Subgroups — #16789556

Order 3328.

438 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×C₄⋊#6092	C₂	1	2	yes	maximal
2	C₂×#11718257	C₂	1	2	yes	maximal
3	#13129875	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#6092	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₄²×Dic₁₃	C₄	1	1	yes	characteristic
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	#11718257	—	2	4	no	—
14	C₄⋊#6092	C₄	1	8	yes	complemented
15	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₁₃	C₈	1	2	yes	—
19	C₂×C₄×Dic₁₃	C₈	1	2	yes	—
20	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
23	C₂²×#1055	C₂³	1	1	yes	characteristic
24	C₂×#6092	—	2	4	no	—
25	C₂×#6092	—	2	4	no	—
26	C₂×#6092	C₂×C₄	1	8	yes	—
27	C₄×C₈×C₁₃	C₈	1	4	yes	—
28	C₈×Dic₁₃	—	2	2	no	—
29	C₈×Dic₁₃	—	2	2	no	—
30	C₈×Dic₁₃	—	2	2	no	—
31	C₈×Dic₁₃	D₄	1	2	yes	—
32	C₈×Dic₁₃	Q₈	1	2	yes	—
33	C₄×#1055	C₂×C₄	1	4	yes	—
34	C₄×C₄⋊C₁₆	—	13	1	no	maximal, Sylow
35	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
39	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₁₃	—	2	1	no	—
41	C₂×C₈×C₁₃	—	2	1	no	—
42	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
46	C₄²×C₁₃	C₄²	1	2	yes	—
47	C₄²×C₁₃	C₂×C₈	1	2	yes	—
48	C₄×Dic₁₃	—	2	1	no	—
49	C₄×Dic₁₃	—	2	1	no	—
50	C₄×Dic₁₃	—	2	4	no	—
51	C₄×Dic₁₃	C₁₆	1	8	yes	complemented
52	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
53	C₄×Dic₁₃	M₁₆	1	2	yes	—
54	C₂×#1055	—	2	1	no	—
55	C₂×#1055	—	2	1	no	—
56	C₂×#1055	C₄²	1	2	yes	—
57	C₂×#1055	C₂²×C₄	1	2	yes	—
58	C₂×#1055	C₂×D₄	1	1	yes	characteristic
59	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
60	C₂×#1055	#41	1	2	yes	—
61	#6092	—	2	8	no	—
62	C₂×C₄⋊C₁₆	—	13	2	no	—
63	C₂×C₄×C₁₆	—	13	2	no	—
64	C₄²×C₈	—	13	1	no	—
65	#1374	—	13	2	no	—
66	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
69	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
70	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	—
71	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×Dic₁₃	—	2	2	no	—
78	C₂×Dic₁₃	—	2	2	no	—
79	C₂×Dic₁₃	C₄⋊C₈	1	4	yes	—
80	C₂×Dic₁₃	C₂×C₁₆	1	4	yes	—
81	C₂×Dic₁₃	C₂×C₁₆	1	4	yes	—
82	C₂×Dic₁₃	M₃₂	1	4	yes	—
83	C₈×C₁₃	—	2	2	no	—
84	C₈×C₁₃	C₄⋊C₈	1	4	yes	—
85	#1055	—	2	2	no	—
86	#1055	C₄×D₄	1	2	yes	—
87	#1055	C₄×Q₈	1	2	yes	—
88	C₂²×C₁₆	—	13	4	no	—
89	C₂×C₄×C₈	—	13	1	no	—
90	C₂×C₄×C₈	—	13	1	no	—
91	C₂×C₄×C₈	—	13	1	no	—
92	C₂×C₄×C₈	—	13	1	no	—
93	C₂×C₄×C₈	—	13	1	no	—
94	C₂×C₄×C₈	#261	1	1	yes	characteristic
95	C₄³	—	13	1	no	—
96	C₄⋊C₁₆	—	13	8	no	—
97	C₄×C₁₆	—	26	4	no	—
98	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
99	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₂²×C₁₃	#431	1	1	yes	characteristic
101	C₂²×C₁₃	C₂²×C₁₆	1	2	yes	—
102	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
103	C₄×C₁₃	—	2	2	no	—
104	C₄×C₁₃	C₄×C₁₆	1	4	yes	complemented
105	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
106	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
107	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
108	C₄×C₁₃	#431	1	1	yes	characteristic
109	Dic₁₃	—	2	4	no	—
110	Dic₁₃	C₄⋊C₁₆	1	8	yes	complemented
111	C₂²×C₈	—	13	1	no	—
112	C₂²×C₈	—	13	1	no	—
113	C₂²×C₈	C₂×#261	1	1	yes	characteristic
