Subgroups — C₂²×C₇×#442 #13237370

Order 1792.

3654 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₇×D₄	C₂	1	2	yes	maximal, complemented
3	C₂×C₇×#442	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₇×#106	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₇×#102	C₂	1	2	yes	maximal, complemented
6	C₂²×C₇×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₇×D₄	C₂²	1	2	yes	—
8	C₂⁴×C₄×C₇	—	2	2	no	—
9	C₂²×C₇×D₈	C₄	1	16	yes	complemented
10	C₂²×C₇×D₈	C₂²	1	12	yes	—
11	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₇×D₄	—	2	12	no	—
14	C₂×C₄×C₇×D₄	C₂²	1	24	yes	complemented
15	C₂²×C₇×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂²×C₇×C₂²⋊C₄	—	2	2	no	—
17	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
18	C₇×#442	C₂²	1	64	yes	direct product factor
19	C₂×C₇×#106	C₂²	1	12	yes	complemented
20	C₂×C₇×#102	C₂²	1	24	yes	complemented
21	C₂×C₇×#97	C₂²	1	12	yes	complemented
22	C₂²×#442	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂⁵×C₇	—	2	2	no	—
24	C₂²×C₇×D₄	—	2	6	no	—
25	C₂²×C₇×D₄	—	2	8	no	—
26	C₂²×C₇×D₄	C₂×C₄	1	16	yes	complemented
27	C₂²×C₇×D₄	C₂³	1	12	yes	—
28	C₂³×C₄×C₇	—	2	2	no	—
29	C₂³×C₄×C₇	—	2	24	no	—
30	C₂³×C₄×C₇	—	4	16	no	—
31	C₂³×C₄×C₇	D₄	1	2	yes	—
32	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
33	C₂×C₇×D₈	C₂×C₄	1	96	yes	complemented
34	C₂×C₇×D₈	C₂³	1	16	yes	—
35	C₂²×C₇×C₈	—	2	4	no	—
36	C₂²×C₇×C₈	C₂×C₄	1	8	yes	complemented
37	C₂²×C₇×C₈	C₂³	1	6	yes	—
38	C₂²×C₇×C₈	C₂³	1	6	yes	—
39	C₄×C₇×D₄	—	2	48	no	—
40	C₄×C₇×D₄	C₂³	1	32	yes	complemented
41	C₂×C₇×C₄⋊C₄	—	2	6	no	—
42	C₂×C₇×C₄⋊C₄	C₂³	1	12	yes	—
43	C₂×C₇×C₂²⋊C₄	—	2	24	no	—
44	C₂×C₄²×C₇	D₄	1	6	yes	—
45	C₂×C₄²×C₇	C₂³	1	6	yes	—
46	C₇×#106	C₂³	1	16	yes	complemented
47	C₇×#102	C₂³	1	32	yes	complemented
48	C₇×#97	C₂³	1	16	yes	complemented
49	C₂³×D₈	C₂×C₇	1	1	yes	characteristic
50	C₂²×C₄×D₄	C₂×C₇	1	2	yes	complemented
51	C₂×#442	C₂×C₇	1	24	yes	direct product factor
52	C₂²×#106	C₂×C₇	1	1	yes	characteristic, complemented
53	C₂²×#102	C₂×C₇	1	2	yes	complemented
54	C₂²×#97	C₂×C₇	1	1	yes	characteristic, complemented
55	C₂⁴×C₇	—	2	16	no	—
56	C₂⁴×C₇	—	2	12	no	—
57	C₂⁴×C₇	—	4	8	no	—
58	C₂⁴×C₇	—	4	8	no	—
59	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₇×D₄	—	2	48	no	—
61	C₂×C₇×D₄	—	2	12	no	—
62	C₂×C₇×D₄	—	2	24	no	—
63	C₂×C₇×D₄	C₂²×C₄	1	48	yes	complemented
64	C₂×C₇×D₄	C₂⁴	1	8	yes	—
65	C₂²×C₄×C₇	—	2	8	no	—
66	C₂²×C₄×C₇	—	2	12	no	—
67	C₂²×C₄×C₇	—	2	32	no	—
68	C₂²×C₄×C₇	—	4	96	no	—
