Subgroups — C₂×C₇×#1831 #13197966

Order 1792.

1246 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₄⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄³×C₇	C₂	1	1	yes	maximal, characteristic
3	C₇×#1831	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₇×#336	C₂	1	4	yes	maximal
5	C₂³×C₇×C₈	C₂²	1	4	yes	—
6	C₂²×C₄²×C₇	C₄	1	2	yes	—
7	C₂²×C₄²×C₇	C₄	1	1	yes	characteristic
8	C₂²×C₄²×C₇	C₄	1	1	yes	characteristic
9	C₂²×C₄²×C₇	C₂²	1	2	yes	—
10	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₂×C₇×C₄⋊C₈	—	2	8	no	—
12	C₂×C₇×C₄⋊C₈	C₂²	1	8	yes	—
13	C₄³×C₇	C₄	1	4	yes	—
14	C₄³×C₇	C₂²	1	4	yes	—
15	C₇×#336	C₂²	1	16	yes	—
16	C₂×#1831	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
22	C₂²×C₇×C₈	—	2	8	no	—
23	C₂²×C₇×C₈	—	2	16	no	—
24	C₂²×C₇×C₈	C₂³	1	8	yes	—
25	C₂×C₄²×C₇	—	2	8	no	—
26	C₂×C₄²×C₇	—	2	4	no	—
27	C₂×C₄²×C₇	—	2	8	no	—
28	C₂×C₄²×C₇	C₈	1	8	yes	complemented
29	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₇	C₂×C₄	1	8	yes	—
31	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₇	D₄	1	4	yes	—
35	C₂×C₄²×C₇	Q₈	1	4	yes	—
36	C₂×C₄²×C₇	C₂³	1	4	yes	—
37	C₂×C₄²×C₇	C₂³	1	2	yes	—
38	C₇×C₄⋊C₈	—	2	16	no	—
39	C₂²×C₄⋊C₈	C₂×C₇	1	2	yes	—
40	C₂×C₄³	C₂×C₇	1	1	yes	characteristic
41	#1831	C₂×C₇	1	8	yes	direct product factor
42	C₂×#336	C₂×C₇	1	4	yes	—
43	C₂⁴×C₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₇	—	2	4	no	—
45	C₂²×C₄×C₇	—	2	4	no	—
46	C₂²×C₄×C₇	—	2	4	no	—
47	C₂²×C₄×C₇	—	2	4	no	—
48	C₂²×C₄×C₇	—	2	4	no	—
49	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₇	C₂×C₈	1	8	yes	—
52	C₂²×C₄×C₇	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₇	M₁₆	1	8	yes	—
54	C₂²×C₄×C₇	M₁₆	1	4	yes	—
55	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₇	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₇	#41	1	8	yes	—
63	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
64	C₂×C₇×C₈	—	2	8	no	—
65	C₂×C₇×C₈	—	2	16	no	—
66	C₂×C₇×C₈	—	4	16	no	—
67	C₄²×C₇	—	2	8	no	—
68	C₄²×C₇	—	2	16	no	—
69	C₄²×C₇	—	2	8	no	—
70	C₄²×C₇	—	2	8	no	—
71	C₄²×C₇	C₄⋊C₄	1	8	yes	—
72	C₄²×C₇	C₂×C₈	1	16	yes	complemented
73	C₄²×C₇	C₂×D₄	1	4	yes	—
74	C₄²×C₇	C₂×Q₈	1	4	yes	—
75	C₂³×C₈	C₂²×C₇	1	4	yes	—
76	C₂²×C₄²	C₄×C₇	1	2	yes	—
77	C₂²×C₄²	C₄×C₇	1	1	yes	characteristic
78	C₂²×C₄²	C₄×C₇	1	1	yes	characteristic
79	C₂²×C₄²	C₂²×C₇	1	2	yes	—
80	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
81	C₂×C₄⋊C₈	—	2	8	no	—
82	C₂×C₄⋊C₈	C₂²×C₇	1	8	yes	—
83	C₄³	C₄×C₇	1	4	yes	—
84	C₄³	C₂²×C₇	1	4	yes	—
85	#336	C₂²×C₇	1	16	yes	—
86	C₂³×C₇	C₂×C₄⋊C₄	1	2	yes	—
87	C₂³×C₇	#117	1	4	yes	—
88	C₂³×C₇	C₂²×C₈	1	2	yes	—
89	C₂³×C₇	C₂×M₁₆	1	2	yes	—
90	C₂³×C₇	C₂×M₁₆	1	4	yes	—
91	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₇	—	2	4	no	—
93	C₂×C₄×C₇	—	2	4	no	—
94	C₂×C₄×C₇	—	2	4	no	—
95	C₂×C₄×C₇	—	2	4	no	—
