Subgroups — C₂×C₇×#2858 #13213843

Order 1792.

4462 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₇×#127	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₇×#124	C₂	1	4	yes	maximal, complemented
4	C₇×#2858	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₇×#390	C₂	1	8	yes	maximal, complemented
6	C₂×C₄³×C₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₇×Q₈	C₂²	1	2	yes	—
8	C₂³×C₇×D₄	C₂²	1	2	yes	—
9	C₂³×C₇×D₄	C₂²	1	4	yes	—
10	C₂×C₇×C₄⋊Q₈	—	2	8	no	—
11	C₂×C₇×C₄⋊Q₈	C₂²	1	8	yes	complemented
12	C₂×C₇×#127	—	2	8	no	—
13	C₂×C₇×#127	C₂²	1	8	yes	—
14	C₂×C₇×#124	—	2	32	no	—
15	C₂×C₇×#124	C₂²	1	32	yes	complemented
16	C₂²×C₇×C₄⋊C₄	C₂²	1	4	yes	—
17	C₂²×C₇×C₂²⋊C₄	C₂²	1	16	yes	—
18	C₂²×C₄²×C₇	C₂²	1	4	yes	—
19	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₇	C₂²	1	2	yes	—
21	C₇×#390	C₂²	1	64	yes	complemented
22	C₄³×C₇	C₂²	1	8	yes	complemented
23	C₂×#2858	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂⁵×C₇	C₂³	1	4	yes	—
25	C₂²×C₇×Q₈	—	2	16	no	—
26	C₂²×C₇×Q₈	—	2	8	no	—
27	C₂²×C₇×Q₈	C₂³	1	8	yes	—
28	C₂²×C₇×D₄	—	2	16	no	—
29	C₂²×C₇×D₄	—	2	16	no	—
30	C₂²×C₇×D₄	—	2	8	no	—
31	C₂²×C₇×D₄	—	2	32	no	—
32	C₂²×C₇×D₄	C₂³	1	8	yes	—
33	C₂²×C₇×D₄	C₂³	1	16	yes	—
34	C₂³×C₄×C₇	C₂³	1	4	yes	—
35	C₂³×C₄×C₇	C₂³	1	4	yes	—
36	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
37	C₂³×C₄×C₇	C₂³	1	2	yes	—
38	C₇×C₄⋊Q₈	—	2	32	no	—
39	C₇×#127	—	2	32	no	—
40	C₇×#124	—	2	128	no	—
41	C₂×C₇×C₄⋊C₄	—	2	16	no	—
42	C₂×C₇×C₄⋊C₄	C₂³	1	16	yes	—
43	C₂×C₇×C₂²⋊C₄	—	2	64	no	—
44	C₂×C₇×C₂²⋊C₄	C₂³	1	64	yes	—
45	C₂×C₄²×C₇	—	2	16	no	—
46	C₂×C₄²×C₇	D₄	1	8	yes	—
47	C₂×C₄²×C₇	D₄	1	16	yes	complemented
48	C₂×C₄²×C₇	C₂³	1	16	yes	—
49	C₂×C₄²×C₇	C₂³	1	4	yes	—
50	C₂×C₄²×C₇	C₂³	1	8	yes	—
51	C₂²×C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
52	C₂²×#127	C₂×C₇	1	1	yes	characteristic
53	C₂²×#124	C₂×C₇	1	4	yes	complemented
54	#2858	C₂×C₇	1	16	yes	direct product factor
55	C₂×#390	C₂×C₇	1	8	yes	complemented
56	C₂×C₄³	C₂×C₇	1	1	yes	characteristic, complemented
57	C₂⁴×C₇	—	2	16	no	—
58	C₂⁴×C₇	—	2	8	no	—
59	C₂⁴×C₇	—	2	32	no	—
60	C₂⁴×C₇	C₂⁴	1	8	yes	—
61	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
62	C₂×C₇×Q₈	—	2	32	no	—
63	C₂×C₇×Q₈	—	2	16	no	—
64	C₂×C₇×Q₈	—	4	32	no	—
