Subgroups — C₄²×C₄⋊#261 #16791509

Order 3328.

2478 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄⋊#261	C₂	1	6	yes	maximal
2	C₂×C₄²×#261	C₂	1	2	yes	maximal
3	C₄×(C₂×#261)⋊C₄	C₂	1	6	yes	maximal
4	C₄³×D₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊#261	C₂²	1	4	yes	—
6	C₂²×C₄×#261	C₂²	1	12	yes	—
7	C₂×C₄²×D₁₃	C₂²	1	3	yes	—
8	C₂×C₄²×D₁₃	C₂²	1	3	yes	—
9	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
10	C₂×(C₂×#261)⋊C₄	C₂²	1	12	yes	—
11	C₄×C₄⋊#261	C₄	1	48	yes	direct product factor
12	C₄²×#261	—	2	8	no	—
13	C₄³×C₁₃	C₄	1	1	yes	characteristic, complemented
14	C₄²×Dic₁₃	C₄	1	1	yes	characteristic, complemented
15	C₄²×Dic₁₃	C₄	1	3	yes	complemented
16	C₄²×Dic₁₃	C₄	1	3	yes	complemented
17	C₂³×#261	C₂³	1	8	yes	—
18	C₂²×C₄×D₁₃	C₂³	1	3	yes	—
19	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
20	C₂²×C₄×D₁₃	C₂³	1	3	yes	—
21	C₂×C₄⋊#261	C₂×C₄	1	48	yes	—
22	C₂×C₄×#261	—	2	24	no	—
23	C₂×C₄×#261	—	2	24	no	—
24	C₂×C₄×#261	C₂×C₄	1	48	yes	—
25	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
26	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
29	C₂×C₄×Dic₁₃	C₂×C₄	1	6	yes	—
30	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
31	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
32	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
33	C₂×C₄×Dic₁₃	C₂×C₄	1	3	yes	—
34	C₄²×D₁₃	—	2	12	no	—
35	C₄²×D₁₃	C₂×C₄	1	24	yes	—
36	C₄²×D₁₃	D₄	1	4	yes	—
37	C₄²×D₁₃	Q₈	1	4	yes	—
38	(C₂×#261)⋊C₄	C₂×C₄	1	48	yes	—
39	C₄²×C₄⋊C₄	—	13	1	no	maximal, Sylow
40	C₂³×D₁₃	C₂⁴	1	1	yes	characteristic
41	C₂²×#261	—	2	8	no	—
42	C₂²×#261	—	2	24	no	—
43	C₂²×#261	C₂²×C₄	1	48	yes	—
44	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
45	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
47	C₂²×Dic₁₃	C₂²×C₄	1	3	yes	—
48	C₂²×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×Dic₁₃	C₂²×C₄	1	3	yes	—
50	C₂×C₄×D₁₃	—	2	12	no	—
51	C₂×C₄×D₁₃	C₂²×C₄	1	12	yes	—
52	C₂×C₄×D₁₃	C₂²×C₄	1	12	yes	—
53	C₂×C₄×D₁₃	C₂²×C₄	1	12	yes	—
54	C₂×C₄×D₁₃	C₂×D₄	1	6	yes	—
55	C₂×C₄×D₁₃	C₂×Q₈	1	6	yes	—
56	C₂×C₄×D₁₃	#41	1	12	yes	—
57	C₄⋊#261	C₄²	1	64	yes	direct product factor
58	C₄×#261	—	2	96	no	—
59	C₄²×C₁₃	—	2	6	no	—
60	C₄²×C₁₃	C₄²	1	12	yes	complemented
61	C₄²×C₁₃	C₄⋊C₄	1	4	yes	complemented
62	C₄×Dic₁₃	—	2	6	no	—
63	C₄×Dic₁₃	—	2	12	no	—
64	C₄×Dic₁₃	C₄²	1	12	yes	complemented
65	C₄×Dic₁₃	C₄²	1	12	yes	complemented
66	C₄×Dic₁₃	C₄²	1	12	yes	complemented
67	C₄×Dic₁₃	C₄⋊C₄	1	12	yes	complemented
68	C₂×C₄×C₄⋊C₄	—	13	6	no	—
69	C₂×C₄³	—	13	1	no	—
70	C₂×C₄³	—	13	2	no	—
71	C₄×#95	—	13	6	no	—
72	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
73	C₂²×D₁₃	C₂³×C₄	1	6	yes	—
74	C₂²×D₁₃	C₂²×D₄	1	1	yes	characteristic
75	C₂²×D₁₃	C₂²×Q₈	1	1	yes	characteristic
76	C₂²×D₁₃	C₂×#41	1	6	yes	—
77	C₂×#261	—	2	48	no	—
78	C₂×#261	—	2	48	no	—
79	C₂×#261	C₂×C₄²	1	32	yes	—
80	C₂×C₄×C₁₃	—	2	6	no	—
81	C₂×C₄×C₁₃	C₂×C₄²	1	6	yes	—
82	C₂×C₄×C₁₃	C₂×C₄²	1	6	yes	—
83	C₂×C₄×C₁₃	C₂×C₄²	1	6	yes	—
84	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
85	C₂×C₄×C₁₃	#117	1	6	yes	—
86	C₂×Dic₁₃	—	2	6	no	—
87	C₂×Dic₁₃	C₂×C₄²	1	6	yes	—
88	C₂×Dic₁₃	C₂×C₄²	1	6	yes	—
89	C₂×Dic₁₃	C₂×C₄²	1	6	yes	—
90	C₂×Dic₁₃	C₂×C₄⋊C₄	1	6	yes	—
91	C₂×Dic₁₃	#117	1	6	yes	—
92	C₄×D₁₃	—	2	24	no	—
93	C₄×D₁₃	C₂×C₄²	1	16	yes	—
94	C₄×D₁₃	C₄×D₄	1	24	yes	—
95	C₄×D₁₃	C₄×Q₈	1	24	yes	—
96	C₂²×C₄⋊C₄	—	13	4	no	—
97	C₂²×C₄²	—	13	3	no	—
98	C₂²×C₄²	—	13	1	no	—
99	C₂²×C₄²	—	13	12	no	—
100	C₂²×C₄²	—	13	3	no	—
101	C₄×C₄⋊C₄	—	13	48	no	—
102	C₂×#95	—	13	12	no	—
103	C₄³	—	13	3	no	—
104	C₄³	—	13	3	no	—
