Subgroups — C₂×C₇×#2856 #13213838

Order 1792.

5694 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	C₂²×C₇×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₇×#121	C₂	1	6	yes	maximal, complemented
3	C₂³×C₄²×C₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#2856	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₇×#390	C₂	1	6	yes	maximal, complemented
6	C₂×C₇×#384	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₇×D₄	C₂²	1	4	yes	—
8	C₂³×C₇×D₄	C₂²	1	6	yes	—
9	C₂⁴×C₄×C₇	C₂²	1	3	yes	—
10	C₂×C₇×#127	—	2	8	no	—
11	C₂×C₇×#127	C₂²	1	8	yes	complemented
12	C₂×C₇×#121	—	2	48	no	—
13	C₂×C₇×#121	C₂²	1	48	yes	complemented
14	C₂²×C₇×C₄⋊C₄	C₂²	1	6	yes	—
15	C₂²×C₇×C₂²⋊C₄	C₂²	1	12	yes	—
16	C₂²×C₄²×C₇	—	2	8	no	—
17	C₂²×C₄²×C₇	C₂²	1	3	yes	—
18	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄²×C₇	C₂²	1	8	yes	complemented
20	C₇×#390	C₂²	1	48	yes	complemented
21	C₇×#384	C₂²	1	8	yes	complemented
22	C₂×#2856	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂⁵×C₇	C₂³	1	4	yes	—
24	C₂⁵×C₇	C₂³	1	1	yes	characteristic
25	C₂²×C₇×D₄	—	2	48	no	—
26	C₂²×C₇×D₄	—	2	24	no	—
27	C₂²×C₇×D₄	—	2	24	no	—
28	C₂²×C₇×D₄	—	2	24	no	—
29	C₂²×C₇×D₄	C₂³	1	16	yes	—
30	C₂²×C₇×D₄	C₂³	1	24	yes	—
31	C₂³×C₄×C₇	—	2	12	no	—
32	C₂³×C₄×C₇	—	2	12	no	—
33	C₂³×C₄×C₇	D₄	1	24	yes	complemented
34	C₂³×C₄×C₇	C₂³	1	4	yes	—
35	C₂³×C₄×C₇	C₂³	1	3	yes	—
36	C₂³×C₄×C₇	C₂³	1	12	yes	—
37	C₂³×C₄×C₇	C₂³	1	3	yes	—
38	C₇×#127	—	2	32	no	—
39	C₇×#121	—	2	192	no	—
40	C₂×C₇×C₄⋊C₄	—	2	24	no	—
41	C₂×C₇×C₄⋊C₄	C₂³	1	24	yes	—
42	C₂×C₇×C₂²⋊C₄	—	2	48	no	—
43	C₂×C₇×C₂²⋊C₄	C₂³	1	48	yes	—
44	C₂×C₄²×C₇	—	2	12	no	—
45	C₂×C₄²×C₇	—	2	32	no	—
46	C₂×C₄²×C₇	D₄	1	8	yes	complemented
47	C₂×C₄²×C₇	C₂³	1	12	yes	—
48	C₂×C₄²×C₇	C₂³	1	4	yes	—
49	C₂²×#127	C₂×C₇	1	1	yes	characteristic, complemented
50	C₂²×#121	C₂×C₇	1	6	yes	complemented
51	C₂³×C₄²	C₂×C₇	1	1	yes	characteristic, complemented
52	#2856	C₂×C₇	1	16	yes	direct product factor
53	C₂×#390	C₂×C₇	1	6	yes	complemented
54	C₂×#384	C₂×C₇	1	1	yes	characteristic, complemented
55	C₂⁴×C₇	—	2	2	no	—
56	C₂⁴×C₇	—	2	6	no	—
57	C₂⁴×C₇	—	2	24	no	—
58	C₂⁴×C₇	—	2	8	no	—
59	C₂⁴×C₇	—	2	24	no	—
60	C₂⁴×C₇	C₂×D₄	1	12	yes	—
61	C₂⁴×C₇	C₂⁴	1	8	yes	—
62	C₂⁴×C₇	C₂⁴	1	2	yes	—
63	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
64	C₂×C₇×D₄	—	2	96	no	—
65	C₂×C₇×D₄	—	2	48	no	—
66	C₂×C₇×D₄	—	2	48	no	—
67	C₂×C₇×D₄	—	2	48	no	—
68	C₂×C₇×D₄	—	4	64	no	—
69	C₂×C₇×D₄	—	4	96	no	—