65	C₂×C₇×D₄	—	2	32	no	—
66	C₂×C₇×D₄	—	2	32	no	—
67	C₂×C₇×D₄	—	2	16	no	—
68	C₂×C₇×D₄	—	2	64	no	—
69	C₂×C₇×D₄	—	4	64	no	—
70	C₂×C₇×D₄	—	4	32	no	—
71	C₂²×C₄×C₇	—	2	16	no	—
72	C₂²×C₄×C₇	—	2	16	no	—
73	C₂²×C₄×C₇	—	2	8	no	—
74	C₂²×C₄×C₇	C₂×D₄	1	16	yes	—
75	C₂²×C₄×C₇	C₂×D₄	1	8	yes	—
76	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₇	C₂×D₄	1	8	yes	—
78	C₂²×C₄×C₇	#41	1	16	yes	—
79	C₂²×C₄×C₇	C₂⁴	1	8	yes	—
80	C₂²×C₄×C₇	C₂⁴	1	8	yes	—
81	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
82	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
83	C₇×C₄⋊C₄	—	2	32	no	—
84	C₇×C₂²⋊C₄	—	2	128	no	—
85	C₄²×C₇	—	2	32	no	—
86	C₄²×C₇	C₂×D₄	1	16	yes	—
87	C₄²×C₇	C₂×D₄	1	32	yes	complemented
88	C₂³×Q₈	C₂²×C₇	1	2	yes	—
89	C₂³×D₄	C₂²×C₇	1	2	yes	—
90	C₂³×D₄	C₂²×C₇	1	4	yes	—
91	C₂×C₄⋊Q₈	—	2	8	no	—
92	C₂×C₄⋊Q₈	C₂²×C₇	1	8	yes	complemented
93	C₂×#127	—	2	8	no	—
94	C₂×#127	C₂²×C₇	1	8	yes	—
95	C₂×#124	—	2	32	no	—
96	C₂×#124	C₂²×C₇	1	32	yes	complemented
97	C₂²×C₄⋊C₄	C₂²×C₇	1	4	yes	—
98	C₂²×C₂²⋊C₄	C₂²×C₇	1	16	yes	—
99	C₂²×C₄²	C₂²×C₇	1	4	yes	—
100	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
101	C₂²×C₄²	C₂²×C₇	1	2	yes	—
102	#390	C₂²×C₇	1	64	yes	complemented
103	C₄³	C₂²×C₇	1	8	yes	complemented
104	C₂³×C₇	—	2	16	no	—
105	C₂³×C₇	—	2	8	no	—
106	C₂³×C₇	—	2	32	no	—
107	C₂³×C₇	—	4	32	no	—
108	C₂³×C₇	—	4	16	no	—
109	C₂³×C₇	—	4	64	no	—
110	C₂³×C₇	C₂²×D₄	1	2	yes	—
111	C₂³×C₇	C₂²×D₄	1	4	yes	—
112	C₂³×C₇	C₂×#41	1	8	yes	—
113	C₂³×C₇	C₂⁵	1	1	yes	characteristic
114	C₇×Q₈	—	4	32	no	—
115	C₇×D₄	—	4	64	no	—
116	C₇×D₄	—	4	32	no	—
117	C₂×C₄×C₇	—	2	32	no	—
118	C₂×C₄×C₇	—	2	16	no	—
119	C₂×C₄×C₇	—	2	16	no	—
120	C₂×C₄×C₇	—	2	8	no	—
121	C₂×C₄×C₇	—	4	16	no	—
122	C₂×C₄×C₇	#124	1	32	yes	complemented
123	C₂×C₄×C₇	#127	1	16	yes	—
124	C₂×C₄×C₇	C₂²×D₄	1	16	yes	—
125	C₂×C₄×C₇	C₂²×D₄	1	8	yes	—
126	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
127	C₂×C₄×C₇	C₂²×D₄	1	8	yes	—
128	C₂×C₄×C₇	C₂×#41	1	16	yes	—
129	C₂⁵	C₂³×C₇	1	4	yes	—
130	C₂²×Q₈	—	2	16	no	—
131	C₂²×Q₈	—	2	8	no	—
132	C₂²×Q₈	C₂³×C₇	1	8	yes	—
133	C₂²×D₄	—	2	16	no	—
134	C₂²×D₄	—	2	16	no	—
135	C₂²×D₄	—	2	8	no	—
136	C₂²×D₄	—	2	32	no	—
137	C₂²×D₄	C₂³×C₇	1	8	yes	—
138	C₂²×D₄	C₂³×C₇	1	16	yes	—
139	C₂³×C₄	C₂³×C₇	1	4	yes	—
140	C₂³×C₄	C₂³×C₇	1	4	yes	—
