Subgroups — C₂×C₇×#2243 #13210914

Order 1792.

634 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#495	C₂	1	1	yes	maximal, characteristic
3	C₇×#2243	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₇×#358	C₂	1	2	yes	maximal, complemented
5	C₂×C₇×#357	C₂	1	2	yes	maximal
6	C₂×C₇×#346	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₇×Q₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₇×SD₁₆	—	2	1	no	—
9	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
11	C₂×C₇×#124	—	2	1	no	—
12	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
13	C₇×#500	—	2	2	no	—
14	C₇×#500	C₂²	1	4	yes	complemented
15	C₇×#495	—	2	2	no	—
16	C₇×#495	C₂²	1	4	yes	—
17	C₂×C₇×#107	—	2	1	no	—
18	C₂×C₇×#107	C₂²	1	2	yes	—
19	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₇×#358	C₂²	1	8	yes	complemented
21	C₇×#357	C₂²	1	8	yes	—
22	C₇×#346	C₂²	1	4	yes	complemented
23	C₂×#2243	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₇×Q₈	—	2	1	no	—
25	C₂²×C₇×Q₈	—	4	1	no	—
26	C₂²×C₇×D₄	—	2	1	no	—
27	C₂×C₇×Q₁₆	—	2	1	no	—
28	C₂×C₇×Q₁₆	—	2	4	no	—
29	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
30	C₂×C₇×SD₁₆	—	2	4	no	—
31	C₂×C₇×SD₁₆	—	4	4	no	—
32	C₂×C₇×D₈	—	2	1	no	—
33	C₂×C₇×D₈	—	2	4	no	—
34	C₂×C₇×D₈	C₂³	1	2	yes	—
35	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
36	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
37	C₇×C₄⋊Q₈	—	2	4	no	—
38	C₇×C₄⋊Q₈	—	4	2	no	—
39	C₇×#124	—	2	4	no	—
40	C₇×#124	—	4	2	no	—
41	C₂×C₇×C₄⋊C₄	—	2	2	no	—
42	C₂×C₇×C₄⋊C₄	—	4	1	no	—
43	C₂×C₇×C₂²⋊C₄	—	4	1	no	—
44	C₂×C₇×C₂²⋊C₄	—	4	1	no	—
45	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
46	C₂×C₇×C₁₆	—	2	4	no	—
47	C₂×C₇×C₁₆	C₂³	1	4	yes	—
48	C₇×#107	—	2	2	no	—
49	C₇×#107	—	2	4	no	—
50	C₇×#107	C₂³	1	4	yes	—
51	C₄×C₇×C₈	D₄	1	2	yes	—
52	C₄×C₇×C₈	C₂³	1	2	yes	—
53	C₂×#500	C₂×C₇	1	1	yes	characteristic, complemented
54	C₂×#495	C₂×C₇	1	1	yes	characteristic
55	#2243	C₂×C₇	1	8	yes	direct product factor
56	C₂×#358	C₂×C₇	1	2	yes	complemented
57	C₂×#357	C₂×C₇	1	2	yes	—
58	C₂×#346	C₂×C₇	1	1	yes	characteristic, complemented
59	C₂⁴×C₇	—	4	1	no	—
60	C₂×C₇×Q₈	—	2	2	no	—
61	C₂×C₇×Q₈	—	4	1	no	—
62	C₂×C₇×Q₈	—	4	4	no	—
63	C₂×C₇×Q₈	—	4	2	no	—
64	C₂×C₇×Q₈	—	4	2	no	—
65	C₂×C₇×Q₈	—	8	4	no	—
66	C₂×C₇×D₄	—	2	2	no	—
67	C₂×C₇×D₄	—	4	2	no	—
68	C₂×C₇×D₄	—	4	1	no	—
69	C₂×C₇×D₄	—	4	2	no	—
70	C₂²×C₄×C₇	—	4	1	no	—
71	C₂²×C₄×C₇	—	4	2	no	—
72	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
75	C₇×Q₁₆	—	2	4	no	—
76	C₇×Q₁₆	—	4	2	no	—
77	C₇×SD₁₆	—	4	8	no	—
78	C₇×D₈	—	2	4	no	—
79	C₇×D₈	—	4	2	no	—
80	C₂×C₇×C₈	SD₁₆	1	4	yes	—
81	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
83	C₂×C₇×C₈	#41	1	4	yes	—
84	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
85	C₇×C₄⋊C₄	—	2	4	no	—
86	C₇×C₄⋊C₄	—	4	2	no	—
87	C₇×C₄⋊C₄	—	4	4	no	—
88	C₇×C₂²⋊C₄	—	4	4	no	—
89	C₇×C₂²⋊C₄	—	4	4	no	—
90	C₄²×C₇	D₈	1	2	yes	—
91	C₄²×C₇	C₂×D₄	1	2	yes	—
92	C₇×C₁₆	—	2	4	no	—