69	C₂²×C₄×C₇	—	4	8	no	—
70	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
72	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
73	C₂²×C₄×C₇	#41	1	4	yes	—
74	C₂²×C₄×C₇	C₂⁴	1	3	yes	—
75	C₇×D₈	C₂²×C₄	1	64	yes	complemented
76	C₂×C₇×C₈	—	2	12	no	—
77	C₂×C₇×C₈	C₂²×C₄	1	24	yes	complemented
78	C₂×C₇×C₈	C₂⁴	1	4	yes	—
79	C₂×C₇×C₈	C₂⁴	1	4	yes	—
80	C₇×C₄⋊C₄	—	2	12	no	—
81	C₇×C₄⋊C₄	C₂⁴	1	8	yes	—
82	C₇×C₂²⋊C₄	—	2	32	no	—
83	C₄²×C₇	C₂×D₄	1	12	yes	—
84	C₄²×C₇	C₂⁴	1	4	yes	—
85	C₂³×D₄	C₂²×C₇	1	2	yes	—
86	C₂⁴×C₄	—	2	2	no	—
87	C₂²×D₈	C₄×C₇	1	16	yes	complemented
88	C₂²×D₈	C₂²×C₇	1	12	yes	—
89	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
90	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
91	C₂×C₄×D₄	—	2	12	no	—
92	C₂×C₄×D₄	C₂²×C₇	1	24	yes	complemented
93	C₂²×C₄⋊C₄	C₂²×C₇	1	2	yes	—
94	C₂²×C₂²⋊C₄	—	2	2	no	—
95	C₂²×C₄²	C₂²×C₇	1	1	yes	characteristic
96	#442	C₂²×C₇	1	64	yes	direct product factor
97	C₂×#106	C₂²×C₇	1	12	yes	complemented
98	C₂×#102	C₂²×C₇	1	24	yes	complemented
99	C₂×#97	C₂²×C₇	1	12	yes	complemented
100	C₂³×C₇	—	2	48	no	—
101	C₂³×C₇	—	2	8	no	—
102	C₂³×C₇	—	4	24	no	—
103	C₂³×C₇	—	4	24	no	—
104	C₂³×C₇	—	4	64	no	—
105	C₂³×C₇	C₄×D₄	1	4	yes	—
106	C₂³×C₇	AGL(1,ℤ/8)	1	8	yes	—
107	C₂³×C₇	C₂²×D₄	1	3	yes	—
108	C₇×D₄	—	2	48	no	—
109	C₇×D₄	—	2	8	no	—
110	C₇×D₄	C₂³×C₄	1	16	yes	complemented
111	C₂×C₄×C₇	—	2	24	no	—
112	C₂×C₄×C₇	—	2	8	no	—
113	C₂×C₄×C₇	—	4	64	no	—
114	C₂×C₄×C₇	—	4	24	no	—
115	C₂×C₄×C₇	C₄×D₄	1	12	yes	—
116	C₂×C₄×C₇	C₂³×C₄	1	6	yes	—
117	C₂×C₄×C₇	C₂²×D₄	1	8	yes	—
118	C₂×C₄×C₇	C₂²×D₄	1	3	yes	—
119	C₂×C₄×C₇	C₂×#41	1	6	yes	—
120	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
121	C₇×C₈	—	2	4	no	—
122	C₇×C₈	C₂³×C₄	1	8	yes	complemented
123	C₂⁵	—	2	2	no	—
124	C₂²×D₄	—	2	6	no	—
125	C₂²×D₄	—	2	8	no	—
126	C₂²×D₄	C₂×C₄×C₇	1	16	yes	complemented
127	C₂²×D₄	C₂³×C₇	1	12	yes	—
128	C₂³×C₄	—	2	2	no	—
129	C₂³×C₄	—	2	24	no	—
130	C₂³×C₄	—	4	16	no	—
131	C₂³×C₄	C₇×D₄	1	2	yes	—
132	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
133	C₂×D₈	C₂×C₄×C₇	1	96	yes	complemented
134	C₂×D₈	C₂³×C₇	1	16	yes	—
135	C₂²×C₈	—	2	4	no	—
136	C₂²×C₈	C₂×C₄×C₇	1	8	yes	complemented
137	C₂²×C₈	C₂³×C₇	1	6	yes	—
138	C₂²×C₈	C₂³×C₇	1	6	yes	—
139	C₄×D₄	—	2	48	no	—
140	C₄×D₄	C₂³×C₇	1	32	yes	complemented
141	C₂×C₄⋊C₄	—	2	6	no	—
142	C₂×C₄⋊C₄	C₂³×C₇	1	12	yes	—
143	C₂×C₂²⋊C₄	—	2	24	no	—
144	C₂×C₄²	C₇×D₄	1	6	yes	—
145	C₂×C₄²	C₂³×C₇	1	6	yes	—
146	#106	C₂³×C₇	1	16	yes	complemented
