Subgroups — C₂×C₇×#2091 #13200938

Order 1792.

1102 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#430	C₂	1	3	yes	maximal, complemented
2	C₂×C₇×#407	C₂	1	3	yes	maximal
3	C₇×#2091	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₇×#338	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₂²×C₇×M₁₆	C₂²	1	3	yes	—
7	C₂²×C₇×M₁₆	C₂²	1	3	yes	—
8	C₇×#430	—	2	6	no	—
9	C₇×#430	C₂²	1	12	yes	complemented
10	C₂×C₇×#108	—	2	6	no	—
11	C₇×#407	—	2	6	no	—
12	C₇×#407	C₂²	1	12	yes	—
13	C₂×C₇×C₂²⋊C₈	—	2	6	no	—
14	C₇×#338	C₂²	1	4	yes	complemented
15	C₂×#2091	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂⁵×C₇	Q₈	1	1	yes	characteristic
17	C₂³×C₄×C₇	—	2	3	no	—
18	C₂³×C₄×C₇	—	2	3	no	—
19	C₂³×C₄×C₇	—	2	3	no	—
20	C₂³×C₄×C₇	—	2	3	no	—
21	C₂³×C₄×C₇	Q₈	1	1	yes	characteristic
22	C₂³×C₄×C₇	Q₈	1	2	yes	—
23	C₂³×C₄×C₇	C₂³	1	2	yes	—
24	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₂×C₇×M₁₆	—	2	6	no	—
26	C₂×C₇×M₁₆	—	2	6	no	—
27	C₂×C₇×M₁₆	—	2	3	no	—
28	C₂×C₇×M₁₆	—	2	6	no	—
29	C₂×C₇×M₁₆	—	2	6	no	—
30	C₂×C₇×M₁₆	D₄	1	6	yes	—
31	C₂×C₇×M₁₆	C₂³	1	6	yes	—
32	C₂×C₇×M₁₆	C₂³	1	6	yes	—
33	C₂²×C₇×C₈	—	2	3	no	—
34	C₂²×C₇×C₈	D₄	1	6	yes	—
35	C₇×#108	—	2	24	no	—
36	C₇×C₂²⋊C₈	—	2	24	no	—
37	C₂×#430	C₂×C₇	1	3	yes	complemented
38	C₂×#407	C₂×C₇	1	3	yes	—
39	#2091	C₂×C₇	1	8	yes	direct product factor
40	C₂×#338	C₂×C₇	1	1	yes	characteristic, complemented
41	C₂⁴×C₇	—	2	3	no	—
42	C₂⁴×C₇	—	2	3	no	—
43	C₂⁴×C₇	—	4	4	no	—
44	C₂⁴×C₇	C₂×Q₈	1	2	yes	—
45	C₂⁴×C₇	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₇	—	2	3	no	—
47	C₂²×C₄×C₇	—	2	3	no	—
48	C₂²×C₄×C₇	—	2	6	no	—
49	C₂²×C₄×C₇	—	2	6	no	—
50	C₂²×C₄×C₇	—	2	6	no	—
51	C₂²×C₄×C₇	—	2	6	no	—
52	C₂²×C₄×C₇	—	4	3	no	—
53	C₂²×C₄×C₇	—	4	3	no	—
54	C₂²×C₄×C₇	—	4	6	no	—
55	C₂²×C₄×C₇	—	4	3	no	—
56	C₂²×C₄×C₇	—	4	1	no	—
57	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
58	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
59	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₇	#41	1	3	yes	—
62	C₂²×C₄×C₇	#41	1	3	yes	—
63	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
64	C₇×M₁₆	—	2	12	no	—
65	C₇×M₁₆	—	2	12	no	—
66	C₇×M₁₆	—	2	12	no	—
67	C₇×M₁₆	—	4	6	no	—
68	C₂×C₇×C₈	—	2	6	no	—
69	C₂×C₇×C₈	—	2	6	no	—
70	C₂×C₇×C₈	—	2	6	no	—
71	C₂×C₇×C₈	—	4	3	no	—
72	C₂×C₇×C₈	—	4	3	no	—
73	C₂×C₇×C₈	C₂×D₄	1	12	yes	—
74	C₂⁴×C₄	C₂²×C₇	1	1	yes	characteristic
75	C₂²×M₁₆	C₂²×C₇	1	3	yes	—
76	C₂²×M₁₆	C₂²×C₇	1	3	yes	—
77	#430	—	2	6	no	—
78	#430	C₂²×C₇	1	12	yes	complemented
79	C₂×#108	—	2	6	no	—
80	#407	—	2	6	no	—
81	#407	C₂²×C₇	1	12	yes	—
82	C₂×C₂²⋊C₈	—	2	6	no	—
83	#338	C₂²×C₇	1	4	yes	complemented