114	C₂²×C₈	C₂×#261	1	1	yes	characteristic
115	C₂×C₄²	—	13	2	no	—
116	C₂×C₄²	—	13	1	no	—
117	C₂×C₄²	—	13	2	no	—
118	C₂×C₄²	—	13	1	no	—
119	C₂×C₄²	C₂×#261	1	1	yes	characteristic
120	C₂×C₁₆	—	13	8	no	—
121	C₂×C₁₆	—	26	4	no	—
122	C₂×C₁₆	—	26	4	no	—
123	C₄×C₈	—	13	2	no	—
124	C₄×C₈	—	13	2	no	—
125	C₄×C₈	—	13	4	no	—
126	C₄×C₈	—	26	2	no	—
127	C₄×C₈	—	26	2	no	—
128	C₄×C₈	—	26	2	no	—
129	C₄×C₈	#1057	1	4	yes	—
130	C₂×C₁₃	C₄×C₄⋊C₈	1	1	yes	characteristic
131	C₂×C₁₃	C₂×C₄×C₁₆	1	1	yes	characteristic
132	C₂×C₁₃	C₄×M₃₂	1	1	yes	characteristic
133	C₂×C₁₃	C₂×C₄⋊C₁₆	1	2	yes	—
134	C₂×C₁₃	#2162	1	2	yes	—
135	C₂²×C₄	—	13	2	no	—
136	C₂²×C₄	—	13	2	no	—
137	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
138	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
139	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	13	2	no	—
143	C₂×C₈	—	13	1	no	—
144	C₂×C₈	—	13	1	no	—
145	C₂×C₈	—	13	2	no	—
146	C₂×C₈	—	13	2	no	—
147	C₂×C₈	—	26	1	no	—
148	C₂×C₈	—	26	1	no	—
149	C₂×C₈	#6118	1	2	yes	—
150	C₂×C₈	C₄⋊#261	1	2	yes	—
151	C₂×C₈	C₂×#1057	1	2	yes	—
152	C₂×C₈	C₂×#1057	1	2	yes	—
153	C₄²	—	13	8	no	—
154	C₄²	—	13	2	no	—
155	C₄²	—	13	2	no	—
156	C₄²	—	26	4	no	—
157	C₄²	—	26	1	no	—
158	C₄²	—	26	1	no	—
159	C₄²	C₄×#261	1	2	yes	—
160	C₄²	C₂×#1057	1	2	yes	—
161	C₁₆	—	26	8	no	—
162	C₁₃	C₄×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
163	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	13	4	no	—
166	C₂×C₄	—	13	4	no	—
167	C₂×C₄	—	13	4	no	—
168	C₂×C₄	—	13	4	no	—
169	C₂×C₄	—	26	2	no	—
170	C₂×C₄	—	26	2	no	—
171	C₂×C₄	D₁₃⋊C₁₆	1	4	yes	—
172	C₂×C₄	C₄×#1057	1	2	yes	—
173	C₂×C₄	C₄×#1057	1	2	yes	—
174	C₂×C₄	C₂×#6118	1	1	yes	characteristic
175	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
176	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
178	C₂×C₄	#88237	1	1	yes	characteristic
179	C₂×C₄	C₂²×#1057	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	13	2	no	—
182	C₈	—	13	2	no	—
183	C₈	—	26	2	no	—
184	C₈	C₄⋊#1057	1	4	yes	—
185	C₂²	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
186	C₂²	#11719051	1	1	yes	characteristic
187	C₂²	C₂×D₁₃⋊C₁₆	1	2	yes	—
188	C₂²	D₁₃⋊M₃₂	1	2	yes	—
189	C₂²	C₂×C₄×#1057	1	1	yes	characteristic
190	C₄	—	2	2	no	—
191	C₄	—	13	8	no	—
192	C₄	—	26	4	no	—
193	C₄	#11718257	1	4	yes	complemented
194	C₄	C₄×#6118	1	1	yes	characteristic
195	C₄	C₄×C₄⋊#261	1	1	yes	characteristic
196	C₄	C₂×C₄⋊#1057	1	1	yes	characteristic
197	C₄	#11719119	1	1	yes	characteristic
198	C₂	C₄×C₄⋊#1057	1	1	yes	characteristic
199	C₂	C₂×#11718257	1	1	yes	characteristic
200	C₂	#13131042	1	1	yes	characteristic
201	C₂	D₁₃⋊(C₄⋊C₁₆)	1	2	yes	—
202	C₂	#13131196	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×#261
commutator	C₂×C₁₃	C₂×C₄×C₁₆
Frattini	C₂²×C₄	C₂²×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle, prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	C₄×C₄⋊C₁₆
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
14. C₄⋊#6092	C₄	semidirect
51. C₄×Dic₁₃	C₁₆	semidirect
104. C₄×C₁₃	C₄×C₁₆	semidirect
110. Dic₁₃	C₄⋊C₁₆	semidirect
162. C₁₃	C₄×C₄⋊C₁₆	semidirect
193. C₄	#11718257	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#6092	2
C₂×#11718257	2
#13129875	2
C₄×C₈×Dic₁₃	2
C₄×C₄⋊C₁₆	13