Subgroups — C₃×C₂²⋊C₄×Dic₇ #13055374

Order 1344.

516 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×Dic₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₇×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₂²⋊Dic₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₄×Dic₇	C₂	1	2	yes	maximal, complemented
5	C₃×C₇⋊#95	C₂	1	2	yes	maximal, complemented
6	C₂²⋊C₄×Dic₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂⁴×C₃×C₇	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
9	C₂²×C₃×Dic₇	—	2	4	no	—
10	C₂²×C₃×Dic₇	C₄	1	4	yes	complemented
11	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
12	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
13	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
14	C₃×C₇×C₂²⋊C₄	C₄	1	4	yes	complemented
15	C₃×C₂²⋊Dic₇	C₄	1	4	yes	complemented
16	C₃×C₄×Dic₇	—	2	4	no	—
17	C₂³×Dic₇	C₂×C₃	1	1	yes	characteristic
18	C₂×C₇×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
19	C₂×C₂²⋊Dic₇	C₂×C₃	1	1	yes	characteristic
20	C₂×C₄×Dic₇	C₂×C₃	1	2	yes	complemented
21	C₇⋊#95	C₂×C₃	1	2	yes	complemented
22	C₃×C₄×C₂²⋊C₄	—	7	1	no	maximal, Hall
23	C₂³×C₃×C₇	—	2	2	no	—
24	C₂³×C₃×C₇	—	2	2	no	—
25	C₂³×C₃×C₇	C₂×C₄	1	2	yes	—
26	C₂³×C₃×C₇	C₂×C₄	1	2	yes	—
27	C₂³×C₃×C₇	C₂×C₄	1	2	yes	—
28	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
29	C₂×C₃×C₄×C₇	—	2	2	no	—
30	C₂×C₃×C₄×C₇	—	2	2	no	—
31	C₂×C₃×C₄×C₇	C₂×C₄	1	4	yes	complemented
32	C₂×C₃×Dic₇	—	2	2	no	—
33	C₂×C₃×Dic₇	—	2	2	no	—
34	C₂×C₃×Dic₇	—	2	2	no	—
35	C₂×C₃×Dic₇	—	2	8	no	—
36	C₂×C₃×Dic₇	C₂×C₄	1	2	yes	—
37	C₂×C₃×Dic₇	C₂×C₄	1	2	yes	—
38	C₂×C₃×Dic₇	C₂×C₄	1	4	yes	complemented
39	C₂×C₃×Dic₇	D₄	1	4	yes	—
40	C₂⁴×C₇	C₂²×C₃	1	1	yes	characteristic
41	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
42	C₂²×Dic₇	—	2	4	no	—
43	C₂²×Dic₇	C₃×C₄	1	4	yes	complemented
44	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
45	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
46	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
47	C₇×C₂²⋊C₄	C₃×C₄	1	4	yes	complemented
48	C₂²⋊Dic₇	C₃×C₄	1	4	yes	complemented
49	C₄×Dic₇	—	2	4	no	—
50	C₂³×C₃×C₄	—	7	1	no	—
51	C₂×C₃×C₂²⋊C₄	—	7	1	no	—
52	C₂×C₃×C₂²⋊C₄	D₇	1	1	yes	characteristic
53	C₂×C₃×C₄²	—	7	2	no	—
54	C₃×#95	—	7	2	no	—
55	C₂²×C₃×C₇	—	2	2	no	—
56	C₂²×C₃×C₇	—	2	2	no	—
57	C₂²×C₃×C₇	—	2	4	no	—
58	C₂²×C₃×C₇	—	2	4	no	—
59	C₂²×C₃×C₇	C₄²	1	4	yes	complemented
60	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
64	C₂²×C₃×C₇	#41	1	2	yes	—
65	C₃×C₄×C₇	—	2	4	no	—
66	C₃×Dic₇	—	2	2	no	—
67	C₃×Dic₇	—	2	4	no	—
68	C₃×Dic₇	C₂²⋊C₄	1	4	yes	direct product factor
69	C₄×C₂²⋊C₄	—	7	1	no	Sylow
70	C₂³×C₇	—	2	2	no	—
71	C₂³×C₇	—	2	2	no	—
72	C₂³×C₇	C₂×C₃×C₄	1	2	yes	—
73	C₂³×C₇	C₂×C₃×C₄	1	2	yes	—
74	C₂³×C₇	C₂×C₃×C₄	1	2	yes	—
75	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
76	C₂×C₄×C₇	—	2	2	no	—
77	C₂×C₄×C₇	—	2	2	no	—
