Subgroups — C₂×C₇×#1844 #13198010

Order 1792.

1358 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#106	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×#1844	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×#106	—	2	16	no	—
10	C₂×C₇×#106	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×#1844	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×#106	—	2	32	no	—
33	C₄×C₇×C₈	C₂×C₄	1	16	yes	complemented
34	C₂²×#106	C₂×C₇	1	4	yes	—
35	C₂²×C₄×C₈	C₂×C₇	1	1	yes	characteristic
36	C₂×#384	C₂×C₇	1	2	yes	—
37	#1844	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	SD₁₆	1	16	yes	—
44	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
47	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
48	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
49	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
50	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
51	C₂×C₇×C₈	C₄⋊C₄	1	32	yes	complemented
52	C₂×C₇×C₈	C₂²×C₄	1	8	yes	—
53	C₂×C₇×C₈	C₂×D₄	1	8	yes	—
54	C₂×C₇×C₈	C₂×Q₈	1	8	yes	—
55	C₇×C₄⋊C₄	—	2	16	no	—
56	C₇×C₄⋊C₄	—	4	8	no	—
57	C₇×C₄⋊C₄	—	4	32	no	—
58	C₄²×C₇	C₄⋊C₄	1	8	yes	—
59	C₄²×C₇	C₂²×C₄	1	4	yes	—
60	C₄²×C₇	C₂×Q₈	1	4	yes	—
61	C₂³×C₈	C₂²×C₇	1	2	yes	—
62	C₂²×C₄⋊C₄	—	2	4	no	—
63	C₂²×C₄⋊C₄	C₂²×C₇	1	4	yes	—
64	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
65	C₂×#106	—	2	16	no	—
66	C₂×#106	C₂²×C₇	1	16	yes	—
67	C₂×C₄×C₈	C₄×C₇	1	8	yes	complemented
68	C₂×C₄×C₈	C₂²×C₇	1	4	yes	—
69	#384	—	2	4	no	—
70	#384	C₂²×C₇	1	8	yes	—
71	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
72	C₂³×C₇	#127	1	2	yes	—
73	C₂³×C₇	C₄⋊Q₈	1	2	yes	—
74	C₂³×C₇	C₂×SD₁₆	1	8	yes	—
75	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
76	C₂×C₄×C₇	—	4	8	no	—
77	C₂×C₄×C₇	—	4	16	no	—
78	C₂×C₄×C₇	—	8	16	no	—
79	C₂×C₄×C₇	#106	1	32	yes	complemented
80	C₂×C₄×C₇	C₂×C₄⋊C₄	1	8	yes	—
81	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₇	C₂×C₄⋊C₄	1	8	yes	—
83	C₂×C₄×C₇	#127	1	2	yes	—
84	C₂×C₄×C₇	C₄⋊Q₈	1	8	yes	—
85	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
86	C₂×C₄×C₇	C₂×SD₁₆	1	16	yes	—
87	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
88	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
89	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
90	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
91	C₇×C₈	C₂×C₄⋊C₄	1	16	yes	complemented
92	C₂³×C₄	—	2	4	no	—
93	C₂³×C₄	C₇×D₄	1	2	yes	—
94	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
95	C₂²×C₈	C₂×C₄×C₇	1	8	yes	—
96	C₂²×C₈	C₇×D₄	1	8	yes	—
97	C₂²×C₈	C₇×Q₈	1	8	yes	—
98	C₂²×C₈	C₂³×C₇	1	4	yes	—
99	C₂×C₄⋊C₄	—	2	16	no	—
100	C₂×C₄⋊C₄	—	2	16	no	—
101	C₂×C₄⋊C₄	—	2	4	no	—
102	C₂×C₄⋊C₄	—	4	16	no	—
103	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
104	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
105	C₂×C₄²	C₇×D₄	1	2	yes	—
106	C₂×C₄²	C₇×Q₈	1	4	yes	—
107	C₂×C₄²	C₂³×C₇	1	2	yes	—
108	#106	—	2	32	no	—
109	C₄×C₈	C₂×C₄×C₇	1	16	yes	complemented
110	C₂²×C₇	#384	1	4	yes	—
111	C₂²×C₇	C₂×#106	1	8	yes	—
112	C₂²×C₇	#492	1	8	yes	—
113	C₂²×C₇	#498	1	8	yes	—
114	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₇	C₂×#127	1	1	yes	characteristic
116	C₂²×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
117	C₂²×C₇	C₂²×SD₁₆	1	4	yes	—
118	C₄×C₇	—	8	8	no	—
119	C₄×C₇	#384	1	4	yes	—
120	C₄×C₇	C₂×#106	1	16	yes	complemented
121	C₄×C₇	C₂²×C₄⋊C₄	1	2	yes	—
