Subgroups — C₃×D₇×C₂²⋊C₈ #13055158

Order 1344.

404 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×D₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₈×D₇	C₂	1	2	yes	maximal, complemented
3	C₃×C₇×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₂²⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×(C₂×D₇)⋊C₈	C₂	1	2	yes	maximal, complemented
6	D₇×C₂²⋊C₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂³×C₃×D₇	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂²×C₃×Dic₇	C₄	1	1	yes	characteristic
10	C₂×C₃×C₄×D₇	—	2	4	no	—
11	C₂×C₃×C₄×D₇	C₄	1	1	yes	characteristic
12	C₂×C₃×C₄×D₇	C₄	1	1	yes	characteristic
13	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄×D₇	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₇×C₈	C₂²	1	2	yes	complemented
16	C₂×C₃×C₇⋊C₈	C₂²	1	2	yes	complemented
17	C₃×C₈×D₇	—	2	4	no	—
18	C₂²×C₄×D₇	C₂×C₃	1	1	yes	characteristic
19	C₂×C₈×D₇	C₂×C₃	1	2	yes	complemented
20	C₇×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂²⋊(C₇⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
22	(C₂×D₇)⋊C₈	C₂×C₃	1	2	yes	complemented
23	C₂×C₃×C₂²⋊C₈	—	7	1	no	maximal, Hall
24	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×D₇	—	2	2	no	—
26	C₂²×C₃×D₇	—	2	2	no	—
27	C₂²×C₃×D₇	C₈	1	4	yes	complemented
28	C₂²×C₃×D₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×D₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₇	—	2	2	no	—
31	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
33	C₂×C₃×Dic₇	—	2	2	no	—
34	C₂×C₃×Dic₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×Dic₇	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄×D₇	—	2	2	no	—
37	C₃×C₄×D₇	—	2	2	no	—
38	C₃×C₄×D₇	—	2	2	no	—
39	C₃×C₄×D₇	D₄	1	4	yes	—
40	C₃×C₇×C₈	—	2	2	no	—
41	C₃×C₇⋊C₈	—	2	2	no	—
42	C₂³×D₇	C₃×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
44	C₂²×Dic₇	C₃×C₄	1	1	yes	characteristic
45	C₂×C₄×D₇	—	2	4	no	—
46	C₂×C₄×D₇	C₃×C₄	1	1	yes	characteristic
47	C₂×C₄×D₇	C₃×C₄	1	1	yes	characteristic
48	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₄×D₇	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₇×C₈	C₂²×C₃	1	2	yes	complemented
51	C₂×C₇⋊C₈	C₂²×C₃	1	2	yes	complemented
52	C₈×D₇	—	2	4	no	—
53	C₂³×C₃×C₄	—	7	1	no	—
54	C₂²×C₃×C₈	—	7	2	no	—
55	C₃×C₂²⋊C₈	—	7	1	no	—
56	C₃×C₂²⋊C₈	—	7	2	no	—
57	C₃×C₂²⋊C₈	D₇	1	1	yes	characteristic, direct product factor
58	C₂²×C₃×C₇	—	2	1	no	—
59	C₂²×C₃×C₇	—	2	1	no	—
60	C₂²×C₃×C₇	C₂×C₈	1	2	yes	complemented
61	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×D₇	—	2	1	no	—
63	C₂×C₃×D₇	—	2	1	no	—
64	C₂×C₃×D₇	—	2	8	no	—
65	C₂×C₃×D₇	C₂²⋊C₄	1	2	yes	—
66	C₂×C₃×D₇	C₂×C₈	1	2	yes	—
67	C₂×C₃×D₇	C₂×C₈	1	2	yes	complemented
68	C₂×C₃×D₇	M₁₆	1	2	yes	—
69	C₃×C₄×C₇	—	2	1	no	—
70	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
71	C₃×Dic₇	—	2	1	no	—
72	C₃×Dic₇	C₂²⋊C₄	1	2	yes	—
73	C₂×C₂²⋊C₈	—	7	1	no	Sylow
74	C₂³×C₇	C₂×C₃×C₄	1	1	yes	characteristic
75	C₂²×D₇	—	2	2	no	—
76	C₂²×D₇	—	2	2	no	—
77	C₂²×D₇	C₃×C₈	1	4	yes	complemented
