Subgroups — C₂×C₇×#904 #13061457

Order 1344.

532 conjugacy classes of subgroups.

The subgroup lattice has 196 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	2	yes	maximal, complemented
2	C₂²×C₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#904	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₇×D₂₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×#204	C₂	1	2	yes	maximal, complemented
7	C₂×C₇×#135	—	3	1	no	maximal, Hall
8	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₇×#67	—	2	2	no	—
10	C₇×#235	—	2	4	no	—
11	C₇×#235	C₂²	1	8	yes	complemented
12	C₂×C₇×D₁₂	C₂²	1	2	yes	—
13	C₂×C₄×D₃×C₇	—	2	2	no	—
14	C₂×C₇×Dic₆	C₂²	1	2	yes	—
15	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₇×C₈	C₂²	1	4	yes	complemented
18	C₇×Dic₁₂	C₂²	1	4	yes	complemented
19	C₇×D₂₄	C₂²	1	4	yes	complemented
20	C₇×#204	C₂²	1	8	yes	complemented
21	C₂×C₇×#41	—	3	2	no	—
22	C₇×#135	—	3	8	no	—
23	C₂×C₇×Q₁₆	—	3	1	no	—
24	C₂×C₇×SD₁₆	—	3	2	no	—
25	C₂×C₇×D₈	—	3	1	no	—
26	C₂²×C₇×C₈	D₃	1	1	yes	characteristic, complemented
27	C₂×#904	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
28	C₂³×C₃×C₇	D₄	1	1	yes	characteristic
29	C₂²×D₃×C₇	—	2	2	no	—
30	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
31	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
33	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
34	C₇×#67	—	2	8	no	—
35	C₂×C₇×Dic₃	—	2	2	no	—
36	C₇×D₁₂	—	2	2	no	—
37	C₇×D₁₂	C₂³	1	4	yes	—
38	C₄×D₃×C₇	—	2	8	no	—
39	C₇×Dic₆	—	2	2	no	—
40	C₇×Dic₆	C₂³	1	4	yes	—
41	C₃×C₇×C₈	C₂³	1	2	yes	—
42	C₃×C₇×C₈	C₂³	1	2	yes	—
43	C₇×#41	—	3	8	no	—
44	C₇×#41	—	6	4	no	—
45	C₂×C₇×Q₈	—	3	2	no	—
46	C₂×C₇×D₄	—	3	2	no	—
47	C₂×C₇×D₄	—	6	2	no	—
48	C₂²×C₄×C₇	—	6	2	no	—
49	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
50	C₇×Q₁₆	—	3	4	no	—
51	C₇×SD₁₆	—	3	8	no	—
52	C₇×D₈	—	3	4	no	—
53	C₂×C₇×C₈	C₂×D₃	1	1	yes	characteristic
54	C₂×C₇×C₈	C₂×D₃	1	1	yes	characteristic
55	C₂×C₇×C₈	C₂×D₃	1	4	yes	complemented
56	C₂×#235	C₂×C₇	1	2	yes	complemented
57	C₂²×C₃×C₈	C₂×C₇	1	1	yes	characteristic, complemented
58	C₂×Dic₁₂	C₂×C₇	1	1	yes	characteristic, complemented
59	#904	C₂×C₇	1	8	yes	direct product factor
60	C₂×D₂₄	C₂×C₇	1	1	yes	characteristic, complemented
61	C₂×#204	C₂×C₇	1	2	yes	complemented
62	C₂²×C₃×C₇	—	2	2	no	—
63	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
64	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
65	C₂×D₃×C₇	—	2	4	no	—
66	C₂×D₃×C₇	—	4	4	no	—
67	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
68	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
69	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
70	C₇×Dic₃	—	2	4	no	—
71	C₂×#135	—	3	1	no	Sylow