78	C₂×C₄×C₇	C₂×C₃×C₄	1	4	yes	complemented
79	C₂×Dic₇	—	2	2	no	—
80	C₂×Dic₇	—	2	2	no	—
81	C₂×Dic₇	—	2	2	no	—
82	C₂×Dic₇	—	2	8	no	—
83	C₂×Dic₇	C₂×C₃×C₄	1	2	yes	—
84	C₂×Dic₇	C₂×C₃×C₄	1	2	yes	—
85	C₂×Dic₇	C₂×C₃×C₄	1	4	yes	complemented
86	C₂×Dic₇	C₃×D₄	1	4	yes	—
87	C₂⁴×C₃	C₂×D₇	1	1	yes	characteristic
88	C₂²×C₃×C₄	—	7	1	no	—
89	C₂²×C₃×C₄	—	7	4	no	—
90	C₂²×C₃×C₄	—	7	1	no	—
91	C₂²×C₃×C₄	—	7	2	no	—
92	C₂²×C₃×C₄	—	14	4	no	—
93	C₂²×C₃×C₄	C₂×D₇	1	2	yes	—
94	C₃×C₂²⋊C₄	—	7	4	no	—
95	C₃×C₂²⋊C₄	Dic₇	1	4	yes	direct product factor
96	C₃×C₄²	—	14	4	no	—
97	C₂×C₃×C₇	—	2	4	no	—
98	C₂×C₃×C₇	C₂×C₄²	1	1	yes	characteristic
99	C₂×C₃×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₃×C₇	#117	1	1	yes	characteristic
101	C₂×C₃×C₇	C₄×D₄	1	2	yes	—
102	C₂×C₃×C₇	C₄×D₄	1	2	yes	—
103	C₂³×C₄	—	7	1	no	—
104	C₂×C₂²⋊C₄	—	7	1	no	—
105	C₂×C₂²⋊C₄	C₃×D₇	1	1	yes	characteristic
106	C₂×C₄²	—	7	2	no	—
107	#95	—	7	2	no	—
108	C₂²×C₇	—	2	2	no	—
109	C₂²×C₇	—	2	2	no	—
110	C₂²×C₇	—	2	4	no	—
111	C₂²×C₇	—	2	4	no	—
112	C₂²×C₇	C₃×C₄²	1	4	yes	complemented
113	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
114	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
115	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
116	C₂²×C₇	C₂×C₃×D₄	1	2	yes	—
117	C₂²×C₇	C₃×#41	1	2	yes	—
118	C₄×C₇	—	2	4	no	—
119	Dic₇	—	2	2	no	—
120	Dic₇	—	2	4	no	—
121	Dic₇	C₃×C₂²⋊C₄	1	4	yes	direct product factor
122	C₂³×C₃	—	2	2	no	—
123	C₂³×C₃	—	2	2	no	—
124	C₂³×C₃	C₄×D₇	1	2	yes	—
125	C₂³×C₃	C₄×D₇	1	2	yes	—
126	C₂³×C₃	C₂×Dic₇	1	2	yes	—
127	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	7	2	no	—
131	C₂×C₃×C₄	—	7	4	no	—
132	C₂×C₃×C₄	—	7	2	no	—
133	C₂×C₃×C₄	—	7	4	no	—
134	C₂×C₃×C₄	—	14	8	no	—
135	C₂×C₃×C₄	—	14	2	no	—
136	C₂×C₃×C₄	—	14	2	no	—
137	C₂×C₃×C₄	—	14	2	no	—
138	C₂×C₃×C₄	C₂×Dic₇	1	4	yes	complemented
139	C₃×C₇	C₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
140	C₂⁴	C₂×C₃×D₇	1	1	yes	characteristic
141	C₂²×C₄	—	7	1	no	—
142	C₂²×C₄	—	7	4	no	—
143	C₂²×C₄	—	7	1	no	—
144	C₂²×C₄	—	7	2	no	—
145	C₂²×C₄	—	14	4	no	—
146	C₂²×C₄	C₂×C₃×D₇	1	2	yes	—
147	C₂²⋊C₄	—	7	4	no	—
148	C₂²⋊C₄	C₃×Dic₇	1	4	yes	direct product factor
149	C₄²	—	14	4	no	—
150	C₂×C₇	—	2	4	no	—
151	C₂×C₇	C₂×C₃×C₄²	1	1	yes	characteristic
152	C₂×C₇	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
153	C₂×C₇	C₃×#117	1	1	yes	characteristic
154	C₂×C₇	C₃×C₄×D₄	1	2	yes	—
155	C₂×C₇	C₃×C₄×D₄	1	2	yes	—
156	C₂²×C₃	—	2	2	no	—
157	C₂²×C₃	—	2	2	no	—
158	C₂²×C₃	—	2	4	no	—
159	C₂²×C₃	—	2	4	no	—
160	C₂²×C₃	C₄×Dic₇	1	4	yes	complemented
161	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
162	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
163	C₂²×C₃	D₄×D₇	1	2	yes	—
164	C₂²×C₃	#1159	1	2	yes	—
165	C₂²×C₃	C₂²×Dic₇	1	1	yes	characteristic
166	C₃×C₄	—	2	4	no	—
167	C₃×C₄	—	7	4	no	—
168	C₃×C₄	—	14	2	no	—
169	C₃×C₄	—	14	4	no	—