93	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
94	C₂²×SD₁₆	—	2	1	no	—
95	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
96	C₂×C₄⋊Q₈	—	2	1	no	—
97	C₂×#124	—	2	1	no	—
98	C₂²×C₁₆	C₂²×C₇	1	2	yes	—
99	#500	—	2	2	no	—
100	#500	C₂²×C₇	1	4	yes	complemented
101	#495	—	2	2	no	—
102	#495	C₂²×C₇	1	4	yes	—
103	C₂×#107	—	2	1	no	—
104	C₂×#107	C₂²×C₇	1	2	yes	—
105	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
106	#358	C₂²×C₇	1	8	yes	complemented
107	#357	C₂²×C₇	1	8	yes	—
108	#346	C₂²×C₇	1	4	yes	complemented
109	C₂³×C₇	—	4	2	no	—
110	C₂³×C₇	—	8	2	no	—
111	C₂³×C₇	—	8	2	no	—
112	C₂³×C₇	—	8	2	no	—
113	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
114	C₇×Q₈	—	4	2	no	—
115	C₇×Q₈	—	4	2	no	—
116	C₇×Q₈	—	8	8	no	—
117	C₇×Q₈	—	8	2	no	—
118	C₇×D₄	—	4	2	no	—
119	C₇×D₄	—	4	2	no	—
120	C₇×D₄	—	8	2	no	—
121	C₂×C₄×C₇	—	2	2	no	—
122	C₂×C₄×C₇	—	2	2	no	—
123	C₂×C₄×C₇	—	4	2	no	—
124	C₂×C₄×C₇	—	4	4	no	—
125	C₂×C₄×C₇	—	8	2	no	—
126	C₂×C₄×C₇	—	8	4	no	—
127	C₂×C₄×C₇	#124	1	2	yes	—
128	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
132	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
133	C₇×C₈	#124	1	2	yes	—
134	C₇×C₈	C₂×SD₁₆	1	4	yes	—
135	C₇×C₈	C₂×#41	1	2	yes	—
136	C₂²×Q₈	—	2	1	no	—
137	C₂²×Q₈	—	4	1	no	—
138	C₂²×D₄	—	2	1	no	—
139	C₂×Q₁₆	—	2	1	no	—
140	C₂×Q₁₆	—	2	4	no	—
141	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
142	C₂×SD₁₆	—	2	4	no	—
143	C₂×SD₁₆	—	4	4	no	—
144	C₂×D₈	—	2	1	no	—
145	C₂×D₈	—	2	4	no	—
146	C₂×D₈	C₂³×C₇	1	2	yes	—
147	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
148	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
149	C₄⋊Q₈	—	2	4	no	—
150	C₄⋊Q₈	—	4	2	no	—
151	#124	—	2	4	no	—
152	#124	—	4	2	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	4	1	no	—
155	C₂×C₂²⋊C₄	—	4	1	no	—
156	C₂×C₂²⋊C₄	—	4	1	no	—
157	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
158	C₂×C₁₆	—	2	4	no	—
159	C₂×C₁₆	C₂³×C₇	1	4	yes	—
160	#107	—	2	4	no	—
161	#107	—	2	2	no	—
162	#107	C₂³×C₇	1	4	yes	—
163	C₄×C₈	C₇×D₄	1	2	yes	—
164	C₄×C₈	C₂³×C₇	1	2	yes	—
165	C₂²×C₇	—	8	2	no	—
166	C₂²×C₇	—	8	2	no	—
167	C₂²×C₇	—	8	2	no	—
168	C₂²×C₇	—	16	4	no	—
169	C₂²×C₇	#486	1	2	yes	—
170	C₂²×C₇	#509	1	2	yes	—
171	C₂²×C₇	#510	1	2	yes	—
172	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
173	C₄×C₇	—	2	2	no	—
174	C₄×C₇	—	2	2	no	—
175	C₄×C₇	—	8	2	no	—
176	C₄×C₇	—	8	4	no	—
177	C₄×C₇	#486	1	2	yes	—
178	C₄×C₇	C₂×#124	1	1	yes	characteristic
179	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
180	C₂⁴	—	4	1	no	—
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	4	1	no	—
183	C₂×Q₈	—	4	4	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	8	4	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂×D₄	—	4	1	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂²×C₄	—	4	1	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
194	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
196	Q₁₆	—	2	4	no	—
