Subgroups — C₃×C₄×Dic₇⋊C₄ #13055362

Order 1344.

388 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×Dic₇⋊C₄	C₂	1	2	yes	maximal
3	C₂×C₃×C₄×Dic₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₄×Dic₇	C₂	1	1	yes	maximal, characteristic
5	C₃×C₇⋊#95	C₂	1	2	yes	maximal
6	C₄×Dic₇⋊C₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
8	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
10	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
11	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
12	C₃×C₄²×C₇	—	2	2	no	—
13	C₃×Dic₇⋊C₄	C₄	1	8	yes	direct product factor
14	C₃×C₄×Dic₇	—	2	2	no	—
15	C₃×C₄×Dic₇	C₄	1	4	yes	complemented
16	C₂×C₄²×C₇	C₂×C₃	1	1	yes	characteristic
17	C₂×Dic₇⋊C₄	C₂×C₃	1	2	yes	—
18	C₂×C₄×Dic₇	C₂×C₃	1	1	yes	characteristic
19	C₂×C₄×Dic₇	C₂×C₃	1	1	yes	characteristic
20	C₇⋊#95	C₂×C₃	1	2	yes	—
21	C₃×C₄×C₄⋊C₄	—	7	1	no	maximal, Hall
22	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
23	C₂×C₃×C₄×C₇	—	2	2	no	—
24	C₂×C₃×C₄×C₇	—	2	2	no	—
25	C₂×C₃×C₄×C₇	C₂×C₄	1	4	yes	—
26	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
28	C₂×C₃×C₄×C₇	Q₈	1	2	yes	—
29	C₂×C₃×Dic₇	—	2	2	no	—
30	C₂×C₃×Dic₇	—	2	2	no	—
31	C₂×C₃×Dic₇	—	2	2	no	—
32	C₂×C₃×Dic₇	C₂×C₄	1	2	yes	—
33	C₂×C₃×Dic₇	C₂×C₄	1	4	yes	—
34	C₂×C₃×Dic₇	C₂×C₄	1	2	yes	—
35	C₂×C₃×Dic₇	C₂×C₄	1	4	yes	—
36	C₂²×C₄×C₇	C₂²×C₃	1	2	yes	—
37	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
38	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
39	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
40	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
41	C₄²×C₇	—	2	2	no	—
42	Dic₇⋊C₄	C₃×C₄	1	8	yes	direct product factor
43	C₄×Dic₇	—	2	2	no	—
44	C₄×Dic₇	C₃×C₄	1	4	yes	complemented
45	C₂×C₃×C₄⋊C₄	—	7	2	no	—
46	C₂×C₃×C₄²	—	7	1	no	—
47	C₂×C₃×C₄²	—	7	1	no	—
48	C₂×C₃×C₄²	D₇	1	1	yes	characteristic
49	C₃×#95	—	7	2	no	—
50	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₃×C₇	C₂²×C₄	1	2	yes	—
52	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₃×C₇	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₃×C₇	#41	1	2	yes	—
55	C₃×C₄×C₇	—	2	4	no	—
56	C₃×C₄×C₇	C₄⋊C₄	1	4	yes	complemented
57	C₃×Dic₇	—	2	2	no	—
58	C₃×Dic₇	—	2	4	no	—
59	C₃×Dic₇	C₄²	1	4	yes	complemented
60	C₄×C₄⋊C₄	—	7	1	no	Sylow
61	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
