Subgroups — C₂×C₇×#1846 #13198012

Order 1792.

1182 conjugacy classes of subgroups.

The subgroup lattice has 236 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	2	yes	maximal
2	C₂²×C₇×#106	C₂	1	2	yes	maximal
3	C₂²×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×#382	C₂	1	2	yes	maximal
5	C₇×#1846	C₂	1	8	yes	maximal, direct product factor
6	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₇×C₄⋊C₄	C₂²	1	4	yes	—
9	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₇×#107	—	2	8	no	—
11	C₂×C₇×#107	C₂²	1	8	yes	—
12	C₂×C₇×#106	—	2	8	no	—
13	C₂×C₇×#106	C₂²	1	8	yes	—
14	C₂×C₄×C₇×C₈	C₄	1	8	yes	complemented
15	C₂×C₄×C₇×C₈	C₂²	1	4	yes	—
16	C₇×#382	—	2	4	no	—
17	C₇×#382	C₂²	1	8	yes	—
18	C₂×C₇×#95	—	2	4	no	—
19	C₂×#1846	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂³×C₄×C₇	—	2	4	no	—
21	C₂³×C₄×C₇	D₄	1	2	yes	—
22	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
23	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₇×C₈	D₄	1	4	yes	—
26	C₂²×C₇×C₈	D₄	1	4	yes	—
27	C₂²×C₇×C₈	Q₈	1	4	yes	—
28	C₂²×C₇×C₈	Q₈	1	4	yes	—
29	C₂²×C₇×C₈	C₂³	1	2	yes	—
30	C₂²×C₇×C₈	C₂³	1	2	yes	—
31	C₂×C₇×C₄⋊C₄	—	2	8	no	—
32	C₂×C₇×C₄⋊C₄	—	2	8	no	—
33	C₂×C₇×C₄⋊C₄	—	2	4	no	—
34	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
35	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₇	D₄	1	2	yes	—
37	C₂×C₄²×C₇	Q₈	1	4	yes	—
38	C₂×C₄²×C₇	C₂³	1	2	yes	—
39	C₇×#107	—	2	16	no	—
40	C₇×#106	—	2	16	no	—
41	C₄×C₇×C₈	C₂×C₄	1	16	yes	complemented
42	C₇×#95	—	2	16	no	—
43	C₂²×#107	C₂×C₇	1	2	yes	—
44	C₂²×#106	C₂×C₇	1	2	yes	—
45	C₂²×C₄×C₈	C₂×C₇	1	1	yes	characteristic
46	C₂×#382	C₂×C₇	1	2	yes	—
47	#1846	C₂×C₇	1	8	yes	direct product factor
48	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇	—	2	8	no	—
50	C₂²×C₄×C₇	—	2	8	no	—
51	C₂²×C₄×C₇	—	4	8	no	—
52	C₂²×C₄×C₇	—	4	16	no	—
53	C₂²×C₄×C₇	C₄⋊C₄	1	8	yes	—
54	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
56	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
63	C₂×C₇×C₈	C₄⋊C₄	1	16	yes	—
64	C₂×C₇×C₈	C₄⋊C₄	1	16	yes	—
65	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
66	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
67	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
68	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
69	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
70	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
71	C₇×C₄⋊C₄	—	2	8	no	—
72	C₇×C₄⋊C₄	—	2	8	no	—
73	C₇×C₄⋊C₄	—	4	8	no	—
74	C₄²×C₇	C₄⋊C₄	1	8	yes	—
75	C₄²×C₇	C₂²×C₄	1	4	yes	—
76	C₄²×C₇	C₂×Q₈	1	4	yes	—
77	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
78	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
79	C₂²×C₄⋊C₄	C₂²×C₇	1	4	yes	—
80	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
81	C₂×#107	—	2	8	no	—
82	C₂×#107	C₂²×C₇	1	8	yes	—
83	C₂×#106	—	2	8	no	—
84	C₂×#106	C₂²×C₇	1	8	yes	—
85	C₂×C₄×C₈	C₄×C₇	1	8	yes	complemented
86	C₂×C₄×C₈	C₂²×C₇	1	4	yes	—
