Subgroups — C₇×#5769 #13061485

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×#1009	C₂	1	2	yes	maximal, complemented
2	C₃×C₇×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#949	C₂	1	2	yes	maximal, complemented
4	C₇×#919	C₂	1	2	yes	maximal, complemented
5	C₇×#914	C₂	1	4	yes	maximal, complemented
6	D₃×C₇×D₈	C₂	1	2	yes	maximal, complemented
7	C₂×C₇×D₂₄	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	C₃×C₇×#41	C₂²	1	2	yes	complemented
11	C₇×#239	C₂²	1	4	yes	complemented
12	D₃×C₇×D₄	—	2	4	no	—
13	D₃×C₇×D₄	C₂²	1	4	yes	complemented
14	C₇×#235	—	2	2	no	—
15	C₇×#235	C₂²	1	1	yes	characteristic
16	C₂×C₇×D₁₂	—	2	2	no	—
17	C₂×C₇×D₁₂	C₂²	1	2	yes	complemented
18	C₃×C₇×Q₁₆	C₂²	1	1	yes	characteristic, complemented
19	C₃×C₇×SD₁₆	C₂²	1	2	yes	complemented
20	C₃×C₇×D₈	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
22	C₇×#215	C₂²	1	4	yes	complemented
23	C₇×#213	C₂²	1	4	yes	complemented
24	C₇×#208	C₂²	1	1	yes	characteristic, complemented
25	C₇×D₂₄	C₂²	1	2	yes	complemented
26	C₇×D₂₄	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	D₃	1	1	yes	characteristic, complemented
34	C₂×C₇×D₈	—	3	2	no	—
35	C₂×C₇×D₈	—	3	1	no	—
36	C₇×#131	—	3	1	no	—
37	#5769	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
38	C₂²×D₃×C₇	—	2	2	no	—
39	C₂²×D₃×C₇	—	2	4	no	—
40	C₃×C₇×Q₈	C₂³	1	2	yes	—
41	C₃×C₇×D₄	—	2	2	no	—
42	C₃×C₇×D₄	C₂³	1	2	yes	—
43	C₂×C₃×C₄×C₇	—	2	2	no	—
44	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
45	C₇×#67	—	2	4	no	—
46	C₇×#67	D₄	1	2	yes	—
47	C₇×D₁₂	—	2	4	no	—
48	C₇×D₁₂	—	2	2	no	—
49	C₇×D₁₂	D₄	1	1	yes	characteristic
50	C₇×D₁₂	C₂³	1	4	yes	—
51	C₄×D₃×C₇	—	2	4	no	—
52	C₄×D₃×C₇	C₂³	1	2	yes	—
53	C₇×Dic₆	D₄	1	1	yes	characteristic
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	—	6	2	no	—
60	C₇×#41	C₂×D₃	1	2	yes	complemented
61	C₂×C₇×D₄	—	3	4	no	—
62	C₂×C₇×D₄	—	3	2	no	—
63	C₂×C₇×D₄	—	6	2	no	—
64	C₂×C₇×D₄	—	6	4	no	—
65	C₇×Q₁₆	C₂×D₃	1	1	yes	characteristic, complemented
66	C₇×SD₁₆	—	3	4	no	—
67	C₇×SD₁₆	C₂×D₃	1	2	yes	complemented
68	C₇×D₈	—	3	4	no	—
69	C₇×D₈	—	3	2	no	—
70	C₇×D₈	—	3	2	no	—
71	C₇×D₈	C₂×D₃	1	1	yes	characteristic, 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	#1009	C₂×C₇	1	2	yes	complemented
77	C₃×#135	C₂×C₇	1	1	yes	characteristic, complemented
78	#949	C₂×C₇	1	2	yes	complemented
79	#919	C₂×C₇	1	2	yes	complemented
80	#914	C₂×C₇	1	4	yes	complemented
81	D₃×D₈	C₂×C₇	1	2	yes	complemented
82	C₂×D₂₄	C₂×C₇	1	1	yes	characteristic, complemented
83	#901	C₂×C₇	1	1	yes	characteristic, complemented
84	C₂²×C₃×C₇	—	2	2	no	—
85	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
86	C₂×D₃×C₇	—	2	4	no	—
87	C₂×D₃×C₇	—	8	4	no	—
88	C₂×D₃×C₇	—	8	2	no	—
89	C₂×D₃×C₇	C₂×D₄	1	2	yes	—
90	C₃×C₄×C₇	—	2	2	no	—
91	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
92	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
93	C₇×Dic₃	C₂×D₄	1	2	yes	—
94	#576	—	3	1	no	Sylow
95	C₂³×C₇	—	6	4	no	—
96	C₂³×C₇	—	6	2	no	—
97	C₇×Q₈	—	3	1	no	—
98	C₇×Q₈	C₂²×D₃	1	2	yes	—
99	C₇×D₄	—	2	2	no	—
100	C₇×D₄	—	3	2	no	—
101	C₇×D₄	—	3	1	no	—
102	C₇×D₄	—	3	4	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	2	yes	—
107	C₂×C₄×C₇	—	2	2	no	—
108	C₂×C₄×C₇	—	3	2	no	—
109	C₂×C₄×C₇	—	6	4	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	C₃×#41	C₂²×C₇	1	2	yes	complemented
115	#239	C₂²×C₇	1	4	yes	complemented
116	D₃×D₄	—	2	4	no	—
117	D₃×D₄	C₂²×C₇	1	4	yes	complemented
118	#235	—	2	2	no	—