78	C₂²×D₇	C₂×C₃×C₄	1	1	yes	characteristic
79	C₂²×D₇	C₂×C₃×C₄	1	1	yes	characteristic
80	C₂×C₄×C₇	—	2	2	no	—
81	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
82	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
83	C₂×Dic₇	—	2	2	no	—
84	C₂×Dic₇	C₂×C₃×C₄	1	1	yes	characteristic
85	C₂×Dic₇	C₂×C₃×C₄	1	1	yes	characteristic
86	C₄×D₇	—	2	2	no	—
87	C₄×D₇	—	2	2	no	—
88	C₄×D₇	—	2	2	no	—
89	C₄×D₇	C₃×D₄	1	4	yes	—
90	C₇×C₈	—	2	2	no	—
91	C₇⋊C₈	—	2	2	no	—
92	C₂⁴×C₃	—	7	1	no	—
93	C₂²×C₃×C₄	—	7	1	no	—
94	C₂²×C₃×C₄	—	7	1	no	—
95	C₂²×C₃×C₄	—	7	1	no	—
96	C₂²×C₃×C₄	—	7	1	no	—
97	C₂²×C₃×C₄	—	7	1	no	—
98	C₂²×C₃×C₄	—	14	4	no	—
99	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
100	C₂×C₃×C₈	—	7	2	no	—
101	C₂×C₃×C₈	—	14	4	no	—
102	C₂×C₃×C₈	C₂×D₇	1	2	yes	complemented
103	C₂×C₃×C₇	—	2	2	no	—
104	C₂×C₃×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₃×C₇	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₃×C₇	C₂×M₁₆	1	1	yes	characteristic
107	C₃×D₇	—	2	2	no	—
108	C₃×D₇	C₂²⋊C₈	1	4	yes	direct product factor
109	C₂³×C₄	—	7	1	no	—
110	C₂²×C₈	—	7	2	no	—
111	C₂²⋊C₈	—	7	1	no	—
112	C₂²⋊C₈	—	7	2	no	—
113	C₂²⋊C₈	C₃×D₇	1	1	yes	characteristic, direct product factor
114	C₂²×C₇	—	2	1	no	—
115	C₂²×C₇	—	2	1	no	—
116	C₂²×C₇	C₂×C₃×C₈	1	2	yes	complemented
117	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
118	C₂×D₇	—	2	1	no	—
119	C₂×D₇	—	2	1	no	—
120	C₂×D₇	—	2	8	no	—
121	C₂×D₇	C₃×C₂²⋊C₄	1	2	yes	—
122	C₂×D₇	C₂×C₃×C₈	1	2	yes	—
123	C₂×D₇	C₂×C₃×C₈	1	2	yes	complemented
124	C₂×D₇	C₃×M₁₆	1	2	yes	—
125	C₄×C₇	—	2	1	no	—
126	C₄×C₇	C₂×C₃×D₄	1	2	yes	—
127	Dic₇	—	2	1	no	—
128	Dic₇	C₃×C₂²⋊C₄	1	2	yes	—
129	C₂³×C₃	—	7	1	no	—
130	C₂³×C₃	—	7	4	no	—
131	C₂³×C₃	—	7	1	no	—
132	C₂³×C₃	—	14	2	no	—
133	C₂³×C₃	—	14	2	no	—
134	C₂³×C₃	C₄×D₇	1	1	yes	characteristic
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	7	1	no	—
137	C₂×C₃×C₄	—	7	4	no	—
138	C₂×C₃×C₄	—	7	1	no	—
139	C₂×C₃×C₄	—	14	2	no	—
140	C₂×C₃×C₄	—	14	2	no	—
141	C₂×C₃×C₄	—	14	2	no	—
142	C₂×C₃×C₄	—	14	2	no	—
143	C₂×C₃×C₄	C₄×D₇	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
145	C₃×C₈	—	2	2	no	—
146	C₃×C₈	—	14	2	no	—
147	C₃×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic, complemented, Hall
148	C₂⁴	—	7	1	no	—
149	C₂²×C₄	—	7	1	no	—
150	C₂²×C₄	—	7	1	no	—
151	C₂²×C₄	—	7	1	no	—
152	C₂²×C₄	—	7	1	no	—
153	C₂²×C₄	—	7	1	no	—
154	C₂²×C₄	—	14	4	no	—
155	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
156	C₂×C₈	—	7	2	no	—
157	C₂×C₈	—	14	4	no	—
158	C₂×C₈	C₂×C₃×D₇	1	2	yes	complemented
159	C₂×C₇	—	2	2	no	—
160	C₂×C₇	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₇	C₂²×C₃×C₈	1	1	yes	characteristic
162	C₂×C₇	C₂×C₃×M₁₆	1	1	yes	characteristic
163	D₇	—	2	2	no	—
164	D₇	C₃×C₂²⋊C₈	1	4	yes	direct product factor
165	C₂²×C₃	—	2	1	no	—
166	C₂²×C₃	—	2	1	no	—
167	C₂²×C₃	—	7	2	no	—
168	C₂²×C₃	—	7	2	no	—
169	C₂²×C₃	—	7	2	no	—
170	C₂²×C₃	—	7	2	no	—
171	C₂²×C₃	—	14	8	no	—
172	C₂²×C₃	—	14	1	no	—
173	C₂²×C₃	—	14	1	no	—
174	C₂²×C₃	C₈×D₇	1	2	yes	complemented