96	C₂×C₄×C₇	—	2	4	no	—
97	C₂×C₄×C₇	—	2	8	no	—
98	C₂×C₄×C₇	—	2	16	no	—
99	C₂×C₄×C₇	—	2	16	no	—
100	C₂×C₄×C₇	C₄⋊C₈	1	16	yes	—
101	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₇	#117	1	4	yes	—
105	C₂×C₄×C₇	#124	1	8	yes	—
106	C₂×C₄×C₇	#125	1	8	yes	—
107	C₂×C₄×C₇	C₂²×C₈	1	8	yes	—
108	C₂×C₄×C₇	C₂²×C₈	1	4	yes	—
109	C₂×C₄×C₇	C₂×M₁₆	1	8	yes	—
110	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
111	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₇	C₂×#41	1	4	yes	—
114	C₇×C₈	—	4	8	no	—
115	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
116	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
117	C₂³×C₄	C₂×C₄×C₇	1	2	yes	—
118	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
119	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
120	C₂²×C₈	—	2	8	no	—
121	C₂²×C₈	—	2	16	no	—
122	C₂²×C₈	C₂³×C₇	1	8	yes	—
123	C₂×C₄²	—	2	8	no	—
124	C₂×C₄²	—	2	4	no	—
125	C₂×C₄²	—	2	8	no	—
126	C₂×C₄²	C₇×C₈	1	8	yes	complemented
127	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
128	C₂×C₄²	C₂×C₄×C₇	1	8	yes	—
129	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
130	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
131	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
132	C₂×C₄²	C₇×D₄	1	4	yes	—
133	C₂×C₄²	C₇×Q₈	1	4	yes	—
134	C₂×C₄²	C₂³×C₇	1	4	yes	—
135	C₂×C₄²	C₂³×C₇	1	2	yes	—
136	C₄⋊C₈	—	2	16	no	—
137	C₂²×C₇	#382	1	4	yes	—
138	C₂²×C₇	C₂×C₄⋊C₈	1	4	yes	—
139	C₂²×C₇	#423	1	4	yes	—
140	C₂²×C₇	#431	1	8	yes	—
141	C₂²×C₇	#432	1	8	yes	—
142	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₇	C₂×#117	1	2	yes	—
144	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
145	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
146	C₂²×C₇	C₂²×M₁₆	1	2	yes	—
147	C₄×C₇	—	2	4	no	—
148	C₄×C₇	—	2	8	no	—
149	C₄×C₇	—	2	8	no	—
150	C₄×C₇	#382	1	4	yes	—
151	C₄×C₇	C₂×C₄⋊C₈	1	8	yes	—
152	C₄×C₇	C₂×#124	1	2	yes	—
153	C₄×C₇	C₂×#125	1	2	yes	—
154	C₂⁴	C₂²×C₄×C₇	1	1	yes	characteristic
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
161	C₂²×C₄	C₇×C₄⋊C₄	1	4	yes	—
162	C₂²×C₄	C₂×C₇×C₈	1	8	yes	—
163	C₂²×C₄	C₂×C₇×C₈	1	4	yes	—
164	C₂²×C₄	C₇×M₁₆	1	8	yes	—
165	C₂²×C₄	C₇×M₁₆	1	4	yes	—
166	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
167	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
168	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
169	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
170	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
171	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
172	C₂²×C₄	C₂×C₇×Q₈	1	2	yes	—
173	C₂²×C₄	C₇×#41	1	8	yes	—
174	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
175	C₂×C₈	—	2	8	no	—
176	C₂×C₈	—	2	16	no	—
177	C₂×C₈	—	4	16	no	—
178	C₄²	—	2	8	no	—
179	C₄²	—	2	16	no	—
