Subgroups — C₇×#5805 #13061521

Order 1344.

668 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#235	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#953	C₂	1	4	yes	maximal, complemented
4	C₇×#940	C₂	1	2	yes	maximal, complemented
5	C₇×#938	C₂	1	4	yes	maximal, complemented
6	C₇×#936	C₂	1	2	yes	maximal, complemented
7	C₇×#926	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#535	—	3	1	no	maximal, Hall
9	C₂⁴×C₃×C₇	C₂²	1	2	yes	—
10	C₂×C₃×C₇×D₄	—	2	4	no	—
11	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
12	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
13	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₂×C₇×#67	C₂²	1	2	yes	—
15	C₂×C₇×#67	C₂²	1	8	yes	complemented
16	C₇×#235	—	2	4	no	—
17	C₂×C₇×D₁₂	C₂²	1	1	yes	characteristic, complemented
18	C₂×C₄×D₃×C₇	C₂²	1	2	yes	complemented
19	C₂×C₇×Dic₆	C₂²	1	1	yes	characteristic, complemented
20	C₇×#217	C₂²	1	2	yes	—
21	C₇×#217	C₂²	1	8	yes	complemented
22	C₇×C₄⋊Dic₃	C₂²	1	2	yes	complemented
23	C₄×C₇×Dic₃	C₂²	1	2	yes	complemented
24	C₂×C₇×#41	—	3	1	no	—
25	C₂²×C₇×D₄	D₃	1	1	yes	characteristic, complemented
26	C₇×#127	—	3	2	no	—
27	C₇×#124	—	3	2	no	—
28	C₇×#121	—	3	4	no	—
29	C₇×#120	—	3	4	no	—
30	C₇×#117	—	3	1	no	—
31	#5805	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
32	C₂³×C₃×C₇	—	2	4	no	—
33	C₂³×C₃×C₇	—	4	4	no	—
34	C₂³×C₃×C₇	C₂³	1	4	yes	—
35	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
36	C₂²×D₃×C₇	C₂³	1	2	yes	—
37	C₃×C₇×D₄	—	2	4	no	—
38	C₃×C₇×D₄	—	2	4	no	—
39	C₂×C₃×C₄×C₇	D₄	1	4	yes	complemented
40	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₇×#67	—	2	4	no	—
43	C₇×#67	—	4	8	no	—
44	C₂×C₇×Dic₃	C₂³	1	4	yes	—
45	C₂×C₇×Dic₃	C₂³	1	2	yes	—
46	C₇×D₁₂	—	2	2	no	—
47	C₄×D₃×C₇	—	2	4	no	—
48	C₇×Dic₆	—	2	2	no	—
49	C₂⁴×C₇	C₂×D₃	1	2	yes	—
50	C₇×#41	—	6	4	no	—
51	C₂×C₇×Q₈	—	3	1	no	—
52	C₂×C₇×D₄	—	2	4	no	—
53	C₂×C₇×D₄	—	3	2	no	—
54	C₂×C₇×D₄	—	3	8	no	—
55	C₂×C₇×D₄	—	3	1	no	—
56	C₂×C₇×D₄	C₂×D₃	1	2	yes	—
57	C₂×C₇×D₄	C₂×D₃	1	2	yes	—
58	C₂²×C₄×C₇	—	3	2	no	—
59	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
60	C₇×C₄⋊C₄	—	3	2	no	—
61	C₇×C₂²⋊C₄	—	3	2	no	—
62	C₇×C₂²⋊C₄	—	3	8	no	—
63	C₄²×C₇	—	3	2	no	—
64	C₂²×C₃×D₄	C₂×C₇	1	1	yes	characteristic, complemented
65	C₂×#235	C₂×C₇	1	1	yes	characteristic, complemented
66	#953	C₂×C₇	1	4	yes	complemented
67	#940	C₂×C₇	1	2	yes	complemented
68	#938	C₂×C₇	1	4	yes	complemented
69	#936	C₂×C₇	1	2	yes	complemented
70	#926	C₂×C₇	1	1	yes	characteristic, complemented
71	C₂²×C₃×C₇	—	2	2	no	—
72	C₂²×C₃×C₇	—	2	4	no	—
73	C₂²×C₃×C₇	—	2	8	no	—
74	C₂²×C₃×C₇	—	4	8	no	—
75	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
76	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
77	C₂×D₃×C₇	—	2	2	no	—
78	C₂×D₃×C₇	—	2	4	no	—
79	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
80	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
81	C₇×Dic₃	—	2	2	no	—
82	C₇×Dic₃	—	2	4	no	—
