Subgroups — C₄×C₇×(C₂×D₃)⋊C₄ #13055973

Order 1344.

516 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×C₄×D₃×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄²×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×(C₂×D₃)⋊C₄	C₂	1	2	yes	maximal
4	C₂×C₄×C₇×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×C₃⋊#95	C₂	1	2	yes	maximal, complemented
6	C₄×C₇×C₂²⋊C₄	—	3	1	no	maximal, Hall
7	C₂³×D₃×C₇	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
9	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
10	C₂²×C₇×Dic₃	C₂²	1	1	yes	characteristic
11	C₂²×C₇×Dic₃	C₂²	1	2	yes	—
12	C₂×C₄×D₃×C₇	—	2	2	no	—
13	C₂×C₄×D₃×C₇	—	2	2	no	—
14	C₂×C₄×D₃×C₇	C₄	1	4	yes	complemented
15	C₃×C₄²×C₇	—	2	2	no	—
16	C₇×(C₂×D₃)⋊C₄	C₄	1	8	yes	direct product factor
17	C₄×C₇×Dic₃	—	2	2	no	—
18	C₂³×C₄×C₇	—	3	1	no	—
19	C₂×C₇×C₂²⋊C₄	—	3	2	no	—
20	C₂×C₄²×C₇	—	3	1	no	—
21	C₂×C₄²×C₇	D₃	1	1	yes	characteristic, complemented
22	C₇×#95	—	3	2	no	—
23	C₄×(C₂×D₃)⋊C₄	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
25	C₂²×D₃×C₇	—	2	1	no	—
26	C₂²×D₃×C₇	—	2	1	no	—
27	C₂²×D₃×C₇	—	2	2	no	—
28	C₂²×D₃×C₇	C₂×C₄	1	4	yes	—
29	C₂²×D₃×C₇	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄×C₇	—	2	2	no	—
31	C₂×C₃×C₄×C₇	—	2	2	no	—
32	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₄×C₇	C₂×C₄	1	4	yes	complemented
34	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
35	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
36	C₂×C₇×Dic₃	—	2	2	no	—
37	C₂×C₇×Dic₃	—	2	1	no	—
38	C₂×C₇×Dic₃	—	2	1	no	—
39	C₂×C₇×Dic₃	—	2	2	no	—
40	C₂×C₇×Dic₃	C₂×C₄	1	2	yes	—
41	C₂×C₇×Dic₃	C₂×C₄	1	4	yes	complemented
42	C₄×D₃×C₇	—	2	8	no	—
43	C₂⁴×C₇	—	3	1	no	—
44	C₂²×C₄×C₇	—	3	1	no	—
45	C₂²×C₄×C₇	—	3	2	no	—
46	C₂²×C₄×C₇	—	3	4	no	—
47	C₂²×C₄×C₇	—	6	2	no	—
48	C₂²×C₄×C₇	—	6	2	no	—
49	C₂²×C₄×C₇	C₂×D₃	1	2	yes	—
50	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
51	C₇×C₂²⋊C₄	—	3	8	no	—
52	C₄²×C₇	—	2	2	no	—
53	C₄²×C₇	—	6	2	no	—
54	C₂²×C₄×D₃	C₂×C₇	1	1	yes	characteristic
55	C₂×C₃×C₄²	C₂×C₇	1	1	yes	characteristic, complemented
56	C₂×(C₂×D₃)⋊C₄	C₂×C₇	1	2	yes	—
57	C₂×C₄×Dic₃	C₂×C₇	1	1	yes	characteristic, complemented
58	C₃⋊#95	C₂×C₇	1	2	yes	complemented
