Subgroups — C₇×#5757 #13061473

Order 1344.

536 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1004	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₇×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#932	C₂	1	2	yes	maximal, complemented
4	C₇×#912	C₂	1	2	yes	maximal, complemented
5	C₇×#911	C₂	1	4	yes	maximal, complemented
6	D₃×C₇×D₈	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×#576	—	3	1	no	maximal, Hall
10	C₂×C₃×C₇×D₄	C₂²	1	2	yes	complemented
11	C₂×C₇×#67	—	2	4	no	—
12	C₇×#237	—	2	2	no	—
13	C₇×#237	C₂²	1	4	yes	complemented
14	D₃×C₇×D₄	—	2	2	no	—
15	D₃×C₇×D₄	C₂²	1	4	yes	complemented
16	C₇×#235	C₂²	1	1	yes	characteristic
17	C₇×#235	C₂²	1	2	yes	complemented
18	C₃×C₇×D₈	C₂²	1	2	yes	complemented
19	C₃×C₇×D₈	C₂²	1	2	yes	complemented
20	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
21	C₇×#214	C₂²	1	4	yes	complemented
22	C₇×#213	C₂²	1	4	yes	complemented
23	C₇×#208	C₂²	1	1	yes	characteristic, complemented
24	C₇×Dic₁₂	C₂²	1	1	yes	characteristic, complemented
25	C₇×D₂₄	C₂²	1	1	yes	characteristic, complemented
26	C₇×#204	C₂²	1	2	yes	complemented
27	C₇×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	complemented
28	D₃×C₇×C₈	C₂²	1	2	yes	complemented
29	C₇×H(5,𝔽₂)	—	3	2	no	—
30	C₇×AGL(1,ℤ/8)	—	3	2	no	—
31	C₇×AGL(1,ℤ/8)	—	3	4	no	—
32	C₇×#135	—	3	2	no	—
33	C₇×#135	—	3	1	no	—
34	C₂×C₇×D₈	—	3	2	no	—
35	C₂×C₇×D₈	D₃	1	1	yes	characteristic, complemented
36	C₇×#131	—	3	1	no	—
37	#5757	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
38	C₂³×C₃×C₇	—	2	2	no	—
39	C₂²×D₃×C₇	—	2	4	no	—
40	C₃×C₇×D₄	—	2	2	no	—
41	C₃×C₇×D₄	C₂³	1	4	yes	—
42	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
43	C₇×#67	—	2	2	no	—
44	C₇×#67	—	2	4	no	—
45	C₇×#67	—	2	4	no	—
46	C₇×#67	D₄	1	2	yes	—
47	C₂×C₇×Dic₃	—	2	4	no	—
48	C₇×D₁₂	D₄	1	1	yes	characteristic
49	C₇×D₁₂	C₂³	1	2	yes	—
50	C₄×D₃×C₇	—	2	2	no	—
51	C₄×D₃×C₇	C₂³	1	2	yes	—
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	2	no	—
61	C₂×C₇×D₄	—	3	4	no	—
62	C₂×C₇×D₄	—	6	2	no	—
63	C₂×C₇×D₄	—	6	4	no	—
64	C₂×C₇×D₄	C₂×D₃	1	2	yes	complemented
65	C₇×Q₁₆	—	3	1	no	—
66	C₇×SD₁₆	—	3	4	no	—
67	C₇×SD₁₆	—	3	2	no	—
68	C₇×D₈	—	3	4	no	—
69	C₇×D₈	—	3	1	no	—
70	C₇×D₈	C₂×D₃	1	2	yes	complemented
71	C₇×D₈	C₂×D₃	1	2	yes	complemented
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, complemented
76	#1004	C₂×C₇	1	2	yes	complemented
77	C₂×C₃×D₈	C₂×C₇	1	1	yes	characteristic, complemented
78	#932	C₂×C₇	1	2	yes	complemented
79	#912	C₂×C₇	1	2	yes	complemented
80	#911	C₂×C₇	1	4	yes	complemented