105	C₄³	—	13	1	no	—
106	C₄³	—	26	8	no	—
107	C₄³	#261	1	1	yes	characteristic, complemented
108	C₂²×C₁₃	C₂²×C₄²	1	3	yes	—
109	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₁₃	C₂×#117	1	3	yes	—
111	C₂×D₁₃	C₂²×C₄²	1	4	yes	—
112	C₂×D₁₃	C₂×C₄×D₄	1	6	yes	—
113	C₂×D₁₃	C₂×C₄×Q₈	1	6	yes	—
114	C₂×D₁₃	C₄×#41	1	12	yes	—
115	#261	—	2	32	no	—
116	C₄×C₁₃	—	2	6	no	—
117	C₄×C₁₃	C₄³	1	4	yes	complemented
118	C₄×C₁₃	C₄×C₄⋊C₄	1	12	yes	complemented
119	Dic₁₃	—	2	6	no	—
120	Dic₁₃	C₄³	1	4	yes	complemented
121	Dic₁₃	C₄×C₄⋊C₄	1	12	yes	complemented
122	C₂³×C₄	—	13	1	no	—
123	C₂³×C₄	—	13	3	no	—
124	C₂³×C₄	—	13	8	no	—
125	C₂³×C₄	—	13	3	no	—
126	C₂×C₄⋊C₄	—	13	48	no	—
127	C₂×C₄²	—	13	3	no	—
128	C₂×C₄²	—	13	3	no	—
129	C₂×C₄²	—	13	3	no	—
130	C₂×C₄²	—	13	3	no	—
131	C₂×C₄²	—	13	24	no	—
132	C₂×C₄²	—	13	4	no	—
133	C₂×C₄²	—	13	4	no	—
134	C₂×C₄²	—	13	48	no	—
135	C₂×C₄²	—	13	6	no	—
136	C₂×C₄²	—	13	3	no	—
137	C₂×C₄²	—	26	12	no	—
138	C₂×C₄²	—	26	24	no	—
139	C₂×C₄²	—	26	24	no	—
140	C₂×C₄²	C₂×#261	1	3	yes	—
141	C₂×C₄²	C₂×#261	1	3	yes	—
142	C₂×C₄²	C₂×#261	1	1	yes	characteristic
143	#95	—	13	48	no	—
144	C₂×C₁₃	C₂×C₄³	1	1	yes	characteristic
145	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	3	yes	—
146	C₂×C₁₃	C₄×#117	1	3	yes	—
147	D₁₃	C₄²×D₄	1	4	yes	—
148	D₁₃	C₄²×Q₈	1	4	yes	—
149	C₂⁴	—	13	1	no	—
150	C₂²×C₄	—	13	1	no	—
151	C₂²×C₄	—	13	3	no	—
152	C₂²×C₄	—	13	12	no	—
153	C₂²×C₄	—	13	12	no	—
154	C₂²×C₄	—	13	6	no	—
155	C₂²×C₄	—	13	6	no	—
156	C₂²×C₄	—	13	12	no	—
157	C₂²×C₄	—	13	48	no	—
158	C₂²×C₄	—	13	3	no	—
159	C₂²×C₄	—	13	12	no	—
160	C₂²×C₄	—	26	12	no	—
161	C₂²×C₄	—	26	8	no	—
162	C₂²×C₄	—	26	24	no	—
163	C₂²×C₄	C₂²×#261	1	3	yes	—
164	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
165	C₂²×C₄	C₂²×#261	1	3	yes	—
166	C₄⋊C₄	—	13	64	no	—
167	C₄²	—	2	6	no	—
168	C₄²	—	13	12	no	—
169	C₄²	—	13	12	no	—
170	C₄²	—	13	12	no	—
171	C₄²	—	13	12	no	—
172	C₄²	—	26	12	no	—
173	C₄²	—	26	96	no	—
174	C₄²	—	26	6	no	—
175	C₄²	C₄×#261	1	12	yes	complemented
176	C₄²	C₄⋊#261	1	4	yes	direct product factor
177	C₁₃	C₄²×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
178	C₂³	—	13	6	no	—
179	C₂³	—	13	1	no	—
180	C₂³	—	13	1	no	—
181	C₂³	—	13	6	no	—
182	C₂³	C₂³×#261	1	1	yes	characteristic
183	C₂×C₄	—	2	6	no	—
184	C₂×C₄	—	13	6	no	—
185	C₂×C₄	—	13	6	no	—
186	C₂×C₄	—	13	6	no	—
187	C₂×C₄	—	13	6	no	—
188	C₂×C₄	—	13	16	no	—
189	C₂×C₄	—	13	24	no	—
190	C₂×C₄	—	13	24	no	—
191	C₂×C₄	—	13	32	no	—
192	C₂×C₄	—	13	6	no	—
193	C₂×C₄	—	26	24	no	—
194	C₂×C₄	—	26	48	no	—
195	C₂×C₄	—	26	48	no	—
196	C₂×C₄	—	26	6	no	—
197	C₂×C₄	C₂×C₄×#261	1	6	yes	—
198	C₂×C₄	C₂×C₄×#261	1	6	yes	—
199	C₂×C₄	C₂×C₄×#261	1	6	yes	—
200	C₂×C₄	C₂×C₄⋊#261	1	6	yes	—
201	C₂×C₄	#88237	1	6	yes	—
202	C₂²	—	13	6	no	—
203	C₂²	—	13	6	no	—
204	C₂²	—	13	12	no	—
205	C₂²	—	13	4	no	—
206	C₂²	C₂²×C₄×#261	1	3	yes	—
207	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
208	C₂²	C₂×#88237	1	3	yes	—
209	C₄	—	2	6	no	—
210	C₄	—	13	12	no	—
211	C₄	—	13	4	no	—
212	C₄	—	26	32	no	—
213	C₄	—	26	6	no	—
214	C₄	C₄²×#261	1	4	yes	complemented
215	C₄	C₄×C₄⋊#261	1	12	yes	direct product factor
216	C₂	—	13	4	no	—
217	C₂	—	13	4	no	—
218	C₂	C₂×C₄²×#261	1	1	yes	characteristic
219	C₂	C₂×C₄×C₄⋊#261	1	3	yes	—
220	C₂	C₄×#88237	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×#261
commutator	C₂×C₁₃	C₂×C₄³
Frattini	C₂³	C₂³×#261
Fitting	C₄³×C₁₃	C₄
socle	C₂³×C₁₃	C₂³×C₄
prime radical, cosocle	C₂³×D₁₃	C₂⁴
hypercenter	C₄³	#261
hypocenter	C₁₃	C₄²×C₄⋊C₄
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄³