119	#235	C₂²×C₇	1	1	yes	characteristic
120	C₂×D₁₂	—	2	2	no	—
121	C₂×D₁₂	C₂²×C₇	1	2	yes	complemented
122	C₃×Q₁₆	C₂²×C₇	1	1	yes	characteristic, complemented
123	C₃×SD₁₆	C₂²×C₇	1	2	yes	complemented
124	C₃×D₈	C₂²×C₇	1	1	yes	characteristic, complemented
125	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic, complemented
126	#215	C₂²×C₇	1	4	yes	complemented
127	#213	C₂²×C₇	1	4	yes	complemented
128	#208	C₂²×C₇	1	1	yes	characteristic, complemented
129	D₂₄	C₂²×C₇	1	2	yes	complemented
130	D₂₄	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	2	no	—
135	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
136	D₃×C₇	—	2	2	no	—
137	D₃×C₇	—	4	4	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	D₃×C₇	1	1	yes	characteristic, complemented
143	C₂×D₈	—	3	2	no	—
144	C₂×D₈	—	3	1	no	—
145	#131	—	3	1	no	—
146	C₂²×C₇	—	2	2	no	—
147	C₂²×C₇	—	3	2	no	—
148	C₂²×C₇	—	6	4	no	—
149	C₂²×C₇	—	24	2	no	—
150	C₂²×C₇	—	24	4	no	—
151	C₂²×C₇	D₃×D₄	1	1	yes	characteristic
152	C₄×C₇	—	2	2	no	—
153	C₄×C₇	—	3	2	no	—
154	C₄×C₇	D₃×D₄	1	1	yes	characteristic
155	C₄×C₇	C₂³×D₃	1	1	yes	characteristic
156	C₂²×D₃	—	2	2	no	—
157	C₂²×D₃	—	2	4	no	—
158	C₃×Q₈	C₂³×C₇	1	2	yes	—
159	C₃×D₄	—	2	2	no	—
160	C₃×D₄	C₂³×C₇	1	2	yes	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
163	#67	—	2	4	no	—
164	#67	C₇×D₄	1	2	yes	—
165	D₁₂	—	2	4	no	—
166	D₁₂	—	2	2	no	—
167	D₁₂	C₇×D₄	1	1	yes	characteristic
168	D₁₂	C₂³×C₇	1	4	yes	—
169	C₄×D₃	—	2	4	no	—
170	C₄×D₃	C₂³×C₇	1	2	yes	—
171	Dic₆	C₇×D₄	1	1	yes	characteristic
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	—	6	2	no	—
179	#41	C₂×D₃×C₇	1	2	yes	complemented
180	C₂×D₄	—	3	4	no	—
181	C₂×D₄	—	3	2	no	—
182	C₂×D₄	—	6	2	no	—
183	C₂×D₄	—	6	4	no	—
184	Q₁₆	C₂×D₃×C₇	1	1	yes	characteristic, complemented
185	SD₁₆	—	3	4	no	—
186	SD₁₆	C₂×D₃×C₇	1	2	yes	complemented
187	D₈	—	3	4	no	—
188	D₈	—	3	2	no	—
189	D₈	—	3	2	no	—
190	D₈	C₂×D₃×C₇	1	1	yes	characteristic, 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	2	no	—
197	C₂×C₇	—	6	2	no	—
198	C₂×C₇	—	12	4	no	—
199	C₂×C₇	C₂×D₃×D₄	1	1	yes	characteristic
200	C₂²×C₃	—	2	2	no	—
201	C₂²×C₃	C₂×C₇×D₄	1	1	yes	characteristic
202	C₂×D₃	—	2	4	no	—
203	C₂×D₃	—	8	4	no	—
204	C₂×D₃	—	8	2	no	—
205	C₂×D₃	C₂×C₇×D₄	1	2	yes	—
206	C₃×C₄	—	2	2	no	—
207	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
208	C₃×C₄	C₂⁴×C₇	1	1	yes	characteristic
209	Dic₃	C₂×C₇×D₄	1	2	yes	—
210	C₂³	—	6	4	no	—
211	C₂³	—	6	2	no	—
212	Q₈	—	3	1	no	—
213	Q₈	C₂²×D₃×C₇	1	2	yes	—
214	D₄	—	2	2	no	—
215	D₄	—	3	2	no	—
216	D₄	—	3	1	no	—
217	D₄	—	3	4	no	—
218	D₄	—	6	4	no	—
219	D₄	—	6	4	no	—
220	D₄	—	6	2	no	—
221	D₄	C₂²×D₃×C₇	1	2	yes	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	3	2	no	—
224	C₂×C₄	—	6	4	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	#5769	1	1	yes	characteristic, direct product factor, Sylow
230	C₂×C₃	—	2	1	no	—
231	C₂×C₃	—	4	2	no	—
232	C₂×C₃	C₂²×C₇×D₄	1	1	yes	characteristic
233	D₃	—	2	2	no	—
234	D₃	—	4	4	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	3	2	no	—
237	C₂²	—	6	4	no	—
238	C₂²	—	24	2	no	—
239	C₂²	—	24	4	no	—
240	C₂²	D₃×C₇×D₄	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	3	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	2	no	—
248	C₂	—	6	2	no	—
249	C₂	—	12	4	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	C₃×C₇×#135	C₂
socle	C₂×C₃×C₇	C₂²×D₄
hypercenter	C₇×#135	D₃
hypocenter	C₃	C₇×#576
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×#135

