Subgroups — C₂×C₇×#2242 #13210912

Order 1792.

610 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#495	C₂	1	1	yes	maximal, characteristic
3	C₇×#2242	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×C₄×C₇×C₁₆	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×#125	—	2	1	no	—
11	C₂×C₇×#124	—	2	1	no	—
12	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
13	C₇×#499	—	2	2	no	—
14	C₇×#499	C₂²	1	4	yes	complemented
15	C₇×#495	—	2	2	no	—
16	C₇×#495	C₂²	1	4	yes	—
17	C₂×C₇×#107	C₂²	1	2	yes	—
18	C₂×C₇×#106	—	2	1	no	—
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	C₄×C₇×C₁₆	C₂²	1	4	yes	complemented
23	C₂×#2242	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₇×Q₈	—	2	1	no	—
25	C₂²×C₇×D₄	—	2	1	no	—
26	C₂×C₇×Q₁₆	—	2	1	no	—
27	C₂×C₇×Q₁₆	—	2	4	no	—
28	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
29	C₂×C₇×SD₁₆	—	2	4	no	—
30	C₂×C₇×SD₁₆	—	4	4	no	—
31	C₂×C₇×D₈	—	2	1	no	—
32	C₂×C₇×D₈	—	2	4	no	—
33	C₂×C₇×D₈	C₂³	1	2	yes	—
34	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
36	C₇×#125	—	2	4	no	—
37	C₇×#125	—	4	2	no	—
38	C₇×#124	—	2	4	no	—
39	C₇×#124	—	4	2	no	—
40	C₂×C₇×C₄⋊C₄	—	2	2	no	—
41	C₂×C₇×C₄⋊C₄	—	4	2	no	—
42	C₂×C₇×C₂²⋊C₄	—	4	2	no	—
43	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
44	C₂×C₇×C₁₆	—	2	4	no	—
45	C₂×C₇×C₁₆	C₂³	1	4	yes	—
46	C₇×#107	—	2	2	no	—
47	C₇×#107	C₂³	1	4	yes	—
48	C₇×#106	—	2	4	no	—
49	C₄×C₇×C₈	D₄	1	2	yes	—
50	C₄×C₇×C₈	C₂³	1	2	yes	—
51	C₂×#499	C₂×C₇	1	1	yes	characteristic, complemented
52	C₂×#495	C₂×C₇	1	1	yes	characteristic
53	#2242	C₂×C₇	1	8	yes	direct product factor
54	C₂×#358	C₂×C₇	1	2	yes	complemented
55	C₂×#357	C₂×C₇	1	2	yes	—
56	C₂×C₄×C₁₆	C₂×C₇	1	1	yes	characteristic, complemented
57	C₂⁴×C₇	—	4	1	no	—
58	C₂×C₇×Q₈	—	2	2	no	—
59	C₂×C₇×Q₈	—	4	1	no	—
60	C₂×C₇×Q₈	—	4	2	no	—
61	C₂×C₇×Q₈	—	4	2	no	—
62	C₂×C₇×D₄	—	2	2	no	—
63	C₂×C₇×D₄	—	4	2	no	—
64	C₂×C₇×D₄	—	4	1	no	—
65	C₂×C₇×D₄	—	4	2	no	—
66	C₂²×C₄×C₇	—	4	1	no	—
67	C₂²×C₄×C₇	—	4	2	no	—
68	C₂²×C₄×C₇	D₈	1	2	yes	—
69	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
70	C₇×Q₁₆	—	2	4	no	—
71	C₇×Q₁₆	—	4	2	no	—
72	C₇×SD₁₆	—	4	8	no	—
73	C₇×D₈	—	2	4	no	—
74	C₇×D₈	—	4	2	no	—
75	C₂×C₇×C₈	SD₁₆	1	4	yes	—
76	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
78	C₂×C₇×C₈	#41	1	4	yes	—
79	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
80	C₇×C₄⋊C₄	—	2	4	no	—
81	C₇×C₄⋊C₄	—	4	2	no	—
82	C₇×C₄⋊C₄	—	4	8	no	—
83	C₇×C₂²⋊C₄	—	4	8	no	—
84	C₄²×C₇	D₈	1	2	yes	—
85	C₄²×C₇	C₂×D₄	1	2	yes	—
86	C₇×C₁₆	—	2	4	no	—
87	C₂²×Q₁₆	C₂²×C₇	1	1	yes	characteristic
88	C₂²×SD₁₆	—	2	1	no	—