81	D₃×D₈	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	—	2	4	no	—
85	C₂²×C₃×C₇	—	8	2	no	—
86	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
87	C₂×D₃×C₇	—	2	2	no	—
88	C₂×D₃×C₇	—	8	4	no	—
89	C₂×D₃×C₇	C₂×D₄	1	2	yes	—
90	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
91	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
92	C₇×Dic₃	—	2	2	no	—
93	C₇×Dic₃	C₂×D₄	1	2	yes	—
94	#576	—	3	1	no	Sylow
95	C₂³×C₇	—	2	2	no	—
96	C₂³×C₇	—	6	4	no	—
97	C₇×Q₈	—	3	1	no	—
98	C₇×Q₈	—	3	2	no	—
99	C₇×D₄	—	2	2	no	—
100	C₇×D₄	—	3	2	no	—
101	C₇×D₄	—	3	1	no	—
102	C₇×D₄	—	3	2	no	—
103	C₇×D₄	—	6	4	no	—
104	C₇×D₄	—	6	4	no	—
105	C₇×D₄	—	6	2	no	—
106	C₇×D₄	C₂²×D₃	1	4	yes	—
107	C₂×C₄×C₇	—	3	2	no	—
108	C₂×C₄×C₇	—	6	4	no	—
109	C₂×C₄×C₇	—	6	2	no	—
110	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
111	C₇×C₈	—	3	2	no	—
112	C₇×C₈	C₂²×D₃	1	1	yes	characteristic
113	C₇×C₈	C₂²×D₃	1	1	yes	characteristic
114	C₂×C₃×D₄	C₂²×C₇	1	2	yes	complemented
115	C₂×#67	—	2	4	no	—
116	#237	—	2	2	no	—
117	#237	C₂²×C₇	1	4	yes	complemented
118	D₃×D₄	—	2	2	no	—
119	D₃×D₄	C₂²×C₇	1	4	yes	complemented
120	#235	C₂²×C₇	1	1	yes	characteristic
121	#235	C₂²×C₇	1	2	yes	complemented
122	C₃×D₈	C₂²×C₇	1	2	yes	complemented
123	C₃×D₈	C₂²×C₇	1	2	yes	complemented
124	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic, complemented
125	#214	C₂²×C₇	1	4	yes	complemented
126	#213	C₂²×C₇	1	4	yes	complemented
127	#208	C₂²×C₇	1	1	yes	characteristic, complemented
128	Dic₁₂	C₂²×C₇	1	1	yes	characteristic, complemented
129	D₂₄	C₂²×C₇	1	1	yes	characteristic, complemented
130	#204	C₂²×C₇	1	2	yes	complemented
131	ΓL(1,𝔽₂₅)	C₂²×C₇	1	2	yes	complemented
132	D₃×C₈	C₂²×C₇	1	2	yes	complemented
133	C₂×C₃×C₇	—	2	1	no	—
134	C₂×C₃×C₇	—	4	4	no	—
135	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
136	D₃×C₇	—	2	2	no	—
137	D₃×C₇	—	4	2	no	—
138	H(5,𝔽₂)	—	3	2	no	—
139	AGL(1,ℤ/8)	—	3	2	no	—
140	AGL(1,ℤ/8)	—	3	4	no	—
141	#135	—	3	2	no	—
142	#135	—	3	1	no	—
143	C₂×D₈	—	3	2	no	—
144	C₂×D₈	D₃×C₇	1	1	yes	characteristic, complemented
145	#131	—	3	1	no	—
146	C₂²×C₇	—	2	4	no	—
147	C₂²×C₇	—	3	2	no	—
148	C₂²×C₇	—	6	2	no	—
149	C₂²×C₇	—	8	2	no	—
150	C₂²×C₇	—	24	4	no	—
151	C₂²×C₇	D₃×D₄	1	1	yes	characteristic
152	C₄×C₇	—	3	2	no	—
153	C₄×C₇	—	6	2	no	—
154	C₄×C₇	D₃×D₄	1	1	yes	characteristic
155	C₄×C₇	C₂³×D₃	1	1	yes	characteristic
156	C₂³×C₃	—	2	2	no	—
157	C₂²×D₃	—	2	4	no	—
158	C₃×D₄	—	2	2	no	—
159	C₃×D₄	C₂³×C₇	1	4	yes	—
160	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
161	#67	—	2	2	no	—
162	#67	—	2	4	no	—
163	#67	—	2	4	no	—