59	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₃×C₇	C₂²×C₄	1	2	yes	—
61	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₃×C₇	#41	1	2	yes	—
64	C₂×D₃×C₇	—	2	2	no	—
65	C₂×D₃×C₇	—	2	2	no	—
66	C₂×D₃×C₇	—	2	4	no	—
67	C₂×D₃×C₇	—	2	4	no	—
68	C₂×D₃×C₇	C₄²	1	4	yes	complemented
69	C₃×C₄×C₇	—	2	4	no	—
70	C₃×C₄×C₇	C₂²⋊C₄	1	4	yes	complemented
71	C₇×Dic₃	—	2	2	no	—
72	C₇×Dic₃	—	2	4	no	—
73	C₄×C₂²⋊C₄	—	3	1	no	Sylow
74	C₂³×C₇	—	3	4	no	—
75	C₂³×C₇	—	3	2	no	—
76	C₂³×C₇	—	6	1	no	—
77	C₂³×C₇	—	6	2	no	—
78	C₂³×C₇	—	6	1	no	—
79	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
80	C₂×C₄×C₇	—	2	2	no	—
81	C₂×C₄×C₇	—	2	2	no	—
82	C₂×C₄×C₇	—	3	4	no	—
83	C₂×C₄×C₇	—	3	2	no	—
84	C₂×C₄×C₇	—	6	2	no	—
85	C₂×C₄×C₇	—	6	1	no	—
86	C₂×C₄×C₇	—	6	2	no	—
87	C₂×C₄×C₇	—	6	1	no	—
88	C₂×C₄×C₇	—	6	8	no	—
89	C₂×C₄×C₇	C₄×D₃	1	2	yes	—
90	C₂×C₄×C₇	C₄×D₃	1	4	yes	complemented
91	C₂×C₄×C₇	D₁₂	1	2	yes	—
92	C₂×C₄×C₇	#67	1	2	yes	—
93	C₂³×D₃	C₂²×C₇	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂²×C₇	1	2	yes	—
95	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
96	C₂²×Dic₃	C₂²×C₇	1	1	yes	characteristic
97	C₂²×Dic₃	C₂²×C₇	1	2	yes	—
98	C₂×C₄×D₃	—	2	2	no	—
99	C₂×C₄×D₃	—	2	2	no	—
100	C₂×C₄×D₃	C₄×C₇	1	4	yes	complemented
101	C₃×C₄²	—	2	2	no	—
102	(C₂×D₃)⋊C₄	C₄×C₇	1	8	yes	direct product factor
103	C₄×Dic₃	—	2	2	no	—
104	C₂×C₃×C₇	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₃×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₃×C₇	#117	1	1	yes	characteristic
107	C₂×C₃×C₇	C₄×D₄	1	2	yes	—
108	C₂×C₃×C₇	C₄×D₄	1	2	yes	—
109	D₃×C₇	—	2	4	no	—
110	C₂³×C₄	—	3	1	no	—
111	C₂×C₂²⋊C₄	—	3	2	no	—
112	C₂×C₄²	—	3	1	no	—
113	C₂×C₄²	D₃×C₇	1	1	yes	characteristic, complemented
114	#95	—	3	2	no	—
115	C₂²×C₇	—	3	4	no	—
116	C₂²×C₇	—	6	2	no	—
117	C₂²×C₇	—	6	4	no	—
118	C₂²×C₇	—	6	2	no	—
119	C₂²×C₇	—	6	4	no	—
120	C₂²×C₇	C₂×C₄×D₃	1	1	yes	characteristic
121	C₂²×C₇	C₂×C₄×D₃	1	2	yes	—
122	C₂²×C₇	C₂×D₁₂	1	1	yes	characteristic
123	C₂²×C₇	#235	1	2	yes	—
124	C₂²×C₇	C₂×#67	1	1	yes	characteristic
125	C₄×C₇	—	2	4	no	—
126	C₄×C₇	—	6	4	no	—
127	C₄×C₇	—	6	2	no	—
128	C₄×C₇	(C₂×D₃)⋊C₄	1	4	yes	direct product factor
129	C₂³×C₃	C₂³×C₇	1	1	yes	characteristic
130	C₂²×D₃	—	2	1	no	—
131	C₂²×D₃	—	2	1	no	—
