Subgroups — #15863536

Order 3328.

758 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123469	C₂	1	2	yes	maximal
2	#13115011	C₂	1	2	yes	maximal, complemented
3	#13114977	C₂	1	2	yes	maximal, complemented
4	#13114878	C₂	1	4	yes	maximal
5	C₄×#88059	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₄×C₁₃×M₁₆	C₄	1	8	yes	—
10	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
13	#11718868	C₂²	1	4	yes	—
14	C₂×#88059	C₂²	1	8	yes	—
15	#11718587	—	2	4	no	—
16	#11718586	—	2	4	no	—
17	#11718585	C₂²	1	4	yes	—
18	C₄×#6099	C₂²	1	4	yes	—
19	#11718144	C₂²	1	8	yes	—
20	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
21	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₁₃×M₁₆	—	2	8	no	—
23	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
25	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
26	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₁₃	D₄	1	2	yes	—
28	C₂×C₄²×C₁₃	D₄	1	2	yes	—
29	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
30	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×#6099	C₂³	1	8	yes	—
33	C₁₃×#97	C₂×C₄	1	4	yes	—
34	C₁₃×#97	C₂×C₄	1	4	yes	—
35	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	—
36	C₂²⋊#1055	—	2	8	no	—
37	#88059	—	2	16	no	—
38	C₄⋊#1055	—	2	8	no	—
39	#88043	—	2	4	no	—
40	C₄×#1055	—	2	4	no	—
41	#20827	—	13	1	no	maximal, Sylow
42	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	—	2	4	no	—
44	C₂²×C₄×C₁₃	—	4	4	no	—
45	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
47	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
53	C₁₃×M₁₆	—	2	16	no	—
54	C₂×C₈×C₁₃	—	2	4	no	—
55	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
56	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
57	C₂×C₈×C₁₃	#41	1	8	yes	—
58	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
59	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
60	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
61	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
62	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₁₃	C₂×D₄	1	2	yes	—
64	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
65	#6099	—	2	16	no	—
66	C₂×#1055	—	2	8	no	—
67	#2920	—	13	2	no	—
68	C₂×C₄×M₁₆	D₁₃	1	1	yes	characteristic
69	#1849	—	13	2	no	—
70	#1838	—	13	2	no	—
71	#1811	—	13	4	no	—
72	C₄×#108	—	13	4	no	—
73	C₂³×C₁₃	—	2	2	no	—
74	C₂³×C₁₃	—	4	2	no	—
75	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
76	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	4	no	—
79	C₂×C₄×C₁₃	—	2	4	no	—
80	C₂×C₄×C₁₃	—	2	8	no	—
81	C₂×C₄×C₁₃	—	2	4	no	—
82	C₂×C₄×C₁₃	—	4	4	no	—
83	C₂×C₄×C₁₃	—	4	2	no	—
84	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
85	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
86	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
87	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
88	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
91	C₈×C₁₃	—	2	4	no	—
92	C₈×C₁₃	#117	1	8	yes	—
93	#1055	—	4	8	no	—
94	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
95	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
96	#432	—	26	4	no	—
97	#431	—	26	4	no	—
98	C₂×#108	—	13	8	no	—
99	#423	—	13	4	no	—
100	#407	—	13	4	no	—
101	C₄×M₁₆	—	13	4	no	—
102	C₄×M₁₆	Dic₁₃	1	8	yes	—
103	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
104	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
106	#341	—	13	8	no	—
107	C₂²×C₁₃	—	2	2	no	—
108	C₂²×C₁₃	—	2	4	no	—
109	C₂²×C₁₃	—	2	2	no	—
110	C₂²×C₁₃	—	4	4	no	—
111	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₁₃	#533	1	2	yes	—
113	C₄×C₁₃	—	2	4	no	—
114	C₄×C₁₃	—	2	2	no	—
115	C₄×C₁₃	—	2	2	no	—
116	C₄×C₁₃	#430	1	4	yes	—
117	C₄×C₁₃	C₂×#117	1	2	yes	—
118	C₄×C₁₃	#532	1	2	yes	—
119	C₂³×C₄	Dic₂₆	1	2	yes	—
120	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
121	C₂×M₁₆	—	2	8	no	—
122	C₂×M₁₆	—	13	8	no	—
123	C₂×M₁₆	C₂×Dic₁₃	1	8	yes	—
124	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
125	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
126	C₂×C₄²	Dic₂₆	1	2	yes	—
127	C₂×C₄²	D₅₂	1	2	yes	—
128	C₂×C₄²	D₅₂	1	2	yes	—
129	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
130	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
131	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
132	#108	—	26	16	no	—
133	C₄⋊C₈	—	26	8	no	—
134	C₂²⋊C₈	—	26	8	no	—
135	#97	—	26	4	no	—
136	#97	C₂×Dic₁₃	1	4	yes	—
137	#97	C₂×Dic₁₃	1	4	yes	—
138	C₄×C₈	—	26	4	no	—
139	C₄×C₈	C₂×Dic₁₃	1	8	yes	—
140	C₂×C₁₃	—	2	2	no	—
141	C₂×C₁₃	—	4	2	no	—
142	C₂×C₁₃	#2285	1	1	yes	characteristic
143	C₂×C₁₃	C₂×#430	1	2	yes	—
144	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	—	4	4	no	—
147	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
148	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
149	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
150	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
151	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
152	C₂²×C₄	C₂×D₅₂	1	2	yes	—
153	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
154	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
155	M₁₆	—	2	16	no	—
156	M₁₆	—	26	16	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	26	8	no	—
159	C₂×C₈	#6127	1	8	yes	—
160	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
161	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
162	C₄²	C₄⋊Dic₁₃	1	4	yes	—
163	C₄²	C₄⋊Dic₁₃	1	4	yes	—
164	C₄²	C₂×D₅₂	1	1	yes	characteristic
165	C₄²	C₂×D₅₂	1	2	yes	—
166	C₄²	C₂×D₅₂	1	1	yes	characteristic
167	C₄²	C₂²×Dic₁₃	1	2	yes	—
168	C₄²	C₂²×Dic₁₃	1	2	yes	—
169	C₁₃	#20827	1	1	yes	characteristic, complemented, Sylow
170	C₂³	—	2	2	no	—
171	C₂³	—	4	2	no	—
172	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
173	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
182	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
183	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
184	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
185	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
186	C₂×C₄	C₂×#6127	1	4	yes	—
187	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
188	C₈	—	2	4	no	—
189	C₈	—	52	8	no	—
190	C₈	#88180	1	8	yes	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	2	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	4	4	no	—
195	C₂²	#11719175	1	2	yes	—
196	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	2	4	no	—
199	C₄	—	2	2	no	—
200	C₄	#11718786	1	4	yes	—
201	C₄	#11719182	1	2	yes	—
202	C₄	C₂×#88180	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	4	2	no	—
205	C₂	#13117243	1	1	yes	characteristic
206	C₂	C₂×#11718786	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₁₃
hypocenter	C₁₃	#20827
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #13115011	C₂	semidirect
3. #13114977	C₂	semidirect
5. C₄×#88059	C₂	semidirect
169. C₁₃	#20827	semidirect

Maximal subgroups

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

subgroup	index
#13123469	2
#13115011	2
#13114977	2
#13114878	2
C₄×#88059	2
C₂×C₄×C₁₃×M₁₆	2
#20827	13