Subgroups — C₄×#11718105 #15778303

Order 3328.

330 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718105	C₂	1	2	yes	maximal
2	C₄×#88059	C₂	1	1	yes	maximal, characteristic
3	C₄×#88057	C₂	1	1	yes	maximal, characteristic
4	#13112935	C₂	1	2	yes	maximal
5	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88059	C₂²	1	2	yes	—
10	C₂×#88057	C₂²	1	2	yes	—
11	C₄×C₄⋊Dic₁₃	—	2	1	no	—
12	C₄×#6099	—	2	1	no	—
13	C₁₃×#346	C₄	1	2	yes	complemented
14	C₄×C₁₃×C₁₆	C₄	1	2	yes	complemented
15	#11718144	—	2	1	no	—
16	#11718143	—	2	1	no	—
17	#11718105	C₄	1	8	yes	direct product factor
18	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₃	—	2	2	no	—
22	C₂×C₄×Dic₁₃	—	4	1	no	—
23	C₂×#6099	—	2	2	no	—
24	C₁₃×M₃₂	—	2	2	no	—
25	C₁₃×M₃₂	C₂×C₄	1	4	yes	—
26	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
32	#88071	—	4	1	no	—
33	#88059	—	2	2	no	—
34	#88059	C₂×C₄	1	4	yes	—
35	#88057	—	2	2	no	—
36	#88057	C₂×C₄	1	4	yes	—
37	#88043	—	4	1	no	—
38	C₄×#1055	—	4	1	no	—
39	C₄×#365	—	13	1	no	maximal, Sylow
40	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₁₃	—	4	2	no	—
44	C₂×C₈×C₁₃	—	2	2	no	—
45	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₁₃	#41	1	2	yes	—
51	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
54	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
55	C₄⋊Dic₁₃	—	2	4	no	—
56	C₄⋊Dic₁₃	—	4	2	no	—
57	C₄×Dic₁₃	—	8	2	no	—
58	#6099	—	2	4	no	—
59	#6099	—	4	2	no	—
60	C₂×#1055	—	4	2	no	—
61	C₁₃×C₁₆	—	2	2	no	—
62	C₁₃×C₁₆	C₄²	1	4	yes	complemented
63	C₂×#365	—	13	2	no	—
64	C₄×M₃₂	D₁₃	1	1	yes	characteristic
65	C₄×#108	—	13	1	no	—
66	C₄×#106	—	13	1	no	—
67	#1383	—	13	2	no	—
68	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	4	1	no	—
71	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
72	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
73	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
74	C₂×C₄×C₁₃	#106	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
77	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₃	#135	1	2	yes	—
79	C₂×Dic₁₃	—	4	4	no	—
80	C₂×Dic₁₃	—	8	2	no	—
81	C₂×Dic₁₃	—	8	2	no	—
82	C₈×C₁₃	—	2	1	no	—
83	C₈×C₁₃	—	2	1	no	—
84	C₈×C₁₃	C₂×C₄²	1	1	yes	characteristic
85	C₈×C₁₃	#117	1	1	yes	characteristic
86	C₈×C₁₃	C₄×Q₈	1	2	yes	—
87	#1055	—	4	4	no	—
88	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
89	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
90	C₂×#108	—	13	2	no	—
91	C₂×#106	—	13	2	no	—
92	C₄×M₁₆	—	26	1	no	—
93	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
94	C₄×C₄⋊C₄	—	26	1	no	—
95	#365	—	13	8	no	—
96	#346	Dic₁₃	1	2	yes	complemented
97	C₄×C₁₆	Dic₁₃	1	2	yes	complemented
98	#341	—	26	1	no	—
99	#340	—	26	1	no	—
100	C₂²×C₁₃	—	2	1	no	—
101	C₂²×C₁₃	—	2	1	no	—
102	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
103	C₂²×C₁₃	C₄×SD₁₆	1	2	yes	—
104	C₄×C₁₃	—	2	1	no	—
105	C₄×C₁₃	—	2	1	no	—
106	C₄×C₁₃	—	2	1	no	—
107	C₄×C₁₃	#365	1	2	yes	complemented
108	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
109	C₄×C₁₃	#427	1	1	yes	characteristic
110	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
111	Dic₁₃	—	8	4	no	—
112	C₂×M₁₆	—	26	2	no	—
113	C₂²×C₈	D₅₂	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	26	2	no	—
116	C₂×C₄²	—	52	1	no	—
117	C₂×C₄²	D₅₂	1	1	yes	characteristic
118	M₃₂	—	2	2	no	—
119	M₃₂	C₄×D₁₃	1	4	yes	—
120	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
