Subgroups — C₂×C₇×#907 #13061460

Order 1344.

532 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×#235	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₇×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#907	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₇×ΓL(1,𝔽₂₅)	C₂	1	2	yes	maximal, complemented
6	C₂×D₃×C₇×C₈	C₂	1	2	yes	maximal, complemented
7	C₂×C₇×#131	—	3	1	no	maximal, Hall
8	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₇×#67	C₄	1	2	yes	—
10	C₇×#235	C₄	1	4	yes	—
11	C₇×#235	C₂²	1	4	yes	—
12	C₂×C₇×D₁₂	C₄	1	1	yes	characteristic
13	C₂×C₄×D₃×C₇	C₂²	1	2	yes	—
14	C₂×C₇×Dic₆	C₄	1	1	yes	characteristic
15	C₃×C₇×M₁₆	C₂²	1	4	yes	complemented
16	C₂×C₃×C₇×C₈	C₂²	1	2	yes	—
17	C₂×C₇×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₇×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₇×C₃⋊C₈	C₂²	1	4	yes	complemented
20	C₇×ΓL(1,𝔽₂₅)	C₂²	1	8	yes	complemented
21	D₃×C₇×C₈	C₂²	1	8	yes	complemented
22	C₂×C₇×#41	—	3	1	no	—
23	C₇×#131	—	3	8	no	—
24	C₂×C₇×M₁₆	—	3	2	no	—
25	C₂×C₇×M₁₆	D₃	1	1	yes	characteristic, complemented
26	C₂²×C₇×C₈	—	3	1	no	—
27	C₂²×C₇×C₈	—	3	2	no	—
28	C₂×#907	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
29	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×D₃×C₇	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
34	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
35	C₇×#67	C₂×C₄	1	8	yes	—
36	C₂×C₇×Dic₃	C₂×C₄	1	2	yes	—
37	C₇×D₁₂	C₂×C₄	1	4	yes	—
38	C₄×D₃×C₇	C₂×C₄	1	4	yes	—
39	C₄×D₃×C₇	C₂³	1	4	yes	—
40	C₇×Dic₆	C₂×C₄	1	4	yes	—
41	C₃×C₇×C₈	C₂³	1	4	yes	—
42	C₇×C₃⋊C₈	C₂³	1	2	yes	—
43	C₇×C₃⋊C₈	C₂³	1	2	yes	—
44	C₇×#41	—	3	4	no	—
45	C₇×#41	—	3	4	no	—
46	C₂×C₇×Q₈	—	3	1	no	—
47	C₂×C₇×D₄	—	3	2	no	—
48	C₂×C₇×D₄	—	3	1	no	—
49	C₂²×C₄×C₇	—	3	2	no	—
50	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
51	C₇×M₁₆	—	3	8	no	—
52	C₇×M₁₆	C₂×D₃	1	4	yes	complemented
53	C₂×C₇×C₈	—	3	4	no	—
54	C₂×C₇×C₈	—	3	1	no	—
55	C₂×C₇×C₈	—	3	1	no	—
56	C₂×C₇×C₈	—	3	8	no	—
57	C₂×C₇×C₈	C₂×D₃	1	2	yes	—
58	C₂×#235	C₂×C₇	1	1	yes	characteristic
59	C₂×C₃×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
60	C₂²×C₃⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
61	#907	C₂×C₇	1	8	yes	direct product factor
62	C₂×ΓL(1,𝔽₂₅)	C₂×C₇	1	2	yes	complemented
63	C₂×D₃×C₈	C₂×C₇	1	2	yes	complemented
64	C₂²×C₃×C₇	—	2	2	no	—
65	C₂²×C₃×C₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×D₃×C₇	—	2	4	no	—
68	C₂×D₃×C₇	C₂²×C₄	1	4	yes	—
69	C₃×C₄×C₇	C₂²×C₄	1	2	yes	—
70	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
71	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
72	C₇×Dic₃	C₂²×C₄	1	4	yes	—
73	C₂×#131	—	3	1	no	Sylow
74	C₂³×C₇	—	3	2	no	—
75	C₂³×C₇	C₄×D₃	1	1	yes	characteristic
