Subgroups — C₂×C₇×#2861 #13213846

Order 1792.

2902 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₇×#124	C₂	1	1	yes	maximal, characteristic
3	C₇×#2861	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₇×#391	C₂	1	2	yes	maximal, complemented
5	C₂×C₇×#390	C₂	1	2	yes	maximal
6	C₂×C₇×#388	C₂	1	8	yes	maximal, complemented
7	C₂×C₄³×C₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₂³×C₇×D₄	C₂²	1	1	yes	characteristic
10	C₂×C₇×#125	—	2	8	no	—
11	C₂×C₇×#125	C₂²	1	8	yes	complemented
12	C₂×C₇×#124	—	2	8	no	—
13	C₂×C₇×#124	C₂²	1	8	yes	—
14	C₂²×C₇×C₄⋊C₄	C₂²	1	4	yes	—
15	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂²×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
17	C₂²×C₇×C₂²⋊C₄	C₂²	1	8	yes	—
18	C₂²×C₄²×C₇	C₂²	1	4	yes	—
19	C₂²×C₄²×C₇	C₂²	1	2	yes	—
20	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
21	C₇×#391	C₂²	1	16	yes	complemented
22	C₇×#390	C₂²	1	16	yes	—
23	C₇×#388	C₂²	1	64	yes	complemented
24	C₂×C₇×#95	C₂²	1	8	yes	—
25	C₄³×C₇	C₂²	1	8	yes	complemented
26	C₂×#2861	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₂⁵×C₇	C₂³	1	2	yes	—
28	C₂²×C₇×Q₈	—	2	4	no	—
29	C₂²×C₇×Q₈	—	2	8	no	—
30	C₂²×C₇×Q₈	C₂³	1	4	yes	—
31	C₂²×C₇×D₄	—	2	4	no	—
32	C₂²×C₇×D₄	—	2	8	no	—
33	C₂²×C₇×D₄	C₂³	1	4	yes	—
34	C₂³×C₄×C₇	C₂³	1	2	yes	—
35	C₂³×C₄×C₇	C₂³	1	4	yes	—
36	C₂³×C₄×C₇	C₂³	1	2	yes	—
37	C₂³×C₄×C₇	C₂³	1	4	yes	—
38	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
39	C₇×#125	—	2	32	no	—
40	C₇×#124	—	2	32	no	—
41	C₂×C₇×C₄⋊C₄	—	2	16	no	—
42	C₂×C₇×C₄⋊C₄	—	2	8	no	—
43	C₂×C₇×C₄⋊C₄	C₂³	1	16	yes	—
44	C₂×C₇×C₄⋊C₄	C₂³	1	8	yes	—
45	C₂×C₇×C₂²⋊C₄	—	2	16	no	—
46	C₂×C₇×C₂²⋊C₄	—	2	32	no	—
47	C₂×C₇×C₂²⋊C₄	C₂³	1	16	yes	—
48	C₂×C₇×C₂²⋊C₄	C₂³	1	32	yes	—
49	C₂×C₄²×C₇	—	2	16	no	—
50	C₂×C₄²×C₇	—	2	8	no	—
51	C₂×C₄²×C₇	D₄	1	8	yes	—
52	C₂×C₄²×C₇	C₂³	1	16	yes	—
53	C₂×C₄²×C₇	C₂³	1	8	yes	—
54	C₂×C₄²×C₇	C₂³	1	4	yes	—
55	C₇×#95	C₂³	1	32	yes	—
56	C₂²×#125	C₂×C₇	1	1	yes	characteristic, complemented
57	C₂²×#124	C₂×C₇	1	1	yes	characteristic
58	#2861	C₂×C₇	1	16	yes	direct product factor
59	C₂×#391	C₂×C₇	1	2	yes	complemented
60	C₂×#390	C₂×C₇	1	2	yes	—
61	C₂×#388	C₂×C₇	1	8	yes	complemented
62	C₂×C₄³	C₂×C₇	1	1	yes	characteristic, complemented
63	C₂⁴×C₇	—	2	4	no	—
64	C₂⁴×C₇	—	2	16	no	—
65	C₂⁴×C₇	—	2	8	no	—
66	C₂⁴×C₇	C₂⁴	1	4	yes	—
67	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
