Subgroups — C₂×C₇×#942 #13061527

Order 1344.

516 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#235	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#216	C₂	1	2	yes	maximal, complemented
4	C₇×#942	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₇×#215	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#208	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×#137	—	3	1	no	maximal, Hall
8	C₂×C₃×C₇×Q₈	—	2	1	no	—
9	C₂×C₃×C₇×Q₈	—	2	2	no	—
10	C₂×C₃×C₇×Q₈	C₂²	1	2	yes	—
11	C₂×C₃×C₇×Q₈	C₂²	1	4	yes	complemented
12	C₂²×C₃×C₄×C₇	—	2	1	no	—
13	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₂×C₇×#67	—	2	1	no	—
15	C₇×#235	—	2	2	no	—
16	C₇×#235	C₂²	1	4	yes	—
17	C₂×C₇×D₁₂	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₃×C₇	—	2	1	no	—
19	C₂×C₇×Dic₆	C₂²	1	1	yes	characteristic
20	C₇×#216	C₂²	1	8	yes	complemented
21	C₇×#215	C₂²	1	8	yes	complemented
22	C₇×#208	C₂²	1	4	yes	complemented
23	C₂×C₇×C₃⋊C₈	C₂²	1	2	yes	—
24	C₂×C₇×#41	—	3	1	no	—
25	C₂²×C₇×Q₈	D₃	1	1	yes	characteristic, complemented
26	C₇×#137	—	3	8	no	—
27	C₂×C₇×Q₁₆	—	3	2	no	—
28	C₂×C₇×SD₁₆	—	3	2	no	—
29	C₂×C₇×M₁₆	—	3	1	no	—
30	C₂×#942	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
31	C₂³×C₃×C₇	D₄	1	1	yes	characteristic
32	C₂²×D₃×C₇	—	2	1	no	—
33	C₃×C₇×Q₈	—	2	2	no	—
34	C₃×C₇×Q₈	—	2	4	no	—
35	C₃×C₇×Q₈	C₂³	1	4	yes	—
36	C₂×C₃×C₄×C₇	—	2	2	no	—
37	C₂×C₃×C₄×C₇	—	2	4	no	—
38	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
40	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
42	C₇×#67	—	2	4	no	—
43	C₂×C₇×Dic₃	—	2	1	no	—
44	C₇×D₁₂	—	2	1	no	—
45	C₇×D₁₂	C₂³	1	2	yes	—
46	C₄×D₃×C₇	—	2	4	no	—
47	C₇×Dic₆	—	2	1	no	—
48	C₇×Dic₆	C₂³	1	2	yes	—
49	C₇×C₃⋊C₈	C₂³	1	4	yes	—
50	C₇×#41	—	3	4	no	—
51	C₇×#41	—	6	2	no	—
52	C₂×C₇×Q₈	—	2	2	no	—
53	C₂×C₇×Q₈	—	2	1	no	—
54	C₂×C₇×Q₈	—	3	1	no	—
55	C₂×C₇×Q₈	C₂×D₃	1	2	yes	—
56	C₂×C₇×Q₈	C₂×D₃	1	4	yes	complemented
57	C₂×C₇×D₄	—	3	1	no	—
58	C₂×C₇×D₄	—	6	1	no	—
59	C₂²×C₄×C₇	—	2	1	no	—
60	C₂²×C₄×C₇	—	6	1	no	—
61	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
62	C₇×Q₁₆	—	3	8	no	—
63	C₇×SD₁₆	—	3	8	no	—
64	C₇×M₁₆	—	3	4	no	—
65	C₂×C₇×C₈	—	3	2	no	—
66	C₂²×C₃×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
67	C₂×#235	C₂×C₇	1	1	yes	characteristic
68	C₂×#216	C₂×C₇	1	2	yes	complemented
69	#942	C₂×C₇	1	8	yes	direct product factor
70	C₂×#215	C₂×C₇	1	2	yes	complemented
71	C₂×#208	C₂×C₇	1	1	yes	characteristic, complemented
72	C₂²×C₃×C₇	—	2	2	no	—
73	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
74	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
75	C₂×D₃×C₇	—	2	2	no	—
76	C₂×D₃×C₇	—	4	2	no	—
77	C₃×C₄×C₇	—	2	4	no	—
78	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
79	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
80	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
81	C₇×Dic₃	—	2	2	no	—
82	C₂×#137	—	3	1	no	Sylow
