Subgroups — C₇×#5825 #13061536

Order 1344.

540 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×#41	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#944	C₂	1	2	yes	maximal, complemented
4	C₇×#936	C₂	1	2	yes	maximal, complemented
5	C₇×#935	C₂	1	4	yes	maximal, complemented
6	C₇×#926	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#924	C₂	1	4	yes	maximal, complemented
8	C₇×#536	—	3	1	no	maximal, Hall
9	C₃×C₇×#41	—	2	4	no	—
10	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
12	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
14	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
15	C₂²×C₇×Dic₃	C₂²	1	2	yes	—
16	C₂×C₇×Dic₆	—	2	4	no	—
17	C₂×C₇×Dic₆	C₂²	1	2	yes	—
18	C₂×C₇×Dic₆	C₂²	1	2	yes	—
19	C₇×#217	C₂²	1	8	yes	—
20	C₇×#217	C₂²	1	2	yes	—
21	C₇×C₄⋊Dic₃	C₂²	1	2	yes	—
22	C₇×Dic₃⋊C₄	C₂²	1	8	yes	—
23	C₄×C₇×Dic₃	C₂²	1	2	yes	—
24	C₂×C₇×#41	D₃	1	1	yes	characteristic
25	C₂²×C₇×Q₈	—	3	1	no	—
26	C₇×C₄⋊Q₈	—	3	2	no	—
27	C₇×#124	—	3	2	no	—
28	C₇×#123	—	3	4	no	—
29	C₇×C₂²⋊Q₈	—	3	4	no	—
30	C₇×#117	—	3	1	no	—
31	#5825	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
32	C₂³×C₃×C₇	C₂³	1	2	yes	—
33	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
34	C₃×C₇×Q₈	—	2	2	no	—
35	C₃×C₇×D₄	—	2	4	no	—
36	C₃×C₇×D₄	—	2	2	no	—
37	C₂×C₃×C₄×C₇	—	2	4	no	—
38	C₂×C₃×C₄×C₇	D₄	1	4	yes	—
39	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
40	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₂×C₇×Dic₃	—	2	4	no	—
43	C₂×C₇×Dic₃	C₂³	1	4	yes	—
44	C₂×C₇×Dic₃	C₂³	1	4	yes	—
45	C₇×Dic₆	—	2	4	no	—
46	C₇×Dic₆	—	2	4	no	—
47	C₇×#41	—	2	4	no	—
48	C₂×C₇×Q₈	—	3	2	no	—
49	C₂×C₇×Q₈	—	3	2	no	—
50	C₂×C₇×Q₈	—	6	4	no	—
51	C₂×C₇×Q₈	C₂×D₃	1	1	yes	characteristic
52	C₂×C₇×D₄	C₂×D₃	1	2	yes	—
53	C₂×C₇×D₄	C₂×D₃	1	1	yes	characteristic
54	C₂²×C₄×C₇	—	3	2	no	—
55	C₂²×C₄×C₇	C₂×D₃	1	2	yes	—
56	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
57	C₇×C₄⋊C₄	—	3	8	no	—
58	C₇×C₄⋊C₄	—	3	2	no	—
59	C₇×C₂²⋊C₄	—	3	2	no	—
60	C₇×C₂²⋊C₄	—	3	8	no	—
61	C₄²×C₇	—	3	2	no	—
62	C₂×C₃×#41	C₂×C₇	1	1	yes	characteristic
63	C₂²×Dic₆	C₂×C₇	1	1	yes	characteristic, complemented
64	#944	C₂×C₇	1	2	yes	complemented
65	#936	C₂×C₇	1	2	yes	complemented
66	#935	C₂×C₇	1	4	yes	complemented
67	#926	C₂×C₇	1	1	yes	characteristic, complemented
68	#924	C₂×C₇	1	4	yes	complemented
69	C₂²×C₃×C₇	—	2	2	no	—
70	C₂²×C₃×C₇	—	2	2	no	—
71	C₂²×C₃×C₇	—	2	4	no	—
72	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
73	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