132	C₂²×D₃	—	2	2	no	—
133	C₂²×D₃	C₂×C₄×C₇	1	4	yes	—
134	C₂²×D₃	C₂×C₄×C₇	1	2	yes	—
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	—
138	C₂×C₃×C₄	C₂×C₄×C₇	1	4	yes	complemented
139	C₂×C₃×C₄	C₇×D₄	1	2	yes	—
140	C₂×C₃×C₄	C₇×D₄	1	2	yes	—
141	C₂×Dic₃	—	2	2	no	—
142	C₂×Dic₃	—	2	1	no	—
143	C₂×Dic₃	—	2	1	no	—
144	C₂×Dic₃	—	2	2	no	—
145	C₂×Dic₃	C₂×C₄×C₇	1	2	yes	—
146	C₂×Dic₃	C₂×C₄×C₇	1	4	yes	complemented
147	C₄×D₃	—	2	8	no	—
148	C₃×C₇	C₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
149	C₂⁴	—	3	1	no	—
150	C₂²×C₄	—	3	1	no	—
151	C₂²×C₄	—	3	2	no	—
152	C₂²×C₄	—	3	4	no	—
153	C₂²×C₄	—	6	2	no	—
154	C₂²×C₄	—	6	2	no	—
155	C₂²×C₄	C₂×D₃×C₇	1	2	yes	—
156	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
157	C₂²⋊C₄	—	3	8	no	—
158	C₄²	—	2	2	no	—
159	C₄²	—	6	2	no	—
160	C₂×C₇	—	6	4	no	—
161	C₂×C₇	C₄²×D₃	1	1	yes	characteristic
162	C₂×C₇	#872	1	1	yes	characteristic
163	C₂×C₇	C₄×D₁₂	1	2	yes	—
164	C₂×C₇	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
165	C₂×C₇	C₄×#67	1	2	yes	—
166	C₂²×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
167	C₂²×C₃	C₂²×C₄×C₇	1	2	yes	—
168	C₂²×C₃	C₂×C₇×D₄	1	1	yes	characteristic
169	C₂²×C₃	C₂×C₇×D₄	1	1	yes	characteristic
170	C₂²×C₃	C₇×#41	1	2	yes	—
171	C₂×D₃	—	2	2	no	—
172	C₂×D₃	—	2	2	no	—
173	C₂×D₃	—	2	4	no	—
174	C₂×D₃	—	2	4	no	—
175	C₂×D₃	C₄²×C₇	1	4	yes	complemented
176	C₃×C₄	—	2	4	no	—
177	C₃×C₄	C₇×C₂²⋊C₄	1	4	yes	complemented
178	Dic₃	—	2	2	no	—
179	Dic₃	—	2	4	no	—
180	C₂³	—	3	4	no	—
181	C₂³	—	3	2	no	—
182	C₂³	—	6	1	no	—
183	C₂³	—	6	2	no	—
184	C₂³	—	6	1	no	—
185	C₂³	C₂²×D₃×C₇	1	1	yes	characteristic
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	3	4	no	—
189	C₂×C₄	—	3	2	no	—
190	C₂×C₄	—	6	2	no	—
191	C₂×C₄	—	6	1	no	—
192	C₂×C₄	—	6	2	no	—
193	C₂×C₄	—	6	1	no	—
194	C₂×C₄	—	6	8	no	—
195	C₂×C₄	C₄×D₃×C₇	1	2	yes	—
196	C₂×C₄	C₄×D₃×C₇	1	4	yes	complemented
197	C₂×C₄	C₇×D₁₂	1	2	yes	—
198	C₂×C₄	C₇×#67	1	2	yes	—
199	C₇	C₄×(C₂×D₃)⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
200	C₂×C₃	C₂×C₄²×C₇	1	1	yes	characteristic
201	C₂×C₃	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
202	C₂×C₃	C₇×#117	1	1	yes	characteristic
203	C₂×C₃	C₄×C₇×D₄	1	2	yes	—
204	C₂×C₃	C₄×C₇×D₄	1	2	yes	—
205	D₃	—	2	4	no	—
