Subgroups — C₂×C₇×#2240 #13210910

Order 1792.

726 conjugacy classes of subgroups.

The subgroup lattice has 236 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×#493	C₂	1	1	yes	maximal, characteristic
3	C₇×#2240	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₇×#357	C₂	1	4	yes	maximal, complemented
5	C₂×C₄×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₇×D₈	—	2	1	no	—
7	C₂²×C₇×D₈	C₂²	1	2	yes	—
8	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
9	C₂×C₇×#127	—	2	1	no	—
10	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
11	C₇×#500	—	2	2	no	—
12	C₇×#500	C₂²	1	4	yes	complemented
13	C₇×#493	—	2	2	no	—
14	C₇×#493	C₂²	1	4	yes	—
15	C₂×C₇×#107	—	2	1	no	—
16	C₂×C₇×#107	C₂²	1	2	yes	—
17	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
18	C₇×#357	C₂²	1	16	yes	complemented
19	C₄×C₇×C₁₆	C₂²	1	4	yes	complemented
20	C₂×#2240	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₇×Q₈	—	4	1	no	—
22	C₂²×C₇×D₄	—	2	2	no	—
23	C₂²×C₇×D₄	—	4	1	no	—
24	C₂²×C₇×D₄	—	4	1	no	—
25	C₂×C₇×D₈	—	2	2	no	—
26	C₂×C₇×D₈	—	2	4	no	—
27	C₂×C₇×D₈	—	2	8	no	—
28	C₂×C₇×D₈	—	4	4	no	—
29	C₂×C₇×D₈	C₂³	1	4	yes	—
30	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
32	C₇×C₄⋊Q₈	—	2	4	no	—
33	C₇×C₄⋊Q₈	—	4	2	no	—
34	C₇×#127	—	2	4	no	—
35	C₇×#127	—	4	2	no	—
36	C₂×C₇×C₄⋊C₄	—	2	2	no	—
37	C₂×C₇×C₄⋊C₄	—	4	1	no	—
38	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
39	C₂×C₇×C₁₆	—	2	4	no	—
40	C₂×C₇×C₁₆	C₂³	1	4	yes	—
41	C₇×#107	—	2	2	no	—
42	C₇×#107	—	2	4	no	—
43	C₇×#107	C₂³	1	4	yes	—
44	C₄×C₇×C₈	D₄	1	2	yes	—
45	C₄×C₇×C₈	C₂³	1	2	yes	—
46	C₂×#500	C₂×C₇	1	1	yes	characteristic, complemented
47	C₂×#493	C₂×C₇	1	1	yes	characteristic
48	#2240	C₂×C₇	1	8	yes	direct product factor
49	C₂×#357	C₂×C₇	1	4	yes	complemented
50	C₂×C₄×C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
51	C₂⁴×C₇	—	4	2	no	—
52	C₂×C₇×Q₈	—	4	4	no	—
53	C₂×C₇×Q₈	—	8	4	no	—
54	C₂×C₇×D₄	—	2	4	no	—
55	C₂×C₇×D₄	—	4	4	no	—
56	C₂×C₇×D₄	—	4	4	no	—
57	C₂×C₇×D₄	—	4	4	no	—
58	C₂×C₇×D₄	—	4	2	no	—
59	C₂×C₇×D₄	—	4	4	no	—
60	C₂×C₇×D₄	—	8	4	no	—
61	C₂×C₇×D₄	—	8	4	no	—
62	C₂²×C₄×C₇	—	4	2	no	—
63	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
66	C₇×D₈	—	2	8	no	—
67	C₇×D₈	—	4	4	no	—
68	C₇×D₈	—	4	8	no	—
69	C₂×C₇×C₈	SD₁₆	1	4	yes	—
70	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
71	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
72	C₂×C₇×C₈	#41	1	4	yes	—
73	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
74	C₇×C₄⋊C₄	—	2	4	no	—
75	C₇×C₄⋊C₄	—	4	2	no	—
76	C₇×C₄⋊C₄	—	4	4	no	—
77	C₄²×C₇	D₈	1	2	yes	—
78	C₄²×C₇	C₂×D₄	1	2	yes	—
79	C₇×C₁₆	—	2	4	no	—
80	C₂²×D₈	—	2	1	no	—
81	C₂²×D₈	C₂²×C₇	1	2	yes	—
