Subgroups — #15831913

Order 3328.

734 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123947	C₂	1	2	yes	maximal
2	#13113588	C₂	1	6	yes	maximal, complemented
3	#13113584	C₂	1	3	yes	maximal, complemented
4	C₂×#11718050	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	C₄	1	1	yes	characteristic
8	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
9	C₁₃×#405	C₄	1	3	yes	—
10	C₁₃×#405	C₄	1	3	yes	—
11	C₄×C₁₃×M₁₆	C₂²	1	3	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
13	Q₈×#1055	—	2	2	no	—
14	D₄×#1055	—	2	6	no	—
15	#11718586	C₂²	1	6	yes	—
16	C₂×C₄⋊#1055	C₂²	1	6	yes	—
17	#11718050	C₂²	1	3	yes	—
18	#11718050	C₂²	1	6	yes	—
19	#11718050	C₂²	1	1	yes	characteristic
20	#11718049	C₂²	1	6	yes	—
21	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
22	C₁₃×#131	—	2	4	no	—
23	C₁₃×#131	C₈	1	8	yes	complemented
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	C₂×C₄	1	3	yes	—
28	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
29	C₄×Q₈×C₁₃	Q₈	1	1	yes	characteristic
30	C₄×D₄×C₁₃	D₄	1	3	yes	—
31	C₄×D₄×C₁₃	Q₈	1	3	yes	—
32	C₁₃×#117	C₂×C₄	1	3	yes	—
33	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
34	C₂²×#1055	—	2	6	no	—
35	C₁₃×#97	C₂×C₄	1	6	yes	—
36	C₄×C₈×C₁₃	C₂×C₄	1	3	yes	—
37	C₄×C₈×C₁₃	C₂×C₄	1	3	yes	—
38	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₄×C₈×C₁₃	C₂³	1	3	yes	—
40	C₂²⋊#1055	—	2	6	no	—
41	C₄⋊#1055	—	2	6	no	—
42	C₄⋊#1055	C₂³	1	2	yes	—
43	C₄⋊#1055	C₂³	1	6	yes	—
44	C₄×#1055	—	2	6	no	—
45	#17789	—	13	1	no	maximal, Sylow
46	C₁₃×#41	C₂×C₈	1	4	yes	—
47	C₁₃×#41	M₁₆	1	4	yes	—
48	C₂×Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂×D₄×C₁₃	C₄⋊C₄	1	3	yes	—
50	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
51	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
52	C₂²×C₄×C₁₃	C₂×Q₈	1	3	yes	—
53	C₁₃×M₁₆	—	2	6	no	—
54	C₁₃×M₁₆	C₂×C₈	1	12	yes	complemented
55	C₂×C₈×C₁₃	—	2	6	no	—
56	C₂×C₈×C₁₃	C₂×C₈	1	12	yes	complemented
57	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
59	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
61	C₁₃×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
62	C₁₃×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
63	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	6	yes	—
64	C₄²×C₁₃	C₂²×C₄	1	3	yes	—
65	C₄²×C₁₃	C₂×D₄	1	3	yes	—
66	C₄²×C₁₃	C₂×Q₈	1	3	yes	—
67	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₂×#1055	—	2	2	no	—
69	C₂×#1055	—	2	6	no	—
70	C₂×#1055	—	8	6	no	—
71	#2988	—	13	2	no	—
72	C₄×#131	D₁₃	1	1	yes	characteristic
73	#1565	—	13	6	no	—
74	#1564	—	13	3	no	—