84	C₂³×C₇	—	2	12	no	—
85	C₂³×C₇	—	2	6	no	—
86	C₂³×C₇	—	4	2	no	—
87	C₂³×C₇	—	4	4	no	—
88	C₂³×C₇	—	4	6	no	—
89	C₂³×C₇	—	4	12	no	—
90	C₂³×C₇	—	4	3	no	—
91	C₂³×C₇	—	4	1	no	—
92	C₂³×C₇	C₂²⋊Q₈	1	3	yes	—
93	C₂³×C₇	C₂²⋊Q₈	1	3	yes	—
94	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₇	—	2	6	no	—
96	C₂×C₄×C₇	—	4	3	no	—
97	C₂×C₄×C₇	—	4	1	no	—
98	C₂×C₄×C₇	—	4	6	no	—
99	C₂×C₄×C₇	—	4	2	no	—
100	C₂×C₄×C₇	—	4	6	no	—
101	C₂×C₄×C₇	—	4	2	no	—
102	C₂×C₄×C₇	C₂²⋊Q₈	1	3	yes	—
103	C₂×C₄×C₇	C₂²⋊Q₈	1	3	yes	—
104	C₂×C₄×C₇	#123	1	3	yes	—
105	C₂×C₄×C₇	#123	1	3	yes	—
106	C₂×C₄×C₇	#123	1	6	yes	—
107	C₂×C₄×C₇	#127	1	3	yes	—
108	C₂×C₄×C₇	#127	1	1	yes	characteristic
109	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
110	C₂×C₄×C₇	C₂×#41	1	3	yes	—
111	C₇×C₈	—	2	12	no	—
112	C₇×C₈	—	4	6	no	—
113	C₂⁵	C₇×Q₈	1	1	yes	characteristic
114	C₂³×C₄	—	2	3	no	—
115	C₂³×C₄	—	2	3	no	—
116	C₂³×C₄	—	2	3	no	—
117	C₂³×C₄	—	2	3	no	—
118	C₂³×C₄	C₇×Q₈	1	1	yes	characteristic
119	C₂³×C₄	C₇×Q₈	1	2	yes	—
120	C₂³×C₄	C₂³×C₇	1	2	yes	—
121	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
122	C₂×M₁₆	—	2	6	no	—
123	C₂×M₁₆	—	2	6	no	—
124	C₂×M₁₆	—	2	3	no	—
125	C₂×M₁₆	—	2	6	no	—
126	C₂×M₁₆	—	2	6	no	—
127	C₂×M₁₆	C₇×D₄	1	6	yes	—
128	C₂×M₁₆	C₂³×C₇	1	6	yes	—
129	C₂×M₁₆	C₂³×C₇	1	6	yes	—
130	C₂²×C₈	—	2	3	no	—
131	C₂²×C₈	C₇×D₄	1	6	yes	—
132	#108	—	2	24	no	—
133	C₂²⋊C₈	—	2	24	no	—
134	C₂²×C₇	—	2	6	no	—
135	C₂²×C₇	—	2	12	no	—
136	C₂²×C₇	—	4	3	no	—
137	C₂²×C₇	—	4	1	no	—
138	C₂²×C₇	—	4	6	no	—
139	C₂²×C₇	—	4	12	no	—
140	C₂²×C₇	—	4	4	no	—
141	C₂²×C₇	—	4	2	no	—
142	C₂²×C₇	#399	1	3	yes	—
143	C₂²×C₇	#399	1	1	yes	characteristic
144	C₂²×C₇	C₂×C₂²⋊Q₈	1	3	yes	—
145	C₄×C₇	—	4	2	no	—
146	C₄×C₇	#399	1	3	yes	—
147	C₄×C₇	#399	1	1	yes	characteristic
148	C₄×C₇	C₂×#123	1	3	yes	—
149	C₄×C₇	C₂×#127	1	1	yes	characteristic
150	C₂⁴	—	2	3	no	—
151	C₂⁴	—	2	3	no	—
152	C₂⁴	—	4	4	no	—
153	C₂⁴	C₂×C₇×Q₈	1	2	yes	—
154	C₂⁴	C₂×C₇×Q₈	1	1	yes	characteristic
155	C₂²×C₄	—	2	3	no	—
156	C₂²×C₄	—	2	3	no	—
157	C₂²×C₄	—	2	6	no	—
158	C₂²×C₄	—	2	6	no	—
159	C₂²×C₄	—	2	6	no	—
160	C₂²×C₄	—	2	6	no	—
161	C₂²×C₄	—	4	3	no	—
162	C₂²×C₄	—	4	3	no	—
163	C₂²×C₄	—	4	6	no	—
164	C₂²×C₄	—	4	1	no	—
165	C₂²×C₄	—	4	3	no	—
166	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
167	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
168	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
169	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
170	C₂²×C₄	C₇×#41	1	3	yes	—
171	C₂²×C₄	C₇×#41	1	3	yes	—
172	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
173	M₁₆	—	2	12	no	—
174	M₁₆	—	2	12	no	—