180	C₄²	—	2	8	no	—
181	C₄²	—	2	8	no	—
182	C₄²	C₇×C₄⋊C₄	1	8	yes	—
183	C₄²	C₂×C₇×C₈	1	16	yes	complemented
184	C₄²	C₂×C₇×D₄	1	4	yes	—
185	C₄²	C₂×C₇×Q₈	1	4	yes	—
186	C₂×C₇	#1831	1	8	yes	direct product factor
187	C₂×C₇	C₂×#382	1	1	yes	characteristic
188	C₂×C₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
189	C₂×C₇	C₂×#423	1	1	yes	characteristic
190	C₂×C₇	C₂×#431	1	2	yes	—
191	C₂×C₇	C₂×#432	1	2	yes	—
192	C₂³	C₂×C₇×C₄⋊C₄	1	2	yes	—
193	C₂³	C₇×#117	1	4	yes	—
194	C₂³	C₂²×C₇×C₈	1	2	yes	—
195	C₂³	C₂×C₇×M₁₆	1	2	yes	—
196	C₂³	C₂×C₇×M₁₆	1	4	yes	—
197	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	8	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	16	no	—
205	C₂×C₄	—	2	16	no	—
206	C₂×C₄	C₇×C₄⋊C₈	1	16	yes	—
207	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
208	C₂×C₄	C₂×C₇×C₄⋊C₄	1	2	yes	—
209	C₂×C₄	C₂×C₇×C₄⋊C₄	1	4	yes	—
210	C₂×C₄	C₇×#117	1	4	yes	—
211	C₂×C₄	C₇×#124	1	8	yes	—
212	C₂×C₄	C₇×#125	1	8	yes	—
213	C₂×C₄	C₂²×C₇×C₈	1	8	yes	—
214	C₂×C₄	C₂²×C₇×C₈	1	4	yes	—
215	C₂×C₄	C₂×C₇×M₁₆	1	8	yes	—
216	C₂×C₄	C₂×C₇×M₁₆	1	4	yes	—
217	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
218	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
219	C₂×C₄	C₂×C₇×#41	1	4	yes	—
220	C₈	—	4	8	no	—
221	C₇	C₂×#1831	1	1	yes	characteristic, direct product factor, Sylow
222	C₂²	C₇×#382	1	4	yes	—
223	C₂²	C₂×C₇×C₄⋊C₈	1	4	yes	—
224	C₂²	C₇×#423	1	4	yes	—
225	C₂²	C₇×#431	1	8	yes	—
226	C₂²	C₇×#432	1	8	yes	—
227	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
228	C₂²	C₂×C₇×#117	1	2	yes	—
229	C₂²	C₂³×C₇×C₈	1	1	yes	characteristic
230	C₂²	C₂²×C₇×M₁₆	1	1	yes	characteristic
231	C₂²	C₂²×C₇×M₁₆	1	2	yes	—
232	C₄	—	2	4	no	—
233	C₄	—	2	8	no	—
234	C₄	—	2	8	no	—
235	C₄	C₇×#382	1	4	yes	—
236	C₄	C₂×C₇×C₄⋊C₈	1	8	yes	—
237	C₄	C₂×C₇×#124	1	2	yes	—
238	C₄	C₂×C₇×#125	1	2	yes	—
239	C₂	C₇×#1831	1	8	yes	direct product factor
240	C₂	C₂×C₇×#382	1	1	yes	characteristic
241	C₂	C₂²×C₇×C₄⋊C₈	1	1	yes	characteristic
242	C₂	C₂×C₇×#423	1	1	yes	characteristic
243	C₂	C₂×C₇×#431	1	2	yes	—
244	C₂	C₂×C₇×#432	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₇	C₂³
commutator	C₂²	C₂³×C₇×C₈
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle, prime radical	C₂⁴×C₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1831	C₂	direct (N × H)
16. C₂×#1831	C₇	direct (N × H)
28. C₂×C₄²×C₇	C₈	semidirect
41. #1831	C₂×C₇	direct (N × H)
72. C₄²×C₇	C₂×C₈	semidirect
126. C₂×C₄²	C₇×C₈	semidirect
183. C₄²	C₂×C₇×C₈	semidirect
186. C₂×C₇	#1831	direct (N × H)
221. C₇	C₂×#1831	direct (N × H)
239. C₂	C₇×#1831	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×C₄⋊C₈	2
C₂×C₄³×C₇	2
C₇×#1831	2
C₂×C₇×#336	2
C₂×#1831	7

Subgroups — C2×C7×#1831 #13197966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#1831 #13197966