72	C₂³×C₇	—	6	2	no	—
73	C₂³×C₇	D₁₂	1	1	yes	characteristic
74	C₇×Q₈	—	3	4	no	—
75	C₇×Q₈	—	6	2	no	—
76	C₇×D₄	—	3	4	no	—
77	C₇×D₄	—	6	8	no	—
78	C₇×D₄	—	6	2	no	—
79	C₂×C₄×C₇	—	6	2	no	—
80	C₂×C₄×C₇	—	6	8	no	—
81	C₂×C₄×C₇	D₁₂	1	2	yes	—
82	C₂×C₄×C₇	D₁₂	1	1	yes	characteristic
83	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
84	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
85	C₇×C₈	C₂²×D₃	1	2	yes	—
86	C₇×C₈	C₂²×D₃	1	2	yes	—
87	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
88	C₂×#67	—	2	2	no	—
89	#235	—	2	4	no	—
90	#235	C₂²×C₇	1	8	yes	complemented
91	C₂×D₁₂	C₂²×C₇	1	2	yes	—
92	C₂×C₄×D₃	—	2	2	no	—
93	C₂×Dic₆	C₂²×C₇	1	2	yes	—
94	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
95	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
96	C₂×C₃×C₈	C₂²×C₇	1	4	yes	complemented
97	Dic₁₂	C₂²×C₇	1	4	yes	complemented
98	D₂₄	C₂²×C₇	1	4	yes	complemented
99	#204	C₂²×C₇	1	8	yes	complemented
100	C₂×C₃×C₇	—	2	2	no	—
101	C₂×C₃×C₇	#135	1	2	yes	complemented
102	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
103	D₃×C₇	—	4	4	no	—
104	C₂×#41	—	3	2	no	—
105	#135	—	3	8	no	—
106	C₂×Q₁₆	—	3	1	no	—
107	C₂×SD₁₆	—	3	2	no	—
108	C₂×D₈	—	3	1	no	—
109	C₂²×C₈	D₃×C₇	1	1	yes	characteristic, complemented
110	C₂²×C₇	—	2	2	no	—
111	C₂²×C₇	—	6	4	no	—
112	C₂²×C₇	—	12	4	no	—
113	C₂²×C₇	C₂×D₁₂	1	2	yes	—
114	C₂²×C₇	C₂×D₁₂	1	1	yes	characteristic
115	C₄×C₇	—	6	4	no	—
116	C₄×C₇	C₂×D₁₂	1	2	yes	—
117	C₄×C₇	C₂×D₁₂	1	1	yes	characteristic
118	C₄×C₇	C₂³×D₃	1	1	yes	characteristic
119	C₂³×C₃	C₇×D₄	1	1	yes	characteristic
120	C₂²×D₃	—	2	2	no	—
121	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
122	C₂×C₃×C₄	C₇×D₄	1	2	yes	—
123	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
125	#67	—	2	8	no	—
126	C₂×Dic₃	—	2	2	no	—
127	D₁₂	—	2	2	no	—
128	D₁₂	C₂³×C₇	1	4	yes	—
129	C₄×D₃	—	2	8	no	—
130	Dic₆	—	2	2	no	—
131	Dic₆	C₂³×C₇	1	4	yes	—
132	C₃×C₈	C₂³×C₇	1	2	yes	—
133	C₃×C₈	C₂³×C₇	1	2	yes	—
134	C₃×C₇	C₂×#135	1	1	yes	characteristic, complemented, Hall
135	#41	—	3	8	no	—
136	#41	—	6	4	no	—
137	C₂×Q₈	—	3	2	no	—
138	C₂×D₄	—	3	2	no	—
139	C₂×D₄	—	6	2	no	—
140	C₂²×C₄	—	6	2	no	—
141	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
142	Q₁₆	—	3	4	no	—
143	SD₁₆	—	3	8	no	—
144	D₈	—	3	4	no	—
145	C₂×C₈	C₂×D₃×C₇	1	1	yes	characteristic
146	C₂×C₈	C₂×D₃×C₇	1	1	yes	characteristic
147	C₂×C₈	C₂×D₃×C₇	1	4	yes	complemented
148	C₂×C₇	—	2	2	no	—
149	C₂×C₇	—	12	4	no	—
150	C₂×C₇	#904	1	2	yes	direct product factor
151	C₂×C₇	C₂²×D₁₂	1	1	yes	characteristic
152	C₂²×C₃	—	2	2	no	—
153	C₂²×C₃	C₂×C₇×D₄	1	2	yes	—
154	C₂²×C₃	C₂×C₇×D₄	1	1	yes	characteristic
155	C₂×D₃	—	2	4	no	—
156	C₂×D₃	—	4	4	no	—
