Subgroups — #16015911

Order 3328.

706 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×#11718801	C₂	1	2	yes	maximal
2	#13116595	C₂	1	6	yes	maximal, complemented
3	#13116592	C₂	1	3	yes	maximal, complemented
4	C₂×C₄⋊#6090	C₂	1	3	yes	maximal, complemented
5	C₄×C₁₃×#131	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
7	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
8	C₄×C₁₃×#41	C₄	1	1	yes	characteristic
9	C₁₃×#405	C₄	1	3	yes	—
10	C₁₃×#405	C₂²	1	3	yes	—
11	C₄×C₁₃×M₁₆	C₄	1	3	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
13	#11718801	—	2	8	no	—
14	C₂×#88052	C₂²	1	6	yes	—
15	C₂²×#6090	C₂²	1	6	yes	—
16	C₄⋊#6090	C₂²	1	1	yes	characteristic
17	C₄⋊#6090	C₂²	1	3	yes	—
18	C₄⋊#6090	C₂²	1	6	yes	—
19	C₄⋊#6090	C₂²	1	6	yes	—
20	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
21	C₁₃×#131	—	2	4	no	—
22	C₁₃×#131	D₄	1	4	yes	—
23	C₁₃×#131	Q₈	1	4	yes	—
24	C₂×C₁₃×M₁₆	C₂×C₄	1	3	yes	—
25	C₂×C₁₃×M₁₆	C₂×C₄	1	3	yes	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	3	yes	—
27	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
28	C₄×Q₈×C₁₃	C₈	1	1	yes	characteristic
29	C₄×Q₈×C₁₃	C₈	1	1	yes	characteristic
30	C₄×D₄×C₁₃	C₈	1	3	yes	—
31	C₄×D₄×C₁₃	C₈	1	3	yes	—
32	C₁₃×#117	C₂×C₄	1	3	yes	—
33	C₂×C₄²×C₁₃	C₂×C₄	1	3	yes	—
34	C₁₃×#97	C₂×C₄	1	6	yes	—
35	C₄×C₈×C₁₃	C₂×C₄	1	3	yes	—
36	C₄×C₈×C₁₃	C₂×C₄	1	3	yes	—
37	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₄×C₈×C₁₃	C₂³	1	3	yes	—
39	#88052	—	2	12	no	—
40	C₂×#6090	—	2	12	no	—
41	C₂×#6090	C₂³	1	2	yes	—
42	C₂×#6090	C₂³	1	6	yes	—
43	#33398	—	13	1	no	maximal, Sylow
44	C₁₃×#41	C₄⋊C₄	1	4	yes	—
45	C₁₃×#41	M₁₆	1	4	yes	—
46	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
47	C₂×D₄×C₁₃	C₂×C₈	1	3	yes	—
48	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
49	C₂²×C₄×C₁₃	C₂×C₈	1	3	yes	—
50	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
51	C₁₃×M₁₆	—	2	6	no	—
52	C₁₃×M₁₆	C₄⋊C₄	1	12	yes	—
53	C₂×C₈×C₁₃	—	2	6	no	—
54	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
56	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
57	C₂×C₈×C₁₃	C₂×D₄	1	6	yes	—
58	C₂×C₈×C₁₃	C₂×Q₈	1	6	yes	—
59	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
60	C₁₃×C₄⋊C₄	C₂×C₈	1	3	yes	—
61	C₁₃×C₄⋊C₄	C₂×C₈	1	3	yes	—
62	C₁₃×C₂²⋊C₄	C₂×C₈	1	6	yes	—
63	C₄²×C₁₃	C₂×C₈	1	3	yes	—
64	C₄²×C₁₃	C₂×C₈	1	3	yes	—
65	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
67	#6090	—	2	2	no	—
68	#6090	—	2	6	no	—
69	C₂×#504	—	13	2	no	—
70	C₄×#131	D₁₃	1	1	yes	characteristic
71	#2151	—	13	6	no	—
72	#2150	—	13	3	no	—
73	C₂×C₄⋊C₁₆	—	13	3	no	—
74	C₂³×C₁₃	C₂²×C₈	1	3	yes	—
75	Q₈×C₁₃	C₄⋊C₈	1	4	yes	—
76	D₄×C₁₃	C₄⋊C₈	1	12	yes	—
77	C₂×C₄×C₁₃	—	2	6	no	—
78	C₂×C₄×C₁₃	—	4	3	no	—
79	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
80	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
82	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
83	C₂×C₄×C₁₃	C₂²×C₈	1	3	yes	—
84	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂×M₁₆	1	6	yes	—
86	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
87	C₈×C₁₃	—	2	1	no	—
88	C₈×C₁₃	—	2	3	no	—
89	C₈×C₁₃	C₂×C₄⋊C₄	1	6	yes	—
90	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
91	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
92	C₂×#131	Dic₁₃	1	1	yes	characteristic
93	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
94	C₄×#41	Dic₁₃	1	1	yes	characteristic
95	#504	—	26	8	no	—
96	C₂×M₃₂	—	13	6	no	—
97	C₂²×C₁₆	—	13	6	no	—
98	#405	Dic₁₃	1	3	yes	—
99	#405	C₂×D₁₃	1	3	yes	—
100	C₄×M₁₆	Dic₁₃	1	3	yes	—
101	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
102	C₄⋊C₁₆	—	13	1	no	—
103	C₄⋊C₁₆	—	13	3	no	—
104	C₄⋊C₁₆	—	13	6	no	—
105	C₄⋊C₁₆	—	13	6	no	—
106	C₂²×C₁₃	—	2	3	no	—
107	C₂²×C₁₃	—	2	3	no	—