197	Q₁₆	—	4	2	no	—
198	SD₁₆	—	4	8	no	—
199	D₈	—	2	4	no	—
200	D₈	—	4	2	no	—
201	C₂×C₈	C₇×SD₁₆	1	4	yes	—
202	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
203	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
204	C₂×C₈	C₇×#41	1	4	yes	—
205	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	4	no	—
209	C₂²⋊C₄	—	4	4	no	—
210	C₂²⋊C₄	—	4	4	no	—
211	C₄²	C₇×D₈	1	2	yes	—
212	C₄²	C₂×C₇×D₄	1	2	yes	—
213	C₁₆	—	2	4	no	—
214	C₂×C₇	—	16	2	no	—
215	C₂×C₇	#2243	1	4	yes	direct product factor
216	C₂×C₇	C₂×#486	1	1	yes	characteristic
217	C₂×C₇	C₂×#509	1	1	yes	characteristic
218	C₂×C₇	C₂×#510	1	1	yes	characteristic
219	C₂³	—	4	2	no	—
220	C₂³	—	8	2	no	—
221	C₂³	—	8	2	no	—
222	C₂³	—	8	2	no	—
223	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
224	Q₈	—	4	2	no	—
225	Q₈	—	4	2	no	—
226	Q₈	—	8	8	no	—
227	Q₈	—	8	2	no	—
228	D₄	—	4	2	no	—
229	D₄	—	4	2	no	—
230	D₄	—	8	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	8	4	no	—
237	C₂×C₄	C₇×#124	1	2	yes	—
238	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
239	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
240	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
241	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
242	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
243	C₈	C₇×#124	1	2	yes	—
244	C₈	C₂×C₇×SD₁₆	1	4	yes	—
245	C₈	C₂×C₇×#41	1	2	yes	—
246	C₇	C₂×#2243	1	1	yes	characteristic, direct product factor, Sylow
247	C₂²	—	8	2	no	—
248	C₂²	—	8	2	no	—
249	C₂²	—	8	2	no	—
250	C₂²	—	16	4	no	—
251	C₂²	C₇×#486	1	2	yes	—
252	C₂²	C₇×#509	1	2	yes	—
253	C₂²	C₇×#510	1	2	yes	—
254	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	8	4	no	—
258	C₄	—	8	2	no	—
259	C₄	C₇×#486	1	2	yes	—
260	C₄	C₂×C₇×#124	1	1	yes	characteristic
261	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
262	C₂	—	16	2	no	—
263	C₂	C₇×#2243	1	4	yes	direct product factor
264	C₂	C₂×C₇×#486	1	1	yes	characteristic
265	C₂	C₂×C₇×#509	1	1	yes	characteristic
266	C₂	C₂×C₇×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇
prime radical	C₂²×C₇×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ C₂×C₄×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×#500	C₂	semidirect
3. C₇×#2243	C₂	direct (N × H)
4. C₂×C₇×#358	C₂	semidirect
6. C₂×C₇×#346	C₂	semidirect
14. C₇×#500	C₂²	semidirect
20. C₇×#358	C₂²	semidirect
22. C₇×#346	C₂²	semidirect
23. C₂×#2243	C₇	direct (N × H)
53. C₂×#500	C₂×C₇	semidirect
55. #2243	C₂×C₇	direct (N × H)
56. C₂×#358	C₂×C₇	semidirect
58. C₂×#346	C₂×C₇	semidirect
100. #500	C₂²×C₇	semidirect
106. #358	C₂²×C₇	semidirect
108. #346	C₂²×C₇	semidirect
215. C₂×C₇	#2243	direct (N × H)
246. C₇	C₂×#2243	direct (N × H)
263. C₂	C₇×#2243	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×#500	2
C₂×C₇×#495	2
C₇×#2243	2
C₂×C₇×#358	2
C₂×C₇×#357	2
C₂×C₇×#346	2
C₂×#2243	7

Subgroups — C2×C7×#2243 #13210914

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2243 #13210914