C₇×#2868	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#2867	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#2858	C₂	1	6	yes	maximal, complemented
5	C₇×#2856	C₂	1	8	yes	maximal, complemented
6	C₄×C₇×#127	C₂	1	8	yes	maximal, complemented
7	C₂×C₄³×C₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₇×#41	C₂²	1	4	yes	—
9	C₇×#532	—	2	48	no	—
10	C₂×C₇×C₄⋊Q₈	C₂²	1	6	yes	—
11	C₂×C₇×#127	C₂²	1	6	yes	—
12	C₂×C₇×#127	C₂²	1	8	yes	—
13	C₂×C₇×#124	C₂²	1	12	yes	—
14	C₂×C₇×#121	C₂²	1	24	yes	—
15	C₂×C₄×C₇×D₄	C₂²	1	24	yes	—
16	C₂²×C₄²×C₇	C₂²	1	4	yes	—
17	C₂²×C₄²×C₇	C₂²	1	3	yes	—
18	C₇×#390	C₂²	1	48	yes	—
19	C₇×#384	C₂²	1	8	yes	—
20	C₄³×C₇	C₂²	1	4	yes	—
21	C₄³×C₇	C₂²	1	3	yes	—
22	C₄³×C₇	C₂²	1	1	yes	characteristic
23	#36644	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂×C₇×#41	—	2	48	no	—
25	C₂²×C₇×Q₈	C₂³	1	4	yes	—
26	C₂²×C₇×D₄	C₂³	1	8	yes	—
27	C₂²×C₇×D₄	C₂³	1	4	yes	—
28	C₂²×C₇×D₄	C₂³	1	24	yes	—
29	C₂³×C₄×C₇	C₂³	1	8	yes	—
30	C₂³×C₄×C₇	C₂³	1	6	yes	—
31	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
32	C₇×C₄⋊Q₈	—	2	24	no	—
33	C₇×#127	—	2	24	no	—
34	C₇×#127	—	2	32	no	—
35	C₇×#124	—	2	48	no	—
36	C₇×#121	—	2	96	no	—
37	C₄×C₇×D₄	—	2	96	no	—
38	C₂×C₇×C₄⋊C₄	C₂³	1	24	yes	—
39	C₂×C₇×C₂²⋊C₄	C₂³	1	48	yes	—
40	C₂×C₄²×C₇	—	2	16	no	—
41	C₂×C₄²×C₇	D₄	1	24	yes	complemented
42	C₂×C₄²×C₇	C₂³	1	12	yes	—
43	C₂×C₄²×C₇	C₂³	1	3	yes	—
44	C₂×C₄²×C₇	C₂³	1	3	yes	—
45	C₂×C₄²×C₇	C₂³	1	6	yes	—
46	C₂×C₄²×C₇	C₂³	1	4	yes	—
47	C₂×#532	C₂×C₇	1	6	yes	complemented
48	#2868	C₂×C₇	1	1	yes	characteristic, complemented
49	#2867	C₂×C₇	1	1	yes	characteristic, complemented
50	#2858	C₂×C₇	1	6	yes	complemented
51	#2856	C₂×C₇	1	8	yes	complemented
52	C₄×#127	C₂×C₇	1	8	yes	complemented
53	C₂×C₄³	C₂×C₇	1	1	yes	characteristic, complemented
54	C₂⁴×C₇	C₂⁴	1	8	yes	—
55	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
56	C₇×#41	—	4	64	no	—
57	C₂×C₇×Q₈	—	2	24	no	—
58	C₂×C₇×D₄	—	2	48	no	—
59	C₂×C₇×D₄	—	2	48	no	—
60	C₂×C₇×D₄	—	2	24	no	—
61	C₂×C₇×D₄	—	2	96	no	—
62	C₂²×C₄×C₇	—	2	24	no	—
63	C₂²×C₄×C₇	—	2	48	no	—
64	C₂²×C₄×C₇	C₂×D₄	1	24	yes	—
65	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
66	C₂²×C₄×C₇	C₂⁴	1	8	yes	—
67	C₂²×C₄×C₇	C₂⁴	1	6	yes	—
68	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
69	C₂²×C₄×C₇	C₂⁴	1	6	yes	—
70	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
71	C₇×C₄⋊C₄	—	2	48	no	—
72	C₇×C₂²⋊C₄	—	2	96	no	—
73	C₄²×C₇	—	2	24	no	—
74	C₄²×C₇	C₂×D₄	1	24	yes	—
