Subgroups — #15781540

Order 3328.

370 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125350	C₂	1	1	yes	maximal, characteristic
2	#13114993	C₂	1	1	yes	maximal, characteristic
3	#13112942	C₂	1	4	yes	maximal
4	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic
5	#11719175	—	2	1	no	—
6	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	#11718767	—	2	1	no	—
9	#11718767	C₂²	1	2	yes	—
10	C₂×#88057	—	2	1	no	—
11	C₂×#88057	C₂²	1	2	yes	—
12	#11718596	—	2	1	no	—
13	#11718344	—	2	1	no	—
14	#11718344	C₄	1	2	yes	complemented
15	#11718343	C₄	1	2	yes	complemented
16	#11718341	—	2	1	no	—
17	C₁₃×#346	—	2	1	no	—
18	C₄×C₁₃×C₁₆	—	2	1	no	—
19	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
22	#88180	—	2	2	no	—
23	C₂×C₄⋊Dic₁₃	—	2	2	no	—
24	C₂×C₄⋊Dic₁₃	—	4	2	no	—
25	#88178	—	4	2	no	—
26	#88124	—	2	2	no	—
27	#88123	—	2	2	no	—
28	C₄×Dic₂₆	—	4	2	no	—
29	C₁₃×M₃₂	—	2	2	no	—
30	C₁₃×M₃₂	Q₈	1	4	yes	complemented
31	C₂×C₁₃×C₁₆	—	2	2	no	—
32	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
35	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
36	#88058	—	2	2	no	—
37	#88058	C₂×C₄	1	4	yes	—
38	#88057	—	2	2	no	—
39	#88057	—	2	2	no	—
40	#88057	—	2	4	no	—
41	#88057	—	4	2	no	—
42	#13287	—	13	1	no	maximal, Sylow
43	C₂²×C₄×C₁₃	D₈	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₁₃	—	4	2	no	—
46	C₂×Dic₂₆	—	4	2	no	—
47	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
50	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
51	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
54	C₂×C₈×C₁₃	#41	1	2	yes	—
55	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₄²×C₁₃	SD₁₆	1	2	yes	—
58	C₂²⋊Dic₁₃	—	4	2	no	—
59	C₄⋊Dic₁₃	—	2	4	no	—
60	C₄⋊Dic₁₃	—	4	2	no	—
61	C₄⋊Dic₁₃	—	4	2	no	—
62	C₄⋊Dic₁₃	—	4	2	no	—
63	C₄⋊Dic₁₃	—	8	4	no	—
64	#6101	—	4	4	no	—
65	C₄×Dic₁₃	—	4	2	no	—
66	C₁₃×C₁₆	—	2	4	no	—
67	#3160	—	13	1	no	—
68	C₄×M₃₂	D₁₃	1	1	yes	characteristic
69	#1844	—	13	1	no	—
70	#1386	—	13	4	no	—
71	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₁₃	—	2	4	no	—
73	C₂×C₄×C₁₃	#102	1	2	yes	—
74	C₂×C₄×C₁₃	#102	1	2	yes	—
75	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×Dic₁₃	—	4	4	no	—
82	C₂×Dic₁₃	—	8	4	no	—
83	C₂×Dic₁₃	—	8	2	no	—
84	Dic₂₆	—	8	4	no	—
85	C₈×C₁₃	#103	1	2	yes	—
86	C₈×C₁₃	#103	1	2	yes	—
87	C₈×C₁₃	C₄×Q₈	1	2	yes	—
88	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
89	#533	—	26	1	no	—
90	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
91	#500	—	13	2	no	—
92	#499	—	13	2	no	—
93	#499	—	26	1	no	—
94	#498	—	26	1	no	—
95	#427	—	13	2	no	—
96	#427	—	26	1	no	—
97	C₂×#106	—	13	2	no	—
98	C₂×#106	—	26	1	no	—
99	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
100	#384	—	26	1	no	—
101	#346	—	2	1	no	—
102	C₄×C₁₆	—	2	1	no	—
103	C₂²×C₁₃	—	2	2	no	—
104	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
105	C₂²×C₁₃	#486	1	1	yes	characteristic
106	C₂²×C₁₃	#500	1	1	yes	characteristic
107	C₄×C₁₃	—	2	2	no	—
108	C₄×C₁₃	#360	1	4	yes	—
109	C₄×C₁₃	#391	1	1	yes	characteristic
110	C₄×C₁₃	C₂×#103	1	1	yes	characteristic
111	C₄×C₁₃	#487	1	1	yes	characteristic
112	C₄×C₁₃	#498	1	1	yes	characteristic
113	Dic₁₃	—	8	4	no	—
114	Dic₁₃	—	16	2	no	—
115	C₂²×C₈	D₅₂	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
