Subgroups — C₂×C₇×#1845 #13198011

Order 1792.

1358 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#107	C₂	1	4	yes	maximal
2	C₂²×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×#384	C₂	1	2	yes	maximal
4	C₇×#1845	C₂	1	8	yes	maximal, direct product factor
5	C₂³×C₇×C₈	C₂²	1	2	yes	—
6	C₂²×C₇×C₄⋊C₄	—	2	4	no	—
7	C₂²×C₇×C₄⋊C₄	C₂²	1	4	yes	—
8	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₂×C₇×#107	—	2	16	no	—
10	C₂×C₇×#107	C₂²	1	16	yes	—
11	C₂×C₄×C₇×C₈	C₄	1	8	yes	complemented
12	C₂×C₄×C₇×C₈	C₂²	1	4	yes	—
13	C₇×#384	—	2	4	no	—
14	C₇×#384	C₂²	1	8	yes	—
15	C₂×#1845	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₄×C₇	—	2	4	no	—
17	C₂³×C₄×C₇	D₄	1	2	yes	—
18	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₂²×C₇×C₈	C₂×C₄	1	8	yes	—
20	C₂²×C₇×C₈	D₄	1	8	yes	—
21	C₂²×C₇×C₈	Q₈	1	8	yes	—
22	C₂²×C₇×C₈	C₂³	1	4	yes	—
23	C₂×C₇×C₄⋊C₄	—	2	16	no	—
24	C₂×C₇×C₄⋊C₄	—	2	16	no	—
25	C₂×C₇×C₄⋊C₄	—	2	4	no	—
26	C₂×C₇×C₄⋊C₄	—	4	16	no	—
27	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
28	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₇	D₄	1	2	yes	—
30	C₂×C₄²×C₇	Q₈	1	4	yes	—
31	C₂×C₄²×C₇	C₂³	1	2	yes	—
32	C₇×#107	—	2	32	no	—
33	C₄×C₇×C₈	C₂×C₄	1	16	yes	complemented
34	C₂²×#107	C₂×C₇	1	4	yes	—
35	C₂²×C₄×C₈	C₂×C₇	1	1	yes	characteristic
36	C₂×#384	C₂×C₇	1	2	yes	—
37	#1845	C₂×C₇	1	8	yes	direct product factor
38	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₇	—	2	8	no	—
40	C₂²×C₄×C₇	—	4	8	no	—
41	C₂²×C₄×C₇	—	4	16	no	—
42	C₂²×C₄×C₇	C₄⋊C₄	1	8	yes	—
43	C₂²×C₄×C₇	D₈	1	8	yes	—
44	C₂²×C₄×C₇	Q₁₆	1	8	yes	—
45	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
48	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
49	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
50	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
52	C₂×C₇×C₈	C₄⋊C₄	1	32	yes	complemented
53	C₂×C₇×C₈	C₂²×C₄	1	8	yes	—
54	C₂×C₇×C₈	C₂×D₄	1	8	yes	—
55	C₂×C₇×C₈	C₂×Q₈	1	8	yes	—
56	C₇×C₄⋊C₄	—	2	16	no	—
57	C₇×C₄⋊C₄	—	4	8	no	—
58	C₇×C₄⋊C₄	—	4	32	no	—
59	C₄²×C₇	C₄⋊C₄	1	8	yes	—
60	C₄²×C₇	C₂²×C₄	1	4	yes	—
61	C₄²×C₇	C₂×Q₈	1	4	yes	—
62	C₂³×C₈	C₂²×C₇	1	2	yes	—
63	C₂²×C₄⋊C₄	—	2	4	no	—
64	C₂²×C₄⋊C₄	C₂²×C₇	1	4	yes	—
65	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
66	C₂×#107	—	2	16	no	—
67	C₂×#107	C₂²×C₇	1	16	yes	—
68	C₂×C₄×C₈	C₄×C₇	1	8	yes	complemented
69	C₂×C₄×C₈	C₂²×C₇	1	4	yes	—
70	#384	—	2	4	no	—
71	#384	C₂²×C₇	1	8	yes	—
72	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
73	C₂³×C₇	#127	1	2	yes	—
74	C₂³×C₇	C₄⋊Q₈	1	2	yes	—
75	C₂³×C₇	C₂×D₈	1	4	yes	—
76	C₂³×C₇	C₂×Q₁₆	1	4	yes	—
77	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
78	C₂×C₄×C₇	—	4	8	no	—