170	C₂³	—	2	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	C₃×C₄×D₇	1	2	yes	—
173	C₂³	C₃×C₄×D₇	1	2	yes	—
174	C₂³	C₂×C₃×Dic₇	1	2	yes	—
175	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	7	2	no	—
179	C₂×C₄	—	7	4	no	—
180	C₂×C₄	—	7	4	no	—
181	C₂×C₄	—	7	2	no	—
182	C₂×C₄	—	14	8	no	—
183	C₂×C₄	—	14	2	no	—
184	C₂×C₄	—	14	2	no	—
185	C₂×C₄	—	14	2	no	—
186	C₂×C₄	C₂×C₃×Dic₇	1	4	yes	complemented
187	C₇	C₃×C₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
188	C₂×C₃	—	2	4	no	—
189	C₂×C₃	#6440	1	1	yes	characteristic
190	C₂×C₃	D₇×C₂²⋊C₄	1	1	yes	characteristic
191	C₂×C₃	(C₂²⋊D₇)⋊C₄	1	2	yes	—
192	C₂×C₃	C₂×C₄×Dic₇	1	1	yes	characteristic
193	C₂×C₃	D₄×Dic₇	1	2	yes	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	4	no	—
197	C₂²	—	2	4	no	—
198	C₂²	C₃×C₄×Dic₇	1	4	yes	complemented
199	C₂²	C₂×C₃×C₄×D₇	1	1	yes	characteristic
200	C₂²	C₂×C₃×C₄×D₇	1	1	yes	characteristic
201	C₂²	C₃×D₄×D₇	1	2	yes	—
202	C₂²	C₃×#1159	1	2	yes	—
203	C₂²	C₂²×C₃×Dic₇	1	1	yes	characteristic
204	C₄	—	2	4	no	—
205	C₄	—	7	4	no	—
206	C₄	—	14	2	no	—
207	C₄	—	14	4	no	—
208	C₃	C₂²⋊C₄×Dic₇	1	1	yes	characteristic, direct product factor, Sylow
209	C₂	—	2	4	no	—
210	C₂	C₃×#6440	1	1	yes	characteristic
211	C₂	C₃×D₇×C₂²⋊C₄	1	1	yes	characteristic
212	C₂	C₃×(C₂²⋊D₇)⋊C₄	1	2	yes	—
213	C₂	C₂×C₃×C₄×Dic₇	1	1	yes	characteristic
214	C₂	C₃×D₄×Dic₇	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂²×D₇
commutator	C₂×C₇	C₂×C₃×C₄²
Frattini	C₂³	C₂²×C₃×D₇
Fitting	C₂×C₃×C₇×C₂²⋊C₄	C₂
socle	C₂³×C₃×C₇	C₂³
prime radical	C₂⁴×C₃×C₇	C₂²
cosocle	C₂³×C₇	C₂³×C₃
hypercenter	C₂×C₃×C₂²⋊C₄	D₇
hypocenter	C₇	C₃×C₄×C₂²⋊C₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₂×C₃×C₄×Dic₇	C₂	semidirect
5. C₃×C₇⋊#95	C₂	semidirect
6. C₂²⋊C₄×Dic₇	C₃	direct (N × H)
10. C₂²×C₃×Dic₇	C₄	semidirect
14. C₃×C₇×C₂²⋊C₄	C₄	semidirect
15. C₃×C₂²⋊Dic₇	C₄	semidirect
20. C₂×C₄×Dic₇	C₂×C₃	semidirect
21. C₇⋊#95	C₂×C₃	semidirect
31. C₂×C₃×C₄×C₇	C₂×C₄	semidirect
38. C₂×C₃×Dic₇	C₂×C₄	semidirect
43. C₂²×Dic₇	C₃×C₄	semidirect
47. C₇×C₂²⋊C₄	C₃×C₄	semidirect
48. C₂²⋊Dic₇	C₃×C₄	semidirect
59. C₂²×C₃×C₇	C₄²	semidirect
68. C₃×Dic₇	C₂²⋊C₄	direct (N × H)
78. C₂×C₄×C₇	C₂×C₃×C₄	semidirect
85. C₂×Dic₇	C₂×C₃×C₄	semidirect
95. C₃×C₂²⋊C₄	Dic₇	direct (N × H)
112. C₂²×C₇	C₃×C₄²	semidirect
121. Dic₇	C₃×C₂²⋊C₄	direct (N × H)
138. C₂×C₃×C₄	C₂×Dic₇	semidirect
139. C₃×C₇	C₄×C₂²⋊C₄	semidirect
148. C₂²⋊C₄	C₃×Dic₇	direct (N × H)
160. C₂²×C₃	C₄×Dic₇	semidirect
186. C₂×C₄	C₂×C₃×Dic₇	semidirect
187. C₇	C₃×C₄×C₂²⋊C₄	semidirect
198. C₂²	C₃×C₄×Dic₇	semidirect
208. C₃	C₂²⋊C₄×Dic₇	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×C₃×Dic₇	2
C₂×C₃×C₇×C₂²⋊C₄	2
C₂×C₃×C₂²⋊Dic₇	2
C₂×C₃×C₄×Dic₇	2
C₃×C₇⋊#95	2
C₂²⋊C₄×Dic₇	3
C₃×C₄×C₂²⋊C₄	7

Subgroups — C3×C22⋊C4×Dic7 #13055374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₂²⋊C₄×Dic₇ #13055374