121	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
122	#108	—	13	4	no	—
123	#108	—	26	2	no	—
124	#106	—	13	4	no	—
125	#106	—	26	2	no	—
126	#97	—	52	1	no	—
127	C₄×C₈	—	52	1	no	—
128	C₄×C₈	Dic₂₆	1	1	yes	characteristic
129	C₄×C₈	Dic₂₆	1	1	yes	characteristic
130	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
131	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
132	#95	—	52	1	no	—
133	C₂×C₁₃	—	2	2	no	—
134	C₂×C₁₃	C₄×#106	1	1	yes	characteristic
135	C₂×C₁₃	C₂×#365	1	1	yes	characteristic
136	C₂×C₁₃	#2155	1	1	yes	characteristic
137	C₂²×C₄	—	52	2	no	—
138	C₂²×C₄	#6095	1	1	yes	characteristic
139	C₂²×C₄	#6095	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
141	M₁₆	—	26	4	no	—
142	M₁₆	—	52	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	52	2	no	—
145	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
146	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
147	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
148	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
149	C₂×C₈	#6127	1	2	yes	—
150	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
151	C₄⋊C₄	—	26	4	no	—
152	C₄⋊C₄	—	52	2	no	—
153	C₄²	—	104	2	no	—
154	C₄²	#6095	1	1	yes	characteristic
155	C₄²	#6095	1	1	yes	characteristic
156	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
157	C₄²	C₄⋊Dic₁₃	1	1	yes	characteristic
158	C₁₆	—	2	2	no	—
159	C₁₆	C₄×Dic₁₃	1	4	yes	complemented
160	C₁₃	C₄×#365	1	1	yes	characteristic, complemented, Sylow
161	C₂³	C₂×#6095	1	1	yes	characteristic
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	52	4	no	—
165	C₂×C₄	—	104	2	no	—
166	C₂×C₄	—	104	2	no	—
167	C₂×C₄	#88057	1	1	yes	characteristic
168	C₂×C₄	#88057	1	1	yes	characteristic
169	C₂×C₄	#88057	1	1	yes	characteristic
170	C₂×C₄	#88057	1	1	yes	characteristic
171	C₂×C₄	C₄×D₅₂	1	2	yes	—
172	C₂×C₄	C₂×#6095	1	1	yes	characteristic
173	C₂×C₄	#88158	1	2	yes	—
174	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	52	4	no	—
178	C₈	C₄×Dic₂₆	1	2	yes	—
179	C₈	C₂×C₄×Dic₁₃	1	1	yes	characteristic
180	C₈	#88180	1	1	yes	characteristic
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	C₄×#6095	1	2	yes	—
184	C₂²	C₂×#88057	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	104	4	no	—
189	C₄	#11718105	1	2	yes	direct product factor
190	C₄	C₄×#6095	1	2	yes	—
191	C₄	C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
192	C₄	#11718767	1	1	yes	characteristic
193	C₂	—	2	2	no	—
194	C₂	C₄×#88057	1	1	yes	characteristic
195	C₂	C₂×#11718105	1	1	yes	characteristic
196	C₂	#13116609	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88057
commutator	C₈×C₁₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂²×C₁₃	C₂×#106
prime radical	C₂³×C₁₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	C₄×#365
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
13. C₁₃×#346	C₄	semidirect
14. C₄×C₁₃×C₁₆	C₄	semidirect
17. #11718105	C₄	direct (N × H)
62. C₁₃×C₁₆	C₄²	semidirect
96. #346	Dic₁₃	semidirect
97. C₄×C₁₆	Dic₁₃	semidirect
107. C₄×C₁₃	#365	semidirect
159. C₁₆	C₄×Dic₁₃	semidirect
160. C₁₃	C₄×#365	semidirect
189. C₄	#11718105	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×#11718105	2
C₄×#88059	2
C₄×#88057	2
#13112935	2
C₄×C₁₃×M₃₂	2
C₄×#365	13

Subgroups — C4×#11718105 #15778303

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11718105 #15778303