89	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
90	C₂×#125	—	2	1	no	—
91	C₂×#124	—	2	1	no	—
92	C₂²×C₁₆	C₂²×C₇	1	2	yes	—
93	#499	—	2	2	no	—
94	#499	C₂²×C₇	1	4	yes	complemented
95	#495	—	2	2	no	—
96	#495	C₂²×C₇	1	4	yes	—
97	C₂×#107	C₂²×C₇	1	2	yes	—
98	C₂×#106	—	2	1	no	—
99	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
100	#358	C₂²×C₇	1	8	yes	complemented
101	#357	C₂²×C₇	1	8	yes	—
102	C₄×C₁₆	C₂²×C₇	1	4	yes	complemented
103	C₂³×C₇	—	4	2	no	—
104	C₂³×C₇	—	8	2	no	—
105	C₂³×C₇	—	8	4	no	—
106	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
107	C₇×Q₈	—	4	2	no	—
108	C₇×Q₈	—	4	2	no	—
109	C₇×Q₈	—	8	2	no	—
110	C₇×D₄	—	4	2	no	—
111	C₇×D₄	—	4	2	no	—
112	C₇×D₄	—	8	2	no	—
113	C₂×C₄×C₇	—	2	4	no	—
114	C₂×C₄×C₇	—	4	2	no	—
115	C₂×C₄×C₇	—	4	4	no	—
116	C₂×C₄×C₇	—	8	2	no	—
117	C₂×C₄×C₇	—	8	4	no	—
118	C₂×C₄×C₇	#124	1	2	yes	—
119	C₂×C₄×C₇	C₂×D₈	1	4	yes	—
120	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
122	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
123	C₇×C₈	#124	1	2	yes	—
124	C₇×C₈	C₂×SD₁₆	1	4	yes	—
125	C₇×C₈	C₂×#41	1	2	yes	—
126	C₂²×Q₈	—	2	1	no	—
127	C₂²×D₄	—	2	1	no	—
128	C₂×Q₁₆	—	2	1	no	—
129	C₂×Q₁₆	—	2	4	no	—
130	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
131	C₂×SD₁₆	—	2	4	no	—
132	C₂×SD₁₆	—	4	4	no	—
133	C₂×D₈	—	2	1	no	—
134	C₂×D₈	—	2	4	no	—
135	C₂×D₈	C₂³×C₇	1	2	yes	—
136	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
137	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
138	#125	—	2	4	no	—
139	#125	—	4	2	no	—
140	#124	—	2	4	no	—
141	#124	—	4	2	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	—	4	2	no	—
144	C₂×C₂²⋊C₄	—	4	2	no	—
145	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
146	C₂×C₁₆	—	2	4	no	—
147	C₂×C₁₆	C₂³×C₇	1	4	yes	—
148	#107	—	2	2	no	—
149	#107	C₂³×C₇	1	4	yes	—
150	#106	—	2	4	no	—
151	C₄×C₈	C₇×D₄	1	2	yes	—
152	C₄×C₈	C₂³×C₇	1	2	yes	—
153	C₂²×C₇	—	8	2	no	—
154	C₂²×C₇	—	8	4	no	—
155	C₂²×C₇	—	16	4	no	—
156	C₂²×C₇	#486	1	2	yes	—
157	C₂²×C₇	#508	1	4	yes	—
158	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
159	C₄×C₇	—	2	4	no	—
160	C₄×C₇	—	8	2	no	—
161	C₄×C₇	—	8	4	no	—
162	C₄×C₇	#486	1	2	yes	—
163	C₄×C₇	C₂×#124	1	1	yes	characteristic
164	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
165	C₂⁴	—	4	1	no	—
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	4	1	no	—
168	C₂×Q₈	—	4	2	no	—
169	C₂×Q₈	—	4	2	no	—
170	C₂×D₄	—	2	2	no	—
171	C₂×D₄	—	4	2	no	—
172	C₂×D₄	—	4	1	no	—
173	C₂×D₄	—	4	2	no	—
174	C₂²×C₄	—	4	1	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	C₇×D₈	1	2	yes	—
177	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
