Subgroups — #15771949

Order 3328.

306 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116635	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic
3	#13112755	C₂	1	2	yes	maximal, complemented
4	C₁₆×#1055	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2162	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×#1055	C₄	1	1	yes	characteristic
8	C₂×C₄×#1055	C₄	1	1	yes	characteristic
9	C₁₃×C₄⋊C₁₆	C₄	1	2	yes	—
10	C₁₃×C₂²⋊C₁₆	C₄	1	2	yes	—
11	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
12	C₂²⋊#6090	C₄	1	2	yes	—
13	C₄⋊#6090	C₄	1	2	yes	—
14	C₄×#6090	C₂²	1	2	yes	—
15	C₈×#1055	C₄	1	1	yes	characteristic
16	C₈×#1055	C₄	1	1	yes	characteristic
17	C₈×#1055	C₂²	1	1	yes	characteristic
18	C₈×#1055	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×#1055	C₈	1	2	yes	—
23	C₂²×#1055	C₈	1	2	yes	—
24	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
25	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×#6090	C₂×C₄	1	4	yes	—
30	C₄×#1055	C₈	1	2	yes	—
31	C₄×#1055	C₈	1	2	yes	—
32	C₄×#1055	C₂×C₄	1	1	yes	characteristic
33	C₄×#1055	C₂×C₄	1	1	yes	characteristic
34	C₄×#1055	C₂×C₄	1	1	yes	characteristic
35	C₄×#1055	C₂×C₄	1	1	yes	characteristic
36	#11410	—	13	1	no	maximal, Sylow
37	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
40	C₂×C₈×C₁₃	—	2	2	no	—
41	C₂×C₈×C₁₃	—	2	2	no	—
42	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
43	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₁₃	C₄²	1	1	yes	characteristic
47	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
48	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
49	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×#1055	—	2	4	no	—
51	C₂×#1055	C₁₆	1	8	yes	complemented
52	C₂×#1055	C₂×C₈	1	2	yes	—
53	C₂×#1055	C₂×C₈	1	2	yes	—
54	C₂×#1055	C₂×C₈	1	2	yes	—
55	C₂×#1055	C₂×C₈	1	2	yes	—
56	C₁₃×C₁₆	—	2	4	no	—
57	#6090	—	2	4	no	—
58	#2162	—	13	1	no	—
59	#2162	D₁₃	1	1	yes	characteristic
60	C₂×C₈²	—	13	1	no	—
61	#1312	—	13	2	no	—
62	C₈×C₁₆	—	13	2	no	—
63	C₂³×C₁₃	C₄×C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₃	—	2	1	no	—
65	C₂×C₄×C₁₃	—	2	1	no	—
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	—
71	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
74	C₈×C₁₃	—	2	1	no	—
75	C₈×C₁₃	—	2	1	no	—
76	C₈×C₁₃	#131	1	2	yes	—
77	C₈×C₁₃	#131	1	2	yes	—
78	#1055	—	2	2	no	—
79	#1055	C₂×C₁₆	1	4	yes	—
80	#1055	C₂×C₁₆	1	4	yes	—
81	#1055	M₃₂	1	4	yes	—
82	C₂×C₄×C₈	—	13	1	no	—
83	C₂×C₄×C₈	—	13	1	no	—
84	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
85	C₄⋊C₁₆	—	13	2	no	—
86	C₄⋊C₁₆	Dic₁₃	1	2	yes	—
87	C₂²⋊C₁₆	—	13	2	no	—
88	C₂²⋊C₁₆	Dic₁₃	1	2	yes	—
89	C₄×C₁₆	—	13	2	no	—
90	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
91	C₈²	—	13	1	no	—
92	C₈²	—	13	1	no	—
93	C₈²	—	13	1	no	—
94	C₈²	—	13	1	no	—
95	C₂²×C₁₃	—	2	1	no	—
96	C₂²×C₁₃	—	2	1	no	—
97	C₂²×C₁₃	C₄×C₁₆	1	2	yes	—
98	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
99	C₄×C₁₃	—	2	1	no	—
100	C₄×C₁₃	—	2	1	no	—
101	C₄×C₁₃	C₄×C₁₆	1	2	yes	—
102	C₄×C₁₃	#405	1	1	yes	characteristic
103	C₄×C₁₃	#405	1	1	yes	characteristic
104	C₄×C₁₃	C₂²×C₁₆	1	2	yes	—
105	C₄×C₁₃	C₂×M₃₂	1	2	yes	—
106	C₂²×C₈	—	13	2	no	—
107	C₂²×C₈	—	13	2	no	—
108	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