62	C₂×C₄×C₇	—	2	2	no	—
63	C₂×C₄×C₇	—	2	2	no	—
64	C₂×C₄×C₇	C₂×C₃×C₄	1	4	yes	—
65	C₂×C₄×C₇	C₂×C₃×C₄	1	2	yes	—
66	C₂×C₄×C₇	C₃×D₄	1	2	yes	—
67	C₂×C₄×C₇	C₃×Q₈	1	2	yes	—
68	C₂×Dic₇	—	2	2	no	—
69	C₂×Dic₇	—	2	2	no	—
70	C₂×Dic₇	—	2	2	no	—
71	C₂×Dic₇	C₂×C₃×C₄	1	2	yes	—
72	C₂×Dic₇	C₂×C₃×C₄	1	4	yes	—
73	C₂×Dic₇	C₂×C₃×C₄	1	2	yes	—
74	C₂×Dic₇	C₂×C₃×C₄	1	4	yes	—
75	C₂²×C₃×C₄	—	7	1	no	—
76	C₂²×C₃×C₄	—	7	1	no	—
77	C₂²×C₃×C₄	—	7	2	no	—
78	C₂²×C₃×C₄	C₂×D₇	1	2	yes	—
79	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
80	C₃×C₄⋊C₄	—	7	8	no	—
81	C₃×C₄²	—	2	2	no	—
82	C₃×C₄²	—	7	4	no	—
83	C₃×C₄²	—	14	2	no	—
84	C₂×C₃×C₇	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₃×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₃×C₇	#117	1	1	yes	characteristic
87	C₂×C₃×C₇	C₄×D₄	1	2	yes	—
88	C₂×C₃×C₇	C₄×Q₈	1	2	yes	—
89	C₂×C₄⋊C₄	—	7	2	no	—
90	C₂×C₄²	—	7	1	no	—
91	C₂×C₄²	—	7	1	no	—
92	C₂×C₄²	C₃×D₇	1	1	yes	characteristic
93	#95	—	7	2	no	—
94	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
95	C₂²×C₇	C₂²×C₃×C₄	1	2	yes	—
96	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
97	C₂²×C₇	C₂×C₃×Q₈	1	1	yes	characteristic
98	C₂²×C₇	C₃×#41	1	2	yes	—
99	C₄×C₇	—	2	4	no	—
100	C₄×C₇	C₃×C₄⋊C₄	1	4	yes	complemented
101	Dic₇	—	2	2	no	—
102	Dic₇	—	2	4	no	—
103	Dic₇	C₃×C₄²	1	4	yes	complemented
104	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	2	2	no	—
107	C₂×C₃×C₄	—	7	2	no	—
108	C₂×C₃×C₄	—	7	4	no	—
109	C₂×C₃×C₄	—	7	2	no	—
110	C₂×C₃×C₄	—	7	4	no	—
111	C₂×C₃×C₄	—	14	2	no	—
112	C₂×C₃×C₄	—	14	2	no	—
113	C₂×C₃×C₄	—	14	2	no	—
114	C₂×C₃×C₄	Dic₁₄	1	2	yes	—
115	C₂×C₃×C₄	C₄×D₇	1	4	yes	—
116	C₂×C₃×C₄	C₄×D₇	1	2	yes	—
117	C₂×C₃×C₄	C₂²⋊D₇	1	2	yes	—
118	C₃×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
119	C₂²×C₄	—	7	1	no	—
120	C₂²×C₄	—	7	1	no	—
121	C₂²×C₄	—	7	2	no	—
122	C₂²×C₄	C₂×C₃×D₇	1	2	yes	—
123	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
124	C₄⋊C₄	—	7	8	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	7	4	no	—
127	C₄²	—	14	2	no	—
128	C₂×C₇	C₂×C₃×C₄²	1	1	yes	characteristic
129	C₂×C₇	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₇	C₃×#117	1	1	yes	characteristic
131	C₂×C₇	C₃×C₄×D₄	1	2	yes	—
132	C₂×C₇	C₃×C₄×Q₈	1	2	yes	—
133	C₂²×C₃	C₂×Dic₁₄	1	1	yes	characteristic
134	C₂²×C₃	C₂×C₄×D₇	1	2	yes	—