178	Q₁₆	—	2	4	no	—
179	Q₁₆	—	4	2	no	—
180	SD₁₆	—	4	8	no	—
181	D₈	—	2	4	no	—
182	D₈	—	4	2	no	—
183	C₂×C₈	C₇×SD₁₆	1	4	yes	—
184	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
185	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
186	C₂×C₈	C₇×#41	1	4	yes	—
187	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	8	no	—
191	C₂²⋊C₄	—	4	8	no	—
192	C₄²	C₇×D₈	1	2	yes	—
193	C₄²	C₂×C₇×D₄	1	2	yes	—
194	C₁₆	—	2	4	no	—
195	C₂×C₇	—	16	2	no	—
196	C₂×C₇	#2242	1	4	yes	direct product factor
197	C₂×C₇	C₂×#486	1	1	yes	characteristic
198	C₂×C₇	C₂×#508	1	2	yes	—
199	C₂³	—	4	2	no	—
200	C₂³	—	8	2	no	—
201	C₂³	—	8	4	no	—
202	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
203	Q₈	—	4	2	no	—
204	Q₈	—	4	2	no	—
205	Q₈	—	8	2	no	—
206	D₄	—	4	2	no	—
207	D₄	—	4	2	no	—
208	D₄	—	8	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	8	2	no	—
213	C₂×C₄	—	8	4	no	—
214	C₂×C₄	C₇×#124	1	2	yes	—
215	C₂×C₄	C₂×C₇×D₈	1	4	yes	—
216	C₂×C₄	C₂×C₇×D₈	1	1	yes	characteristic
217	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
218	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
219	C₈	C₇×#124	1	2	yes	—
220	C₈	C₂×C₇×SD₁₆	1	4	yes	—
221	C₈	C₂×C₇×#41	1	2	yes	—
222	C₇	C₂×#2242	1	1	yes	characteristic, direct product factor, Sylow
223	C₂²	—	8	4	no	—
224	C₂²	—	8	2	no	—
225	C₂²	—	16	4	no	—
226	C₂²	C₇×#486	1	2	yes	—
227	C₂²	C₇×#508	1	4	yes	—
228	C₂²	C₂²×C₇×D₈	1	1	yes	characteristic
229	C₄	—	2	4	no	—
230	C₄	—	8	4	no	—
231	C₄	—	8	2	no	—
232	C₄	C₇×#486	1	2	yes	—
233	C₄	C₂×C₇×#124	1	1	yes	characteristic
234	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
235	C₂	—	16	2	no	—
236	C₂	C₇×#2242	1	4	yes	direct product factor
237	C₂	C₂×C₇×#486	1	1	yes	characteristic
238	C₂	C₂×C₇×#508	1	2	yes	—

	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×#499	C₂	semidirect
3. C₇×#2242	C₂	direct (N × H)
4. C₂×C₇×#358	C₂	semidirect
6. C₂×C₄×C₇×C₁₆	C₂	semidirect
14. C₇×#499	C₂²	semidirect
20. C₇×#358	C₂²	semidirect
22. C₄×C₇×C₁₆	C₂²	semidirect
23. C₂×#2242	C₇	direct (N × H)
51. C₂×#499	C₂×C₇	semidirect
53. #2242	C₂×C₇	direct (N × H)
54. C₂×#358	C₂×C₇	semidirect
56. C₂×C₄×C₁₆	C₂×C₇	semidirect
94. #499	C₂²×C₇	semidirect
100. #358	C₂²×C₇	semidirect
102. C₄×C₁₆	C₂²×C₇	semidirect
196. C₂×C₇	#2242	direct (N × H)
222. C₇	C₂×#2242	direct (N × H)
236. C₂	C₇×#2242	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×#499	2
C₂×C₇×#495	2
C₇×#2242	2
C₂×C₇×#358	2
C₂×C₇×#357	2
C₂×C₄×C₇×C₁₆	2
C₂×#2242	7

Subgroups — C2×C7×#2242 #13210912

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2242 #13210912