82	C₂×C₄⋊Q₈	—	2	1	no	—
83	C₂×#127	—	2	1	no	—
84	C₂²×C₁₆	C₂²×C₇	1	2	yes	—
85	#500	—	2	2	no	—
86	#500	C₂²×C₇	1	4	yes	complemented
87	#493	—	2	2	no	—
88	#493	C₂²×C₇	1	4	yes	—
89	C₂×#107	—	2	1	no	—
90	C₂×#107	C₂²×C₇	1	2	yes	—
91	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
92	#357	C₂²×C₇	1	16	yes	complemented
93	C₄×C₁₆	C₂²×C₇	1	4	yes	complemented
94	C₂³×C₇	—	4	4	no	—
95	C₂³×C₇	—	8	4	no	—
96	C₂³×C₇	—	8	4	no	—
97	C₂³×C₇	—	8	4	no	—
98	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
99	C₇×Q₈	—	8	8	no	—
100	C₇×D₄	—	4	4	no	—
101	C₇×D₄	—	4	4	no	—
102	C₇×D₄	—	8	4	no	—
103	C₇×D₄	—	8	8	no	—
104	C₇×D₄	—	8	8	no	—
105	C₂×C₄×C₇	—	4	4	no	—
106	C₂×C₄×C₇	—	8	4	no	—
107	C₂×C₄×C₇	D₁₆	1	4	yes	—
108	C₂×C₄×C₇	SD₃₂	1	4	yes	—
109	C₂×C₄×C₇	#124	1	2	yes	—
110	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
112	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
114	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
115	C₇×C₈	#124	1	2	yes	—
116	C₇×C₈	C₂×SD₁₆	1	4	yes	—
117	C₇×C₈	C₂×#41	1	2	yes	—
118	C₂²×Q₈	—	4	1	no	—
119	C₂²×D₄	—	2	2	no	—
120	C₂²×D₄	—	4	1	no	—
121	C₂²×D₄	—	4	1	no	—
122	C₂×D₈	—	2	4	no	—
123	C₂×D₈	—	2	2	no	—
124	C₂×D₈	—	2	8	no	—
125	C₂×D₈	—	4	4	no	—
126	C₂×D₈	C₂³×C₇	1	4	yes	—
127	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
128	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
129	C₄⋊Q₈	—	2	4	no	—
130	C₄⋊Q₈	—	4	2	no	—
131	#127	—	2	4	no	—
132	#127	—	4	2	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	4	1	no	—
135	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
136	C₂×C₁₆	—	2	4	no	—
137	C₂×C₁₆	C₂³×C₇	1	4	yes	—
138	#107	—	2	4	no	—
139	#107	—	2	2	no	—
140	#107	C₂³×C₇	1	4	yes	—
141	C₄×C₈	C₇×D₄	1	2	yes	—
142	C₄×C₈	C₂³×C₇	1	2	yes	—
143	C₂²×C₇	—	8	4	no	—
144	C₂²×C₇	—	8	4	no	—
145	C₂²×C₇	—	8	4	no	—
146	C₂²×C₇	—	16	8	no	—
147	C₂²×C₇	#486	1	2	yes	—
148	C₂²×C₇	C₂×D₁₆	1	2	yes	—
149	C₂²×C₇	C₂×SD₃₂	1	2	yes	—
150	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
151	C₄×C₇	—	8	4	no	—
152	C₄×C₇	#486	1	2	yes	—
153	C₄×C₇	C₂×D₁₆	1	4	yes	—
154	C₄×C₇	C₂×SD₃₂	1	4	yes	—
155	C₄×C₇	C₂×#124	1	1	yes	characteristic
156	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
157	C₂⁴	—	4	2	no	—
158	C₂×Q₈	—	4	4	no	—
159	C₂×Q₈	—	8	4	no	—
160	C₂×D₄	—	2	4	no	—
161	C₂×D₄	—	4	4	no	—
162	C₂×D₄	—	4	4	no	—
163	C₂×D₄	—	4	4	no	—
164	C₂×D₄	—	4	2	no	—
165	C₂×D₄	—	4	4	no	—
166	C₂×D₄	—	8	4	no	—
167	C₂×D₄	—	8	4	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
170	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
172	D₈	—	2	8	no	—
173	D₈	—	4	4	no	—
174	D₈	—	4	8	no	—
175	C₂×C₈	C₇×SD₁₆	1	4	yes	—
176	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