70	C₂²×C₄×C₇	—	2	6	no	—
71	C₂²×C₄×C₇	—	2	6	no	—
72	C₂²×C₄×C₇	—	2	96	no	—
73	C₂²×C₄×C₇	—	2	24	no	—
74	C₂²×C₄×C₇	—	2	24	no	—
75	C₂²×C₄×C₇	—	2	24	no	—
76	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
77	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
78	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
79	C₂²×C₄×C₇	C₂×D₄	1	48	yes	complemented
80	C₂²×C₄×C₇	#41	1	12	yes	—
81	C₂²×C₄×C₇	C₂⁴	1	8	yes	—
82	C₂²×C₄×C₇	C₂⁴	1	6	yes	—
83	C₂²×C₄×C₇	C₂⁴	1	6	yes	—
84	C₇×C₄⋊C₄	—	2	48	no	—
85	C₇×C₂²⋊C₄	—	2	96	no	—
86	C₄²×C₇	—	2	24	no	—
87	C₄²×C₇	C₂×D₄	1	16	yes	complemented
88	C₂³×D₄	C₂²×C₇	1	4	yes	—
89	C₂³×D₄	C₂²×C₇	1	6	yes	—
90	C₂⁴×C₄	C₂²×C₇	1	3	yes	—
91	C₂×#127	—	2	8	no	—
92	C₂×#127	C₂²×C₇	1	8	yes	complemented
93	C₂×#121	—	2	48	no	—
94	C₂×#121	C₂²×C₇	1	48	yes	complemented
95	C₂²×C₄⋊C₄	C₂²×C₇	1	6	yes	—
96	C₂²×C₂²⋊C₄	C₂²×C₇	1	12	yes	—
97	C₂²×C₄²	—	2	8	no	—
98	C₂²×C₄²	C₂²×C₇	1	3	yes	—
99	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
100	C₂²×C₄²	C₂²×C₇	1	8	yes	complemented
101	#390	C₂²×C₇	1	48	yes	complemented
102	#384	C₂²×C₇	1	8	yes	complemented
103	C₂³×C₇	—	2	24	no	—
104	C₂³×C₇	—	2	24	no	—
105	C₂³×C₇	—	2	2	no	—
106	C₂³×C₇	—	2	6	no	—
107	C₂³×C₇	—	2	24	no	—
108	C₂³×C₇	—	2	8	no	—
109	C₂³×C₇	—	2	24	no	—
110	C₂³×C₇	—	4	48	no	—
111	C₂³×C₇	—	4	16	no	—
112	C₂³×C₇	—	4	48	no	—
113	C₂³×C₇	#127	1	16	yes	complemented
114	C₂³×C₇	C₂²×D₄	1	12	yes	—
115	C₂³×C₇	C₂²×D₄	1	2	yes	—
116	C₂³×C₇	C₂²×D₄	1	6	yes	—
117	C₂³×C₇	C₂×#41	1	6	yes	—
118	C₂³×C₇	C₂⁵	1	1	yes	characteristic
119	C₇×D₄	—	4	64	no	—
120	C₇×D₄	—	4	96	no	—
121	C₂×C₄×C₇	—	2	24	no	—
122	C₂×C₄×C₇	—	2	6	no	—
123	C₂×C₄×C₇	—	2	6	no	—
124	C₂×C₄×C₇	—	2	96	no	—
125	C₂×C₄×C₇	—	2	24	no	—
126	C₂×C₄×C₇	—	4	16	no	—
127	C₂×C₄×C₇	#121	1	48	yes	complemented
128	C₂×C₄×C₇	C₂²×D₄	1	12	yes	—
129	C₂×C₄×C₇	C₂²×D₄	1	12	yes	—
130	C₂×C₄×C₇	C₂²×D₄	1	12	yes	—
131	C₂×C₄×C₇	C₂×#41	1	12	yes	—
132	C₂⁵	C₂³×C₇	1	4	yes	—
133	C₂⁵	C₂³×C₇	1	1	yes	characteristic
134	C₂²×D₄	—	2	48	no	—
135	C₂²×D₄	—	2	24	no	—
136	C₂²×D₄	—	2	24	no	—
137	C₂²×D₄	—	2	24	no	—
138	C₂²×D₄	C₂³×C₇	1	16	yes	—
139	C₂²×D₄	C₂³×C₇	1	24	yes	—
140	C₂³×C₄	—	2	12	no	—
141	C₂³×C₄	—	2	12	no	—
142	C₂³×C₄	C₇×D₄	1	24	yes	complemented
143	C₂³×C₄	C₂³×C₇	1	4	yes	—
144	C₂³×C₄	C₂³×C₇	1	3	yes	—
145	C₂³×C₄	C₂³×C₇	1	12	yes	—
146	C₂³×C₄	C₂³×C₇	1	3	yes	—
147	#127	—	2	32	no	—
148	#121	—	2	192	no	—
149	C₂×C₄⋊C₄	—	2	24	no	—
150	C₂×C₄⋊C₄	C₂³×C₇	1	24	yes	—
151	C₂×C₂²⋊C₄	—	2	48	no	—