83	#535	—	3	1	no	Sylow
84	C₂³×C₇	—	2	4	no	—
85	C₂³×C₇	—	3	2	no	—
86	C₂³×C₇	—	4	4	no	—
87	C₂³×C₇	C₂²×D₃	1	4	yes	—
88	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
89	C₇×Q₈	—	6	2	no	—
90	C₇×D₄	—	2	4	no	—
91	C₇×D₄	—	2	4	no	—
92	C₇×D₄	—	6	4	no	—
93	C₇×D₄	—	6	2	no	—
94	C₇×D₄	—	12	8	no	—
95	C₂×C₄×C₇	—	3	4	no	—
96	C₂×C₄×C₇	—	3	2	no	—
97	C₂×C₄×C₇	—	6	4	no	—
98	C₂×C₄×C₇	#67	1	4	yes	complemented
99	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
100	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
101	C₂⁴×C₃	C₂²×C₇	1	2	yes	—
102	C₂×C₃×D₄	—	2	4	no	—
103	C₂×C₃×D₄	C₂²×C₇	1	2	yes	—
104	C₂×C₃×D₄	C₂²×C₇	1	2	yes	—
105	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
106	C₂×#67	C₂²×C₇	1	2	yes	—
107	C₂×#67	C₂²×C₇	1	8	yes	complemented
108	#235	—	2	4	no	—
109	C₂×D₁₂	C₂²×C₇	1	1	yes	characteristic, complemented
110	C₂×C₄×D₃	C₂²×C₇	1	2	yes	complemented
111	C₂×Dic₆	C₂²×C₇	1	1	yes	characteristic, complemented
112	#217	C₂²×C₇	1	2	yes	—
113	#217	C₂²×C₇	1	8	yes	complemented
114	C₄⋊Dic₃	C₂²×C₇	1	2	yes	complemented
115	C₄×Dic₃	C₂²×C₇	1	2	yes	complemented
116	C₂×C₃×C₇	—	2	2	no	—
117	C₂×C₃×C₇	—	4	4	no	—
118	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₃×C₇	H(5,𝔽₂)	1	2	yes	—
120	D₃×C₇	—	4	2	no	—
121	C₂×#41	—	3	1	no	—
122	C₂²×D₄	D₃×C₇	1	1	yes	characteristic, complemented
123	#127	—	3	2	no	—
124	#124	—	3	2	no	—
125	#121	—	3	4	no	—
126	#120	—	3	4	no	—
127	#117	—	3	1	no	—
128	C₂²×C₇	—	2	2	no	—
129	C₂²×C₇	—	2	8	no	—
130	C₂²×C₇	—	2	4	no	—
131	C₂²×C₇	—	4	8	no	—
132	C₂²×C₇	—	6	4	no	—
133	C₂²×C₇	—	6	2	no	—
134	C₂²×C₇	C₂×#67	1	2	yes	—
135	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
136	C₄×C₇	—	6	4	no	—
137	C₄×C₇	—	6	2	no	—
138	C₄×C₇	C₂×#67	1	2	yes	—
139	C₄×C₇	C₂×#67	1	2	yes	—
140	C₂³×C₃	—	2	4	no	—
141	C₂³×C₃	—	4	4	no	—
142	C₂³×C₃	C₂³×C₇	1	4	yes	—
143	C₂³×C₃	C₂³×C₇	1	1	yes	characteristic
144	C₂²×D₃	C₂³×C₇	1	2	yes	—
145	C₃×D₄	—	2	4	no	—
146	C₃×D₄	—	2	4	no	—
147	C₂×C₃×C₄	C₇×D₄	1	4	yes	complemented
148	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
150	#67	—	2	4	no	—
151	#67	—	4	8	no	—
152	C₂×Dic₃	C₂³×C₇	1	4	yes	—
153	C₂×Dic₃	C₂³×C₇	1	2	yes	—
154	D₁₂	—	2	2	no	—
155	C₄×D₃	—	2	4	no	—
156	Dic₆	—	2	2	no	—
157	C₃×C₇	#535	1	1	yes	characteristic, complemented, Hall
158	C₂⁴	C₂×D₃×C₇	1	2	yes	—
159	#41	—	6	4	no	—
160	C₂×Q₈	—	3	1	no	—
161	C₂×D₄	—	2	4	no	—
162	C₂×D₄	—	3	2	no	—
163	C₂×D₄	—	3	8	no	—
164	C₂×D₄	—	3	1	no	—
165	C₂×D₄	C₂×D₃×C₇	1	2	yes	—
166	C₂×D₄	C₂×D₃×C₇	1	2	yes	—
167	C₂²×C₄	—	3	2	no	—
168	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
169	C₄⋊C₄	—	3	2	no	—
170	C₂²⋊C₄	—	3	2	no	—
171	C₂²⋊C₄	—	3	8	no	—
172	C₄²	—	3	2	no	—
173	C₂×C₇	—	2	2	no	—
174	C₂×C₇	—	4	4	no	—
175	C₂×C₇	—	12	2	no	—
176	C₂×C₇	#1004	1	2	yes	—
177	C₂×C₇	C₂²×#67	1	1	yes	characteristic
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	—	2	4	no	—
180	C₂²×C₃	—	2	8	no	—
181	C₂²×C₃	—	4	8	no	—
182	C₂²×C₃	C₂×C₇×D₄	1	2	yes	—
