Subgroups — C₄⋊C₄×Dic₂₁ #13056605

Order 1344.

388 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₂₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₂₁	C₂	1	2	yes	maximal
4	C₂×C₄×Dic₂₁	C₂	1	1	yes	maximal, characteristic
5	#10620670	C₂	1	2	yes	maximal
6	C₄⋊C₄×Dic₇	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₄×C₇	C₂²	1	2	yes	—
8	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₂²×Dic₂₁	C₂²	1	2	yes	—
10	C₂²×Dic₂₁	C₂²	1	1	yes	characteristic
11	C₂²×Dic₂₁	C₂²	1	1	yes	characteristic
12	C₃×C₇×C₄⋊C₄	C₄	1	4	yes	complemented
13	C₄⋊Dic₂₁	C₄	1	4	yes	complemented
14	C₄×Dic₂₁	—	2	4	no	—
15	C₄×Dic₂₁	C₄	1	4	yes	complemented
16	C₂×C₇×C₄⋊C₄	D₃	1	1	yes	characteristic
17	C₂×C₄⋊Dic₇	—	3	1	no	—
18	C₂×C₄×Dic₇	—	3	1	no	—
19	C₂×C₄×Dic₇	—	3	2	no	—
20	C₇⋊#95	—	3	2	no	—
21	Dic₃×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	2	yes	—
26	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₄×C₇	C₂×C₄	1	4	yes	—
28	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
29	C₂×Dic₂₁	—	2	2	no	—
30	C₂×Dic₂₁	—	2	1	no	—
31	C₂×Dic₂₁	—	2	1	no	—
32	C₂×Dic₂₁	—	2	2	no	—
33	C₂×Dic₂₁	C₂×C₄	1	2	yes	—
34	C₂×Dic₂₁	C₂×C₄	1	4	yes	—
35	C₂×Dic₂₁	C₂×C₄	1	2	yes	—
36	C₂×Dic₂₁	D₄	1	2	yes	—
37	C₂×Dic₂₁	Q₈	1	2	yes	—
38	C₂²×C₄×C₇	C₂×D₃	1	2	yes	—
39	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
40	C₂²×Dic₇	—	3	1	no	—
41	C₂²×Dic₇	—	3	1	no	—
42	C₂²×Dic₇	—	3	2	no	—
43	C₇×C₄⋊C₄	Dic₃	1	4	yes	complemented
44	C₄⋊Dic₇	—	3	4	no	—
45	C₄×Dic₇	—	3	4	no	—
46	C₄×Dic₇	—	6	4	no	—
47	C₂×C₃×C₄⋊C₄	D₇	1	1	yes	characteristic
48	C₂×C₄⋊Dic₃	—	7	1	no	—
49	C₂×C₄×Dic₃	—	7	1	no	—
50	C₂×C₄×Dic₃	—	7	2	no	—
51	C₃⋊#95	—	7	2	no	—
52	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₃×C₇	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₃×C₇	#41	1	1	yes	characteristic
58	C₂²×C₃×C₇	#41	1	1	yes	characteristic
59	C₃×C₄×C₇	—	2	4	no	—
60	C₃×C₄×C₇	C₄²	1	4	yes	complemented
61	Dic₂₁	—	2	2	no	—
62	Dic₂₁	—	2	4	no	—
63	Dic₂₁	C₄⋊C₄	1	4	yes	direct product factor
64	C₄×C₄⋊C₄	—	21	1	no	Sylow
65	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
66	C₂×C₄×C₇	—	2	2	no	—
67	C₂×C₄×C₇	—	2	2	no	—
68	C₂×C₄×C₇	C₄×D₃	1	2	yes	—
69	C₂×C₄×C₇	C₄×D₃	1	2	yes	—
70	C₂×C₄×C₇	C₂×Dic₃	1	4	yes	—
71	C₂×C₄×C₇	C₂×Dic₃	1	2	yes	—
72	C₂×Dic₇	—	3	2	no	—
73	C₂×Dic₇	—	3	2	no	—
74	C₂×Dic₇	—	3	2	no	—
75	C₂×Dic₇	—	3	2	no	—
76	C₂×Dic₇	—	3	4	no	—
77	C₂×Dic₇	—	6	2	no	—
78	C₂×Dic₇	—	6	1	no	—
79	C₂×Dic₇	—	6	1	no	—
80	C₂×Dic₇	—	6	2	no	—
81	C₂²×C₃×C₄	C₂×D₇	1	2	yes	—