75	C₄²×C₇	C₂×D₄	1	12	yes	—
76	C₄²×C₇	C₂×D₄	1	12	yes	—
77	C₄²×C₇	#41	1	16	yes	complemented
78	C₂²×#41	C₂²×C₇	1	4	yes	—
79	#532	—	2	48	no	—
80	C₂×C₄⋊Q₈	C₂²×C₇	1	6	yes	—
81	C₂×#127	C₂²×C₇	1	8	yes	—
82	C₂×#127	C₂²×C₇	1	6	yes	—
83	C₂×#124	C₂²×C₇	1	12	yes	—
84	C₂×#121	C₂²×C₇	1	24	yes	—
85	C₂×C₄×D₄	C₂²×C₇	1	24	yes	—
86	C₂²×C₄²	C₂²×C₇	1	4	yes	—
87	C₂²×C₄²	C₂²×C₇	1	3	yes	—
88	#390	C₂²×C₇	1	48	yes	—
89	#384	C₂²×C₇	1	8	yes	—
90	C₄³	C₂²×C₇	1	4	yes	—
91	C₄³	C₂²×C₇	1	3	yes	—
92	C₄³	C₂²×C₇	1	1	yes	characteristic
93	C₂³×C₇	—	2	4	no	—
94	C₂³×C₇	—	2	24	no	—
95	C₂³×C₇	—	2	32	no	—
96	C₂³×C₇	C₂²×D₄	1	6	yes	—
97	C₂³×C₇	C₂⁵	1	1	yes	characteristic
98	C₇×Q₈	—	4	16	no	—
99	C₇×D₄	—	4	32	no	—
100	C₇×D₄	—	4	16	no	—
101	C₇×D₄	—	4	96	no	—
102	C₂×C₄×C₇	—	2	12	no	—
103	C₂×C₄×C₇	—	2	48	no	—
104	C₂×C₄×C₇	—	2	24	no	—
105	C₂×C₄×C₇	—	4	32	no	—
106	C₂×C₄×C₇	#127	1	16	yes	complemented
107	C₂×C₄×C₇	C₂²×D₄	1	12	yes	—
108	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
109	C₂×C₄×C₇	C₂²×D₄	1	6	yes	—
110	C₂×C₄×C₇	C₂²×D₄	1	12	yes	—
111	C₂×C₄×C₇	C₂×#41	1	24	yes	—
112	C₂×#41	—	2	48	no	—
113	C₂²×Q₈	C₂³×C₇	1	4	yes	—
114	C₂²×D₄	C₂³×C₇	1	24	yes	—
115	C₂²×D₄	C₂³×C₇	1	8	yes	—
116	C₂²×D₄	C₂³×C₇	1	4	yes	—
117	C₂³×C₄	C₂³×C₇	1	8	yes	—
118	C₂³×C₄	C₂³×C₇	1	6	yes	—
119	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
120	C₄⋊Q₈	—	2	24	no	—
121	#127	—	2	24	no	—
122	#127	—	2	32	no	—
123	#124	—	2	48	no	—
124	#121	—	2	96	no	—
125	C₄×D₄	—	2	96	no	—
126	C₂×C₄⋊C₄	C₂³×C₇	1	24	yes	—
127	C₂×C₂²⋊C₄	C₂³×C₇	1	48	yes	—
128	C₂×C₄²	—	2	16	no	—
129	C₂×C₄²	C₇×D₄	1	24	yes	complemented
130	C₂×C₄²	C₂³×C₇	1	12	yes	—
131	C₂×C₄²	C₂³×C₇	1	6	yes	—
132	C₂×C₄²	C₂³×C₇	1	3	yes	—
133	C₂×C₄²	C₂³×C₇	1	3	yes	—
134	C₂×C₄²	C₂³×C₇	1	4	yes	—
135	C₂²×C₇	—	2	12	no	—
136	C₂²×C₇	—	4	8	no	—
137	C₂²×C₇	—	4	48	no	—
138	C₂²×C₇	C₂×#127	1	4	yes	—
139	C₂²×C₇	C₂³×D₄	1	3	yes	—
140	C₂²×C₇	C₂²×#41	1	4	yes	—
141	C₄×C₇	—	2	12	no	—
142	C₄×C₇	—	4	8	no	—
143	C₄×C₇	C₂×#127	1	4	yes	—
144	C₄×C₇	C₂×#127	1	4	yes	—
145	C₄×C₇	#532	1	24	yes	complemented
146	C₂⁴	C₂⁴×C₇	1	8	yes	—
147	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
148	#41	—	4	64	no	—
149	C₂×Q₈	—	2	24	no	—
150	C₂×D₄	—	2	48	no	—
151	C₂×D₄	—	2	48	no	—
152	C₂×D₄	—	2	24	no	—
153	C₂×D₄	—	2	96	no	—
154	C₂²×C₄	—	2	24	no	—
155	C₂²×C₄	—	2	48	no	—
156	C₂²×C₄	C₂×C₇×D₄	1	24	yes	—
157	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
158	C₂²×C₄	C₂⁴×C₇	1	8	yes	—
159	C₂²×C₄	C₂⁴×C₇	1	6	yes	—
