Subgroups — C₇×#4839 #13055876

Order 1344.

228 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
2	C₂×D₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#872	C₂	1	1	yes	maximal, characteristic
4	C₃×C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×Dic₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×Dic₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#424	—	3	1	no	maximal, Hall
9	C₂×C₄×D₃×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
13	C₇×(C₂×D₃)⋊C₄	C₄	1	1	yes	characteristic
14	C₇×(C₂×D₃)⋊C₄	C₄	1	1	yes	characteristic
15	C₇×Dic₃⋊C₄	C₄	1	1	yes	characteristic
16	C₇×Dic₃⋊C₄	C₄	1	1	yes	characteristic
17	C₄×C₇×Dic₃	C₂²	1	1	yes	characteristic
18	C₂×C₇×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₇×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₇×ΓL(1,𝔽₂₅)	—	2	2	no	—
21	D₃×C₇×C₈	—	2	2	no	—
22	C₂×C₇×M₁₆	—	3	1	no	—
23	C₂²×C₇×C₈	—	3	1	no	—
24	C₇×#117	—	3	1	no	—
25	C₇×C₄⋊C₈	—	3	1	no	—
26	C₇×C₄⋊C₈	—	3	1	no	—
27	C₇×C₄⋊C₈	—	3	1	no	—
28	C₇×C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
29	#4839	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
30	C₂²×D₃×C₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
34	C₂×C₇×Dic₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₇×Dic₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₇×Dic₃	C₂×C₄	1	1	yes	characteristic
37	C₄×D₃×C₇	D₄	1	2	yes	—
38	C₄×D₃×C₇	Q₈	1	2	yes	—
39	C₃×C₇×C₈	—	2	1	no	—
40	C₃×C₇×C₈	—	2	1	no	—
41	C₇×C₃⋊C₈	—	2	1	no	—
42	C₇×C₃⋊C₈	—	2	1	no	—
43	C₂²×C₄×C₇	—	3	1	no	—
44	C₇×M₁₆	—	6	2	no	—
45	C₂×C₇×C₈	—	3	1	no	—
46	C₂×C₇×C₈	—	3	1	no	—
47	C₂×C₇×C₈	—	6	2	no	—
48	C₂×C₇×C₈	C₂×D₃	1	1	yes	characteristic
49	C₂×C₇×C₈	C₂×D₃	1	1	yes	characteristic
50	C₇×C₄⋊C₄	—	3	1	no	—
51	C₇×C₄⋊C₄	—	3	1	no	—
52	C₇×C₂²⋊C₄	—	3	1	no	—
53	C₇×C₂²⋊C₄	—	3	1	no	—
54	C₄²×C₇	—	3	1	no	—
55	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
56	C₂×ΓL(1,𝔽₂₅)	C₂×C₇	1	1	yes	characteristic
57	C₂×D₃×C₈	C₂×C₇	1	1	yes	characteristic
58	#872	C₂×C₇	1	1	yes	characteristic
59	C₃×C₄⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
60	Dic₃⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
61	Dic₃⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
62	C₄⋊(C₃⋊C₈)	C₂×C₇	1	1	yes	characteristic, complemented
63	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×D₃×C₇	—	2	1	no	—
65	C₂×D₃×C₇	—	2	1	no	—
66	C₂×D₃×C₇	C₄⋊C₄	1	2	yes	—
67	C₃×C₄×C₇	—	2	1	no	—
68	C₃×C₄×C₇	—	2	1	no	—
69	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