83	C₂³×C₇	—	6	1	no	—
84	C₂³×C₇	#67	1	1	yes	characteristic
85	C₇×Q₈	—	2	2	no	—
86	C₇×Q₈	—	2	4	no	—
87	C₇×Q₈	—	3	2	no	—
88	C₇×Q₈	—	6	1	no	—
89	C₇×Q₈	C₂²×D₃	1	4	yes	—
90	C₇×D₄	—	3	2	no	—
91	C₇×D₄	—	6	4	no	—
92	C₇×D₄	—	6	1	no	—
93	C₂×C₄×C₇	—	2	4	no	—
94	C₂×C₄×C₇	—	2	2	no	—
95	C₂×C₄×C₇	—	6	1	no	—
96	C₂×C₄×C₇	—	6	4	no	—
97	C₂×C₄×C₇	#67	1	2	yes	—
98	C₂×C₄×C₇	#67	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
100	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
101	C₇×C₈	—	3	4	no	—
102	C₂×C₃×Q₈	—	2	1	no	—
103	C₂×C₃×Q₈	—	2	2	no	—
104	C₂×C₃×Q₈	C₂²×C₇	1	2	yes	—
105	C₂×C₃×Q₈	C₂²×C₇	1	4	yes	complemented
106	C₂²×C₃×C₄	—	2	1	no	—
107	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
108	C₂×#67	—	2	1	no	—
109	#235	—	2	2	no	—
110	#235	C₂²×C₇	1	4	yes	—
111	C₂×D₁₂	C₂²×C₇	1	1	yes	characteristic
112	C₂×C₄×D₃	—	2	1	no	—
113	C₂×Dic₆	C₂²×C₇	1	1	yes	characteristic
114	#216	C₂²×C₇	1	8	yes	complemented
115	#215	C₂²×C₇	1	8	yes	complemented
116	#208	C₂²×C₇	1	4	yes	complemented
117	C₂×C₃⋊C₈	C₂²×C₇	1	2	yes	—
118	C₂×C₃×C₇	—	2	2	no	—
119	C₂×C₃×C₇	#137	1	2	yes	complemented
120	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
121	D₃×C₇	—	4	2	no	—
122	C₂×#41	—	3	1	no	—
123	C₂²×Q₈	D₃×C₇	1	1	yes	characteristic, complemented
124	#137	—	3	8	no	—
125	C₂×Q₁₆	—	3	2	no	—
126	C₂×SD₁₆	—	3	2	no	—
127	C₂×M₁₆	—	3	1	no	—
128	C₂²×C₇	—	2	2	no	—
129	C₂²×C₇	—	6	2	no	—
130	C₂²×C₇	—	12	2	no	—
131	C₂²×C₇	C₂×#67	1	2	yes	—
132	C₂²×C₇	C₂×#67	1	1	yes	characteristic
133	C₄×C₇	—	2	4	no	—
134	C₄×C₇	—	6	2	no	—
135	C₄×C₇	C₂×#67	1	2	yes	—
136	C₄×C₇	C₂×#67	1	1	yes	characteristic
137	C₄×C₇	C₂³×D₃	1	1	yes	characteristic
138	C₂³×C₃	C₇×D₄	1	1	yes	characteristic
139	C₂²×D₃	—	2	1	no	—
140	C₃×Q₈	—	2	2	no	—
141	C₃×Q₈	—	2	4	no	—
142	C₃×Q₈	C₂³×C₇	1	4	yes	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	4	no	—
145	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
146	C₂×C₃×C₄	C₇×D₄	1	2	yes	—
147	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
149	#67	—	2	4	no	—
150	C₂×Dic₃	—	2	1	no	—
151	D₁₂	—	2	1	no	—
152	D₁₂	C₂³×C₇	1	2	yes	—
153	C₄×D₃	—	2	4	no	—
154	Dic₆	—	2	1	no	—
155	Dic₆	C₂³×C₇	1	2	yes	—
156	C₃⋊C₈	C₂³×C₇	1	4	yes	—
157	C₃×C₇	C₂×#137	1	1	yes	characteristic, complemented, Hall
158	#41	—	3	4	no	—
159	#41	—	6	2	no	—
160	C₂×Q₈	—	2	1	no	—
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	3	1	no	—
163	C₂×Q₈	C₂×D₃×C₇	1	2	yes	—
164	C₂×Q₈	C₂×D₃×C₇	1	4	yes	complemented
165	C₂×D₄	—	3	1	no	—
166	C₂×D₄	—	6	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	6	1	no	—
169	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
170	Q₁₆	—	3	8	no	—
171	SD₁₆	—	3	8	no	—
172	M₁₆	—	3	4	no	—
173	C₂×C₈	—	3	2	no	—
174	C₂×C₇	—	2	2	no	—
175	C₂×C₇	—	12	2	no	—
176	C₂×C₇	#942	1	2	yes	direct product factor
177	C₂×C₇	C₂²×#67	1	1	yes	characteristic
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	C₂×C₇×D₄	1	2	yes	—