141	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
142	C₂³×C₄	C₂³×C₇	1	2	yes	—
143	C₄⋊Q₈	—	2	32	no	—
144	#127	—	2	32	no	—
145	#124	—	2	128	no	—
146	C₂×C₄⋊C₄	—	2	16	no	—
147	C₂×C₄⋊C₄	C₂³×C₇	1	16	yes	—
148	C₂×C₂²⋊C₄	—	2	64	no	—
149	C₂×C₂²⋊C₄	C₂³×C₇	1	64	yes	—
150	C₂×C₄²	—	2	16	no	—
151	C₂×C₄²	C₇×D₄	1	8	yes	—
152	C₂×C₄²	C₇×D₄	1	16	yes	complemented
153	C₂×C₄²	C₂³×C₇	1	16	yes	—
154	C₂×C₄²	C₂³×C₇	1	4	yes	—
155	C₂×C₄²	C₂³×C₇	1	8	yes	—
156	C₂²×C₇	—	4	16	no	—
157	C₂²×C₇	—	4	8	no	—
158	C₂²×C₇	—	4	32	no	—
159	C₂²×C₇	—	8	32	no	—
160	C₂²×C₇	C₂×#124	1	8	yes	—
161	C₂²×C₇	C₂×#127	1	4	yes	—
162	C₂²×C₇	#532	1	16	yes	—
163	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
164	C₂²×C₇	C₂³×D₄	1	2	yes	—
165	C₂²×C₇	C₂²×#41	1	4	yes	—
166	C₄×C₇	—	2	16	no	—
167	C₄×C₇	—	4	8	no	—
168	C₄×C₇	C₂×#124	1	16	yes	complemented
169	C₄×C₇	C₂×#127	1	8	yes	—
170	C₂⁴	—	2	16	no	—
171	C₂⁴	—	2	8	no	—
172	C₂⁴	—	2	32	no	—
173	C₂⁴	C₂⁴×C₇	1	8	yes	—
174	C₂⁴	C₂⁴×C₇	1	1	yes	characteristic
175	C₂×Q₈	—	2	32	no	—
176	C₂×Q₈	—	2	16	no	—
177	C₂×Q₈	—	4	32	no	—
178	C₂×D₄	—	2	32	no	—
179	C₂×D₄	—	2	32	no	—
180	C₂×D₄	—	2	16	no	—
181	C₂×D₄	—	2	64	no	—
182	C₂×D₄	—	4	64	no	—
183	C₂×D₄	—	4	32	no	—
184	C₂²×C₄	—	2	16	no	—
185	C₂²×C₄	—	2	16	no	—
186	C₂²×C₄	—	2	8	no	—
187	C₂²×C₄	C₂×C₇×D₄	1	16	yes	—
188	C₂²×C₄	C₂×C₇×D₄	1	8	yes	—
189	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
190	C₂²×C₄	C₂×C₇×D₄	1	8	yes	—
191	C₂²×C₄	C₇×#41	1	16	yes	—
192	C₂²×C₄	C₂⁴×C₇	1	8	yes	—
193	C₂²×C₄	C₂⁴×C₇	1	8	yes	—
194	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
195	C₂²×C₄	C₂⁴×C₇	1	4	yes	—
196	C₄⋊C₄	—	2	32	no	—
197	C₂²⋊C₄	—	2	128	no	—
198	C₄²	—	2	32	no	—
199	C₄²	C₂×C₇×D₄	1	16	yes	—
200	C₄²	C₂×C₇×D₄	1	32	yes	complemented
201	C₂×C₇	—	8	8	no	—
202	C₂×C₇	#2858	1	8	yes	direct product factor
203	C₂×C₇	C₂²×#124	1	2	yes	—
204	C₂×C₇	C₂²×#127	1	1	yes	characteristic
205	C₂×C₇	C₂×#532	1	4	yes	—
206	C₂³	—	2	16	no	—
207	C₂³	—	2	8	no	—
208	C₂³	—	2	32	no	—
209	C₂³	—	4	32	no	—
210	C₂³	—	4	16	no	—
211	C₂³	—	4	64	no	—
212	C₂³	C₂²×C₇×D₄	1	2	yes	—
213	C₂³	C₂²×C₇×D₄	1	4	yes	—
214	C₂³	C₂×C₇×#41	1	8	yes	—
215	C₂³	C₂⁵×C₇	1	1	yes	characteristic
216	Q₈	—	4	32	no	—
217	D₄	—	4	64	no	—
218	D₄	—	4	32	no	—
219	C₂×C₄	—	2	16	no	—
