Subgroups — C₄⋊C₄×#672 #12014644

Order 1280.

398 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#672	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#672	C₂	1	2	yes	maximal
4	C₂×C₄×#672	C₂	1	1	yes	maximal, characteristic
5	#10598148	C₂	1	2	yes	maximal
6	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₄×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
10	C₂²×#672	C₂²	1	1	yes	characteristic
11	C₂²×#672	C₂²	1	2	yes	—
12	C₂²×#672	C₂²	1	1	yes	characteristic
13	C₅×#336	C₄	1	2	yes	—
14	C₄⋊#672	C₄	1	4	yes	complemented
15	C₄×#672	—	2	4	no	—
16	C₄×#672	C₄	1	4	yes	complemented
17	C₁₆×C₄⋊C₄	—	5	1	no	maximal, Sylow
18	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
20	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₅×C₄⋊C₄	C₈	1	2	yes	—
22	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
24	C₅×C₄⋊C₈	C₈	1	4	yes	—
25	C₄×C₅×C₈	—	2	2	no	—
26	C₄×C₅×C₈	—	2	2	no	—
27	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₅×#95	C₈	1	2	yes	—
30	C₂×#672	—	2	2	no	—
31	C₂×#672	—	2	1	no	—
32	C₂×#672	—	2	1	no	—
33	C₂×#672	—	2	2	no	—
34	C₂×#672	C₂×C₄	1	2	yes	—
35	C₂×#672	C₂×C₄	1	4	yes	—
36	C₂×#672	C₂×C₄	1	2	yes	—
37	C₂×#672	D₄	1	2	yes	—
38	C₂×#672	Q₈	1	2	yes	—
39	C₂×C₄⋊C₁₆	—	5	1	no	—
40	C₂×C₄×C₁₆	—	5	1	no	—
41	C₂×C₄×C₁₆	—	5	2	no	—
42	C₈×C₄⋊C₄	D₅	1	1	yes	characteristic
43	#1374	—	5	2	no	—
44	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
46	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₅×C₈	—	2	2	no	—
49	C₂×C₅×C₈	—	2	1	no	—
50	C₂×C₅×C₈	—	2	1	no	—
51	C₂×C₅×C₈	—	2	2	no	—
52	C₂×C₅×C₈	C₄²	1	2	yes	—
53	C₂×C₅×C₈	C₂×C₈	1	4	yes	—
54	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₅×C₈	#41	1	1	yes	characteristic
59	C₂×C₅×C₈	#41	1	1	yes	characteristic
60	C₅×C₄⋊C₄	C₁₆	1	8	yes	complemented
61	C₄²×C₅	—	2	2	no	—
62	C₄²×C₅	—	2	2	no	—
63	C₄²×C₅	C₄²	1	2	yes	—
64	C₄²×C₅	C₂×C₈	1	2	yes	—
65	#672	—	2	2	no	—
66	#672	—	2	4	no	—
67	#672	C₄⋊C₄	1	4	yes	direct product factor
68	C₂²×C₁₆	—	5	1	no	—
69	C₂²×C₁₆	—	5	1	no	—
70	C₂²×C₁₆	—	5	2	no	—
71	C₂×C₄⋊C₈	Dic₅	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂×D₅	1	2	yes	—
73	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
74	C₄×C₄⋊C₄	Dic₅	1	1	yes	characteristic
75	C₄⋊C₁₆	—	5	4	no	—
76	C₄×C₁₆	—	5	4	no	—
77	C₄×C₁₆	—	10	4	no	—
78	#336	Dic₅	1	2	yes	—
79	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₅	—	2	4	no	—
81	C₂×C₄×C₅	—	2	2	no	—
82	C₂×C₄×C₅	—	2	4	no	—
83	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
84	C₂×C₄×C₅	C₄×C₈	1	2	yes	—
85	C₂×C₄×C₅	C₂×C₁₆	1	8	yes	—
86	C₂×C₄×C₅	C₂×C₁₆	1	4	yes	—
87	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₅	#131	1	1	yes	characteristic
92	C₂×C₄×C₅	#131	1	1	yes	characteristic
93	C₅×C₈	—	2	2	no	—
94	C₅×C₈	—	2	4	no	—
95	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₅×C₈	#117	1	1	yes	characteristic
97	C₅×C₈	C₄×D₄	1	1	yes	characteristic
98	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
99	C₂²×C₈	C₂×Dic₅	1	2	yes	—
100	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
102	C₂×C₄⋊C₄	#168	1	2	yes	—
103	C₂×C₄²	C₂×Dic₅	1	2	yes	—
104	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
105	C₂×C₁₆	—	5	2	no	—
106	C₂×C₁₆	—	5	2	no	—
107	C₂×C₁₆	—	5	2	no	—
108	C₂×C₁₆	—	5	2	no	—
109	C₂×C₁₆	—	5	4	no	—
110	C₂×C₁₆	—	10	2	no	—
111	C₂×C₁₆	—	10	1	no	—
112	C₂×C₁₆	—	10	1	no	—
113	C₂×C₁₆	—	10	2	no	—
114	C₄⋊C₈	#168	1	4	yes	—
115	C₄×C₈	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	C₄×C₈	C₄×D₅	1	2	yes	—
118	C₄×C₈	C₄×D₅	1	2	yes	—