164	#67	C₇×D₄	1	2	yes	—
165	C₂×Dic₃	—	2	4	no	—
166	D₁₂	C₇×D₄	1	1	yes	characteristic
167	D₁₂	C₂³×C₇	1	2	yes	—
168	C₄×D₃	—	2	2	no	—
169	C₄×D₃	C₂³×C₇	1	2	yes	—
170	Dic₆	C₇×D₄	1	1	yes	characteristic
171	Dic₆	C₂³×C₇	1	2	yes	—
172	C₃×C₈	C₂³×C₇	1	1	yes	characteristic
173	C₃×C₈	C₂³×C₇	1	1	yes	characteristic
174	C₃⋊C₈	C₂³×C₇	1	2	yes	—
175	C₃×C₇	#576	1	1	yes	characteristic, complemented, Hall
176	#41	—	3	1	no	—
177	#41	—	3	4	no	—
178	#41	—	3	2	no	—
179	#41	—	6	2	no	—
180	C₂×D₄	—	3	4	no	—
181	C₂×D₄	—	6	2	no	—
182	C₂×D₄	—	6	4	no	—
183	C₂×D₄	C₂×D₃×C₇	1	2	yes	complemented
184	Q₁₆	—	3	1	no	—
185	SD₁₆	—	3	4	no	—
186	SD₁₆	—	3	2	no	—
187	D₈	—	3	4	no	—
188	D₈	—	3	1	no	—
189	D₈	C₂×D₃×C₇	1	2	yes	complemented
190	D₈	C₂×D₃×C₇	1	2	yes	complemented
191	M₁₆	—	3	1	no	—
192	M₁₆	—	3	2	no	—
193	C₂×C₈	—	3	2	no	—
194	C₂×C₈	C₂×D₃×C₇	1	1	yes	characteristic, complemented
195	C₂×C₇	—	2	1	no	—
196	C₂×C₇	—	4	4	no	—
197	C₂×C₇	—	6	2	no	—
198	C₂×C₇	—	12	2	no	—
199	C₂×C₇	C₂×D₃×D₄	1	1	yes	characteristic
200	C₂²×C₃	—	2	4	no	—
201	C₂²×C₃	—	8	2	no	—
202	C₂²×C₃	C₂×C₇×D₄	1	1	yes	characteristic
203	C₂×D₃	—	2	2	no	—
204	C₂×D₃	—	8	4	no	—
205	C₂×D₃	C₂×C₇×D₄	1	2	yes	—
206	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
207	C₃×C₄	C₂⁴×C₇	1	1	yes	characteristic
208	Dic₃	—	2	2	no	—
209	Dic₃	C₂×C₇×D₄	1	2	yes	—
210	C₂³	—	2	2	no	—
211	C₂³	—	6	4	no	—
212	Q₈	—	3	1	no	—
213	Q₈	—	3	2	no	—
214	D₄	—	2	2	no	—
215	D₄	—	3	2	no	—
216	D₄	—	3	1	no	—
217	D₄	—	3	2	no	—
218	D₄	—	6	4	no	—
219	D₄	—	6	4	no	—
220	D₄	—	6	2	no	—
221	D₄	C₂²×D₃×C₇	1	4	yes	—
222	C₂×C₄	—	3	2	no	—
223	C₂×C₄	—	6	4	no	—
224	C₂×C₄	—	6	2	no	—
225	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
226	C₈	—	3	2	no	—
227	C₈	C₂²×D₃×C₇	1	1	yes	characteristic
228	C₈	C₂²×D₃×C₇	1	1	yes	characteristic
229	C₇	#5757	1	1	yes	characteristic, direct product factor, Sylow
230	C₂×C₃	—	2	1	no	—
231	C₂×C₃	—	4	4	no	—
232	C₂×C₃	C₂²×C₇×D₄	1	1	yes	characteristic
233	D₃	—	2	2	no	—
234	D₃	—	4	2	no	—
235	C₂²	—	2	4	no	—
236	C₂²	—	3	2	no	—
237	C₂²	—	6	2	no	—
238	C₂²	—	8	2	no	—
239	C₂²	—	24	4	no	—
240	C₂²	D₃×C₇×D₄	1	1	yes	characteristic
241	C₄	—	3	2	no	—
242	C₄	—	6	2	no	—
243	C₄	D₃×C₇×D₄	1	1	yes	characteristic
244	C₄	C₂³×D₃×C₇	1	1	yes	characteristic
245	C₃	C₇×#576	1	1	yes	characteristic, complemented, Sylow
246	C₂	—	2	1	no	—
247	C₂	—	4	4	no	—
248	C₂	—	6	2	no	—
249	C₂	—	12	2	no	—
250	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×D₈	C₂
socle	C₂×C₃×C₇	C₂²×D₄
hypercenter	C₂×C₇×D₈	D₃
hypocenter	C₃	C₇×#576
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×D₈

