Subgroups — #15754108

Order 3328.

438 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	#13114689	C₂	1	2	yes	maximal
2	C₂×#11718049	C₂	1	2	yes	maximal
3	#13112671	C₂	1	2	yes	maximal
4	C₄²×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
6	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₄³×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊#1055	C₂²	1	4	yes	—
11	C₂×#88043	—	2	2	no	—
12	#11718116	—	2	2	no	—
13	#11718049	C₄	1	8	yes	complemented
14	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
21	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
22	C₂²×#1055	—	2	4	no	—
23	C₄×C₈×C₁₃	—	2	4	no	—
24	C₄×C₈×C₁₃	—	2	4	no	—
25	C₄×C₈×C₁₃	—	2	2	no	—
26	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
27	C₄⋊#1055	—	2	4	no	—
28	C₄⋊#1055	C₂×C₄	1	8	yes	—
29	#88043	—	4	4	no	—
30	#8273	—	13	1	no	maximal, Sylow
31	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
33	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₁₃	—	2	2	no	—
37	C₂×C₈×C₁₃	—	2	2	no	—
38	C₂×C₈×C₁₃	—	2	2	no	—
39	C₂×C₈×C₁₃	C₄²	1	2	yes	—
40	C₂×C₈×C₁₃	C₄²	1	2	yes	—
41	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
42	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
43	C₄²×C₁₃	—	2	4	no	—
44	C₄²×C₁₃	—	2	2	no	—
45	C₄²×C₁₃	—	2	2	no	—
46	C₄²×C₁₃	—	2	2	no	—
47	C₄²×C₁₃	SD₁₆	1	4	yes	—
48	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
49	C₄²×C₁₃	#41	1	2	yes	—
50	C₂×#1055	—	2	8	no	—
51	C₂×#1055	—	4	4	no	—
52	C₂×#1055	—	4	4	no	—
53	#1767	—	13	2	no	—
54	C₄²×C₈	D₁₃	1	1	yes	characteristic
55	C₂×#334	—	13	2	no	—
56	#1277	—	13	2	no	—
57	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃	—	2	2	no	—
59	C₂×C₄×C₁₃	—	2	2	no	—
60	C₂×C₄×C₁₃	—	2	2	no	—
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	2	no	—
63	C₂×C₄×C₁₃	#106	1	4	yes	—
64	C₂×C₄×C₁₃	#108	1	4	yes	—
65	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
68	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
69	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
70	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
71	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
72	C₂×C₄×C₁₃	#135	1	2	yes	—
73	C₈×C₁₃	—	2	2	no	—
74	C₈×C₁₃	—	2	4	no	—
75	C₈×C₁₃	#97	1	4	yes	complemented
76	#1055	—	4	8	no	—
77	C₂×C₄⋊C₈	—	13	4	no	—
78	C₂×#97	—	26	2	no	—
79	C₂×C₄×C₈	Dic₁₃	1	2	yes	—
80	C₂×C₄×C₈	Dic₁₃	1	2	yes	—
81	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
82	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
83	C₄³	C₂×D₁₃	1	1	yes	characteristic
84	#336	—	26	2	no	—
85	#334	—	13	8	no	—
86	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₁₃	#423	1	2	yes	—
88	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
89	C₂²×C₁₃	#427	1	1	yes	characteristic
90	C₂²×C₁₃	C₂×#108	1	2	yes	—
91	C₄×C₁₃	—	2	4	no	—
92	C₄×C₁₃	—	2	4	no	—
93	C₄×C₁₃	—	2	2	no	—
94	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
95	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
96	C₄×C₁₃	#432	1	2	yes	—
97	C₄×C₁₃	C₄×SD₁₆	1	4	yes	—
98	C₂²×C₈	—	26	4	no	—
99	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
100	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
101	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
102	C₂×C₄²	Dic₂₆	1	2	yes	—
103	C₂×C₄²	D₅₂	1	1	yes	characteristic
104	C₂×C₄²	D₅₂	1	1	yes	characteristic
105	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
106	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
107	C₄⋊C₈	—	13	8	no	—
108	C₄⋊C₈	—	26	4	no	—
109	#97	—	52	4	no	—
110	C₄×C₈	—	2	4	no	—
111	C₄×C₈	—	2	4	no	—
112	C₄×C₈	—	2	2	no	—
113	C₄×C₈	C₄×D₁₃	1	4	yes	—
114	C₂×C₁₃	#1566	1	2	yes	—
115	C₂×C₁₃	#1568	1	2	yes	—
116	C₂×C₁₃	#1767	1	1	yes	characteristic
117	C₂×C₁₃	C₄×#106	1	1	yes	characteristic
118	C₂×C₁₃	C₄×#108	1	1	yes	characteristic
119	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
120	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
121	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
122	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
123	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	26	8	no	—
128	C₂×C₈	—	52	4	no	—
129	C₂×C₈	—	52	4	no	—
130	C₂×C₈	#6099	1	4	yes	—
131	C₂×C₈	#6099	1	4	yes	—
132	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
133	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
134	C₄²	—	2	2	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	2	4	no	—
138	C₄²	#6095	1	4	yes	—
139	C₄²	C₂×C₄×D₁₃	1	2	yes	—
140	C₄²	#6127	1	2	yes	—
141	C₁₃	#8273	1	1	yes	characteristic, complemented, Sylow
142	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	#88057	1	4	yes	—
149	C₂×C₄	#88059	1	4	yes	—
150	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
151	C₂×C₄	C₄×D₅₂	1	2	yes	—
152	C₂×C₄	C₂×#6095	1	2	yes	—
153	C₂×C₄	#88158	1	2	yes	—
154	C₂×C₄	C₂×#6099	1	2	yes	—
155	C₂×C₄	C₂×#6099	1	2	yes	—
156	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
157	C₂×C₄	#88180	1	1	yes	characteristic
158	C₈	—	2	2	no	—
159	C₈	—	2	4	no	—
160	C₈	—	52	8	no	—
161	C₈	#88043	1	4	yes	complemented
162	C₂²	#11718585	1	2	yes	—
163	C₂²	C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
164	C₂²	C₂×#88057	1	1	yes	characteristic
165	C₂²	#11718767	1	1	yes	characteristic
166	C₂²	C₂×#88059	1	2	yes	—
167	C₄	—	2	2	no	—
168	C₄	—	2	4	no	—
169	C₄	—	2	4	no	—
170	C₄	C₄×#6095	1	4	yes	—
171	C₄	C₂×#88043	1	1	yes	characteristic
172	C₄	C₄×#6099	1	1	yes	characteristic
173	C₄	#11718587	1	2	yes	—
174	C₂	#13113592	1	2	yes	—
175	C₂	#13113599	1	2	yes	—
176	C₂	#13114689	1	1	yes	characteristic
177	C₂	C₄×#88057	1	1	yes	characteristic
178	C₂	C₄×#88059	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄²×C₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄²×C₈	D₁₃
hypocenter	C₁₃	#8273
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
13. #11718049	C₄	semidirect
75. C₈×C₁₃	#97	semidirect
141. C₁₃	#8273	semidirect
161. C₈	#88043	semidirect

Maximal subgroups

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

subgroup	index
#13114689	2
C₂×#11718049	2
#13112671	2
C₄²×C₈×C₁₃	2
#8273	13