175	M₁₆	—	2	12	no	—
176	M₁₆	—	4	6	no	—
177	C₂×C₈	—	2	6	no	—
178	C₂×C₈	—	2	6	no	—
179	C₂×C₈	—	2	6	no	—
180	C₂×C₈	—	4	3	no	—
181	C₂×C₈	—	4	3	no	—
182	C₂×C₈	C₂×C₇×D₄	1	12	yes	—
183	C₂×C₇	—	2	3	no	—
184	C₂×C₇	—	2	3	no	—
185	C₂×C₇	—	4	4	no	—
186	C₂×C₇	#2091	1	2	yes	direct product factor
187	C₂×C₇	C₂×#399	1	1	yes	characteristic
188	C₂³	—	2	12	no	—
189	C₂³	—	2	6	no	—
190	C₂³	—	4	2	no	—
191	C₂³	—	4	4	no	—
192	C₂³	—	4	6	no	—
193	C₂³	—	4	12	no	—
194	C₂³	—	4	3	no	—
195	C₂³	—	4	1	no	—
196	C₂³	C₇×C₂²⋊Q₈	1	3	yes	—
197	C₂³	C₇×C₂²⋊Q₈	1	3	yes	—
198	C₂³	C₂²×C₇×Q₈	1	1	yes	characteristic
199	C₂×C₄	—	2	6	no	—
200	C₂×C₄	—	4	3	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	6	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	6	no	—
206	C₂×C₄	C₇×C₂²⋊Q₈	1	3	yes	—
207	C₂×C₄	C₇×C₂²⋊Q₈	1	3	yes	—
208	C₂×C₄	C₇×#123	1	3	yes	—
209	C₂×C₄	C₇×#123	1	3	yes	—
210	C₂×C₄	C₇×#123	1	6	yes	—
211	C₂×C₄	C₇×#127	1	1	yes	characteristic
212	C₂×C₄	C₇×#127	1	3	yes	—
213	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
214	C₂×C₄	C₂×C₇×#41	1	3	yes	—
215	C₈	—	2	12	no	—
216	C₈	—	4	6	no	—
217	C₇	C₂×#2091	1	1	yes	characteristic, direct product factor, Sylow
218	C₂²	—	2	6	no	—
219	C₂²	—	2	12	no	—
220	C₂²	—	4	3	no	—
221	C₂²	—	4	1	no	—
222	C₂²	—	4	6	no	—
223	C₂²	—	4	12	no	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	4	no	—
226	C₂²	C₇×#399	1	3	yes	—
227	C₂²	C₇×#399	1	1	yes	characteristic
228	C₂²	C₂×C₇×C₂²⋊Q₈	1	3	yes	—
229	C₄	—	4	2	no	—
230	C₄	C₇×#399	1	1	yes	characteristic
231	C₄	C₇×#399	1	3	yes	—
232	C₄	C₂×C₇×#123	1	3	yes	—
233	C₄	C₂×C₇×#127	1	1	yes	characteristic
234	C₂	—	2	3	no	—
235	C₂	—	2	3	no	—
236	C₂	—	4	4	no	—
237	C₂	C₇×#2091	1	2	yes	direct product factor
238	C₂	C₂×C₇×#399	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	#127
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂²×C₇	#399
prime radical	C₂⁵×C₇	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂³×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#430	C₂	semidirect
3. C₇×#2091	C₂	direct (N × H)
4. C₂×C₇×#338	C₂	semidirect
9. C₇×#430	C₂²	semidirect
14. C₇×#338	C₂²	semidirect
15. C₂×#2091	C₇	direct (N × H)
37. C₂×#430	C₂×C₇	semidirect
39. #2091	C₂×C₇	direct (N × H)
40. C₂×#338	C₂×C₇	semidirect
78. #430	C₂²×C₇	semidirect
83. #338	C₂²×C₇	semidirect
186. C₂×C₇	#2091	direct (N × H)
217. C₇	C₂×#2091	direct (N × H)
237. C₂	C₇×#2091	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×#430	2
C₂×C₇×#407	2
C₇×#2091	2
C₂×C₇×#338	2
C₂×#2091	7

Subgroups — C2×C7×#2091 #13200938

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2091 #13200938