76	C₇×Q₈	—	3	4	no	—
77	C₇×D₄	—	3	8	no	—
78	C₇×D₄	—	3	4	no	—
79	C₂×C₄×C₇	—	3	2	no	—
80	C₂×C₄×C₇	—	3	4	no	—
81	C₂×C₄×C₇	—	3	4	no	—
82	C₂×C₄×C₇	C₄×D₃	1	1	yes	characteristic
83	C₂×C₄×C₇	C₄×D₃	1	2	yes	—
84	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
85	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
86	C₇×C₈	—	3	2	no	—
87	C₇×C₈	—	3	2	no	—
88	C₇×C₈	C₂²×D₃	1	4	yes	—
89	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
90	C₂×#67	C₄×C₇	1	2	yes	—
91	#235	C₄×C₇	1	4	yes	—
92	#235	C₂²×C₇	1	4	yes	—
93	C₂×D₁₂	C₄×C₇	1	1	yes	characteristic
94	C₂×C₄×D₃	C₂²×C₇	1	2	yes	—
95	C₂×Dic₆	C₄×C₇	1	1	yes	characteristic
96	C₃×M₁₆	C₂²×C₇	1	4	yes	complemented
97	C₂×C₃×C₈	C₂²×C₇	1	2	yes	—
98	C₂×C₃⋊C₈	C₂²×C₇	1	1	yes	characteristic
99	C₂×C₃⋊C₈	C₂²×C₇	1	1	yes	characteristic
100	C₂×C₃⋊C₈	C₂²×C₇	1	4	yes	complemented
101	ΓL(1,𝔽₂₅)	C₂²×C₇	1	8	yes	complemented
102	D₃×C₈	C₂²×C₇	1	8	yes	complemented
103	C₂×C₃×C₇	—	2	2	no	—
104	C₂×C₃×C₇	#131	1	2	yes	complemented
105	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
106	D₃×C₇	—	2	4	no	—
107	C₂×#41	—	3	1	no	—
108	#131	—	3	8	no	—
109	C₂×M₁₆	—	3	2	no	—
110	C₂×M₁₆	D₃×C₇	1	1	yes	characteristic, complemented
111	C₂²×C₈	—	3	1	no	—
112	C₂²×C₈	—	3	2	no	—
113	C₂²×C₇	—	2	2	no	—
114	C₂²×C₇	—	3	4	no	—
115	C₂²×C₇	—	6	4	no	—
116	C₂²×C₇	C₂×C₄×D₃	1	2	yes	—
117	C₂²×C₇	C₂×C₄×D₃	1	1	yes	characteristic
118	C₄×C₇	—	3	4	no	—
119	C₄×C₇	C₂×C₄×D₃	1	1	yes	characteristic
120	C₄×C₇	C₂×C₄×D₃	1	2	yes	—
121	C₄×C₇	C₂³×D₃	1	1	yes	characteristic
122	C₂³×C₃	C₂×C₄×C₇	1	1	yes	characteristic
123	C₂²×D₃	C₂×C₄×C₇	1	2	yes	—
124	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	—
126	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
128	#67	C₂×C₄×C₇	1	8	yes	—
129	C₂×Dic₃	C₂×C₄×C₇	1	2	yes	—
130	D₁₂	C₂×C₄×C₇	1	4	yes	—
131	C₄×D₃	C₂×C₄×C₇	1	4	yes	—
132	C₄×D₃	C₂³×C₇	1	4	yes	—
133	Dic₆	C₂×C₄×C₇	1	4	yes	—
134	C₃×C₈	C₂³×C₇	1	4	yes	—
135	C₃⋊C₈	C₂³×C₇	1	2	yes	—
136	C₃⋊C₈	C₂³×C₇	1	2	yes	—
137	C₃×C₇	C₂×#131	1	1	yes	characteristic, complemented, Hall
138	#41	—	3	4	no	—
139	#41	—	3	4	no	—
140	C₂×Q₈	—	3	1	no	—
141	C₂×D₄	—	3	2	no	—
142	C₂×D₄	—	3	1	no	—
143	C₂²×C₄	—	3	2	no	—
144	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
145	M₁₆	—	3	8	no	—
146	M₁₆	C₂×D₃×C₇	1	4	yes	complemented
147	C₂×C₈	—	3	1	no	—
148	C₂×C₈	—	3	1	no	—
149	C₂×C₈	—	3	4	no	—
150	C₂×C₈	—	3	8	no	—
151	C₂×C₈	C₂×D₃×C₇	1	2	yes	—
152	C₂×C₇	—	2	2	no	—
153	C₂×C₇	—	6	4	no	—
154	C₂×C₇	#907	1	2	yes	direct product factor
155	C₂×C₇	C₂²×C₄×D₃	1	1	yes	characteristic
156	C₂²×C₃	—	2	2	no	—
157	C₂²×C₃	C₂²×C₄×C₇	1	2	yes	—
158	C₂²×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
159	C₂×D₃	—	2	4	no	—
160	C₂×D₃	C₂²×C₄×C₇	1	4	yes	—
161	C₃×C₄	C₂²×C₄×C₇	1	2	yes	—