152	C₂×C₂²⋊C₄	C₂³×C₇	1	48	yes	—
153	C₂×C₄²	—	2	12	no	—
154	C₂×C₄²	—	2	32	no	—
155	C₂×C₄²	C₇×D₄	1	8	yes	complemented
156	C₂×C₄²	C₂³×C₇	1	12	yes	—
157	C₂×C₄²	C₂³×C₇	1	4	yes	—
158	C₂²×C₇	—	2	32	no	—
159	C₂²×C₇	—	2	12	no	—
160	C₂²×C₇	—	2	12	no	—
161	C₂²×C₇	—	4	24	no	—
162	C₂²×C₇	—	4	8	no	—
163	C₂²×C₇	—	4	24	no	—
164	C₂²×C₇	—	8	32	no	—
165	C₂²×C₇	C₂×#121	1	12	yes	—
166	C₂²×C₇	C₂×#127	1	4	yes	—
167	C₂²×C₇	C₂×#127	1	8	yes	complemented
168	C₂²×C₇	#532	1	12	yes	—
169	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
170	C₂²×C₇	C₂³×D₄	1	3	yes	—
171	C₂²×C₇	C₂²×#41	1	3	yes	—
172	C₄×C₇	—	2	12	no	—
173	C₄×C₇	—	4	8	no	—
174	C₄×C₇	C₂×#121	1	24	yes	complemented
175	C₂⁴	—	2	2	no	—
176	C₂⁴	—	2	6	no	—
177	C₂⁴	—	2	24	no	—
178	C₂⁴	—	2	8	no	—
179	C₂⁴	—	2	24	no	—
180	C₂⁴	C₂×C₇×D₄	1	12	yes	—
181	C₂⁴	C₂⁴×C₇	1	8	yes	—
182	C₂⁴	C₂⁴×C₇	1	2	yes	—
183	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
184	C₂×D₄	—	2	96	no	—
185	C₂×D₄	—	2	48	no	—
186	C₂×D₄	—	2	48	no	—
187	C₂×D₄	—	2	48	no	—
188	C₂×D₄	—	4	64	no	—
189	C₂×D₄	—	4	96	no	—
190	C₂²×C₄	—	2	6	no	—
191	C₂²×C₄	—	2	6	no	—
192	C₂²×C₄	—	2	96	no	—
193	C₂²×C₄	—	2	24	no	—
194	C₂²×C₄	—	2	24	no	—
195	C₂²×C₄	—	2	24	no	—
196	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
197	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
198	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
199	C₂²×C₄	C₂×C₇×D₄	1	48	yes	complemented
200	C₂²×C₄	C₇×#41	1	12	yes	—
201	C₂²×C₄	C₂⁴×C₇	1	8	yes	—
202	C₂²×C₄	C₂⁴×C₇	1	6	yes	—
203	C₂²×C₄	C₂⁴×C₇	1	6	yes	—
204	C₄⋊C₄	—	2	48	no	—
205	C₂²⋊C₄	—	2	96	no	—
206	C₄²	—	2	24	no	—
207	C₄²	C₂×C₇×D₄	1	16	yes	complemented
208	C₂×C₇	—	2	8	no	—
209	C₂×C₇	—	8	8	no	—
210	C₂×C₇	#2856	1	8	yes	direct product factor
211	C₂×C₇	C₂²×#121	1	3	yes	—
212	C₂×C₇	C₂²×#127	1	1	yes	characteristic
213	C₂×C₇	C₂×#532	1	3	yes	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	6	no	—
216	C₂³	—	2	24	no	—
217	C₂³	—	2	24	no	—
218	C₂³	—	2	24	no	—
219	C₂³	—	2	8	no	—
220	C₂³	—	2	24	no	—
221	C₂³	—	4	48	no	—
222	C₂³	—	4	16	no	—
223	C₂³	—	4	48	no	—
224	C₂³	C₇×#127	1	16	yes	complemented
225	C₂³	C₂²×C₇×D₄	1	12	yes	—
226	C₂³	C₂²×C₇×D₄	1	2	yes	—
227	C₂³	C₂²×C₇×D₄	1	6	yes	—
228	C₂³	C₂×C₇×#41	1	6	yes	—
229	C₂³	C₂⁵×C₇	1	1	yes	characteristic
230	D₄	—	4	64	no	—
231	D₄	—	4	96	no	—
232	C₂×C₄	—	2	6	no	—
233	C₂×C₄	—	2	6	no	—
234	C₂×C₄	—	2	24	no	—
235	C₂×C₄	—	2	96	no	—
236	C₂×C₄	—	2	24	no	—
237	C₂×C₄	—	4	16	no	—