109	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
110	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
111	C₂×C₁₆	—	13	4	no	—
112	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	—
113	C₄×C₈	—	13	2	no	—
114	C₄×C₈	—	13	2	no	—
115	C₄×C₈	—	13	1	no	—
116	C₄×C₈	—	13	1	no	—
117	C₄×C₈	—	13	1	no	—
118	C₄×C₈	—	13	1	no	—
119	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
120	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
121	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
122	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
123	C₂×C₁₃	—	2	2	no	—
124	C₂×C₁₃	#1450	1	1	yes	characteristic
125	C₂×C₁₃	C₂×C₄×C₁₆	1	1	yes	characteristic
126	C₂×C₁₃	C₄×M₃₂	1	1	yes	characteristic
127	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
128	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
129	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	13	2	no	—
133	C₂×C₈	—	13	2	no	—
134	C₂×C₈	—	13	2	no	—
135	C₂×C₈	—	13	8	no	—
136	C₂×C₈	—	13	2	no	—
137	C₂×C₈	—	26	4	no	—
138	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
139	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
140	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
141	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
142	C₄²	C₈×D₁₃	1	1	yes	characteristic
143	C₄²	C₈×D₁₃	1	1	yes	characteristic
144	C₄²	C₄×Dic₁₃	1	1	yes	characteristic
145	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
146	C₁₆	—	2	4	no	—
147	C₁₆	—	26	4	no	—
148	C₁₃	#11410	1	1	yes	characteristic, complemented, Sylow
149	C₂³	C₈×Dic₁₃	1	1	yes	characteristic
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	C₁₆×D₁₃	1	4	yes	—
154	C₂×C₄	C₈×Dic₁₃	1	1	yes	characteristic
155	C₂×C₄	C₈×Dic₁₃	1	1	yes	characteristic
156	C₂×C₄	C₈×Dic₁₃	1	1	yes	characteristic
157	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
158	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
159	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	13	4	no	—
163	C₈	—	13	4	no	—
164	C₈	—	13	4	no	—
165	C₈	—	26	2	no	—
166	C₈	#88161	1	2	yes	—
167	C₈	#88202	1	2	yes	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	C₁₆×Dic₁₃	1	2	yes	—
171	C₂²	C₂×C₈×Dic₁₃	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	C₁₆×Dic₁₃	1	2	yes	—
175	C₄	C₂×C₁₆×D₁₃	1	2	yes	—
176	C₄	D₁₃×M₃₂	1	2	yes	—
177	C₄	#11718658	1	1	yes	characteristic
178	C₄	#11718783	1	1	yes	characteristic
179	C₂	—	2	2	no	—
180	C₂	#13113120	1	1	yes	characteristic
181	C₂	C₂×C₁₆×Dic₁₃	1	1	yes	characteristic
182	C₂	M₃₂×Dic₁₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₁₃
commutator	C₂×C₁₃	C₂×C₄×C₁₆
Fitting	C₁₃×#2162	C₂
socle	C₂²×C₁₃	C₂×C₄×C₈
prime radical	C₂³×C₁₃	C₄×C₈
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	#2162	D₁₃
hypocenter	C₁₃	#11410
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ #2162

Complements

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

N	H ≅ G/N	type
3. #13112755	C₂	semidirect
4. C₁₆×#1055	C₂	semidirect
51. C₂×#1055	C₁₆	semidirect
148. C₁₃	#11410	semidirect

Maximal subgroups

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

subgroup	index
#13116635	2
C₂×C₈×#1055	2
#13112755	2
C₁₆×#1055	2
C₁₃×#2162	2
#11410	13