70	C₃×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
71	C₇×Dic₃	—	2	1	no	—
72	C₇×Dic₃	—	2	1	no	—
73	C₇×Dic₃	C₄⋊C₄	1	2	yes	—
74	#424	—	3	1	no	Sylow
75	C₂³×C₇	—	3	1	no	—
76	C₂×C₄×C₇	—	3	1	no	—
77	C₂×C₄×C₇	—	3	2	no	—
78	C₂×C₄×C₇	—	3	2	no	—
79	C₂×C₄×C₇	—	3	1	no	—
80	C₂×C₄×C₇	—	3	1	no	—
81	C₂×C₄×C₇	C₄×D₃	1	1	yes	characteristic
82	C₂×C₄×C₇	C₄×D₃	1	1	yes	characteristic
83	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
84	C₇×C₈	—	2	1	no	—
85	C₇×C₈	—	2	1	no	—
86	C₇×C₈	—	6	1	no	—
87	C₇×C₈	—	6	1	no	—
88	C₂×C₄×D₃	C₂²×C₇	1	1	yes	characteristic
89	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
90	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
91	C₃×C₄²	C₂²×C₇	1	1	yes	characteristic
92	(C₂×D₃)⋊C₄	C₄×C₇	1	1	yes	characteristic
93	(C₂×D₃)⋊C₄	C₄×C₇	1	1	yes	characteristic
94	Dic₃⋊C₄	C₄×C₇	1	1	yes	characteristic
95	Dic₃⋊C₄	C₄×C₇	1	1	yes	characteristic
96	C₄×Dic₃	C₂²×C₇	1	1	yes	characteristic
97	C₂×C₃⋊C₈	C₂²×C₇	1	1	yes	characteristic
98	C₂×C₃⋊C₈	C₂²×C₇	1	1	yes	characteristic
99	ΓL(1,𝔽₂₅)	—	2	2	no	—
100	D₃×C₈	—	2	2	no	—
101	C₂×C₃×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₃×C₇	#131	1	1	yes	characteristic
103	C₂×C₃×C₇	#131	1	1	yes	characteristic
104	D₃×C₇	—	2	2	no	—
105	C₂×M₁₆	—	3	1	no	—
106	C₂²×C₈	—	3	1	no	—
107	#117	—	3	1	no	—
108	C₄⋊C₈	—	3	1	no	—
109	C₄⋊C₈	—	3	1	no	—
110	C₄⋊C₈	—	3	1	no	—
111	C₄⋊C₈	D₃×C₇	1	1	yes	characteristic, complemented
112	C₂²×C₇	—	3	2	no	—
113	C₂²×C₇	—	6	1	no	—
114	C₂²×C₇	—	6	1	no	—
115	C₂²×C₇	C₂×C₄×D₃	1	1	yes	characteristic
116	C₄×C₇	—	2	1	no	—
117	C₄×C₇	—	2	1	no	—
118	C₄×C₇	—	3	2	no	—
119	C₄×C₇	—	6	1	no	—
120	C₄×C₇	—	6	1	no	—
121	C₄×C₇	D₃×D₄	1	1	yes	characteristic
122	C₄×C₇	D₃×Q₈	1	1	yes	characteristic
123	C₂²×D₃	C₂×C₄×C₇	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
127	C₂×Dic₃	C₂×C₄×C₇	1	1	yes	characteristic
128	C₂×Dic₃	C₂×C₄×C₇	1	1	yes	characteristic
129	C₂×Dic₃	C₂×C₄×C₇	1	1	yes	characteristic
130	C₄×D₃	C₇×D₄	1	2	yes	—
131	C₄×D₃	C₇×Q₈	1	2	yes	—
132	C₃×C₈	—	2	1	no	—
133	C₃×C₈	—	2	1	no	—
134	C₃⋊C₈	—	2	1	no	—
135	C₃⋊C₈	—	2	1	no	—
136	C₃×C₇	#424	1	1	yes	characteristic, complemented, Hall
137	C₂²×C₄	—	3	1	no	—
138	M₁₆	—	6	2	no	—
139	C₂×C₈	—	3	1	no	—
140	C₂×C₈	—	3	1	no	—
141	C₂×C₈	—	6	2	no	—
142	C₂×C₈	C₂×D₃×C₇	1	1	yes	characteristic
143	C₂×C₈	C₂×D₃×C₇	1	1	yes	characteristic
144	C₄⋊C₄	—	3	1	no	—
145	C₄⋊C₄	—	3	1	no	—
146	C₂²⋊C₄	—	3	1	no	—
147	C₂²⋊C₄	—	3	1	no	—
148	C₄²	—	3	1	no	—
149	C₄²	C₂×D₃×C₇	1	1	yes	characteristic
150	C₂×C₇	—	6	2	no	—