87	#382	—	2	4	no	—
88	#382	C₂²×C₇	1	8	yes	—
89	C₂×#95	—	2	4	no	—
90	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
91	C₂³×C₇	#127	1	2	yes	—
92	C₂³×C₇	C₄⋊Q₈	1	2	yes	—
93	C₂³×C₇	#135	1	8	yes	—
94	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₇	—	2	8	no	—
96	C₂×C₄×C₇	—	2	16	no	—
97	C₂×C₄×C₇	—	4	8	no	—
98	C₂×C₄×C₇	—	4	16	no	—
99	C₂×C₄×C₇	—	8	16	no	—
100	C₂×C₄×C₇	C₂×C₄⋊C₄	1	8	yes	—
101	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
103	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₇	#127	1	2	yes	—
105	C₂×C₄×C₇	C₄⋊Q₈	1	4	yes	—
106	C₂×C₄×C₇	C₄⋊Q₈	1	4	yes	—
107	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
108	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
114	C₇×C₈	C₂×C₄⋊C₄	1	8	yes	—
115	C₇×C₈	C₂×C₄⋊C₄	1	8	yes	—
116	C₂³×C₄	—	2	4	no	—
117	C₂³×C₄	C₇×D₄	1	2	yes	—
118	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
119	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
120	C₂²×C₈	C₂×C₄×C₇	1	4	yes	—
121	C₂²×C₈	C₇×D₄	1	4	yes	—
122	C₂²×C₈	C₇×D₄	1	4	yes	—
123	C₂²×C₈	C₇×Q₈	1	4	yes	—
124	C₂²×C₈	C₇×Q₈	1	4	yes	—
125	C₂²×C₈	C₂³×C₇	1	2	yes	—
126	C₂²×C₈	C₂³×C₇	1	2	yes	—
127	C₂×C₄⋊C₄	—	2	8	no	—
128	C₂×C₄⋊C₄	—	2	8	no	—
129	C₂×C₄⋊C₄	—	2	4	no	—
130	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
131	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
132	C₂×C₄²	C₇×D₄	1	2	yes	—
133	C₂×C₄²	C₇×Q₈	1	4	yes	—
134	C₂×C₄²	C₂³×C₇	1	2	yes	—
135	#107	—	2	16	no	—
136	#106	—	2	16	no	—
137	C₄×C₈	C₂×C₄×C₇	1	16	yes	complemented
138	#95	—	2	16	no	—
139	C₂²×C₇	#384	1	4	yes	—
140	C₂²×C₇	#427	1	8	yes	—
141	C₂²×C₇	#495	1	8	yes	—
142	C₂²×C₇	#499	1	8	yes	—
143	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×C₇	C₂×#127	1	1	yes	characteristic
145	C₂²×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
146	C₂²×C₇	C₂×#135	1	4	yes	—
147	C₄×C₇	—	2	8	no	—
148	C₄×C₇	—	8	8	no	—
149	C₄×C₇	#384	1	4	yes	—
150	C₄×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
153	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
154	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
155	C₂²×C₄	—	2	8	no	—
156	C₂²×C₄	—	2	8	no	—
157	C₂²×C₄	—	4	8	no	—
158	C₂²×C₄	—	4	16	no	—
159	C₂²×C₄	C₇×C₄⋊C₄	1	8	yes	—
160	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
161	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
162	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
164	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
165	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
166	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
167	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
168	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
169	C₂×C₈	C₇×C₄⋊C₄	1	16	yes	—
170	C₂×C₈	C₇×C₄⋊C₄	1	16	yes	—
171	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
172	C₂×C₈	C₂²×C₄×C₇	1	4	yes	—
173	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