119	#95	#168	1	2	yes	—
120	C₂²×C₅	C₂×C₄×C₈	1	1	yes	characteristic
121	C₂²×C₅	C₈×D₄	1	1	yes	characteristic
122	C₂²×C₅	C₈×Q₈	1	1	yes	characteristic
123	C₂²×C₅	C₂²×C₁₆	1	2	yes	—
124	C₂²×C₅	#504	1	2	yes	—
125	C₄×C₅	—	2	2	no	—
126	C₄×C₅	—	2	8	no	—
127	C₄×C₅	C₄×C₁₆	1	4	yes	complemented
128	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
129	C₄×C₅	#405	1	1	yes	characteristic
130	C₄×C₅	C₈×D₄	1	1	yes	characteristic
131	C₄×C₅	C₈×Q₈	1	1	yes	characteristic
132	C₂²×C₄	C₂×#168	1	1	yes	characteristic
133	C₂²×C₄	C₂×#168	1	4	yes	—
134	C₂²×C₄	C₂×#168	1	1	yes	characteristic
135	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	C₂×#168	1	4	yes	—
141	C₂×C₈	C₄×Dic₅	1	2	yes	—
142	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
143	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
144	C₂×C₈	D₄×D₅	1	1	yes	characteristic
145	C₂×C₈	#711	1	1	yes	characteristic
146	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
147	C₂×C₈	#713	1	1	yes	characteristic
148	C₄⋊C₄	#672	1	8	yes	direct product factor
149	C₄²	—	2	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	C₂×#168	1	2	yes	—
152	C₄²	C₄×Dic₅	1	2	yes	—
153	C₁₆	—	5	4	no	—
154	C₁₆	—	10	2	no	—
155	C₁₆	—	10	4	no	—
156	C₂×C₅	C₈×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₅	C₂×C₄×C₁₆	1	1	yes	characteristic
158	C₂×C₅	#2108	1	1	yes	characteristic
159	C₂×C₅	D₄×C₁₆	1	2	yes	—
160	C₂×C₅	Q₈×C₁₆	1	2	yes	—
161	C₂³	C₂²×#168	1	1	yes	characteristic
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	C₄×#168	1	2	yes	—
166	C₂×C₄	C₄×#168	1	2	yes	—
167	C₂×C₄	C₂×#672	1	8	yes	—
168	C₂×C₄	C₂×#672	1	4	yes	—
169	C₂×C₄	C₂²×#168	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
171	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
172	C₂×C₄	Q₈×Dic₅	1	1	yes	characteristic
173	C₂×C₄	#4094	1	1	yes	characteristic
174	C₂×C₄	#4094	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	2	4	no	—
177	C₈	#4033	1	1	yes	characteristic
178	C₈	D₅×C₄⋊C₄	1	1	yes	characteristic
179	C₈	#4038	1	1	yes	characteristic
180	C₈	#4039	1	1	yes	characteristic
181	C₅	C₁₆×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
182	C₂²	C₂×C₄×#168	1	1	yes	characteristic
183	C₂²	D₄×#168	1	1	yes	characteristic
184	C₂²	Q₈×#168	1	1	yes	characteristic
185	C₂²	C₂²×#672	1	2	yes	—
186	C₂²	#65429	1	2	yes	—
187	C₄	—	2	8	no	—
188	C₄	—	2	2	no	—
189	C₄	C₄×#672	1	4	yes	complemented
190	C₄	C₄⋊C₄×Dic₅	1	1	yes	characteristic
191	C₄	#65286	1	1	yes	characteristic
192	C₄	D₄×#168	1	1	yes	characteristic
193	C₄	Q₈×#168	1	1	yes	characteristic
194	C₂	C₄⋊C₄×#168	1	1	yes	characteristic
195	C₂	C₂×C₄×#672	1	1	yes	characteristic
196	C₂	#10600674	1	1	yes	characteristic
197	C₂	D₄×#672	1	2	yes	—
198	C₂	Q₈×#672	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator	C₂×C₅	C₂×C₄×C₁₆
Fitting	C₅×C₈×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₅	C₂²×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₈×C₄⋊C₄	D₅
hypocenter	C₅	C₁₆×C₄⋊C₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
14. C₄⋊#672	C₄	semidirect
16. C₄×#672	C₄	semidirect
60. C₅×C₄⋊C₄	C₁₆	semidirect
67. #672	C₄⋊C₄	direct (N × H)
127. C₄×C₅	C₄×C₁₆	semidirect
148. C₄⋊C₄	#672	direct (N × H)
181. C₅	C₁₆×C₄⋊C₄	semidirect
189. C₄	C₄×#672	semidirect

Maximal subgroups

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

subgroup	index
C₅×C₈×C₄⋊C₄	2
C₂×C₄⋊#672	2
C₂×C₄×#672	2
#10598148	2
C₁₆×C₄⋊C₄	5

Subgroups — C4⋊C4×#672 #12014644

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊C₄×#672 #12014644