157	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
158	C₃×C₄	C₂×C₇×D₄	1	2	yes	—
159	C₃×C₄	C₂⁴×C₇	1	1	yes	characteristic
160	Dic₃	—	2	4	no	—
161	C₂³	—	6	2	no	—
162	C₂³	C₇×D₁₂	1	1	yes	characteristic
163	Q₈	—	3	4	no	—
164	Q₈	—	6	2	no	—
165	D₄	—	3	4	no	—
166	D₄	—	6	8	no	—
167	D₄	—	6	2	no	—
168	C₂×C₄	—	6	2	no	—
169	C₂×C₄	—	6	8	no	—
170	C₂×C₄	C₇×D₁₂	1	2	yes	—
171	C₂×C₄	C₇×D₁₂	1	1	yes	characteristic
172	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
173	C₂×C₄	C₂²×D₃×C₇	1	2	yes	—
174	C₈	C₂²×D₃×C₇	1	2	yes	—
175	C₈	C₂²×D₃×C₇	1	2	yes	—
176	C₇	C₂×#904	1	1	yes	characteristic, direct product factor, Sylow
177	C₂×C₃	—	2	2	no	—
178	C₂×C₃	C₇×#135	1	2	yes	complemented
179	C₂×C₃	C₂²×C₇×D₄	1	1	yes	characteristic
180	D₃	—	4	4	no	—
181	C₂²	—	2	2	no	—
182	C₂²	—	6	4	no	—
183	C₂²	—	12	4	no	—
184	C₂²	C₂×C₇×D₁₂	1	2	yes	—
185	C₂²	C₂×C₇×D₁₂	1	1	yes	characteristic
186	C₄	—	6	4	no	—
187	C₄	C₂×C₇×D₁₂	1	2	yes	—
188	C₄	C₂×C₇×D₁₂	1	1	yes	characteristic
189	C₄	C₂³×D₃×C₇	1	1	yes	characteristic
190	C₃	C₂×C₇×#135	1	1	yes	characteristic, complemented, Sylow
191	C₂	—	2	2	no	—
192	C₂	—	12	4	no	—
193	C₂	C₇×#904	1	2	yes	direct product factor
194	C₂	C₂²×C₇×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	D₁₂
commutator, cosocle	C₃×C₄	C₂⁴×C₇
Frattini	C₄	C₂³×D₃×C₇
Fitting	C₂²×C₃×C₇×C₈	C₂
socle	C₂²×C₃×C₇	C₂×D₄
hypercenter	C₂²×C₇×C₈	D₃
hypocenter	C₃	C₂×C₇×#135
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#235	C₂	semidirect
2. C₂²×C₃×C₇×C₈	C₂	semidirect
3. C₂×C₇×Dic₁₂	C₂	semidirect
4. C₇×#904	C₂	direct (N × H)
5. C₂×C₇×D₂₄	C₂	semidirect
6. C₂×C₇×#204	C₂	semidirect
11. C₇×#235	C₂²	semidirect
17. C₂×C₃×C₇×C₈	C₂²	semidirect
18. C₇×Dic₁₂	C₂²	semidirect
19. C₇×D₂₄	C₂²	semidirect
20. C₇×#204	C₂²	semidirect
26. C₂²×C₇×C₈	D₃	semidirect
27. C₂×#904	C₇	direct (N × H)
55. C₂×C₇×C₈	C₂×D₃	semidirect
56. C₂×#235	C₂×C₇	semidirect
57. C₂²×C₃×C₈	C₂×C₇	semidirect
58. C₂×Dic₁₂	C₂×C₇	semidirect
59. #904	C₂×C₇	direct (N × H)
60. C₂×D₂₄	C₂×C₇	semidirect
61. C₂×#204	C₂×C₇	semidirect
90. #235	C₂²×C₇	semidirect
96. C₂×C₃×C₈	C₂²×C₇	semidirect
97. Dic₁₂	C₂²×C₇	semidirect
98. D₂₄	C₂²×C₇	semidirect
99. #204	C₂²×C₇	semidirect
101. C₂×C₃×C₇	#135	semidirect
109. C₂²×C₈	D₃×C₇	semidirect
134. C₃×C₇	C₂×#135	semidirect
147. C₂×C₈	C₂×D₃×C₇	semidirect
150. C₂×C₇	#904	direct (N × H)
176. C₇	C₂×#904	direct (N × H)
178. C₂×C₃	C₇×#135	semidirect
190. C₃	C₂×C₇×#135	semidirect
193. C₂	C₇×#904	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
C₂²×C₃×C₇×C₈	2
C₂×C₇×Dic₁₂	2
C₇×#904	2
C₂×C₇×D₂₄	2
C₂×C₇×#204	2
C₂×C₇×#135	3
C₂×#904	7

Subgroups — C2×C7×#904 #13061457

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#904 #13061457