82	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
83	C₂²×Dic₃	—	7	1	no	—
84	C₂²×Dic₃	—	7	1	no	—
85	C₂²×Dic₃	—	7	2	no	—
86	C₃×C₄⋊C₄	Dic₇	1	4	yes	complemented
87	C₄⋊Dic₃	—	7	4	no	—
88	C₄×Dic₃	—	7	4	no	—
89	C₄×Dic₃	—	14	4	no	—
90	C₂×C₃×C₇	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₃×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₃×C₇	#117	1	1	yes	characteristic
93	C₂×C₃×C₇	C₄×D₄	1	1	yes	characteristic
94	C₂×C₃×C₇	C₄×D₄	1	1	yes	characteristic
95	C₂×C₃×C₇	C₄×Q₈	1	1	yes	characteristic
96	C₂×C₃×C₇	C₄×Q₈	1	1	yes	characteristic
97	C₂×C₄⋊C₄	—	21	1	no	—
98	C₂×C₄⋊C₄	D₂₁	1	1	yes	characteristic
99	C₂×C₄²	—	21	1	no	—
100	C₂×C₄²	—	21	2	no	—
101	#95	—	21	2	no	—
102	C₂²×C₇	C₂×C₄×D₃	1	1	yes	characteristic
103	C₂²×C₇	C₂×C₄×D₃	1	1	yes	characteristic
104	C₂²×C₇	D₃×D₄	1	1	yes	characteristic
105	C₂²×C₇	#237	1	1	yes	characteristic
106	C₂²×C₇	D₃×Q₈	1	1	yes	characteristic
107	C₂²×C₇	#239	1	1	yes	characteristic
108	C₂²×C₇	C₂²×Dic₃	1	1	yes	characteristic
109	C₄×C₇	—	2	4	no	—
110	C₄×C₇	C₄×Dic₃	1	4	yes	complemented
111	Dic₇	—	3	4	no	—
112	Dic₇	—	6	2	no	—
113	Dic₇	—	6	4	no	—
114	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	C₄×D₇	1	2	yes	—
118	C₂×C₃×C₄	C₄×D₇	1	2	yes	—
119	C₂×C₃×C₄	C₂×Dic₇	1	4	yes	—
120	C₂×C₃×C₄	C₂×Dic₇	1	2	yes	—
121	C₂×Dic₃	—	7	2	no	—
122	C₂×Dic₃	—	7	2	no	—
123	C₂×Dic₃	—	7	2	no	—
124	C₂×Dic₃	—	7	2	no	—
125	C₂×Dic₃	—	7	4	no	—
126	C₂×Dic₃	—	14	2	no	—
127	C₂×Dic₃	—	14	1	no	—
128	C₂×Dic₃	—	14	1	no	—
129	C₂×Dic₃	—	14	2	no	—
130	C₃×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
131	C₂²×C₄	—	21	1	no	—
132	C₂²×C₄	—	21	1	no	—
133	C₂²×C₄	—	21	2	no	—
134	C₂²×C₄	C₂×D₂₁	1	2	yes	—
135	C₂²×C₄	C₂×D₂₁	1	1	yes	characteristic
136	C₄⋊C₄	—	21	4	no	—
137	C₄⋊C₄	Dic₂₁	1	4	yes	direct product factor
138	C₄²	—	21	4	no	—
139	C₄²	—	42	4	no	—
140	C₂×C₇	#887	1	1	yes	characteristic
141	C₂×C₇	D₃×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₇	#892	1	1	yes	characteristic
143	C₂×C₇	#893	1	1	yes	characteristic
144	C₂×C₇	C₂×C₄×Dic₃	1	1	yes	characteristic
145	C₂×C₇	D₄×Dic₃	1	1	yes	characteristic
146	C₂×C₇	Q₈×Dic₃	1	1	yes	characteristic
147	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
148	C₂²×C₃	C₂×C₄×D₇	1	1	yes	characteristic
149	C₂²×C₃	D₄×D₇	1	1	yes	characteristic
150	C₂²×C₃	#1159	1	1	yes	characteristic
151	C₂²×C₃	Q₈×D₇	1	1	yes	characteristic
152	C₂²×C₃	#1161	1	1	yes	characteristic
153	C₂²×C₃	C₂²×Dic₇	1	1	yes	characteristic
154	C₃×C₄	—	2	4	no	—
155	C₃×C₄	C₄×Dic₇	1	4	yes	complemented