108	C₂²×C₁₃	C₂×C₄⋊C₈	1	6	yes	—
109	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
110	C₄×C₁₃	—	2	3	no	—
111	C₄×C₁₃	—	2	1	no	—
112	C₄×C₁₃	—	2	3	no	—
113	C₄×C₁₃	C₂×C₄⋊C₈	1	6	yes	—
114	C₄×C₁₃	#504	1	2	yes	complemented
115	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
116	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
117	C₂×#41	C₂×Dic₁₃	1	1	yes	characteristic
118	#131	—	2	4	no	—
119	#131	Dic₂₆	1	4	yes	—
120	#131	D₅₂	1	4	yes	—
121	C₂×M₁₆	C₂×Dic₁₃	1	3	yes	—
122	C₂×M₁₆	C₂×Dic₁₃	1	3	yes	—
123	C₂²×C₈	C₂×Dic₁₃	1	3	yes	—
124	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
125	C₄×Q₈	#1055	1	1	yes	characteristic
126	C₄×Q₈	#1055	1	1	yes	characteristic
127	C₄×D₄	#1055	1	3	yes	—
128	C₄×D₄	#1055	1	3	yes	—
129	#117	C₂×Dic₁₃	1	3	yes	—
130	C₂×C₄²	C₂×Dic₁₃	1	3	yes	—
131	M₃₂	—	26	12	no	—
132	C₂×C₁₆	—	13	2	no	—
133	C₂×C₁₆	—	13	6	no	—
134	C₂×C₁₆	—	26	12	no	—
135	#97	C₂×Dic₁₃	1	6	yes	—
136	C₄×C₈	C₂×Dic₁₃	1	3	yes	—
137	C₄×C₈	C₂×Dic₁₃	1	3	yes	—
138	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
139	C₄×C₈	C₂²×D₁₃	1	3	yes	—
140	C₂×C₁₃	—	2	6	no	—
141	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
142	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
143	C₂×C₁₃	#3407	1	1	yes	characteristic
144	#41	#6099	1	4	yes	—
145	#41	C₄⋊Dic₁₃	1	4	yes	—
146	C₂×Q₈	C₂×#1055	1	1	yes	characteristic
147	C₂×D₄	C₂×#1055	1	3	yes	—
148	C₂²×C₄	C₂×#1055	1	3	yes	—
149	C₂²×C₄	C₂×#1055	1	3	yes	—
150	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
151	M₁₆	—	2	6	no	—
152	M₁₆	C₄⋊Dic₁₃	1	12	yes	—
153	C₂×C₈	—	2	6	no	—
154	C₂×C₈	C₂×Dic₂₆	1	6	yes	—
155	C₂×C₈	C₂×D₅₂	1	6	yes	—
156	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
157	C₂×C₈	C₂²×Dic₁₃	1	3	yes	—
158	C₂×C₈	C₂²×Dic₁₃	1	3	yes	—
159	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
160	C₄⋊C₄	C₂×#1055	1	3	yes	—
161	C₄⋊C₄	C₂×#1055	1	3	yes	—
162	C₂²⋊C₄	C₂×#1055	1	6	yes	—
163	C₄²	C₂×#1055	1	3	yes	—
164	C₄²	C₂×#1055	1	3	yes	—
165	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
166	C₄²	C₂²×Dic₁₃	1	3	yes	—
167	C₁₆	—	26	2	no	—
168	C₁₆	—	26	6	no	—
169	C₁₃	#33398	1	1	yes	characteristic, complemented, Sylow
170	C₂³	C₂²×#1055	1	3	yes	—
171	Q₈	C₄⋊#1055	1	4	yes	—
172	D₄	C₄⋊#1055	1	12	yes	—
173	C₂×C₄	—	2	6	no	—
174	C₂×C₄	—	4	3	no	—
175	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
176	C₂×C₄	C₂²×#1055	1	3	yes	—
177	C₂×C₄	C₂²×#1055	1	3	yes	—
178	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
179	C₂×C₄	C₂²×#1055	1	3	yes	—
180	C₂×C₄	C₂×#6099	1	6	yes	—
181	C₂×C₄	C₂×C₄⋊Dic₁₃	1	6	yes	—
182	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	3	no	—
185	C₈	C₂×C₄⋊Dic₁₃	1	6	yes	—
186	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
187	C₈	C₂²×D₅₂	1	1	yes	characteristic
188	C₂²	—	2	3	no	—
189	C₂²	—	2	3	no	—
190	C₂²	C₂×C₄⋊#1055	1	6	yes	—
191	C₂²	C₂³×#1055	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	3	no	—
194	C₄	—	2	3	no	—
195	C₄	C₂×C₄⋊#1055	1	6	yes	—
196	C₄	#11718801	1	2	yes	complemented
197	C₄	C₂²×#6099	1	1	yes	characteristic
198	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
199	C₂	—	2	6	no	—
200	C₂	C₂²×C₄⋊#1055	1	1	yes	characteristic
201	C₂	C₂×#11718801	1	1	yes	characteristic
202	C₂	#13127939	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₄×C₁₃×#131	C₂
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₄×#131	D₁₃
hypocenter	C₁₃	#33398
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #13116595	C₂	semidirect
3. #13116592	C₂	semidirect
4. C₂×C₄⋊#6090	C₂	semidirect
114. C₄×C₁₃	#504	semidirect
169. C₁₃	#33398	semidirect
196. C₄	#11718801	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718801	2
#13116595	2
#13116592	2
C₂×C₄⋊#6090	2
C₄×C₁₃×#131	2
#33398	13