220	C₂×C₄	—	2	32	no	—
221	C₂×C₄	—	2	16	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	4	16	no	—
224	C₂×C₄	C₇×#124	1	32	yes	complemented
225	C₂×C₄	C₇×#127	1	16	yes	—
226	C₂×C₄	C₂²×C₇×D₄	1	16	yes	—
227	C₂×C₄	C₂²×C₇×D₄	1	8	yes	—
228	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
229	C₂×C₄	C₂²×C₇×D₄	1	8	yes	—
230	C₂×C₄	C₂×C₇×#41	1	16	yes	—
231	C₇	C₂×#2858	1	1	yes	characteristic, direct product factor, Sylow
232	C₂²	—	4	16	no	—
233	C₂²	—	4	8	no	—
234	C₂²	—	4	32	no	—
235	C₂²	—	8	32	no	—
236	C₂²	C₂×C₇×#124	1	8	yes	—
237	C₂²	C₂×C₇×#127	1	4	yes	—
238	C₂²	C₇×#532	1	16	yes	—
239	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
240	C₂²	C₂³×C₇×D₄	1	2	yes	—
241	C₂²	C₂²×C₇×#41	1	4	yes	—
242	C₄	—	2	16	no	—
243	C₄	—	4	8	no	—
244	C₄	C₂×C₇×#124	1	16	yes	complemented
245	C₄	C₂×C₇×#127	1	8	yes	—
246	C₂	—	8	8	no	—
247	C₂	C₇×#2858	1	8	yes	direct product factor
248	C₂	C₂²×C₇×#124	1	2	yes	—
249	C₂	C₂²×C₇×#127	1	1	yes	characteristic
250	C₂	C₂×C₇×#532	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₇
prime radical	C₂²×C₇×#127	C₂
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×C₄⋊Q₈	C₂	semidirect
3. C₂²×C₇×#124	C₂	semidirect
4. C₇×#2858	C₂	direct (N × H)
5. C₂×C₇×#390	C₂	semidirect
6. C₂×C₄³×C₇	C₂	semidirect
11. C₂×C₇×C₄⋊Q₈	C₂²	semidirect
15. C₂×C₇×#124	C₂²	semidirect
21. C₇×#390	C₂²	semidirect
22. C₄³×C₇	C₂²	semidirect
23. C₂×#2858	C₇	direct (N × H)
47. C₂×C₄²×C₇	D₄	semidirect
51. C₂²×C₄⋊Q₈	C₂×C₇	semidirect
53. C₂²×#124	C₂×C₇	semidirect
54. #2858	C₂×C₇	direct (N × H)
55. C₂×#390	C₂×C₇	semidirect
56. C₂×C₄³	C₂×C₇	semidirect
87. C₄²×C₇	C₂×D₄	semidirect
92. C₂×C₄⋊Q₈	C₂²×C₇	semidirect
96. C₂×#124	C₂²×C₇	semidirect
102. #390	C₂²×C₇	semidirect
103. C₄³	C₂²×C₇	semidirect
122. C₂×C₄×C₇	#124	semidirect
152. C₂×C₄²	C₇×D₄	semidirect
168. C₄×C₇	C₂×#124	semidirect
200. C₄²	C₂×C₇×D₄	semidirect
202. C₂×C₇	#2858	direct (N × H)
224. C₂×C₄	C₇×#124	semidirect
231. C₇	C₂×#2858	direct (N × H)
244. C₄	C₂×C₇×#124	semidirect
247. C₂	C₇×#2858	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⋊Q₈	2
C₂²×C₇×#127	2
C₂²×C₇×#124	2
C₇×#2858	2
C₂×C₇×#390	2
C₂×C₄³×C₇	2
C₂×#2858	7

Subgroups — C2×C7×#2858 #13213843

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2858 #13213843