147	#102	C₂³×C₇	1	32	yes	complemented
148	#97	C₂³×C₇	1	16	yes	complemented
149	C₂²×C₇	—	2	16	no	—
150	C₂²×C₇	—	4	96	no	—
151	C₂²×C₇	—	4	8	no	—
152	C₂²×C₇	—	4	8	no	—
153	C₂²×C₇	#442	1	16	yes	direct product factor
154	C₂²×C₇	C₂×C₄×D₄	1	6	yes	—
155	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	12	yes	—
156	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
157	C₄×C₇	—	2	8	no	—
158	C₄×C₇	—	4	8	no	—
159	C₄×C₇	C₂×C₄×D₄	1	6	yes	—
160	C₄×C₇	C₂⁴×C₄	1	1	yes	characteristic
161	C₄×C₇	C₂²×#41	1	1	yes	characteristic
162	C₂⁴	—	2	16	no	—
163	C₂⁴	—	2	12	no	—
164	C₂⁴	—	4	8	no	—
165	C₂⁴	—	4	8	no	—
166	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
167	C₂×D₄	—	2	48	no	—
168	C₂×D₄	—	2	12	no	—
169	C₂×D₄	—	2	24	no	—
170	C₂×D₄	C₂²×C₄×C₇	1	48	yes	complemented
171	C₂×D₄	C₂⁴×C₇	1	8	yes	—
172	C₂²×C₄	—	2	8	no	—
173	C₂²×C₄	—	2	12	no	—
174	C₂²×C₄	—	2	32	no	—
175	C₂²×C₄	—	4	96	no	—
176	C₂²×C₄	—	4	8	no	—
177	C₂²×C₄	C₂²×C₄×C₇	1	4	yes	—
178	C₂²×C₄	C₂×C₇×D₄	1	12	yes	—
179	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
180	C₂²×C₄	C₇×#41	1	4	yes	—
181	C₂²×C₄	C₂⁴×C₇	1	3	yes	—
182	D₈	C₂²×C₄×C₇	1	64	yes	complemented
183	C₂×C₈	—	2	12	no	—
184	C₂×C₈	C₂²×C₄×C₇	1	24	yes	complemented
185	C₂×C₈	C₂⁴×C₇	1	4	yes	—
186	C₂×C₈	C₂⁴×C₇	1	4	yes	—
187	C₄⋊C₄	—	2	12	no	—
188	C₄⋊C₄	C₂⁴×C₇	1	8	yes	—
189	C₂²⋊C₄	—	2	32	no	—
190	C₄²	C₂×C₇×D₄	1	12	yes	—
191	C₄²	C₂⁴×C₇	1	4	yes	—
192	C₂×C₇	—	4	16	no	—
193	C₂×C₇	C₂×#442	1	12	yes	direct product factor
194	C₂×C₇	C₂²×C₄×D₄	1	1	yes	characteristic
195	C₂×C₇	C₂²×AGL(1,ℤ/8)	1	2	yes	—
196	C₂³	—	2	8	no	—
197	C₂³	—	2	48	no	—
198	C₂³	—	4	24	no	—
199	C₂³	—	4	24	no	—
200	C₂³	—	4	64	no	—
201	C₂³	C₄×C₇×D₄	1	4	yes	—
202	C₂³	C₇×AGL(1,ℤ/8)	1	8	yes	—
203	C₂³	C₂²×C₇×D₄	1	3	yes	—
204	D₄	—	2	8	no	—
205	D₄	—	2	48	no	—
206	D₄	C₂³×C₄×C₇	1	16	yes	complemented
207	C₂×C₄	—	2	24	no	—
208	C₂×C₄	—	2	8	no	—
209	C₂×C₄	—	4	64	no	—
210	C₂×C₄	—	4	24	no	—
211	C₂×C₄	C₄×C₇×D₄	1	12	yes	—
212	C₂×C₄	C₂³×C₄×C₇	1	6	yes	—
213	C₂×C₄	C₂²×C₇×D₄	1	8	yes	—
214	C₂×C₄	C₂²×C₇×D₄	1	3	yes	—
215	C₂×C₄	C₂×C₇×#41	1	6	yes	—
216	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
217	C₈	—	2	4	no	—
218	C₈	C₂³×C₄×C₇	1	8	yes	complemented
219	C₇	C₂²×#442	1	1	yes	characteristic, direct product factor, Sylow
220	C₂²	—	2	16	no	—
221	C₂²	—	4	8	no	—
222	C₂²	—	4	8	no	—
223	C₂²	—	4	96	no	—
224	C₂²	C₇×#442	1	16	yes	direct product factor
225	C₂²	C₂×C₄×C₇×D₄	1	6	yes	—
226	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	12	yes	—