156	Dic₃	—	7	4	no	—
157	Dic₃	—	14	2	no	—
158	Dic₃	—	14	4	no	—
159	C₂³	C₂²×D₂₁	1	1	yes	characteristic
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	21	2	no	—
163	C₂×C₄	—	21	2	no	—
164	C₂×C₄	—	21	2	no	—
165	C₂×C₄	—	21	2	no	—
166	C₂×C₄	—	21	4	no	—
167	C₂×C₄	—	42	2	no	—
168	C₂×C₄	—	42	1	no	—
169	C₂×C₄	—	42	1	no	—
170	C₂×C₄	—	42	2	no	—
171	C₂×C₄	C₄×D₂₁	1	2	yes	—
172	C₂×C₄	C₄×D₂₁	1	2	yes	—
173	C₂×C₄	C₂×Dic₂₁	1	4	yes	—
174	C₂×C₄	C₂×Dic₂₁	1	2	yes	—
175	C₇	Dic₃×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
176	C₂×C₃	#6450	1	1	yes	characteristic
177	C₂×C₃	D₇×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₃	#6455	1	1	yes	characteristic
179	C₂×C₃	#6456	1	1	yes	characteristic
180	C₂×C₃	C₂×C₄×Dic₇	1	1	yes	characteristic
181	C₂×C₃	D₄×Dic₇	1	1	yes	characteristic
182	C₂×C₃	Q₈×Dic₇	1	1	yes	characteristic
183	C₂²	C₂×C₄×D₂₁	1	1	yes	characteristic
184	C₂²	C₂×C₄×D₂₁	1	1	yes	characteristic
185	C₂²	D₄×D₂₁	1	1	yes	characteristic
186	C₂²	#66730	1	1	yes	characteristic
187	C₂²	Q₈×D₂₁	1	1	yes	characteristic
188	C₂²	#66732	1	1	yes	characteristic
189	C₂²	C₂²×Dic₂₁	1	1	yes	characteristic
190	C₄	—	2	4	no	—
191	C₄	—	21	4	no	—
192	C₄	—	42	2	no	—
193	C₄	—	42	4	no	—
194	C₄	C₄×Dic₂₁	1	4	yes	complemented
195	C₃	C₄⋊C₄×Dic₇	1	1	yes	characteristic, complemented, Sylow
196	C₂	#10621370	1	1	yes	characteristic
197	C₂	C₄⋊C₄×D₂₁	1	1	yes	characteristic
198	C₂	#10621375	1	1	yes	characteristic
199	C₂	#10621376	1	1	yes	characteristic
200	C₂	C₂×C₄×Dic₂₁	1	1	yes	characteristic
201	C₂	D₄×Dic₂₁	1	1	yes	characteristic
202	C₂	Q₈×Dic₂₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₁
commutator	C₂×C₃×C₇	C₂×C₄²
Fitting	C₂×C₃×C₇×C₄⋊C₄	C₂
socle, prime radical, cosocle	C₂³×C₃×C₇	C₂³
hypercenter	C₂×C₄⋊C₄	D₂₁
hypocenter	C₃×C₇	C₄×C₄⋊C₄
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
12. C₃×C₇×C₄⋊C₄	C₄	semidirect
13. C₄⋊Dic₂₁	C₄	semidirect
15. C₄×Dic₂₁	C₄	semidirect
43. C₇×C₄⋊C₄	Dic₃	semidirect
60. C₃×C₄×C₇	C₄²	semidirect
63. Dic₂₁	C₄⋊C₄	direct (N × H)
86. C₃×C₄⋊C₄	Dic₇	semidirect
110. C₄×C₇	C₄×Dic₃	semidirect
130. C₃×C₇	C₄×C₄⋊C₄	semidirect
137. C₄⋊C₄	Dic₂₁	direct (N × H)
155. C₃×C₄	C₄×Dic₇	semidirect
175. C₇	Dic₃×C₄⋊C₄	semidirect
194. C₄	C₄×Dic₂₁	semidirect
195. C₃	C₄⋊C₄×Dic₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₇×C₄⋊C₄	2
C₂×C₄⋊Dic₂₁	2
C₂×C₄×Dic₂₁	2
#10620670	2
C₄⋊C₄×Dic₇	3
Dic₃×C₄⋊C₄	7

Subgroups — C4⋊C4×Dic21 #13056605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊C₄×Dic₂₁ #13056605