Complements

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

N	H ≅ G/N	type
1. C₇×#1009	C₂	semidirect
2. C₃×C₇×#135	C₂	semidirect
3. C₇×#949	C₂	semidirect
4. C₇×#919	C₂	semidirect
5. C₇×#914	C₂	semidirect
6. D₃×C₇×D₈	C₂	semidirect
7. C₂×C₇×D₂₄	C₂	semidirect
8. C₇×#901	C₂	semidirect
10. C₃×C₇×#41	C₂²	semidirect
11. C₇×#239	C₂²	semidirect
13. D₃×C₇×D₄	C₂²	semidirect
17. C₂×C₇×D₁₂	C₂²	semidirect
18. C₃×C₇×Q₁₆	C₂²	semidirect
19. C₃×C₇×SD₁₆	C₂²	semidirect
20. C₃×C₇×D₈	C₂²	semidirect
21. C₂×C₃×C₇×C₈	C₂²	semidirect
22. C₇×#215	C₂²	semidirect
23. C₇×#213	C₂²	semidirect
24. C₇×#208	C₂²	semidirect
25. C₇×D₂₄	C₂²	semidirect
26. C₇×D₂₄	C₂²	semidirect
27. C₇×ΓL(1,𝔽₂₅)	C₂²	semidirect
28. D₃×C₇×C₈	C₂²	semidirect
33. C₇×#135	D₃	semidirect
37. #5769	C₇	direct (N × H)
60. C₇×#41	C₂×D₃	semidirect
65. C₇×Q₁₆	C₂×D₃	semidirect
67. C₇×SD₁₆	C₂×D₃	semidirect
71. C₇×D₈	C₂×D₃	semidirect
75. C₂×C₇×C₈	C₂×D₃	semidirect
76. #1009	C₂×C₇	semidirect
77. C₃×#135	C₂×C₇	semidirect
78. #949	C₂×C₇	semidirect
79. #919	C₂×C₇	semidirect
80. #914	C₂×C₇	semidirect
81. D₃×D₈	C₂×C₇	semidirect
82. C₂×D₂₄	C₂×C₇	semidirect
83. #901	C₂×C₇	semidirect
114. C₃×#41	C₂²×C₇	semidirect
115. #239	C₂²×C₇	semidirect
117. D₃×D₄	C₂²×C₇	semidirect
121. C₂×D₁₂	C₂²×C₇	semidirect
122. C₃×Q₁₆	C₂²×C₇	semidirect
123. C₃×SD₁₆	C₂²×C₇	semidirect
124. C₃×D₈	C₂²×C₇	semidirect
125. C₂×C₃×C₈	C₂²×C₇	semidirect
126. #215	C₂²×C₇	semidirect
127. #213	C₂²×C₇	semidirect
128. #208	C₂²×C₇	semidirect
129. D₂₄	C₂²×C₇	semidirect
130. D₂₄	C₂²×C₇	semidirect
131. ΓL(1,𝔽₂₅)	C₂²×C₇	semidirect
132. D₃×C₈	C₂²×C₇	semidirect
142. #135	D₃×C₇	semidirect
175. C₃×C₇	#576	semidirect
179. #41	C₂×D₃×C₇	semidirect
184. Q₁₆	C₂×D₃×C₇	semidirect
186. SD₁₆	C₂×D₃×C₇	semidirect
190. D₈	C₂×D₃×C₇	semidirect
194. C₂×C₈	C₂×D₃×C₇	semidirect
229. C₇	#5769	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×#1009	2
C₃×C₇×#135	2
C₇×#949	2
C₇×#919	2
C₇×#914	2
D₃×C₇×D₈	2
C₂×C₇×D₂₄	2
C₇×#901	2
C₇×#576	3
#5769	7

Subgroups — C7×#5769 #13061485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#5769 #13061485