151	C₂×C₇	D₃×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₇	#901	1	1	yes	characteristic
153	C₂×C₇	#907	1	1	yes	characteristic
154	C₂²×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
155	C₂×D₃	—	2	1	no	—
156	C₂×D₃	—	2	1	no	—
157	C₂×D₃	C₇×C₄⋊C₄	1	2	yes	—
158	C₃×C₄	—	2	1	no	—
159	C₃×C₄	—	2	1	no	—
160	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
161	C₃×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
162	Dic₃	—	2	1	no	—
163	Dic₃	—	2	1	no	—
164	Dic₃	C₇×C₄⋊C₄	1	2	yes	—
165	C₂³	—	3	1	no	—
166	C₂×C₄	—	3	1	no	—
167	C₂×C₄	—	3	2	no	—
168	C₂×C₄	—	3	2	no	—
169	C₂×C₄	—	3	1	no	—
170	C₂×C₄	—	3	1	no	—
171	C₂×C₄	C₄×D₃×C₇	1	1	yes	characteristic
172	C₂×C₄	C₄×D₃×C₇	1	1	yes	characteristic
173	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	6	1	no	—
177	C₈	—	6	1	no	—
178	C₇	#4839	1	1	yes	characteristic, direct product factor, Sylow
179	C₂×C₃	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
180	C₂×C₃	C₇×#131	1	1	yes	characteristic
181	C₂×C₃	C₇×#131	1	1	yes	characteristic
182	D₃	—	2	2	no	—
183	C₂²	—	3	2	no	—
184	C₂²	—	6	1	no	—
185	C₂²	—	6	1	no	—
186	C₂²	C₂×C₄×D₃×C₇	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	3	2	no	—
190	C₄	—	6	1	no	—
191	C₄	—	6	1	no	—
192	C₄	D₃×C₇×D₄	1	1	yes	characteristic
193	C₄	D₃×C₇×Q₈	1	1	yes	characteristic
194	C₃	C₇×#424	1	1	yes	characteristic, complemented, Sylow
195	C₂	—	6	2	no	—
196	C₂	D₃×C₇×C₄⋊C₄	1	1	yes	characteristic
197	C₂	C₇×#901	1	1	yes	characteristic
198	C₂	C₇×#907	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄×C₇
Frattini	C₂×C₄	C₂²×D₃×C₇
Fitting	C₃×C₇×C₄⋊C₈	C₂
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂²×D₃×C₇	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³×C₇
hypercenter	C₇×C₄⋊C₈	D₃
hypocenter	C₃	C₇×#424
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₇×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. C₃×C₇×C₄⋊C₈	C₂	semidirect
5. C₇×Dic₃⋊C₈	C₂	semidirect
6. C₇×Dic₃⋊C₈	C₂	semidirect
7. C₇×C₄⋊(C₃⋊C₈)	C₂	semidirect
28. C₇×C₄⋊C₈	D₃	semidirect
29. #4839	C₇	direct (N × H)
59. C₃×C₄⋊C₈	C₂×C₇	semidirect
60. Dic₃⋊C₈	C₂×C₇	semidirect
61. Dic₃⋊C₈	C₂×C₇	semidirect
62. C₄⋊(C₃⋊C₈)	C₂×C₇	semidirect
111. C₄⋊C₈	D₃×C₇	semidirect
136. C₃×C₇	#424	semidirect
178. C₇	#4839	direct (N × H)
194. C₃	C₇×#424	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×ΓL(1,𝔽₂₅)	2
C₂×D₃×C₇×C₈	2
C₇×#872	2
C₃×C₇×C₄⋊C₈	2
C₇×Dic₃⋊C₈	2
C₇×C₄⋊(C₃⋊C₈)	2
C₇×#424	3
#4839	7

Subgroups — C7×#4839 #13055876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#4839 #13055876