Subgroups — C₃×C₄×(C₂×D₇)⋊C₄ #13055371

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×C₃×C₄×D₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄²×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×(C₂×D₇)⋊C₄	C₂	1	2	yes	maximal
4	C₂×C₃×C₄×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₇⋊#95	C₂	1	2	yes	maximal, complemented
6	C₄×(C₂×D₇)⋊C₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂³×C₃×D₇	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×C₃×Dic₇	C₂²	1	1	yes	characteristic
11	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
12	C₂×C₃×C₄×D₇	—	2	2	no	—
13	C₂×C₃×C₄×D₇	—	2	2	no	—
14	C₂×C₃×C₄×D₇	C₄	1	4	yes	complemented
15	C₃×C₄²×C₇	—	2	2	no	—
16	C₃×(C₂×D₇)⋊C₄	C₄	1	8	yes	direct product factor
17	C₃×C₄×Dic₇	—	2	2	no	—
18	C₂²×C₄×D₇	C₂×C₃	1	1	yes	characteristic
19	C₂×C₄²×C₇	C₂×C₃	1	1	yes	characteristic, complemented
20	C₂×(C₂×D₇)⋊C₄	C₂×C₃	1	2	yes	—
21	C₂×C₄×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
22	C₇⋊#95	C₂×C₃	1	2	yes	complemented
23	C₃×C₄×C₂²⋊C₄	—	7	1	no	maximal, Hall
24	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
25	C₂²×C₃×D₇	—	2	1	no	—
26	C₂²×C₃×D₇	—	2	1	no	—
27	C₂²×C₃×D₇	—	2	2	no	—
28	C₂²×C₃×D₇	C₂×C₄	1	4	yes	—
29	C₂²×C₃×D₇	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×C₃×Dic₇	—	2	2	no	—
37	C₂×C₃×Dic₇	—	2	1	no	—
38	C₂×C₃×Dic₇	—	2	1	no	—
39	C₂×C₃×Dic₇	—	2	2	no	—
40	C₂×C₃×Dic₇	C₂×C₄	1	2	yes	—
41	C₂×C₃×Dic₇	C₂×C₄	1	4	yes	complemented
42	C₃×C₄×D₇	—	2	8	no	—
43	C₂³×D₇	C₂²×C₃	1	1	yes	characteristic
44	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
45	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
46	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
47	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
48	C₂×C₄×D₇	—	2	2	no	—
49	C₂×C₄×D₇	—	2	2	no	—
50	C₂×C₄×D₇	C₃×C₄	1	4	yes	complemented
51	C₄²×C₇	—	2	2	no	—
52	(C₂×D₇)⋊C₄	C₃×C₄	1	8	yes	direct product factor
53	C₄×Dic₇	—	2	2	no	—
54	C₂³×C₃×C₄	—	7	1	no	—
55	C₂×C₃×C₂²⋊C₄	—	7	2	no	—
56	C₂×C₃×C₄²	—	7	1	no	—
57	C₂×C₃×C₄²	D₇	1	1	yes	characteristic, complemented
58	C₃×#95	—	7	2	no	—
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×C₃×D₇	—	2	2	no	—
65	C₂×C₃×D₇	—	2	2	no	—
66	C₂×C₃×D₇	—	2	4	no	—
67	C₂×C₃×D₇	—	2	4	no	—
68	C₂×C₃×D₇	C₄²	1	4	yes	complemented
69	C₃×C₄×C₇	—	2	4	no	—
70	C₃×C₄×C₇	C₂²⋊C₄	1	4	yes	complemented
71	C₃×Dic₇	—	2	2	no	—
72	C₃×Dic₇	—	2	4	no	—
73	C₄×C₂²⋊C₄	—	7	1	no	Sylow
74	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
75	C₂²×D₇	—	2	1	no	—
76	C₂²×D₇	—	2	1	no	—
77	C₂²×D₇	—	2	2	no	—
78	C₂²×D₇	C₂×C₃×C₄	1	4	yes	—
79	C₂²×D₇	C₂×C₃×C₄	1	2	yes	—
80	C₂×C₄×C₇	—	2	2	no	—
81	C₂×C₄×C₇	—	2	2	no	—
82	C₂×C₄×C₇	C₂×C₃×C₄	1	2	yes	—
83	C₂×C₄×C₇	C₂×C₃×C₄	1	4	yes	complemented
84	C₂×C₄×C₇	C₃×D₄	1	2	yes	—
85	C₂×C₄×C₇	C₃×D₄	1	2	yes	—