79	C₂×C₄×C₇	—	4	16	no	—
80	C₂×C₄×C₇	—	8	16	no	—
81	C₂×C₄×C₇	#107	1	32	yes	complemented
82	C₂×C₄×C₇	C₂×C₄⋊C₄	1	8	yes	—
83	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
84	C₂×C₄×C₇	C₂×C₄⋊C₄	1	8	yes	—
85	C₂×C₄×C₇	#127	1	2	yes	—
86	C₂×C₄×C₇	C₄⋊Q₈	1	8	yes	—
87	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
88	C₂×C₄×C₇	C₂×D₈	1	8	yes	—
89	C₂×C₄×C₇	C₂×Q₁₆	1	8	yes	—
90	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
92	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
93	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
94	C₇×C₈	C₂×C₄⋊C₄	1	16	yes	complemented
95	C₂³×C₄	—	2	4	no	—
96	C₂³×C₄	C₇×D₄	1	2	yes	—
97	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
98	C₂²×C₈	C₂×C₄×C₇	1	8	yes	—
99	C₂²×C₈	C₇×D₄	1	8	yes	—
100	C₂²×C₈	C₇×Q₈	1	8	yes	—
101	C₂²×C₈	C₂³×C₇	1	4	yes	—
102	C₂×C₄⋊C₄	—	2	16	no	—
103	C₂×C₄⋊C₄	—	2	16	no	—
104	C₂×C₄⋊C₄	—	2	4	no	—
105	C₂×C₄⋊C₄	—	4	16	no	—
106	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
107	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
108	C₂×C₄²	C₇×D₄	1	2	yes	—
109	C₂×C₄²	C₇×Q₈	1	4	yes	—
110	C₂×C₄²	C₂³×C₇	1	2	yes	—
111	#107	—	2	32	no	—
112	C₄×C₈	C₂×C₄×C₇	1	16	yes	complemented
113	C₂²×C₇	#384	1	4	yes	—
114	C₂²×C₇	C₂×#107	1	8	yes	—
115	C₂²×C₇	#493	1	4	yes	—
116	C₂²×C₇	#494	1	4	yes	—
117	C₂²×C₇	#500	1	8	yes	—
118	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₇	C₂×#127	1	1	yes	characteristic
120	C₂²×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
121	C₂²×C₇	C₂²×D₈	1	2	yes	—
122	C₂²×C₇	C₂²×Q₁₆	1	2	yes	—
123	C₄×C₇	—	8	8	no	—
124	C₄×C₇	#384	1	4	yes	—
125	C₄×C₇	C₂×#107	1	16	yes	complemented
126	C₄×C₇	C₂²×C₄⋊C₄	1	2	yes	—
127	C₄×C₇	C₂×C₄⋊Q₈	1	2	yes	—
128	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
129	C₂²×C₄	—	2	8	no	—
130	C₂²×C₄	—	4	8	no	—
131	C₂²×C₄	—	4	16	no	—
132	C₂²×C₄	C₇×C₄⋊C₄	1	8	yes	—
133	C₂²×C₄	C₇×D₈	1	8	yes	—
134	C₂²×C₄	C₇×Q₁₆	1	8	yes	—
135	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
136	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
138	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
139	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
140	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
141	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
142	C₂×C₈	C₇×C₄⋊C₄	1	32	yes	complemented
143	C₂×C₈	C₂²×C₄×C₇	1	8	yes	—
144	C₂×C₈	C₂×C₇×D₄	1	8	yes	—
145	C₂×C₈	C₂×C₇×Q₈	1	8	yes	—
146	C₄⋊C₄	—	2	16	no	—
147	C₄⋊C₄	—	4	8	no	—
148	C₄⋊C₄	—	4	32	no	—
149	C₄²	C₇×C₄⋊C₄	1	8	yes	—
150	C₄²	C₂²×C₄×C₇	1	4	yes	—
151	C₄²	C₂×C₇×Q₈	1	4	yes	—
152	C₂×C₇	#1845	1	8	yes	direct product factor
153	C₂×C₇	C₂×#384	1	1	yes	characteristic