68	C₂×C₇×Q₈	—	2	8	no	—
69	C₂×C₇×Q₈	—	2	16	no	—
70	C₂×C₇×Q₈	—	4	16	no	—
71	C₂×C₇×D₄	—	2	8	no	—
72	C₂×C₇×D₄	—	2	16	no	—
73	C₂×C₇×D₄	—	4	16	no	—
74	C₂²×C₄×C₇	—	2	8	no	—
75	C₂²×C₄×C₇	—	2	8	no	—
76	C₂²×C₄×C₇	—	2	8	no	—
77	C₂²×C₄×C₇	—	2	4	no	—
78	C₂²×C₄×C₇	—	2	8	no	—
79	C₂²×C₄×C₇	—	2	16	no	—
80	C₂²×C₄×C₇	—	2	8	no	—
81	C₂²×C₄×C₇	C₂×D₄	1	8	yes	—
82	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
83	C₂²×C₄×C₇	#41	1	16	yes	—
84	C₂²×C₄×C₇	#41	1	8	yes	—
85	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
86	C₂²×C₄×C₇	C₂⁴	1	8	yes	—
87	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
88	C₂²×C₄×C₇	C₂⁴	1	8	yes	—
89	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
90	C₇×C₄⋊C₄	—	2	32	no	—
91	C₇×C₄⋊C₄	—	2	16	no	—
92	C₇×C₂²⋊C₄	—	2	32	no	—
93	C₇×C₂²⋊C₄	—	2	64	no	—
94	C₄²×C₇	—	2	32	no	—
95	C₄²×C₇	—	2	16	no	—
96	C₄²×C₇	C₂×D₄	1	16	yes	—
97	C₂³×Q₈	C₂²×C₇	1	1	yes	characteristic
98	C₂³×D₄	C₂²×C₇	1	1	yes	characteristic
99	C₂×#125	—	2	8	no	—
100	C₂×#125	C₂²×C₇	1	8	yes	complemented
101	C₂×#124	—	2	8	no	—
102	C₂×#124	C₂²×C₇	1	8	yes	—
103	C₂²×C₄⋊C₄	C₂²×C₇	1	4	yes	—
104	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
105	C₂²×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
106	C₂²×C₂²⋊C₄	C₂²×C₇	1	8	yes	—
107	C₂²×C₄²	C₂²×C₇	1	4	yes	—
108	C₂²×C₄²	C₂²×C₇	1	2	yes	—
109	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
110	#391	C₂²×C₇	1	16	yes	complemented
111	#390	C₂²×C₇	1	16	yes	—
112	#388	C₂²×C₇	1	64	yes	complemented
113	C₂×#95	C₂²×C₇	1	8	yes	—
114	C₄³	C₂²×C₇	1	8	yes	complemented
115	C₂³×C₇	—	2	4	no	—
116	C₂³×C₇	—	2	16	no	—
117	C₂³×C₇	—	2	8	no	—
118	C₂³×C₇	—	4	8	no	—
119	C₂³×C₇	—	4	32	no	—
120	C₂³×C₇	—	4	16	no	—
121	C₂³×C₇	C₂²×D₄	1	2	yes	—
122	C₂³×C₇	C₂×#41	1	4	yes	—
123	C₂³×C₇	C₂×#41	1	8	yes	—
124	C₂³×C₇	C₂⁵	1	1	yes	characteristic
125	C₇×Q₈	—	4	16	no	—
126	C₇×D₄	—	4	16	no	—
127	C₂×C₄×C₇	—	2	32	no	—
128	C₂×C₄×C₇	—	2	16	no	—
129	C₂×C₄×C₇	—	2	8	no	—
130	C₂×C₄×C₇	—	2	8	no	—
131	C₂×C₄×C₇	—	2	8	no	—
132	C₂×C₄×C₇	—	2	4	no	—
133	C₂×C₄×C₇	—	2	8	no	—
134	C₂×C₄×C₇	—	2	16	no	—
135	C₂×C₄×C₇	—	2	8	no	—
136	C₂×C₄×C₇	—	4	8	no	—
137	C₂×C₄×C₇	—	4	16	no	—
138	C₂×C₄×C₇	#124	1	16	yes	—
139	C₂×C₄×C₇	C₂²×D₄	1	8	yes	—
140	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
141	C₂×C₄×C₇	C₂×#41	1	16	yes	—
142	C₂×C₄×C₇	C₂×#41	1	8	yes	—
143	C₂⁵	C₂³×C₇	1	2	yes	—