174	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
175	C₂×C₈	C₂×C₇×Q₈	1	4	yes	—
176	C₂×C₈	C₂×C₇×Q₈	1	4	yes	—
177	C₄⋊C₄	—	2	8	no	—
178	C₄⋊C₄	—	2	8	no	—
179	C₄⋊C₄	—	4	8	no	—
180	C₄²	C₇×C₄⋊C₄	1	8	yes	—
181	C₄²	C₂²×C₄×C₇	1	4	yes	—
182	C₄²	C₂×C₇×Q₈	1	4	yes	—
183	C₂×C₇	#1846	1	8	yes	direct product factor
184	C₂×C₇	C₂×#384	1	1	yes	characteristic
185	C₂×C₇	C₂×#427	1	2	yes	—
186	C₂×C₇	C₂×#495	1	2	yes	—
187	C₂×C₇	C₂×#499	1	2	yes	—
188	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
189	C₂³	C₇×#127	1	2	yes	—
190	C₂³	C₇×C₄⋊Q₈	1	2	yes	—
191	C₂³	C₇×#135	1	8	yes	—
192	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
193	C₂×C₄	—	2	16	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	4	8	no	—
196	C₂×C₄	—	4	16	no	—
197	C₂×C₄	—	8	16	no	—
198	C₂×C₄	C₂×C₇×C₄⋊C₄	1	8	yes	—
199	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
200	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
201	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
202	C₂×C₄	C₇×#127	1	2	yes	—
203	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
204	C₂×C₄	C₇×C₄⋊Q₈	1	4	yes	—
205	C₂×C₄	C₇×C₄⋊Q₈	1	4	yes	—
206	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
207	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
208	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
209	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
210	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
211	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
212	C₈	C₂×C₇×C₄⋊C₄	1	8	yes	—
213	C₈	C₂×C₇×C₄⋊C₄	1	8	yes	—
214	C₇	C₂×#1846	1	1	yes	characteristic, direct product factor, Sylow
215	C₂²	C₇×#384	1	4	yes	—
216	C₂²	C₇×#427	1	8	yes	—
217	C₂²	C₇×#495	1	8	yes	—
218	C₂²	C₇×#499	1	8	yes	—
219	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
220	C₂²	C₂×C₇×#127	1	1	yes	characteristic
221	C₂²	C₂×C₇×C₄⋊Q₈	1	1	yes	characteristic
222	C₂²	C₂×C₇×#135	1	4	yes	—
223	C₄	—	2	8	no	—
224	C₄	—	8	8	no	—
225	C₄	C₇×#384	1	4	yes	—
226	C₄	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
227	C₄	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
228	C₄	C₂×C₇×C₄⋊Q₈	1	1	yes	characteristic
229	C₄	C₂×C₇×C₄⋊Q₈	1	1	yes	characteristic
230	C₂	C₇×#1846	1	8	yes	direct product factor
231	C₂	C₂×C₇×#384	1	1	yes	characteristic
232	C₂	C₂×C₇×#427	1	2	yes	—
233	C₂	C₂×C₇×#495	1	2	yes	—
234	C₂	C₂×C₇×#499	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
5. C₇×#1846	C₂	direct (N × H)
14. C₂×C₄×C₇×C₈	C₄	semidirect
19. C₂×#1846	C₇	direct (N × H)
41. C₄×C₇×C₈	C₂×C₄	semidirect
47. #1846	C₂×C₇	direct (N × H)
85. C₂×C₄×C₈	C₄×C₇	semidirect
137. C₄×C₈	C₂×C₄×C₇	semidirect
183. C₂×C₇	#1846	direct (N × H)
214. C₇	C₂×#1846	direct (N × H)
230. C₂	C₇×#1846	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×C₇×#106	2
C₂²×C₄×C₇×C₈	2
C₂×C₇×#382	2
C₇×#1846	2
C₂×#1846	7

Subgroups — C2×C7×#1846 #13198012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1846 #13198012