86	C₂×Dic₇	—	2	2	no	—
87	C₂×Dic₇	—	2	1	no	—
88	C₂×Dic₇	—	2	1	no	—
89	C₂×Dic₇	—	2	2	no	—
90	C₂×Dic₇	C₂×C₃×C₄	1	2	yes	—
91	C₂×Dic₇	C₂×C₃×C₄	1	4	yes	complemented
92	C₄×D₇	—	2	8	no	—
93	C₂⁴×C₃	—	7	1	no	—
94	C₂²×C₃×C₄	—	7	1	no	—
95	C₂²×C₃×C₄	—	7	2	no	—
96	C₂²×C₃×C₄	—	7	4	no	—
97	C₂²×C₃×C₄	—	14	2	no	—
98	C₂²×C₃×C₄	—	14	2	no	—
99	C₂²×C₃×C₄	C₂×D₇	1	2	yes	—
100	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
101	C₃×C₂²⋊C₄	—	7	8	no	—
102	C₃×C₄²	—	2	2	no	—
103	C₃×C₄²	—	14	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	C₃×D₇	—	2	4	no	—
110	C₂³×C₄	—	7	1	no	—
111	C₂×C₂²⋊C₄	—	7	2	no	—
112	C₂×C₄²	—	7	1	no	—
113	C₂×C₄²	C₃×D₇	1	1	yes	characteristic, complemented
114	#95	—	7	2	no	—
115	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
116	C₂²×C₇	C₂²×C₃×C₄	1	2	yes	—
117	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
118	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
119	C₂²×C₇	C₃×#41	1	2	yes	—
120	C₂×D₇	—	2	2	no	—
121	C₂×D₇	—	2	2	no	—
122	C₂×D₇	—	2	4	no	—
123	C₂×D₇	—	2	4	no	—
124	C₂×D₇	C₃×C₄²	1	4	yes	complemented
125	C₄×C₇	—	2	4	no	—
126	C₄×C₇	C₃×C₂²⋊C₄	1	4	yes	complemented
127	Dic₇	—	2	2	no	—
128	Dic₇	—	2	4	no	—
129	C₂³×C₃	—	7	4	no	—
130	C₂³×C₃	—	7	2	no	—
131	C₂³×C₃	—	14	1	no	—
132	C₂³×C₃	—	14	2	no	—
133	C₂³×C₃	—	14	1	no	—
134	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	7	2	no	—
138	C₂×C₃×C₄	—	7	4	no	—
139	C₂×C₃×C₄	—	14	2	no	—
140	C₂×C₃×C₄	—	14	1	no	—
141	C₂×C₃×C₄	—	14	2	no	—
142	C₂×C₃×C₄	—	14	1	no	—
143	C₂×C₃×C₄	—	14	8	no	—
144	C₂×C₃×C₄	C₄×D₇	1	2	yes	—
145	C₂×C₃×C₄	C₄×D₇	1	4	yes	complemented
146	C₂×C₃×C₄	D₂₈	1	2	yes	—
147	C₂×C₃×C₄	C₂²⋊D₇	1	2	yes	—
148	C₃×C₇	C₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
149	C₂⁴	—	7	1	no	—
150	C₂²×C₄	—	7	1	no	—
151	C₂²×C₄	—	7	2	no	—
152	C₂²×C₄	—	7	4	no	—
153	C₂²×C₄	—	14	2	no	—
154	C₂²×C₄	—	14	2	no	—
155	C₂²×C₄	C₂×C₃×D₇	1	2	yes	—
156	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
157	C₂²⋊C₄	—	7	8	no	—
158	C₄²	—	2	2	no	—
159	C₄²	—	14	2	no	—
160	C₂×C₇	C₂×C₃×C₄²	1	1	yes	characteristic
161	C₂×C₇	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₇	C₃×#117	1	1	yes	characteristic
163	C₂×C₇	C₃×C₄×D₄	1	2	yes	—
164	C₂×C₇	C₃×C₄×D₄	1	2	yes	—
165	D₇	—	2	4	no	—
166	C₂²×C₃	—	7	4	no	—
167	C₂²×C₃	—	14	2	no	—
168	C₂²×C₃	—	14	4	no	—
169	C₂²×C₃	—	14	2	no	—
170	C₂²×C₃	—	14	4	no	—
171	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
172	C₂²×C₃	C₂×C₄×D₇	1	2	yes	—
173	C₂²×C₃	C₂×D₂₈	1	1	yes	characteristic
174	C₂²×C₃	#1157	1	2	yes	—
175	C₂²×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
176	C₃×C₄	—	2	4	no	—
177	C₃×C₄	—	14	4	no	—
178	C₃×C₄	—	14	2	no	—
179	C₃×C₄	(C₂×D₇)⋊C₄	1	4	yes	direct product factor
180	C₂³	—	7	4	no	—
181	C₂³	—	7	2	no	—
182	C₂³	—	14	1	no	—
183	C₂³	—	14	2	no	—
184	C₂³	—	14	1	no	—