144	C₂²×Q₈	—	2	4	no	—
145	C₂²×Q₈	—	2	8	no	—
146	C₂²×Q₈	C₂³×C₇	1	4	yes	—
147	C₂²×D₄	—	2	4	no	—
148	C₂²×D₄	—	2	8	no	—
149	C₂²×D₄	C₂³×C₇	1	4	yes	—
150	C₂³×C₄	C₂³×C₇	1	4	yes	—
151	C₂³×C₄	C₂³×C₇	1	2	yes	—
152	C₂³×C₄	C₂³×C₇	1	2	yes	—
153	C₂³×C₄	C₂³×C₇	1	4	yes	—
154	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
155	#125	—	2	32	no	—
156	#124	—	2	32	no	—
157	C₂×C₄⋊C₄	—	2	16	no	—
158	C₂×C₄⋊C₄	—	2	8	no	—
159	C₂×C₄⋊C₄	C₂³×C₇	1	16	yes	—
160	C₂×C₄⋊C₄	C₂³×C₇	1	8	yes	—
161	C₂×C₂²⋊C₄	—	2	16	no	—
162	C₂×C₂²⋊C₄	—	2	32	no	—
163	C₂×C₂²⋊C₄	C₂³×C₇	1	16	yes	—
164	C₂×C₂²⋊C₄	C₂³×C₇	1	32	yes	—
165	C₂×C₄²	—	2	16	no	—
166	C₂×C₄²	—	2	8	no	—
167	C₂×C₄²	C₇×D₄	1	8	yes	—
168	C₂×C₄²	C₂³×C₇	1	16	yes	—
169	C₂×C₄²	C₂³×C₇	1	8	yes	—
170	C₂×C₄²	C₂³×C₇	1	4	yes	—
171	#95	C₂³×C₇	1	32	yes	—
172	C₂²×C₇	—	4	4	no	—
173	C₂²×C₇	—	4	16	no	—
174	C₂²×C₇	—	4	8	no	—
175	C₂²×C₇	—	8	16	no	—
176	C₂²×C₇	C₂×#124	1	4	yes	—
177	C₂²×C₇	#529	1	16	yes	—
178	C₂²×C₇	#532	1	8	yes	—
179	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
180	C₂²×C₇	C₂²×#41	1	2	yes	—
181	C₂²×C₇	C₂²×#41	1	4	yes	—
182	C₄×C₇	—	2	16	no	—
183	C₄×C₇	—	2	8	no	—
184	C₄×C₇	—	4	4	no	—
185	C₄×C₇	—	4	8	no	—
186	C₄×C₇	C₂×#124	1	8	yes	—
187	C₂⁴	—	2	4	no	—
188	C₂⁴	—	2	16	no	—
189	C₂⁴	—	2	8	no	—
190	C₂⁴	C₂⁴×C₇	1	4	yes	—
191	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
192	C₂×Q₈	—	2	8	no	—
193	C₂×Q₈	—	2	16	no	—
194	C₂×Q₈	—	4	16	no	—
195	C₂×D₄	—	2	8	no	—
196	C₂×D₄	—	2	16	no	—
197	C₂×D₄	—	4	16	no	—
198	C₂²×C₄	—	2	8	no	—
199	C₂²×C₄	—	2	8	no	—
200	C₂²×C₄	—	2	8	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	8	no	—
203	C₂²×C₄	—	2	16	no	—
204	C₂²×C₄	—	2	8	no	—
205	C₂²×C₄	C₂×C₇×D₄	1	8	yes	—
206	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
207	C₂²×C₄	C₇×#41	1	16	yes	—
208	C₂²×C₄	C₇×#41	1	8	yes	—
209	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
210	C₂²×C₄	C₂⁴×C₇	1	8	yes	—
211	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
212	C₂²×C₄	C₂⁴×C₇	1	8	yes	—
213	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
214	C₄⋊C₄	—	2	32	no	—
215	C₄⋊C₄	—	2	16	no	—
216	C₂²⋊C₄	—	2	32	no	—
217	C₂²⋊C₄	—	2	64	no	—
218	C₄²	—	2	32	no	—
219	C₄²	—	2	16	no	—
220	C₄²	C₂×C₇×D₄	1	16	yes	—
221	C₂×C₇	—	8	4	no	—
222	C₂×C₇	#2861	1	8	yes	direct product factor
223	C₂×C₇	C₂²×#124	1	1	yes	characteristic
224	C₂×C₇	C₂×#529	1	4	yes	—
225	C₂×C₇	C₂×#532	1	2	yes	—