Complements

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

N	H ≅ G/N	type
11. C₄×C₄⋊#261	C₄	direct (N × H)
13. C₄³×C₁₃	C₄	semidirect
14. C₄²×Dic₁₃	C₄	semidirect
15. C₄²×Dic₁₃	C₄	semidirect
16. C₄²×Dic₁₃	C₄	semidirect
57. C₄⋊#261	C₄²	direct (N × H)
60. C₄²×C₁₃	C₄²	semidirect
61. C₄²×C₁₃	C₄⋊C₄	semidirect
64. C₄×Dic₁₃	C₄²	semidirect
65. C₄×Dic₁₃	C₄²	semidirect
66. C₄×Dic₁₃	C₄²	semidirect
67. C₄×Dic₁₃	C₄⋊C₄	semidirect
107. C₄³	#261	semidirect
117. C₄×C₁₃	C₄³	semidirect
118. C₄×C₁₃	C₄×C₄⋊C₄	semidirect
120. Dic₁₃	C₄³	semidirect
121. Dic₁₃	C₄×C₄⋊C₄	semidirect
175. C₄²	C₄×#261	semidirect
176. C₄²	C₄⋊#261	direct (N × H)
177. C₁₃	C₄²×C₄⋊C₄	semidirect
214. C₄	C₄²×#261	semidirect
215. C₄	C₄×C₄⋊#261	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×C₄×C₄⋊#261	2
C₂×C₄²×#261	2
C₄×(C₂×#261)⋊C₄	2
C₄³×D₁₃	2
C₄²×C₄⋊C₄	13

Subgroups — C42×C4⋊#261 #16791509

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₄⋊#261 #16791509