183	C₂²×C₃	C₂⁴×C₇	1	1	yes	characteristic
184	C₂×D₃	—	2	2	no	—
185	C₂×D₃	—	2	4	no	—
186	C₃×C₄	C₂×C₇×D₄	1	2	yes	—
187	C₃×C₄	C₂×C₇×D₄	1	2	yes	—
188	Dic₃	—	2	2	no	—
189	Dic₃	—	2	4	no	—
190	C₂³	—	2	4	no	—
191	C₂³	—	3	2	no	—
192	C₂³	—	4	4	no	—
193	C₂³	C₂²×D₃×C₇	1	4	yes	—
194	C₂³	C₂²×D₃×C₇	1	1	yes	characteristic
195	Q₈	—	6	2	no	—
196	D₄	—	2	4	no	—
197	D₄	—	2	4	no	—
198	D₄	—	6	4	no	—
199	D₄	—	6	2	no	—
200	D₄	—	12	8	no	—
201	C₂×C₄	—	3	4	no	—
202	C₂×C₄	—	3	2	no	—
203	C₂×C₄	—	6	4	no	—
204	C₂×C₄	C₇×#67	1	4	yes	complemented
205	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
206	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
207	C₇	#5805	1	1	yes	characteristic, direct product factor, Sylow
208	C₂×C₃	—	2	2	no	—
209	C₂×C₃	—	4	4	no	—
210	C₂×C₃	C₂²×C₇×D₄	1	1	yes	characteristic
211	C₂×C₃	C₇×H(5,𝔽₂)	1	2	yes	—
212	D₃	—	4	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	8	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	4	8	no	—
217	C₂²	—	6	4	no	—
218	C₂²	—	6	2	no	—
219	C₂²	C₂×C₇×#67	1	2	yes	—
220	C₂²	C₂³×D₃×C₇	1	1	yes	characteristic
221	C₄	—	6	4	no	—
222	C₄	—	6	2	no	—
223	C₄	C₂×C₇×#67	1	2	yes	—
224	C₄	C₂×C₇×#67	1	2	yes	—
225	C₃	C₇×#535	1	1	yes	characteristic, complemented, Sylow
226	C₂	—	2	2	no	—
227	C₂	—	4	4	no	—
228	C₂	—	12	2	no	—
229	C₂	C₇×#1004	1	2	yes	—
230	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	C₂²×C₃×C₇×D₄	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₂²×C₇×D₄	D₃
hypocenter	C₃	C₇×#535
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₇×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₇×D₄	C₂	semidirect
2. C₂×C₇×#235	C₂	semidirect
3. C₇×#953	C₂	semidirect
4. C₇×#940	C₂	semidirect
5. C₇×#938	C₂	semidirect
6. C₇×#936	C₂	semidirect
7. C₇×#926	C₂	semidirect
15. C₂×C₇×#67	C₂²	semidirect
17. C₂×C₇×D₁₂	C₂²	semidirect
18. C₂×C₄×D₃×C₇	C₂²	semidirect
19. C₂×C₇×Dic₆	C₂²	semidirect
21. C₇×#217	C₂²	semidirect
22. C₇×C₄⋊Dic₃	C₂²	semidirect
23. C₄×C₇×Dic₃	C₂²	semidirect
25. C₂²×C₇×D₄	D₃	semidirect
31. #5805	C₇	direct (N × H)
39. C₂×C₃×C₄×C₇	D₄	semidirect
64. C₂²×C₃×D₄	C₂×C₇	semidirect
65. C₂×#235	C₂×C₇	semidirect
66. #953	C₂×C₇	semidirect
67. #940	C₂×C₇	semidirect
68. #938	C₂×C₇	semidirect
69. #936	C₂×C₇	semidirect
70. #926	C₂×C₇	semidirect
98. C₂×C₄×C₇	#67	semidirect
107. C₂×#67	C₂²×C₇	semidirect
109. C₂×D₁₂	C₂²×C₇	semidirect
110. C₂×C₄×D₃	C₂²×C₇	semidirect
111. C₂×Dic₆	C₂²×C₇	semidirect
113. #217	C₂²×C₇	semidirect
114. C₄⋊Dic₃	C₂²×C₇	semidirect
115. C₄×Dic₃	C₂²×C₇	semidirect
122. C₂²×D₄	D₃×C₇	semidirect
147. C₂×C₃×C₄	C₇×D₄	semidirect
157. C₃×C₇	#535	semidirect
204. C₂×C₄	C₇×#67	semidirect
207. C₇	#5805	direct (N × H)
225. C₃	C₇×#535	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₇×D₄	2
C₂×C₇×#235	2
C₇×#953	2
C₇×#940	2
C₇×#938	2
C₇×#936	2
C₇×#926	2
C₇×#535	3
#5805	7

Subgroups — C7×#5805 #13061521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#5805 #13061521