74	C₃×C₄×C₇	—	2	2	no	—
75	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
76	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
77	C₇×Dic₃	—	2	4	no	—
78	C₇×Dic₃	—	2	4	no	—
79	#536	—	3	1	no	Sylow
80	C₂³×C₇	C₂²×D₃	1	2	yes	—
81	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
82	C₇×Q₈	—	2	2	no	—
83	C₇×Q₈	—	6	4	no	—
84	C₇×Q₈	—	6	4	no	—
85	C₇×D₄	—	2	4	no	—
86	C₇×D₄	—	2	2	no	—
87	C₂×C₄×C₇	—	2	4	no	—
88	C₂×C₄×C₇	—	3	4	no	—
89	C₂×C₄×C₇	—	3	4	no	—
90	C₂×C₄×C₇	—	6	4	no	—
91	C₂×C₄×C₇	#67	1	4	yes	—
92	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
93	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
95	C₃×#41	—	2	4	no	—
96	C₂×C₃×Q₈	C₂²×C₇	1	1	yes	characteristic
97	C₂×C₃×D₄	C₂²×C₇	1	2	yes	—
98	C₂×C₃×D₄	C₂²×C₇	1	1	yes	characteristic
99	C₂²×C₃×C₄	C₂²×C₇	1	2	yes	—
100	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
101	C₂²×Dic₃	C₂²×C₇	1	2	yes	—
102	C₂×Dic₆	—	2	4	no	—
103	C₂×Dic₆	C₂²×C₇	1	2	yes	—
104	C₂×Dic₆	C₂²×C₇	1	2	yes	—
105	#217	C₂²×C₇	1	8	yes	—
106	#217	C₂²×C₇	1	2	yes	—
107	C₄⋊Dic₃	C₂²×C₇	1	2	yes	—
108	Dic₃⋊C₄	C₂²×C₇	1	8	yes	—
109	C₄×Dic₃	C₂²×C₇	1	2	yes	—
110	C₂×C₃×C₇	—	2	2	no	—
111	C₂×C₃×C₇	—	4	2	no	—
112	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₃×C₇	2₋¹⁺⁴	1	2	yes	—
114	C₂×#41	D₃×C₇	1	1	yes	characteristic
115	C₂²×Q₈	—	3	1	no	—
116	C₄⋊Q₈	—	3	2	no	—
117	#124	—	3	2	no	—
118	#123	—	3	4	no	—
119	C₂²⋊Q₈	—	3	4	no	—
120	#117	—	3	1	no	—
121	C₂²×C₇	—	2	2	no	—
122	C₂²×C₇	—	2	4	no	—
123	C₂²×C₇	—	2	2	no	—
124	C₂²×C₇	C₂×#67	1	2	yes	—
125	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
126	C₄×C₇	—	2	2	no	—
127	C₄×C₇	—	6	4	no	—
128	C₄×C₇	—	6	4	no	—
129	C₄×C₇	C₂×#67	1	2	yes	—
130	C₄×C₇	C₂×#67	1	2	yes	—
131	C₂³×C₃	C₂³×C₇	1	2	yes	—
132	C₂³×C₃	C₂³×C₇	1	1	yes	characteristic
133	C₃×Q₈	—	2	2	no	—
134	C₃×D₄	—	2	4	no	—
135	C₃×D₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	4	no	—
137	C₂×C₃×C₄	C₇×D₄	1	4	yes	—
138	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
139	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
141	C₂×Dic₃	—	2	4	no	—
142	C₂×Dic₃	C₂³×C₇	1	4	yes	—
143	C₂×Dic₃	C₂³×C₇	1	4	yes	—
144	Dic₆	—	2	4	no	—
145	Dic₆	—	2	4	no	—
146	C₃×C₇	#536	1	1	yes	characteristic, complemented, Hall
147	#41	—	2	4	no	—
148	C₂×Q₈	—	3	2	no	—
149	C₂×Q₈	—	3	2	no	—
150	C₂×Q₈	—	6	4	no	—
151	C₂×Q₈	C₂×D₃×C₇	1	1	yes	characteristic
152	C₂×D₄	C₂×D₃×C₇	1	2	yes	—
153	C₂×D₄	C₂×D₃×C₇	1	1	yes	characteristic
154	C₂²×C₄	—	3	2	no	—
155	C₂²×C₄	C₂×D₃×C₇	1	2	yes	—
156	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