122	C₄×C₇	C₂×C₄⋊Q₈	1	2	yes	—
123	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
124	C₂²×C₄	—	2	8	no	—
125	C₂²×C₄	—	4	8	no	—
126	C₂²×C₄	—	4	16	no	—
127	C₂²×C₄	C₇×C₄⋊C₄	1	8	yes	—
128	C₂²×C₄	C₇×SD₁₆	1	16	yes	—
129	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
130	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
132	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
133	C₂²×C₄	C₂×C₇×Q₈	1	4	yes	—
134	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
135	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
136	C₂×C₈	C₇×C₄⋊C₄	1	32	yes	complemented
137	C₂×C₈	C₂²×C₄×C₇	1	8	yes	—
138	C₂×C₈	C₂×C₇×D₄	1	8	yes	—
139	C₂×C₈	C₂×C₇×Q₈	1	8	yes	—
140	C₄⋊C₄	—	2	16	no	—
141	C₄⋊C₄	—	4	8	no	—
142	C₄⋊C₄	—	4	32	no	—
143	C₄²	C₇×C₄⋊C₄	1	8	yes	—
144	C₄²	C₂²×C₄×C₇	1	4	yes	—
145	C₄²	C₂×C₇×Q₈	1	4	yes	—
146	C₂×C₇	#1844	1	8	yes	direct product factor
147	C₂×C₇	C₂×#384	1	1	yes	characteristic
148	C₂×C₇	C₂²×#106	1	2	yes	—
149	C₂×C₇	C₂×#492	1	2	yes	—
150	C₂×C₇	C₂×#498	1	2	yes	—
151	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
152	C₂³	C₇×#127	1	2	yes	—
153	C₂³	C₇×C₄⋊Q₈	1	2	yes	—
154	C₂³	C₂×C₇×SD₁₆	1	8	yes	—
155	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
156	C₂×C₄	—	4	8	no	—
157	C₂×C₄	—	4	16	no	—
158	C₂×C₄	—	8	16	no	—
159	C₂×C₄	C₇×#106	1	32	yes	complemented
160	C₂×C₄	C₂×C₇×C₄⋊C₄	1	8	yes	—
161	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
162	C₂×C₄	C₂×C₇×C₄⋊C₄	1	8	yes	—
163	C₂×C₄	C₇×#127	1	2	yes	—
164	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
165	C₂×C₄	C₇×C₄⋊Q₈	1	8	yes	—
166	C₂×C₄	C₂×C₇×SD₁₆	1	16	yes	—
167	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
168	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
169	C₂×C₄	C₂²×C₇×Q₈	1	2	yes	—
170	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
171	C₈	C₂×C₇×C₄⋊C₄	1	16	yes	complemented
172	C₇	C₂×#1844	1	1	yes	characteristic, direct product factor, Sylow
173	C₂²	C₇×#384	1	4	yes	—
174	C₂²	C₂×C₇×#106	1	8	yes	—
175	C₂²	C₇×#492	1	8	yes	—
176	C₂²	C₇×#498	1	8	yes	—
177	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
178	C₂²	C₂×C₇×#127	1	1	yes	characteristic
179	C₂²	C₂×C₇×C₄⋊Q₈	1	1	yes	characteristic
180	C₂²	C₂²×C₇×SD₁₆	1	4	yes	—
181	C₄	—	8	8	no	—
182	C₄	C₇×#384	1	4	yes	—
183	C₄	C₂×C₇×#106	1	16	yes	complemented
184	C₄	C₂²×C₇×C₄⋊C₄	1	2	yes	—
185	C₄	C₂×C₇×C₄⋊Q₈	1	2	yes	—
186	C₂	C₇×#1844	1	8	yes	direct product factor
187	C₂	C₂×C₇×#384	1	1	yes	characteristic
188	C₂	C₂²×C₇×#106	1	2	yes	—
189	C₂	C₂×C₇×#492	1	2	yes	—
190	C₂	C₂×C₇×#498	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×#1844	C₂	direct (N × H)
11. C₂×C₄×C₇×C₈	C₄	semidirect
15. C₂×#1844	C₇	direct (N × H)
33. C₄×C₇×C₈	C₂×C₄	semidirect
37. #1844	C₂×C₇	direct (N × H)
51. C₂×C₇×C₈	C₄⋊C₄	semidirect
67. C₂×C₄×C₈	C₄×C₇	semidirect
79. C₂×C₄×C₇	#106	semidirect
91. C₇×C₈	C₂×C₄⋊C₄	semidirect
109. C₄×C₈	C₂×C₄×C₇	semidirect
120. C₄×C₇	C₂×#106	semidirect
136. C₂×C₈	C₇×C₄⋊C₄	semidirect
146. C₂×C₇	#1844	direct (N × H)
159. C₂×C₄	C₇×#106	semidirect
171. C₈	C₂×C₇×C₄⋊C₄	semidirect
172. C₇	C₂×#1844	direct (N × H)
183. C₄	C₂×C₇×#106	semidirect
186. C₂	C₇×#1844	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×#106	2
C₂²×C₄×C₇×C₈	2
C₂×C₇×#384	2
C₇×#1844	2
C₂×#1844	7

Subgroups — C2×C7×#1844 #13198010

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1844 #13198010