226	C₂³	—	2	4	no	—
227	C₂³	—	2	16	no	—
228	C₂³	—	2	8	no	—
229	C₂³	—	4	8	no	—
230	C₂³	—	4	32	no	—
231	C₂³	—	4	16	no	—
232	C₂³	C₂²×C₇×D₄	1	2	yes	—
233	C₂³	C₂×C₇×#41	1	4	yes	—
234	C₂³	C₂×C₇×#41	1	8	yes	—
235	C₂³	C₂⁵×C₇	1	1	yes	characteristic
236	Q₈	—	4	16	no	—
237	D₄	—	4	16	no	—
238	C₂×C₄	—	2	8	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	2	32	no	—
242	C₂×C₄	—	2	16	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	16	no	—
246	C₂×C₄	—	2	8	no	—
247	C₂×C₄	—	4	8	no	—
248	C₂×C₄	—	4	16	no	—
249	C₂×C₄	C₇×#124	1	16	yes	—
250	C₂×C₄	C₂²×C₇×D₄	1	8	yes	—
251	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
252	C₂×C₄	C₂×C₇×#41	1	16	yes	—
253	C₂×C₄	C₂×C₇×#41	1	8	yes	—
254	C₇	C₂×#2861	1	1	yes	characteristic, direct product factor, Sylow
255	C₂²	—	4	4	no	—
256	C₂²	—	4	16	no	—
257	C₂²	—	4	8	no	—
258	C₂²	—	8	16	no	—
259	C₂²	C₂×C₇×#124	1	4	yes	—
260	C₂²	C₇×#529	1	16	yes	—
261	C₂²	C₇×#532	1	8	yes	—
262	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
263	C₂²	C₂²×C₇×#41	1	2	yes	—
264	C₂²	C₂²×C₇×#41	1	4	yes	—
265	C₄	—	2	16	no	—
266	C₄	—	2	8	no	—
267	C₄	—	4	4	no	—
268	C₄	—	4	8	no	—
269	C₄	C₂×C₇×#124	1	8	yes	—
270	C₂	—	8	4	no	—
271	C₂	C₇×#2861	1	8	yes	direct product factor
272	C₂	C₂²×C₇×#124	1	1	yes	characteristic
273	C₂	C₂×C₇×#529	1	4	yes	—
274	C₂	C₂×C₇×#532	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₇
prime radical	C₂³×C₇×D₄	C₂²
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×#125	C₂	semidirect
3. C₇×#2861	C₂	direct (N × H)
4. C₂×C₇×#391	C₂	semidirect
6. C₂×C₇×#388	C₂	semidirect
7. C₂×C₄³×C₇	C₂	semidirect
11. C₂×C₇×#125	C₂²	semidirect
21. C₇×#391	C₂²	semidirect
23. C₇×#388	C₂²	semidirect
25. C₄³×C₇	C₂²	semidirect
26. C₂×#2861	C₇	direct (N × H)
56. C₂²×#125	C₂×C₇	semidirect
58. #2861	C₂×C₇	direct (N × H)
59. C₂×#391	C₂×C₇	semidirect
61. C₂×#388	C₂×C₇	semidirect
62. C₂×C₄³	C₂×C₇	semidirect
100. C₂×#125	C₂²×C₇	semidirect
110. #391	C₂²×C₇	semidirect
112. #388	C₂²×C₇	semidirect
114. C₄³	C₂²×C₇	semidirect
222. C₂×C₇	#2861	direct (N × H)
254. C₇	C₂×#2861	direct (N × H)
271. C₂	C₇×#2861	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×#125	2
C₂²×C₇×#124	2
C₇×#2861	2
C₂×C₇×#391	2
C₂×C₇×#390	2
C₂×C₇×#388	2
C₂×C₄³×C₇	2
C₂×#2861	7

Subgroups — C2×C7×#2861 #13213846

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2861 #13213846