157	C₄⋊C₄	—	3	2	no	—
158	C₄⋊C₄	—	3	8	no	—
159	C₂²⋊C₄	—	3	2	no	—
160	C₂²⋊C₄	—	3	8	no	—
161	C₄²	—	3	2	no	—
162	C₂×C₇	—	2	2	no	—
163	C₂×C₇	—	4	2	no	—
164	C₂×C₇	#1010	1	2	yes	—
165	C₂×C₇	C₂²×#67	1	1	yes	characteristic
166	C₂²×C₃	—	2	2	no	—
167	C₂²×C₃	—	2	2	no	—
168	C₂²×C₃	—	2	4	no	—
169	C₂²×C₃	C₂×C₇×D₄	1	2	yes	—
170	C₂²×C₃	C₂⁴×C₇	1	1	yes	characteristic
171	C₃×C₄	—	2	2	no	—
172	C₃×C₄	C₂×C₇×D₄	1	2	yes	—
173	C₃×C₄	C₂×C₇×D₄	1	2	yes	—
174	Dic₃	—	2	4	no	—
175	Dic₃	—	2	4	no	—
176	C₂³	C₂²×D₃×C₇	1	2	yes	—
177	C₂³	C₂²×D₃×C₇	1	1	yes	characteristic
178	Q₈	—	2	2	no	—
179	Q₈	—	6	4	no	—
180	Q₈	—	6	4	no	—
181	D₄	—	2	4	no	—
182	D₄	—	2	2	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	3	4	no	—
185	C₂×C₄	—	3	4	no	—
186	C₂×C₄	—	6	4	no	—
187	C₂×C₄	C₇×#67	1	4	yes	—
188	C₂×C₄	C₂²×D₃×C₇	1	2	yes	—
189	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
190	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
191	C₇	#5825	1	1	yes	characteristic, direct product factor, Sylow
192	C₂×C₃	—	2	2	no	—
193	C₂×C₃	—	4	2	no	—
194	C₂×C₃	C₂²×C₇×D₄	1	1	yes	characteristic
195	C₂×C₃	C₇×2₋¹⁺⁴	1	2	yes	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	4	no	—
198	C₂²	—	2	2	no	—
199	C₂²	C₂×C₇×#67	1	2	yes	—
200	C₂²	C₂³×D₃×C₇	1	1	yes	characteristic
201	C₄	—	2	2	no	—
202	C₄	—	6	4	no	—
203	C₄	—	6	4	no	—
204	C₄	C₂×C₇×#67	1	2	yes	—
205	C₄	C₂×C₇×#67	1	2	yes	—
206	C₃	C₇×#536	1	1	yes	characteristic, complemented, Sylow
207	C₂	—	2	2	no	—
208	C₂	—	4	2	no	—
209	C₂	C₇×#1010	1	2	yes	—
210	C₂	C₂²×C₇×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₇
Frattini	C₂²	C₂³×D₃×C₇
Fitting, prime radical	C₂×C₃×C₇×#41	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₂×C₇×#41	D₃
hypocenter	C₃	C₇×#536
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₇×#41

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇×Dic₆	C₂	semidirect
3. C₇×#944	C₂	semidirect
4. C₇×#936	C₂	semidirect
5. C₇×#935	C₂	semidirect
6. C₇×#926	C₂	semidirect
7. C₇×#924	C₂	semidirect
31. #5825	C₇	direct (N × H)
63. C₂²×Dic₆	C₂×C₇	semidirect
64. #944	C₂×C₇	semidirect
65. #936	C₂×C₇	semidirect
66. #935	C₂×C₇	semidirect
67. #926	C₂×C₇	semidirect
68. #924	C₂×C₇	semidirect
146. C₃×C₇	#536	semidirect
191. C₇	#5825	direct (N × H)
206. C₃	C₇×#536	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₇×#41	2
C₂²×C₇×Dic₆	2
C₇×#944	2
C₇×#936	2
C₇×#935	2
C₇×#926	2
C₇×#924	2
C₇×#536	3
#5825	7

Subgroups — C7×#5825 #13061536

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#5825 #13061536