227	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
228	C₄	—	2	8	no	—
229	C₄	—	4	8	no	—
230	C₄	C₂×C₄×C₇×D₄	1	6	yes	—
231	C₄	C₂⁴×C₄×C₇	1	1	yes	characteristic
232	C₄	C₂²×C₇×#41	1	1	yes	characteristic
233	C₂	—	4	16	no	—
234	C₂	C₂×C₇×#442	1	12	yes	direct product factor
235	C₂	C₂²×C₄×C₇×D₄	1	1	yes	characteristic
236	C₂	C₂²×C₇×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₇	C₂×D₄
commutator	C₄	C₂⁴×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₂³×C₇×D₈	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇ ◁ C₂²×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×C₇×D₄	C₂	semidirect
3. C₂×C₇×#442	C₂	direct (N × H)
4. C₂²×C₇×#106	C₂	semidirect
5. C₂²×C₇×#102	C₂	semidirect
6. C₂²×C₇×#97	C₂	semidirect
9. C₂²×C₇×D₈	C₄	semidirect
14. C₂×C₄×C₇×D₄	C₂²	semidirect
18. C₇×#442	C₂²	direct (N × H)
19. C₂×C₇×#106	C₂²	semidirect
20. C₂×C₇×#102	C₂²	semidirect
21. C₂×C₇×#97	C₂²	semidirect
22. C₂²×#442	C₇	direct (N × H)
26. C₂²×C₇×D₄	C₂×C₄	semidirect
33. C₂×C₇×D₈	C₂×C₄	semidirect
36. C₂²×C₇×C₈	C₂×C₄	semidirect
40. C₄×C₇×D₄	C₂³	semidirect
46. C₇×#106	C₂³	semidirect
47. C₇×#102	C₂³	semidirect
48. C₇×#97	C₂³	semidirect
50. C₂²×C₄×D₄	C₂×C₇	semidirect
51. C₂×#442	C₂×C₇	direct (N × H)
52. C₂²×#106	C₂×C₇	semidirect
53. C₂²×#102	C₂×C₇	semidirect
54. C₂²×#97	C₂×C₇	semidirect
63. C₂×C₇×D₄	C₂²×C₄	semidirect
75. C₇×D₈	C₂²×C₄	semidirect
77. C₂×C₇×C₈	C₂²×C₄	semidirect
87. C₂²×D₈	C₄×C₇	semidirect
92. C₂×C₄×D₄	C₂²×C₇	semidirect
96. #442	C₂²×C₇	direct (N × H)
97. C₂×#106	C₂²×C₇	semidirect
98. C₂×#102	C₂²×C₇	semidirect
99. C₂×#97	C₂²×C₇	semidirect
110. C₇×D₄	C₂³×C₄	semidirect
122. C₇×C₈	C₂³×C₄	semidirect
126. C₂²×D₄	C₂×C₄×C₇	semidirect
133. C₂×D₈	C₂×C₄×C₇	semidirect
136. C₂²×C₈	C₂×C₄×C₇	semidirect
140. C₄×D₄	C₂³×C₇	semidirect
146. #106	C₂³×C₇	semidirect
147. #102	C₂³×C₇	semidirect
148. #97	C₂³×C₇	semidirect
153. C₂²×C₇	#442	direct (N × H)
170. C₂×D₄	C₂²×C₄×C₇	semidirect
182. D₈	C₂²×C₄×C₇	semidirect
184. C₂×C₈	C₂²×C₄×C₇	semidirect
193. C₂×C₇	C₂×#442	direct (N × H)
206. D₄	C₂³×C₄×C₇	semidirect
218. C₈	C₂³×C₄×C₇	semidirect
219. C₇	C₂²×#442	direct (N × H)
224. C₂²	C₇×#442	direct (N × H)
234. C₂	C₂×C₇×#442	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₇×D₈	2
C₂²×C₄×C₇×D₄	2
C₂×C₇×#442	2
C₂²×C₇×#106	2
C₂²×C₇×#102	2
C₂²×C₇×#97	2
C₂²×#442	7

Subgroups — C22×C7×#442 #13237370

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×#442 #13237370