154	C₂×C₇	C₂²×#107	1	2	yes	—
155	C₂×C₇	C₂×#493	1	1	yes	characteristic
156	C₂×C₇	C₂×#494	1	1	yes	characteristic
157	C₂×C₇	C₂×#500	1	2	yes	—
158	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
159	C₂³	C₇×#127	1	2	yes	—
160	C₂³	C₇×C₄⋊Q₈	1	2	yes	—
161	C₂³	C₂×C₇×D₈	1	4	yes	—
162	C₂³	C₂×C₇×Q₁₆	1	4	yes	—
163	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
164	C₂×C₄	—	4	8	no	—
165	C₂×C₄	—	4	16	no	—
166	C₂×C₄	—	8	16	no	—
167	C₂×C₄	C₇×#107	1	32	yes	complemented
168	C₂×C₄	C₂×C₇×C₄⋊C₄	1	8	yes	—
169	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
170	C₂×C₄	C₂×C₇×C₄⋊C₄	1	8	yes	—
171	C₂×C₄	C₇×#127	1	2	yes	—
172	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
173	C₂×C₄	C₇×C₄⋊Q₈	1	8	yes	—
174	C₂×C₄	C₂×C₇×D₈	1	8	yes	—
175	C₂×C₄	C₂×C₇×Q₁₆	1	8	yes	—
176	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
177	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
178	C₂×C₄	C₂²×C₇×Q₈	1	2	yes	—
179	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
180	C₈	C₂×C₇×C₄⋊C₄	1	16	yes	complemented
181	C₇	C₂×#1845	1	1	yes	characteristic, direct product factor, Sylow
182	C₂²	C₇×#384	1	4	yes	—
183	C₂²	C₂×C₇×#107	1	8	yes	—
184	C₂²	C₇×#493	1	4	yes	—
185	C₂²	C₇×#494	1	4	yes	—
186	C₂²	C₇×#500	1	8	yes	—
187	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
188	C₂²	C₂×C₇×#127	1	1	yes	characteristic
189	C₂²	C₂×C₇×C₄⋊Q₈	1	1	yes	characteristic
190	C₂²	C₂²×C₇×D₈	1	2	yes	—
191	C₂²	C₂²×C₇×Q₁₆	1	2	yes	—
192	C₄	—	8	8	no	—
193	C₄	C₇×#384	1	4	yes	—
194	C₄	C₂×C₇×#107	1	16	yes	complemented
195	C₄	C₂²×C₇×C₄⋊C₄	1	2	yes	—
196	C₄	C₂×C₇×C₄⋊Q₈	1	2	yes	—
197	C₂	C₇×#1845	1	8	yes	direct product factor
198	C₂	C₂×C₇×#384	1	1	yes	characteristic
199	C₂	C₂²×C₇×#107	1	2	yes	—
200	C₂	C₂×C₇×#493	1	1	yes	characteristic
201	C₂	C₂×C₇×#494	1	1	yes	characteristic
202	C₂	C₂×C₇×#500	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇×#1845	C₂	direct (N × H)
11. C₂×C₄×C₇×C₈	C₄	semidirect
15. C₂×#1845	C₇	direct (N × H)
33. C₄×C₇×C₈	C₂×C₄	semidirect
37. #1845	C₂×C₇	direct (N × H)
52. C₂×C₇×C₈	C₄⋊C₄	semidirect
68. C₂×C₄×C₈	C₄×C₇	semidirect
81. C₂×C₄×C₇	#107	semidirect
94. C₇×C₈	C₂×C₄⋊C₄	semidirect
112. C₄×C₈	C₂×C₄×C₇	semidirect
125. C₄×C₇	C₂×#107	semidirect
142. C₂×C₈	C₇×C₄⋊C₄	semidirect
152. C₂×C₇	#1845	direct (N × H)
167. C₂×C₄	C₇×#107	semidirect
180. C₈	C₂×C₇×C₄⋊C₄	semidirect
181. C₇	C₂×#1845	direct (N × H)
194. C₄	C₂×C₇×#107	semidirect
197. C₂	C₇×#1845	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×C₇×#107	2
C₂²×C₄×C₇×C₈	2
C₂×C₇×#384	2
C₇×#1845	2
C₂×#1845	7

Subgroups — C2×C7×#1845 #13198011

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1845 #13198011