Complements

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

N	H ≅ G/N	type
1. C₇×#1004	C₂	semidirect
2. C₂×C₃×C₇×D₈	C₂	semidirect
3. C₇×#932	C₂	semidirect
4. C₇×#912	C₂	semidirect
5. C₇×#911	C₂	semidirect
6. D₃×C₇×D₈	C₂	semidirect
7. C₇×#904	C₂	semidirect
8. C₇×#901	C₂	semidirect
10. C₂×C₃×C₇×D₄	C₂²	semidirect
13. C₇×#237	C₂²	semidirect
15. D₃×C₇×D₄	C₂²	semidirect
17. C₇×#235	C₂²	semidirect
18. C₃×C₇×D₈	C₂²	semidirect
19. C₃×C₇×D₈	C₂²	semidirect
20. C₂×C₃×C₇×C₈	C₂²	semidirect
21. C₇×#214	C₂²	semidirect
22. C₇×#213	C₂²	semidirect
23. C₇×#208	C₂²	semidirect
24. C₇×Dic₁₂	C₂²	semidirect
25. C₇×D₂₄	C₂²	semidirect
26. C₇×#204	C₂²	semidirect
27. C₇×ΓL(1,𝔽₂₅)	C₂²	semidirect
28. D₃×C₇×C₈	C₂²	semidirect
35. C₂×C₇×D₈	D₃	semidirect
37. #5757	C₇	direct (N × H)
64. C₂×C₇×D₄	C₂×D₃	semidirect
70. C₇×D₈	C₂×D₃	semidirect
71. C₇×D₈	C₂×D₃	semidirect
75. C₂×C₇×C₈	C₂×D₃	semidirect
76. #1004	C₂×C₇	semidirect
77. C₂×C₃×D₈	C₂×C₇	semidirect
78. #932	C₂×C₇	semidirect
79. #912	C₂×C₇	semidirect
80. #911	C₂×C₇	semidirect
81. D₃×D₈	C₂×C₇	semidirect
82. #904	C₂×C₇	semidirect
83. #901	C₂×C₇	semidirect
114. C₂×C₃×D₄	C₂²×C₇	semidirect
117. #237	C₂²×C₇	semidirect
119. D₃×D₄	C₂²×C₇	semidirect
121. #235	C₂²×C₇	semidirect
122. C₃×D₈	C₂²×C₇	semidirect
123. C₃×D₈	C₂²×C₇	semidirect
124. C₂×C₃×C₈	C₂²×C₇	semidirect
125. #214	C₂²×C₇	semidirect
126. #213	C₂²×C₇	semidirect
127. #208	C₂²×C₇	semidirect
128. Dic₁₂	C₂²×C₇	semidirect
129. D₂₄	C₂²×C₇	semidirect
130. #204	C₂²×C₇	semidirect
131. ΓL(1,𝔽₂₅)	C₂²×C₇	semidirect
132. D₃×C₈	C₂²×C₇	semidirect
144. C₂×D₈	D₃×C₇	semidirect
175. C₃×C₇	#576	semidirect
183. C₂×D₄	C₂×D₃×C₇	semidirect
189. D₈	C₂×D₃×C₇	semidirect
190. D₈	C₂×D₃×C₇	semidirect
194. C₂×C₈	C₂×D₃×C₇	semidirect
229. C₇	#5757	direct (N × H)
245. C₃	C₇×#576	semidirect

Maximal subgroups

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

subgroup	index
C₇×#1004	2
C₂×C₃×C₇×D₈	2
C₇×#932	2
C₇×#912	2
C₇×#911	2
D₃×C₇×D₈	2
C₇×#904	2
C₇×#901	2
C₇×#576	3
#5757	7

Subgroups — C7×#5757 #13061473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#5757 #13061473