135	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
136	C₂²×C₃	#1157	1	2	yes	—
137	C₂²×C₃	C₂×C₂²⋊D₇	1	1	yes	characteristic
138	C₃×C₄	—	2	4	no	—
139	C₃×C₄	—	7	4	no	—
140	C₃×C₄	—	14	2	no	—
141	C₃×C₄	—	14	4	no	—
142	C₃×C₄	Dic₇⋊C₄	1	4	yes	direct product factor
143	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	7	2	no	—
147	C₂×C₄	—	7	4	no	—
148	C₂×C₄	—	7	2	no	—
149	C₂×C₄	—	7	4	no	—
150	C₂×C₄	—	14	2	no	—
151	C₂×C₄	—	14	2	no	—
152	C₂×C₄	—	14	2	no	—
153	C₂×C₄	C₃×Dic₁₄	1	2	yes	—
154	C₂×C₄	C₃×C₄×D₇	1	4	yes	—
155	C₂×C₄	C₃×C₄×D₇	1	2	yes	—
156	C₂×C₄	C₃×C₂²⋊D₇	1	2	yes	—
157	C₇	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
158	C₂×C₃	C₄×Dic₁₄	1	2	yes	—
159	C₂×C₃	C₄²×D₇	1	1	yes	characteristic
160	C₂×C₃	#6435	1	1	yes	characteristic
161	C₂×C₃	C₂×Dic₇⋊C₄	1	1	yes	characteristic
162	C₂×C₃	C₄×C₂²⋊D₇	1	2	yes	—
163	C₂²	C₂×C₃×Dic₁₄	1	1	yes	characteristic
164	C₂²	C₂×C₃×C₄×D₇	1	1	yes	characteristic
165	C₂²	C₂×C₃×C₄×D₇	1	2	yes	—
166	C₂²	C₃×#1157	1	2	yes	—
167	C₂²	C₂×C₃×C₂²⋊D₇	1	1	yes	characteristic
168	C₄	—	2	4	no	—
169	C₄	—	7	4	no	—
170	C₄	—	14	4	no	—
171	C₄	—	14	2	no	—
172	C₄	C₃×Dic₇⋊C₄	1	4	yes	direct product factor
173	C₃	C₄×Dic₇⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
174	C₂	C₃×C₄×Dic₁₄	1	2	yes	—
175	C₂	C₃×C₄²×D₇	1	1	yes	characteristic
176	C₂	C₃×#6435	1	1	yes	characteristic
177	C₂	C₂×C₃×Dic₇⋊C₄	1	1	yes	characteristic
178	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, prime radical	C₂³×C₃×C₇	C₂³
cosocle	C₂³×C₇	C₂³×C₃
hypercenter	C₂×C₃×C₄²	D₇
hypocenter	C₇	C₃×C₄×C₄⋊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
6. C₄×Dic₇⋊C₄	C₃	direct (N × H)
13. C₃×Dic₇⋊C₄	C₄	direct (N × H)
15. C₃×C₄×Dic₇	C₄	semidirect
42. Dic₇⋊C₄	C₃×C₄	direct (N × H)
44. C₄×Dic₇	C₃×C₄	semidirect
56. C₃×C₄×C₇	C₄⋊C₄	semidirect
59. C₃×Dic₇	C₄²	semidirect
100. C₄×C₇	C₃×C₄⋊C₄	semidirect
103. Dic₇	C₃×C₄²	semidirect
118. C₃×C₇	C₄×C₄⋊C₄	semidirect
142. C₃×C₄	Dic₇⋊C₄	direct (N × H)
157. C₇	C₃×C₄×C₄⋊C₄	semidirect
172. C₄	C₃×Dic₇⋊C₄	direct (N × H)
173. C₃	C₄×Dic₇⋊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
C₂×C₃×C₄²×C₇	2
C₂×C₃×Dic₇⋊C₄	2
C₂×C₃×C₄×Dic₇	2
C₃×C₇⋊#95	2
C₄×Dic₇⋊C₄	3
C₃×C₄×C₄⋊C₄	7

Subgroups — C3×C4×Dic7⋊C4 #13055362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×Dic₇⋊C₄ #13055362