180	C₂²×C₃	C₂×C₇×D₄	1	1	yes	characteristic
181	C₂×D₃	—	2	2	no	—
182	C₂×D₃	—	4	2	no	—
183	C₃×C₄	—	2	4	no	—
184	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
185	C₃×C₄	C₂×C₇×D₄	1	2	yes	—
186	C₃×C₄	C₂⁴×C₇	1	1	yes	characteristic
187	Dic₃	—	2	2	no	—
188	C₂³	—	6	1	no	—
189	C₂³	C₇×#67	1	1	yes	characteristic
190	Q₈	—	2	2	no	—
191	Q₈	—	2	4	no	—
192	Q₈	—	3	2	no	—
193	Q₈	—	6	1	no	—
194	Q₈	C₂²×D₃×C₇	1	4	yes	—
195	D₄	—	3	2	no	—
196	D₄	—	6	4	no	—
197	D₄	—	6	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	6	1	no	—
201	C₂×C₄	—	6	4	no	—
202	C₂×C₄	C₇×#67	1	1	yes	characteristic
203	C₂×C₄	C₇×#67	1	2	yes	—
204	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
205	C₂×C₄	C₂²×D₃×C₇	1	2	yes	—
206	C₈	—	3	4	no	—
207	C₇	C₂×#942	1	1	yes	characteristic, direct product factor, Sylow
208	C₂×C₃	—	2	2	no	—
209	C₂×C₃	C₇×#137	1	2	yes	complemented
210	C₂×C₃	C₂²×C₇×D₄	1	1	yes	characteristic
211	D₃	—	4	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	6	2	no	—
214	C₂²	—	12	2	no	—
215	C₂²	C₂×C₇×#67	1	2	yes	—
216	C₂²	C₂×C₇×#67	1	1	yes	characteristic
217	C₄	—	2	4	no	—
218	C₄	—	6	2	no	—
219	C₄	C₂×C₇×#67	1	2	yes	—
220	C₄	C₂×C₇×#67	1	1	yes	characteristic
221	C₄	C₂³×D₃×C₇	1	1	yes	characteristic
222	C₃	C₂×C₇×#137	1	1	yes	characteristic, complemented, Sylow
223	C₂	—	2	2	no	—
224	C₂	—	12	2	no	—
225	C₂	C₇×#942	1	2	yes	direct product factor
226	C₂	C₂²×C₇×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇	C₂×#67
commutator, cosocle	C₃×C₄	C₂⁴×C₇
Frattini	C₄	C₂³×D₃×C₇
Fitting	C₂²×C₃×C₇×Q₈	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂×C₇×#235	C₂
hypercenter	C₂²×C₇×Q₈	D₃
hypocenter	C₃	C₂×C₇×#137
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₇×Q₈	C₂	semidirect
3. C₂×C₇×#216	C₂	semidirect
4. C₇×#942	C₂	direct (N × H)
5. C₂×C₇×#215	C₂	semidirect
6. C₂×C₇×#208	C₂	semidirect
11. C₂×C₃×C₇×Q₈	C₂²	semidirect
20. C₇×#216	C₂²	semidirect
21. C₇×#215	C₂²	semidirect
22. C₇×#208	C₂²	semidirect
25. C₂²×C₇×Q₈	D₃	semidirect
30. C₂×#942	C₇	direct (N × H)
56. C₂×C₇×Q₈	C₂×D₃	semidirect
66. C₂²×C₃×Q₈	C₂×C₇	semidirect
68. C₂×#216	C₂×C₇	semidirect
69. #942	C₂×C₇	direct (N × H)
70. C₂×#215	C₂×C₇	semidirect
71. C₂×#208	C₂×C₇	semidirect
105. C₂×C₃×Q₈	C₂²×C₇	semidirect
114. #216	C₂²×C₇	semidirect
115. #215	C₂²×C₇	semidirect
116. #208	C₂²×C₇	semidirect
119. C₂×C₃×C₇	#137	semidirect
123. C₂²×Q₈	D₃×C₇	semidirect
157. C₃×C₇	C₂×#137	semidirect
164. C₂×Q₈	C₂×D₃×C₇	semidirect
176. C₂×C₇	#942	direct (N × H)
207. C₇	C₂×#942	direct (N × H)
209. C₂×C₃	C₇×#137	semidirect
222. C₃	C₂×C₇×#137	semidirect
225. C₂	C₇×#942	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₇×Q₈	2
C₂×C₇×#235	2
C₂×C₇×#216	2
C₇×#942	2
C₂×C₇×#215	2
C₂×C₇×#208	2
C₂×C₇×#137	3
C₂×#942	7

Subgroups — C2×C7×#942 #13061527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#942 #13061527