185	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	7	4	no	—
189	C₂×C₄	—	7	2	no	—
190	C₂×C₄	—	14	2	no	—
191	C₂×C₄	—	14	1	no	—
192	C₂×C₄	—	14	2	no	—
193	C₂×C₄	—	14	1	no	—
194	C₂×C₄	—	14	8	no	—
195	C₂×C₄	C₃×C₄×D₇	1	2	yes	—
196	C₂×C₄	C₃×C₄×D₇	1	4	yes	complemented
197	C₂×C₄	C₃×D₂₈	1	2	yes	—
198	C₂×C₄	C₃×C₂²⋊D₇	1	2	yes	—
199	C₇	C₃×C₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
200	C₂×C₃	—	14	4	no	—
201	C₂×C₃	C₄²×D₇	1	1	yes	characteristic
202	C₂×C₃	#6435	1	1	yes	characteristic
203	C₂×C₃	C₄×D₂₈	1	2	yes	—
204	C₂×C₃	C₂×(C₂×D₇)⋊C₄	1	1	yes	characteristic
205	C₂×C₃	C₄×C₂²⋊D₇	1	2	yes	—
206	C₂²	—	7	4	no	—
207	C₂²	—	14	4	no	—
208	C₂²	—	14	2	no	—
209	C₂²	—	14	4	no	—
210	C₂²	—	14	2	no	—
211	C₂²	C₂×C₃×C₄×D₇	1	2	yes	—
212	C₂²	C₂×C₃×C₄×D₇	1	1	yes	characteristic
213	C₂²	C₂×C₃×D₂₈	1	1	yes	characteristic
214	C₂²	C₃×#1157	1	2	yes	—
215	C₂²	C₂×C₃×C₂²⋊D₇	1	1	yes	characteristic
216	C₄	—	2	4	no	—
217	C₄	—	14	4	no	—
218	C₄	—	14	2	no	—
219	C₄	C₃×(C₂×D₇)⋊C₄	1	4	yes	direct product factor
220	C₃	C₄×(C₂×D₇)⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
221	C₂	—	14	4	no	—
222	C₂	C₃×C₄²×D₇	1	1	yes	characteristic
223	C₂	C₃×#6435	1	1	yes	characteristic
224	C₂	C₃×C₄×D₂₈	1	2	yes	—
225	C₂	C₂×C₃×(C₂×D₇)⋊C₄	1	1	yes	characteristic
226	C₂	C₃×C₄×C₂²⋊D₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂×D₇
commutator	C₂×C₇	C₂×C₃×C₄²
Frattini	C₂³	C₂²×C₃×D₇
Fitting	C₂×C₃×C₄²×C₇	C₂
socle	C₂³×C₃×C₇	C₂³
prime radical	C₂³×C₃×D₇	C₂²
cosocle	C₂³×C₇	C₂³×C₃
hypercenter	C₂×C₃×C₄²	D₇
hypocenter	C₇	C₃×C₄×C₂²⋊C₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄²

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×C₃×C₄×Dic₇	C₂	semidirect
5. C₃×C₇⋊#95	C₂	semidirect
6. C₄×(C₂×D₇)⋊C₄	C₃	direct (N × H)
14. C₂×C₃×C₄×D₇	C₄	semidirect
16. C₃×(C₂×D₇)⋊C₄	C₄	direct (N × H)
19. C₂×C₄²×C₇	C₂×C₃	semidirect
21. C₂×C₄×Dic₇	C₂×C₃	semidirect
22. C₇⋊#95	C₂×C₃	semidirect
33. C₂×C₃×C₄×C₇	C₂×C₄	semidirect
41. C₂×C₃×Dic₇	C₂×C₄	semidirect
50. C₂×C₄×D₇	C₃×C₄	semidirect
52. (C₂×D₇)⋊C₄	C₃×C₄	direct (N × H)
57. C₂×C₃×C₄²	D₇	semidirect
68. C₂×C₃×D₇	C₄²	semidirect
70. C₃×C₄×C₇	C₂²⋊C₄	semidirect
83. C₂×C₄×C₇	C₂×C₃×C₄	semidirect
91. C₂×Dic₇	C₂×C₃×C₄	semidirect
113. C₂×C₄²	C₃×D₇	semidirect
124. C₂×D₇	C₃×C₄²	semidirect
126. C₄×C₇	C₃×C₂²⋊C₄	semidirect
145. C₂×C₃×C₄	C₄×D₇	semidirect
148. C₃×C₇	C₄×C₂²⋊C₄	semidirect
179. C₃×C₄	(C₂×D₇)⋊C₄	direct (N × H)
196. C₂×C₄	C₃×C₄×D₇	semidirect
199. C₇	C₃×C₄×C₂²⋊C₄	semidirect
219. C₄	C₃×(C₂×D₇)⋊C₄	direct (N × H)
220. C₃	C₄×(C₂×D₇)⋊C₄	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×C₄²×C₇	2
C₂×C₃×(C₂×D₇)⋊C₄	2
C₂×C₃×C₄×Dic₇	2
C₃×C₇⋊#95	2
C₄×(C₂×D₇)⋊C₄	3
C₃×C₄×C₂²⋊C₄	7

Subgroups — C3×C4×(C2×D7)⋊C4 #13055371

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×(C₂×D₇)⋊C₄ #13055371