238	C₂×C₄	C₇×#121	1	48	yes	complemented
239	C₂×C₄	C₂²×C₇×D₄	1	12	yes	—
240	C₂×C₄	C₂²×C₇×D₄	1	12	yes	—
241	C₂×C₄	C₂²×C₇×D₄	1	12	yes	—
242	C₂×C₄	C₂×C₇×#41	1	12	yes	—
243	C₇	C₂×#2856	1	1	yes	characteristic, direct product factor, Sylow
244	C₂²	—	2	12	no	—
245	C₂²	—	2	12	no	—
246	C₂²	—	2	32	no	—
247	C₂²	—	4	24	no	—
248	C₂²	—	4	8	no	—
249	C₂²	—	4	24	no	—
250	C₂²	—	8	32	no	—
251	C₂²	C₂×C₇×#121	1	12	yes	—
252	C₂²	C₂×C₇×#127	1	4	yes	—
253	C₂²	C₂×C₇×#127	1	8	yes	complemented
254	C₂²	C₇×#532	1	12	yes	—
255	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
256	C₂²	C₂³×C₇×D₄	1	3	yes	—
257	C₂²	C₂²×C₇×#41	1	3	yes	—
258	C₄	—	2	12	no	—
259	C₄	—	4	8	no	—
260	C₄	C₂×C₇×#121	1	24	yes	complemented
261	C₂	—	2	8	no	—
262	C₂	—	8	8	no	—
263	C₂	C₇×#2856	1	8	yes	direct product factor
264	C₂	C₂²×C₇×#121	1	3	yes	—
265	C₂	C₂²×C₇×#127	1	1	yes	characteristic
266	C₂	C₂×C₇×#532	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇×#127	C₂	semidirect
2. C₂²×C₇×#121	C₂	semidirect
3. C₂³×C₄²×C₇	C₂	semidirect
4. C₇×#2856	C₂	direct (N × H)
5. C₂×C₇×#390	C₂	semidirect
6. C₂×C₇×#384	C₂	semidirect
11. C₂×C₇×#127	C₂²	semidirect
13. C₂×C₇×#121	C₂²	semidirect
19. C₂²×C₄²×C₇	C₂²	semidirect
20. C₇×#390	C₂²	semidirect
21. C₇×#384	C₂²	semidirect
22. C₂×#2856	C₇	direct (N × H)
33. C₂³×C₄×C₇	D₄	semidirect
46. C₂×C₄²×C₇	D₄	semidirect
49. C₂²×#127	C₂×C₇	semidirect
50. C₂²×#121	C₂×C₇	semidirect
51. C₂³×C₄²	C₂×C₇	semidirect
52. #2856	C₂×C₇	direct (N × H)
53. C₂×#390	C₂×C₇	semidirect
54. C₂×#384	C₂×C₇	semidirect
79. C₂²×C₄×C₇	C₂×D₄	semidirect
87. C₄²×C₇	C₂×D₄	semidirect
92. C₂×#127	C₂²×C₇	semidirect
94. C₂×#121	C₂²×C₇	semidirect
100. C₂²×C₄²	C₂²×C₇	semidirect
101. #390	C₂²×C₇	semidirect
102. #384	C₂²×C₇	semidirect
113. C₂³×C₇	#127	semidirect
127. C₂×C₄×C₇	#121	semidirect
142. C₂³×C₄	C₇×D₄	semidirect
155. C₂×C₄²	C₇×D₄	semidirect
167. C₂²×C₇	C₂×#127	semidirect
174. C₄×C₇	C₂×#121	semidirect
199. C₂²×C₄	C₂×C₇×D₄	semidirect
207. C₄²	C₂×C₇×D₄	semidirect
210. C₂×C₇	#2856	direct (N × H)
224. C₂³	C₇×#127	semidirect
238. C₂×C₄	C₇×#121	semidirect
243. C₇	C₂×#2856	direct (N × H)
253. C₂²	C₂×C₇×#127	semidirect
260. C₄	C₂×C₇×#121	semidirect
263. C₂	C₇×#2856	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×#127	2
C₂²×C₇×#121	2
C₂³×C₄²×C₇	2
C₇×#2856	2
C₂×C₇×#390	2
C₂×C₇×#384	2
C₂×#2856	7

Subgroups — C2×C7×#2856 #13213838

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2856 #13213838