175	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	—	7	2	no	—
178	C₃×C₄	—	14	1	no	—
179	C₃×C₄	D₄×D₇	1	2	yes	—
180	C₂³	—	7	1	no	—
181	C₂³	—	7	4	no	—
182	C₂³	—	7	1	no	—
183	C₂³	—	14	2	no	—
184	C₂³	—	14	2	no	—
185	C₂³	C₃×C₄×D₇	1	1	yes	characteristic
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	7	1	no	—
188	C₂×C₄	—	7	4	no	—
189	C₂×C₄	—	7	1	no	—
190	C₂×C₄	—	14	2	no	—
191	C₂×C₄	—	14	2	no	—
192	C₂×C₄	—	14	2	no	—
193	C₂×C₄	—	14	2	no	—
194	C₂×C₄	C₃×C₄×D₇	1	1	yes	characteristic
195	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
196	C₈	—	2	2	no	—
197	C₈	—	14	2	no	—
198	C₇	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
199	C₂×C₃	—	2	2	no	—
200	C₂×C₃	—	7	4	no	—
201	C₂×C₃	—	14	2	no	—
202	C₂×C₃	D₇×C₂²⋊C₄	1	1	yes	characteristic
203	C₂×C₃	C₂×C₈×D₇	1	1	yes	characteristic
204	C₂×C₃	D₇×M₁₆	1	1	yes	characteristic
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	7	2	no	—
208	C₂²	—	7	2	no	—
209	C₂²	—	7	2	no	—
210	C₂²	—	7	2	no	—
211	C₂²	—	14	8	no	—
212	C₂²	—	14	1	no	—
213	C₂²	—	14	1	no	—
214	C₂²	C₃×C₈×D₇	1	2	yes	complemented
215	C₂²	C₂×C₃×C₄×D₇	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	7	2	no	—
218	C₄	—	14	1	no	—
219	C₄	C₃×D₄×D₇	1	2	yes	—
220	C₃	D₇×C₂²⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
221	C₂	—	2	2	no	—
222	C₂	—	7	4	no	—
223	C₂	—	14	2	no	—
224	C₂	C₃×D₇×C₂²⋊C₄	1	1	yes	characteristic
225	C₂	C₂×C₃×C₈×D₇	1	1	yes	characteristic
226	C₂	C₃×D₇×M₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₇
commutator	C₂×C₇	C₂²×C₃×C₈
Frattini	C₂×C₄	C₂²×C₃×D₇
Fitting	C₃×C₇×C₂²⋊C₈	C₂
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂³×C₃×D₇	C₄
cosocle	C₂×C₄×C₇	C₂³×C₃
hypercenter	C₃×C₂²⋊C₈	D₇
hypocenter	C₇	C₂×C₃×C₂²⋊C₈
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₈×D₇	C₂	semidirect
3. C₃×C₇×C₂²⋊C₈	C₂	semidirect
4. C₃×C₂²⋊(C₇⋊C₈)	C₂	semidirect
5. C₃×(C₂×D₇)⋊C₈	C₂	semidirect
6. D₇×C₂²⋊C₈	C₃	direct (N × H)
15. C₂×C₃×C₇×C₈	C₂²	semidirect
16. C₂×C₃×C₇⋊C₈	C₂²	semidirect
19. C₂×C₈×D₇	C₂×C₃	semidirect
20. C₇×C₂²⋊C₈	C₂×C₃	semidirect
21. C₂²⋊(C₇⋊C₈)	C₂×C₃	semidirect
22. (C₂×D₇)⋊C₈	C₂×C₃	semidirect
27. C₂²×C₃×D₇	C₈	semidirect
50. C₂×C₇×C₈	C₂²×C₃	semidirect
51. C₂×C₇⋊C₈	C₂²×C₃	semidirect
57. C₃×C₂²⋊C₈	D₇	direct (N × H)
60. C₂²×C₃×C₇	C₂×C₈	semidirect
67. C₂×C₃×D₇	C₂×C₈	semidirect
77. C₂²×D₇	C₃×C₈	semidirect
102. C₂×C₃×C₈	C₂×D₇	semidirect
108. C₃×D₇	C₂²⋊C₈	direct (N × H)
113. C₂²⋊C₈	C₃×D₇	direct (N × H)
116. C₂²×C₇	C₂×C₃×C₈	semidirect
123. C₂×D₇	C₂×C₃×C₈	semidirect
147. C₃×C₇	C₂×C₂²⋊C₈	semidirect
158. C₂×C₈	C₂×C₃×D₇	semidirect
164. D₇	C₃×C₂²⋊C₈	direct (N × H)
174. C₂²×C₃	C₈×D₇	semidirect
198. C₇	C₂×C₃×C₂²⋊C₈	semidirect
214. C₂²	C₃×C₈×D₇	semidirect
220. C₃	D₇×C₂²⋊C₈	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×C₃×C₄×D₇	2
C₂×C₃×C₈×D₇	2
C₃×C₇×C₂²⋊C₈	2
C₃×C₂²⋊(C₇⋊C₈)	2
C₃×(C₂×D₇)⋊C₈	2
D₇×C₂²⋊C₈	3
C₂×C₃×C₂²⋊C₈	7

Subgroups — C3×D7×C22⋊C8 #13055158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₇×C₂²⋊C₈ #13055158