75	C₂×#335	—	13	3	no	—
76	C₂³×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
77	Q₈×C₁₃	C₄⋊C₈	1	4	yes	—
78	D₄×C₁₃	C₄⋊C₈	1	12	yes	—
79	C₂×C₄×C₁₃	—	2	3	no	—
80	C₂×C₄×C₁₃	—	2	3	no	—
81	C₂×C₄×C₁₃	—	4	3	no	—
82	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
83	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
84	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	3	yes	—
85	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
86	C₂×C₄×C₁₃	C₂×M₁₆	1	6	yes	—
87	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
90	C₈×C₁₃	—	2	1	no	—
91	C₈×C₁₃	—	2	3	no	—
92	C₈×C₁₃	C₂²×C₈	1	2	yes	—
93	C₈×C₁₃	C₂²×C₈	1	6	yes	—
94	#1055	—	4	6	no	—
95	#1055	—	4	2	no	—
96	C₂×#131	Dic₁₃	1	1	yes	characteristic
97	C₂×#131	Dic₁₃	1	1	yes	characteristic
98	C₄×#41	C₂×D₁₃	1	1	yes	characteristic
99	C₈×Q₈	—	26	2	no	—
100	C₈×D₄	—	26	6	no	—
101	#431	—	13	6	no	—
102	C₂×C₄⋊C₈	—	13	6	no	—
103	#405	Dic₁₃	1	3	yes	—
104	#405	Dic₁₃	1	3	yes	—
105	C₄×M₁₆	C₂×D₁₃	1	3	yes	—
106	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
107	#335	—	13	3	no	—
108	#335	—	13	6	no	—
109	#335	—	13	1	no	—
110	#334	—	13	6	no	—
111	C₂²×C₁₃	—	2	3	no	—
112	C₂²×C₁₃	—	2	3	no	—
113	C₂²×C₁₃	C₂×C₄⋊C₈	1	6	yes	—
114	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₄×C₁₃	—	2	3	no	—
116	C₄×C₁₃	—	2	1	no	—
117	C₄×C₁₃	—	2	3	no	—
118	C₄×C₁₃	C₂×C₄⋊C₈	1	6	yes	—
119	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
120	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
121	C₄×C₁₃	#576	1	1	yes	characteristic
122	C₄×C₁₃	#578	1	1	yes	characteristic
123	C₂×#41	C₂×Dic₁₃	1	1	yes	characteristic
124	#131	—	2	4	no	—
125	#131	#1055	1	8	yes	complemented
126	C₂×M₁₆	C₂×Dic₁₃	1	3	yes	—
127	C₂×M₁₆	C₂×Dic₁₃	1	3	yes	—
128	C₂²×C₈	—	26	6	no	—
129	C₂²×C₈	C₂×Dic₁₃	1	3	yes	—
130	C₂²×C₈	C₂×Dic₁₃	1	3	yes	—
131	C₄×Q₈	Dic₂₆	1	1	yes	characteristic
132	C₄×Q₈	D₅₂	1	1	yes	characteristic
133	C₄×D₄	Dic₂₆	1	3	yes	—
134	C₄×D₄	D₅₂	1	3	yes	—
135	#117	C₂×Dic₁₃	1	3	yes	—
136	C₂×C₄²	C₂²×D₁₃	1	3	yes	—
137	C₄⋊C₈	—	13	2	no	—
138	C₄⋊C₈	—	13	6	no	—
139	C₄⋊C₈	—	26	6	no	—
140	C₂²⋊C₈	—	26	6	no	—
141	#97	C₂×Dic₁₃	1	6	yes	—
142	C₄×C₈	—	26	6	no	—
143	C₄×C₈	C₂×Dic₁₃	1	3	yes	—
144	C₄×C₈	C₂×Dic₁₃	1	3	yes	—
145	C₄×C₈	C₂²×D₁₃	1	3	yes	—
146	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
147	C₂×C₁₃	—	2	6	no	—
148	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
149	C₂×C₁₃	#2913	1	1	yes	characteristic
150	C₂×C₁₃	#2916	1	1	yes	characteristic
151	#41	C₂×#1055	1	4	yes	—
152	#41	#6099	1	4	yes	—
153	C₂×Q₈	C₄⋊Dic₁₃	1	1	yes	characteristic
154	C₂×D₄	C₄⋊Dic₁₃	1	3	yes	—
