Subgroups — #15774489

Order 3328.

270 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116592	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic
3	#13112905	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×M₃₂	—	2	1	no	—
6	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×#88052	—	2	1	no	—
8	C₂×C₄×#1055	C₄	1	1	yes	characteristic
9	C₂×C₄×#1055	C₄	1	1	yes	characteristic
10	C₁₃×C₄⋊C₁₆	—	2	1	no	—
11	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
12	C₄⋊#6090	—	2	1	no	—
13	C₄⋊#6090	C₂²	1	2	yes	—
14	C₈×#1055	C₄	1	2	yes	—
15	C₈×#1055	C₂²	1	2	yes	—
16	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₃	Q₈	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×#1055	—	2	1	no	—
20	C₂²×#1055	C₈	1	2	yes	—
21	C₁₃×M₃₂	—	4	2	no	—
22	C₂×C₁₃×C₁₆	—	2	2	no	—
23	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
26	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	#88052	—	4	2	no	—
28	C₂×#6090	—	2	2	no	—
29	C₄×#1055	—	2	1	no	—
30	C₄×#1055	C₈	1	1	yes	characteristic
31	C₄×#1055	C₈	1	1	yes	characteristic
32	C₄×#1055	C₂×C₄	1	2	yes	—
33	C₄×#1055	C₂×C₄	1	2	yes	—
34	#12004	—	13	1	no	maximal, Sylow
35	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
38	C₂×C₈×C₁₃	—	2	2	no	—
39	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₃	D₈	1	2	yes	—
42	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
43	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
45	C₄²×C₁₃	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
48	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×#1055	—	2	4	no	—
50	C₂×#1055	—	2	2	no	—
51	C₂×#1055	—	2	4	no	—
52	C₂×#1055	C₂×C₈	1	2	yes	—
53	C₂×#1055	C₂×C₈	1	2	yes	—
54	C₁₃×C₁₆	—	4	2	no	—
55	#6090	—	4	2	no	—
56	#2150	—	13	1	no	—
57	#2150	D₁₃	1	1	yes	characteristic
58	C₂×C₈²	—	13	1	no	—
59	#1370	—	13	4	no	—
60	C₂³×C₁₃	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	2	no	—
63	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₁₃	C₄⋊C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₃	#107	1	2	yes	—
67	C₂×C₄×C₁₃	#108	1	2	yes	—
68	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
71	C₈×C₁₃	—	2	1	no	—
72	C₈×C₁₃	—	2	1	no	—
73	C₈×C₁₃	#107	1	2	yes	—
74	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
75	C₈×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
76	#1055	—	2	4	no	—
77	#1055	—	2	2	no	—
78	#1055	M₃₂	1	4	yes	complemented
79	C₂×M₃₂	—	2	1	no	—
80	C₂×M₃₂	—	26	1	no	—
81	C₂×C₄×C₈	—	13	1	no	—
82	C₂×C₄×C₈	—	13	1	no	—
83	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
84	C₄⋊C₁₆	—	2	1	no	—
85	C₄⋊C₁₆	—	13	2	no	—
86	C₄⋊C₁₆	—	26	1	no	—
87	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
88	C₈²	—	13	2	no	—
89	C₈²	—	13	2	no	—
90	C₂²×C₁₃	—	2	1	no	—
91	C₂²×C₁₃	—	2	1	no	—
92	C₂²×C₁₃	#335	1	2	yes	—
93	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
94	C₄×C₁₃	—	2	1	no	—
95	C₄×C₁₃	—	2	1	no	—
96	C₄×C₁₃	#335	1	2	yes	—
97	C₄×C₁₃	#364	1	2	yes	—
98	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
99	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
100	C₄×C₁₃	C₂×M₃₂	1	2	yes	—
101	C₂²×C₈	—	13	2	no	—
102	C₂²×C₈	—	26	1	no	—
103	C₂²×C₈	Dic₂₆	1	1	yes	characteristic
104	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
105	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
106	M₃₂	—	4	2	no	—
107	M₃₂	—	52	2	no	—
108	C₂×C₁₆	—	2	2	no	—
109	C₂×C₁₆	—	26	2	no	—
110	C₄×C₈	—	13	1	no	—
111	C₄×C₈	—	13	1	no	—
112	C₄×C₈	—	13	2	no	—
113	C₄×C₈	—	13	2	no	—
114	C₄×C₈	—	26	1	no	—
115	C₄×C₈	Dic₂₆	1	1	yes	characteristic
116	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
117	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
118	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
119	C₂×C₁₃	—	2	2	no	—
120	C₂×C₁₃	C₂×#335	1	1	yes	characteristic
121	C₂×C₁₃	#2150	1	1	yes	characteristic
122	C₂×C₁₃	C₂×#364	1	1	yes	characteristic
123	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
124	C₂²×C₄	#6094	1	1	yes	characteristic
125	C₂²×C₄	#6101	1	1	yes	characteristic
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	13	2	no	—
128	C₂×C₈	—	13	2	no	—
129	C₂×C₈	—	26	4	no	—
130	C₂×C₈	—	26	4	no	—
131	C₂×C₈	—	26	2	no	—
132	C₂×C₈	#6101	1	1	yes	characteristic
133	C₂×C₈	#6101	1	1	yes	characteristic
134	C₂×C₈	#6104	1	2	yes	—
135	C₂×C₈	#6107	1	2	yes	—
136	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
137	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
138	C₄²	C₈×D₁₃	1	1	yes	characteristic
139	C₄²	#6094	1	1	yes	characteristic
140	C₄²	#6101	1	1	yes	characteristic
141	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
142	C₁₆	—	4	2	no	—
143	C₁₆	—	52	2	no	—
144	C₁₃	#12004	1	1	yes	characteristic, complemented, Sylow
145	C₂³	#88054	1	1	yes	characteristic
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	#88047	1	2	yes	—
149	C₂×C₄	#88054	1	1	yes	characteristic
150	C₂×C₄	#88054	1	1	yes	characteristic
151	C₂×C₄	#88054	1	1	yes	characteristic
152	C₂×C₄	#88062	1	2	yes	—
153	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
154	C₂×C₄	C₂×#6094	1	1	yes	characteristic
155	C₂×C₄	C₂×#6101	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	13	4	no	—
159	C₈	—	26	4	no	—
160	C₈	—	26	2	no	—
161	C₈	#88047	1	2	yes	—
162	C₈	C₂×#6104	1	1	yes	characteristic
163	C₈	C₂×#6107	1	1	yes	characteristic
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#11718071	1	2	yes	—
167	C₂²	C₂×#88054	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	#11718071	1	2	yes	—
171	C₄	#11718103	1	2	yes	—
172	C₄	D₁₃×M₃₂	1	2	yes	—
173	C₄	C₂×#88047	1	1	yes	characteristic
174	C₄	C₂×#88062	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	C₂×#11718071	1	1	yes	characteristic
177	C₂	#13116591	1	1	yes	characteristic
178	C₂	C₂×#11718103	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#2150	C₂
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂³×C₁₃	C₄⋊C₈
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	#2150	D₁₃
hypocenter	C₁₃	#12004
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
3. #13112905	C₂	semidirect
78. #1055	M₃₂	semidirect
144. C₁₃	#12004	semidirect

Maximal subgroups

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

subgroup	index
#13116592	2
C₂×C₈×#1055	2
#13112905	2
C₁₃×#2150	2
#12004	13