117	C₄⋊Q₈	—	26	2	no	—
118	#125	—	26	2	no	—
119	C₂²⋊Q₈	—	52	2	no	—
120	C₄×Q₈	—	52	2	no	—
121	#117	—	26	2	no	—
122	C₂×C₄⋊C₄	—	26	2	no	—
123	C₂×C₄⋊C₄	—	52	2	no	—
124	C₂×C₄²	D₅₂	1	1	yes	characteristic
125	M₃₂	—	2	2	no	—
126	M₃₂	Dic₂₆	1	4	yes	complemented
127	C₂×C₁₆	—	2	2	no	—
128	#107	—	13	4	no	—
129	#107	—	26	2	no	—
130	#106	—	26	2	no	—
131	#106	—	26	2	no	—
132	#106	—	26	4	no	—
133	#106	—	52	2	no	—
134	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
135	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
136	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
137	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
138	C₂×C₁₃	—	2	1	no	—
139	C₂×C₁₃	—	2	1	no	—
140	C₂×C₁₃	#1990	1	1	yes	characteristic
141	C₂×C₁₃	C₂×#360	1	2	yes	—
142	C₂×Q₈	—	52	2	no	—
143	C₂²×C₄	—	52	2	no	—
144	C₂²×C₄	D₁₀₄	1	2	yes	—
145	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
146	C₂×C₈	#6095	1	2	yes	—
147	C₂×C₈	Dic₅₂	1	2	yes	—
148	C₂×C₈	#6103	1	1	yes	characteristic
149	C₂×C₈	#6103	1	1	yes	characteristic
150	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
151	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
152	C₂×C₈	#6127	1	2	yes	—
153	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
154	C₄⋊C₄	—	26	4	no	—
155	C₄⋊C₄	—	52	4	no	—
156	C₄⋊C₄	—	52	2	no	—
157	C₄⋊C₄	—	52	2	no	—
158	C₄⋊C₄	—	52	2	no	—
159	C₄⋊C₄	—	104	4	no	—
160	C₂²⋊C₄	—	52	2	no	—
161	C₄²	—	52	2	no	—
162	C₄²	#6095	1	2	yes	—
163	C₄²	#6103	1	1	yes	characteristic
164	C₄²	#6103	1	1	yes	characteristic
165	C₁₆	—	2	4	no	—
166	C₁₃	#13287	1	1	yes	characteristic, complemented, Sylow
167	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
168	Q₈	—	104	4	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	52	4	no	—
171	C₂×C₄	—	104	4	no	—
172	C₂×C₄	—	104	2	no	—
173	C₂×C₄	#88061	1	2	yes	—
174	C₂×C₄	#88061	1	2	yes	—
175	C₂×C₄	#88123	1	1	yes	characteristic
176	C₂×C₄	#88129	1	1	yes	characteristic
177	C₂×C₄	C₂×#6095	1	1	yes	characteristic
178	C₂×C₄	C₂×#6095	1	1	yes	characteristic
179	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
180	C₂×C₄	C₂×#6103	1	1	yes	characteristic
181	C₈	#88056	1	2	yes	—
182	C₈	#88056	1	2	yes	—
183	C₈	C₄×Dic₂₆	1	2	yes	—
184	C₈	#88178	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	#11718343	1	1	yes	characteristic
187	C₂²	#11718355	1	1	yes	characteristic
188	C₂²	C₂×#88061	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	104	4	no	—
191	C₄	—	208	2	no	—
192	C₄	#11718110	1	4	yes	—
193	C₄	#11718341	1	1	yes	characteristic
194	C₄	#11718342	1	1	yes	characteristic
195	C₄	#11718590	1	1	yes	characteristic
196	C₄	C₂×#88056	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#13115797	1	1	yes	characteristic
200	C₂	C₂×#11718110	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#88061
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#13287
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
14. #11718344	C₄	semidirect
15. #11718343	C₄	semidirect
30. C₁₃×M₃₂	Q₈	semidirect
126. M₃₂	Dic₂₆	semidirect
166. C₁₃	#13287	semidirect

Maximal subgroups

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

subgroup	index
#13125350	2
#13114993	2
#13112942	2
C₄×C₁₃×M₃₂	2
#13287	13