160	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
161	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
162	C₂²×C₄	C₂⁴×C₇	1	6	yes	—
163	C₄⋊C₄	—	2	48	no	—
164	C₂²⋊C₄	—	2	96	no	—
165	C₄²	—	2	24	no	—
166	C₄²	C₂×C₇×D₄	1	24	yes	—
167	C₄²	C₂×C₇×D₄	1	12	yes	—
168	C₄²	C₂×C₇×D₄	1	12	yes	—
169	C₄²	C₇×#41	1	16	yes	complemented
170	C₂×C₇	—	2	4	no	—
171	C₂×C₇	—	8	8	no	—
172	C₂×C₇	C₂²×#127	1	1	yes	characteristic
173	C₂×C₇	C₂×#532	1	6	yes	—
174	C₂³	—	2	4	no	—
175	C₂³	—	2	24	no	—
176	C₂³	—	2	32	no	—
177	C₂³	C₂²×C₇×D₄	1	6	yes	—
178	C₂³	C₂⁵×C₇	1	1	yes	characteristic
179	Q₈	—	4	16	no	—
180	D₄	—	4	32	no	—
181	D₄	—	4	16	no	—
182	D₄	—	4	96	no	—
183	C₂×C₄	—	2	12	no	—
184	C₂×C₄	—	2	48	no	—
185	C₂×C₄	—	2	24	no	—
186	C₂×C₄	—	4	32	no	—
187	C₂×C₄	C₇×#127	1	16	yes	complemented
188	C₂×C₄	C₂²×C₇×D₄	1	12	yes	—
189	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
190	C₂×C₄	C₂²×C₇×D₄	1	6	yes	—
191	C₂×C₄	C₂²×C₇×D₄	1	12	yes	—
192	C₂×C₄	C₂×C₇×#41	1	24	yes	—
193	C₇	#36644	1	1	yes	characteristic, direct product factor, Sylow
194	C₂²	—	2	12	no	—
195	C₂²	—	4	8	no	—
196	C₂²	—	4	48	no	—
197	C₂²	C₂×C₇×#127	1	4	yes	—
198	C₂²	C₂³×C₇×D₄	1	3	yes	—
199	C₂²	C₂²×C₇×#41	1	4	yes	—
200	C₄	—	2	12	no	—
201	C₄	—	4	8	no	—
202	C₄	C₂×C₇×#127	1	4	yes	—
203	C₄	C₂×C₇×#127	1	4	yes	—
204	C₄	C₇×#532	1	24	yes	complemented
205	C₂	—	2	4	no	—
206	C₂	—	8	8	no	—
207	C₂	C₂²×C₇×#127	1	1	yes	characteristic
208	C₂	C₂×C₇×#532	1	6	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₇
socle	C₂³×C₇	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#532	C₂	semidirect
2. C₇×#2868	C₂	semidirect
3. C₇×#2867	C₂	semidirect
4. C₇×#2858	C₂	semidirect
5. C₇×#2856	C₂	semidirect
6. C₄×C₇×#127	C₂	semidirect
7. C₂×C₄³×C₇	C₂	semidirect
23. #36644	C₇	direct (N × H)
41. C₂×C₄²×C₇	D₄	semidirect
47. C₂×#532	C₂×C₇	semidirect
48. #2868	C₂×C₇	semidirect
49. #2867	C₂×C₇	semidirect
50. #2858	C₂×C₇	semidirect
51. #2856	C₂×C₇	semidirect
52. C₄×#127	C₂×C₇	semidirect
53. C₂×C₄³	C₂×C₇	semidirect
77. C₄²×C₇	#41	semidirect
106. C₂×C₄×C₇	#127	semidirect
129. C₂×C₄²	C₇×D₄	semidirect
145. C₄×C₇	#532	semidirect
169. C₄²	C₇×#41	semidirect
187. C₂×C₄	C₇×#127	semidirect
193. C₇	#36644	direct (N × H)
204. C₄	C₇×#532	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#532	2
C₇×#2868	2
C₇×#2867	2
C₇×#2858	2
C₇×#2856	2
C₄×C₇×#127	2
C₂×C₄³×C₇	2
#36644	7

Subgroups — C7×#36644 #13213860

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#36644 #13213860