162	C₃×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
163	C₃×C₄	C₂⁴×C₇	1	1	yes	characteristic
164	Dic₃	C₂²×C₄×C₇	1	4	yes	—
165	C₂³	—	3	2	no	—
166	C₂³	C₄×D₃×C₇	1	1	yes	characteristic
167	Q₈	—	3	4	no	—
168	D₄	—	3	8	no	—
169	D₄	—	3	4	no	—
170	C₂×C₄	—	3	2	no	—
171	C₂×C₄	—	3	4	no	—
172	C₂×C₄	—	3	4	no	—
173	C₂×C₄	C₄×D₃×C₇	1	1	yes	characteristic
174	C₂×C₄	C₄×D₃×C₇	1	2	yes	—
175	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
176	C₂×C₄	C₂²×D₃×C₇	1	2	yes	—
177	C₈	—	3	2	no	—
178	C₈	—	3	2	no	—
179	C₈	C₂²×D₃×C₇	1	4	yes	—
180	C₇	C₂×#907	1	1	yes	characteristic, direct product factor, Sylow
181	C₂×C₃	—	2	2	no	—
182	C₂×C₃	C₇×#131	1	2	yes	complemented
183	C₂×C₃	C₂³×C₄×C₇	1	1	yes	characteristic
184	D₃	—	2	4	no	—
185	C₂²	—	2	2	no	—
186	C₂²	—	3	4	no	—
187	C₂²	—	6	4	no	—
188	C₂²	C₂×C₄×D₃×C₇	1	2	yes	—
189	C₂²	C₂×C₄×D₃×C₇	1	1	yes	characteristic
190	C₄	—	3	4	no	—
191	C₄	C₂×C₄×D₃×C₇	1	1	yes	characteristic
192	C₄	C₂×C₄×D₃×C₇	1	2	yes	—
193	C₄	C₂³×D₃×C₇	1	1	yes	characteristic
194	C₃	C₂×C₇×#131	1	1	yes	characteristic, complemented, Sylow
195	C₂	—	2	2	no	—
196	C₂	—	6	4	no	—
197	C₂	C₇×#907	1	2	yes	direct product factor
198	C₂	C₂²×C₄×D₃×C₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₃
commutator	C₂×C₃	C₂³×C₄×C₇
Frattini	C₄	C₂³×D₃×C₇
Fitting	C₂×C₃×C₇×M₁₆	C₂
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂×C₇×#235	C₂
cosocle	C₃×C₄	C₂⁴×C₇
hypercenter	C₂×C₇×M₁₆	D₃
hypocenter	C₃	C₂×C₇×#131
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₇×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₇×M₁₆	C₂	semidirect
3. C₂²×C₇×C₃⋊C₈	C₂	semidirect
4. C₇×#907	C₂	direct (N × H)
5. C₂×C₇×ΓL(1,𝔽₂₅)	C₂	semidirect
6. C₂×D₃×C₇×C₈	C₂	semidirect
15. C₃×C₇×M₁₆	C₂²	semidirect
19. C₂×C₇×C₃⋊C₈	C₂²	semidirect
20. C₇×ΓL(1,𝔽₂₅)	C₂²	semidirect
21. D₃×C₇×C₈	C₂²	semidirect
25. C₂×C₇×M₁₆	D₃	semidirect
28. C₂×#907	C₇	direct (N × H)
52. C₇×M₁₆	C₂×D₃	semidirect
59. C₂×C₃×M₁₆	C₂×C₇	semidirect
60. C₂²×C₃⋊C₈	C₂×C₇	semidirect
61. #907	C₂×C₇	direct (N × H)
62. C₂×ΓL(1,𝔽₂₅)	C₂×C₇	semidirect
63. C₂×D₃×C₈	C₂×C₇	semidirect
96. C₃×M₁₆	C₂²×C₇	semidirect
100. C₂×C₃⋊C₈	C₂²×C₇	semidirect
101. ΓL(1,𝔽₂₅)	C₂²×C₇	semidirect
102. D₃×C₈	C₂²×C₇	semidirect
104. C₂×C₃×C₇	#131	semidirect
110. C₂×M₁₆	D₃×C₇	semidirect
137. C₃×C₇	C₂×#131	semidirect
146. M₁₆	C₂×D₃×C₇	semidirect
154. C₂×C₇	#907	direct (N × H)
180. C₇	C₂×#907	direct (N × H)
182. C₂×C₃	C₇×#131	semidirect
194. C₃	C₂×C₇×#131	semidirect
197. C₂	C₇×#907	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#235	2
C₂×C₃×C₇×M₁₆	2
C₂²×C₇×C₃⋊C₈	2
C₇×#907	2
C₂×C₇×ΓL(1,𝔽₂₅)	2
C₂×D₃×C₇×C₈	2
C₂×C₇×#131	3
C₂×#907	7

Subgroups — C2×C7×#907 #13061460

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#907 #13061460