177	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
178	C₂×C₈	C₇×#41	1	4	yes	—
179	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₄²	C₇×D₈	1	2	yes	—
184	C₄²	C₂×C₇×D₄	1	2	yes	—
185	C₁₆	—	2	4	no	—
186	C₂×C₇	—	16	4	no	—
187	C₂×C₇	#2240	1	4	yes	direct product factor
188	C₂×C₇	C₂×#486	1	1	yes	characteristic
189	C₂×C₇	C₂²×D₁₆	1	1	yes	characteristic
190	C₂×C₇	C₂²×SD₃₂	1	1	yes	characteristic
191	C₂³	—	4	4	no	—
192	C₂³	—	8	4	no	—
193	C₂³	—	8	4	no	—
194	C₂³	—	8	4	no	—
195	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
196	Q₈	—	8	8	no	—
197	D₄	—	4	4	no	—
198	D₄	—	4	4	no	—
199	D₄	—	8	4	no	—
200	D₄	—	8	8	no	—
201	D₄	—	8	8	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	8	4	no	—
204	C₂×C₄	C₇×D₁₆	1	4	yes	—
205	C₂×C₄	C₇×SD₃₂	1	4	yes	—
206	C₂×C₄	C₇×#124	1	2	yes	—
207	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
208	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
209	C₂×C₄	C₂×C₇×D₈	1	2	yes	—
210	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
211	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
212	C₈	C₇×#124	1	2	yes	—
213	C₈	C₂×C₇×SD₁₆	1	4	yes	—
214	C₈	C₂×C₇×#41	1	2	yes	—
215	C₇	C₂×#2240	1	1	yes	characteristic, direct product factor, Sylow
216	C₂²	—	8	4	no	—
217	C₂²	—	8	4	no	—
218	C₂²	—	8	4	no	—
219	C₂²	—	16	8	no	—
220	C₂²	C₇×#486	1	2	yes	—
221	C₂²	C₂×C₇×D₁₆	1	2	yes	—
222	C₂²	C₂×C₇×SD₃₂	1	2	yes	—
223	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
224	C₄	—	8	4	no	—
225	C₄	C₇×#486	1	2	yes	—
226	C₄	C₂×C₇×D₁₆	1	4	yes	—
227	C₄	C₂×C₇×SD₃₂	1	4	yes	—
228	C₄	C₂×C₇×#124	1	1	yes	characteristic
229	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
230	C₂	—	16	4	no	—
231	C₂	C₇×#2240	1	4	yes	direct product factor
232	C₂	C₂×C₇×#486	1	1	yes	characteristic
233	C₂	C₂²×C₇×D₁₆	1	1	yes	characteristic
234	C₂	C₂²×C₇×SD₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇
prime radical	C₂×C₇×#493	C₂
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×#2240	C₂	direct (N × H)
4. C₂×C₇×#357	C₂	semidirect
5. C₂×C₄×C₇×C₁₆	C₂	semidirect
12. C₇×#500	C₂²	semidirect
18. C₇×#357	C₂²	semidirect
19. C₄×C₇×C₁₆	C₂²	semidirect
20. C₂×#2240	C₇	direct (N × H)
46. C₂×#500	C₂×C₇	semidirect
48. #2240	C₂×C₇	direct (N × H)
49. C₂×#357	C₂×C₇	semidirect
50. C₂×C₄×C₁₆	C₂×C₇	semidirect
86. #500	C₂²×C₇	semidirect
92. #357	C₂²×C₇	semidirect
93. C₄×C₁₆	C₂²×C₇	semidirect
187. C₂×C₇	#2240	direct (N × H)
215. C₇	C₂×#2240	direct (N × H)
231. C₂	C₇×#2240	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×#493	2
C₇×#2240	2
C₂×C₇×#357	2
C₂×C₄×C₇×C₁₆	2
C₂×#2240	7

Subgroups — C2×C7×#2240 #13210910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2240 #13210910