155	C₂²×C₄	C₂×Dic₂₆	1	3	yes	—
156	C₂²×C₄	C₂×D₅₂	1	3	yes	—
157	C₂²×C₄	C₂²×Dic₁₃	1	3	yes	—
158	M₁₆	—	2	6	no	—
159	M₁₆	C₂×#1055	1	12	yes	complemented
160	C₂×C₈	—	2	6	no	—
161	C₂×C₈	—	26	2	no	—
162	C₂×C₈	—	26	6	no	—
163	C₂×C₈	—	104	6	no	—
164	C₂×C₈	C₂×#1055	1	12	yes	complemented
165	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
166	C₂×C₈	C₂²×Dic₁₃	1	3	yes	—
167	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
168	C₂×C₈	C₂²×Dic₁₃	1	3	yes	—
169	C₄⋊C₄	C₄⋊Dic₁₃	1	3	yes	—
170	C₄⋊C₄	C₄⋊Dic₁₃	1	3	yes	—
171	C₂²⋊C₄	C₄⋊Dic₁₃	1	6	yes	—
172	C₄²	C₂×Dic₂₆	1	3	yes	—
173	C₄²	C₂×D₅₂	1	3	yes	—
174	C₄²	C₂²×Dic₁₃	1	3	yes	—
175	C₄²	C₂³×D₁₃	1	1	yes	characteristic
176	C₁₃	#17789	1	1	yes	characteristic, complemented, Sylow
177	C₂³	C₂×C₄⋊Dic₁₃	1	3	yes	—
178	Q₈	C₄⋊#1055	1	4	yes	—
179	D₄	C₄⋊#1055	1	12	yes	—
180	C₂×C₄	—	2	3	no	—
181	C₂×C₄	—	2	3	no	—
182	C₂×C₄	—	4	3	no	—
183	C₂×C₄	C₂²×#1055	1	6	yes	—
184	C₂×C₄	C₂×#6099	1	6	yes	—
185	C₂×C₄	C₂×C₄⋊Dic₁₃	1	3	yes	—
186	C₂×C₄	C₂×C₄⋊Dic₁₃	1	3	yes	—
187	C₂×C₄	C₂×C₄⋊Dic₁₃	1	3	yes	—
188	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
189	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
190	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
191	C₈	—	2	1	no	—
192	C₈	—	2	3	no	—
193	C₈	—	52	6	no	—
194	C₈	—	52	2	no	—
195	C₈	C₂²×#1055	1	2	yes	—
196	C₈	C₂²×#1055	1	6	yes	—
197	C₂²	—	2	3	no	—
198	C₂²	—	2	3	no	—
199	C₂²	C₂×C₄⋊#1055	1	6	yes	—
200	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
201	C₄	—	2	3	no	—
202	C₄	—	2	1	no	—
203	C₄	—	2	3	no	—
204	C₄	C₂×C₄⋊#1055	1	6	yes	—
205	C₄	#11719457	1	1	yes	characteristic
206	C₄	#11719458	1	1	yes	characteristic
207	C₄	C₂³×#1055	1	1	yes	characteristic
208	C₄	C₂²×#6099	1	1	yes	characteristic
209	C₂	—	2	6	no	—
210	C₂	C₂²×C₄⋊#1055	1	1	yes	characteristic
211	C₂	#13123436	1	1	yes	characteristic
212	C₂	#13123447	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₄²	C₂³×D₁₃
Fitting	C₄×C₁₃×#131	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₄×#131	D₁₃
hypocenter	C₁₃	#17789
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #13113588	C₂	semidirect
3. #13113584	C₂	semidirect
4. C₂×#11718050	C₂	semidirect
23. C₁₃×#131	C₈	semidirect
54. C₁₃×M₁₆	C₂×C₈	semidirect
56. C₂×C₈×C₁₃	C₂×C₈	semidirect
125. #131	#1055	semidirect
159. M₁₆	C₂×#1055	semidirect
164. C₂×C₈	C₂×#1055	semidirect
176. C₁₃	#17789	semidirect

Maximal subgroups

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

subgroup	index
#13123947	2
#13113588	2
#13113584	2
C₂×#11718050	2
C₄×C₁₃×#131	2
#17789	13