206	C₂²	—	3	4	no	—
207	C₂²	—	6	2	no	—
208	C₂²	—	6	4	no	—
209	C₂²	—	6	2	no	—
210	C₂²	—	6	4	no	—
211	C₂²	C₂×C₄×D₃×C₇	1	1	yes	characteristic
212	C₂²	C₂×C₄×D₃×C₇	1	2	yes	—
213	C₂²	C₂×C₇×D₁₂	1	1	yes	characteristic
214	C₂²	C₇×#235	1	2	yes	—
215	C₂²	C₂×C₇×#67	1	1	yes	characteristic
216	C₄	—	2	4	no	—
217	C₄	—	6	4	no	—
218	C₄	—	6	2	no	—
219	C₄	C₇×(C₂×D₃)⋊C₄	1	4	yes	direct product factor
220	C₃	C₄×C₇×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
221	C₂	—	6	4	no	—
222	C₂	C₄²×D₃×C₇	1	1	yes	characteristic
223	C₂	C₇×#872	1	1	yes	characteristic
224	C₂	C₄×C₇×D₁₂	1	2	yes	—
225	C₂	C₂×C₇×(C₂×D₃)⋊C₄	1	1	yes	characteristic
226	C₂	C₄×C₇×#67	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₃
commutator	C₂×C₃	C₂×C₄²×C₇
Frattini	C₂³	C₂²×D₃×C₇
Fitting	C₂×C₃×C₄²×C₇	C₂
socle	C₂³×C₃×C₇	C₂³
prime radical	C₂³×D₃×C₇	C₂²
cosocle	C₂³×C₃	C₂³×C₇
hypercenter	C₂×C₄²×C₇	D₃
hypocenter	C₃	C₄×C₇×C₂²⋊C₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₂×C₄²×C₇

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄²×C₇	C₂	semidirect
4. C₂×C₄×C₇×Dic₃	C₂	semidirect
5. C₇×C₃⋊#95	C₂	semidirect
14. C₂×C₄×D₃×C₇	C₄	semidirect
16. C₇×(C₂×D₃)⋊C₄	C₄	direct (N × H)
21. C₂×C₄²×C₇	D₃	semidirect
23. C₄×(C₂×D₃)⋊C₄	C₇	direct (N × H)
33. C₂×C₃×C₄×C₇	C₂×C₄	semidirect
41. C₂×C₇×Dic₃	C₂×C₄	semidirect
55. C₂×C₃×C₄²	C₂×C₇	semidirect
57. C₂×C₄×Dic₃	C₂×C₇	semidirect
58. C₃⋊#95	C₂×C₇	semidirect
68. C₂×D₃×C₇	C₄²	semidirect
70. C₃×C₄×C₇	C₂²⋊C₄	semidirect
90. C₂×C₄×C₇	C₄×D₃	semidirect
100. C₂×C₄×D₃	C₄×C₇	semidirect
102. (C₂×D₃)⋊C₄	C₄×C₇	direct (N × H)
113. C₂×C₄²	D₃×C₇	semidirect
128. C₄×C₇	(C₂×D₃)⋊C₄	direct (N × H)
138. C₂×C₃×C₄	C₂×C₄×C₇	semidirect
146. C₂×Dic₃	C₂×C₄×C₇	semidirect
148. C₃×C₇	C₄×C₂²⋊C₄	semidirect
175. C₂×D₃	C₄²×C₇	semidirect
177. C₃×C₄	C₇×C₂²⋊C₄	semidirect
196. C₂×C₄	C₄×D₃×C₇	semidirect
199. C₇	C₄×(C₂×D₃)⋊C₄	direct (N × H)
219. C₄	C₇×(C₂×D₃)⋊C₄	direct (N × H)
220. C₃	C₄×C₇×C₂²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₃×C₇	2
C₂×C₃×C₄²×C₇	2
C₂×C₇×(C₂×D₃)⋊C₄	2
C₂×C₄×C₇×Dic₃	2
C₇×C₃⋊#95	2
C₄×C₇×C₂²⋊C₄	3
C₄×(C₂×D₃)⋊C₄	7

Subgroups — C4×C7×(C2×D3)⋊C4 #13055973

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₇×(C₂×D₃)⋊C₄ #13055973