Subgroups — #15754223

Order 3328.

358 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13113584	C₂	1	1	yes	maximal, characteristic
2	M₁₆×#1055	C₂	1	2	yes	maximal
3	C₂×#11718047	C₂	1	1	yes	maximal, characteristic
4	#13112674	C₂	1	2	yes	maximal
5	C₄×C₁₃×#108	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#108	C₄	1	2	yes	—
7	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	#11718586	C₂²	1	2	yes	—
10	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
11	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
12	C₁₃×#341	C₄	1	2	yes	—
13	#11718050	C₄	1	2	yes	—
14	#11718049	C₄	1	2	yes	—
15	#11718047	—	2	2	no	—
16	#11718047	C₄	1	2	yes	—
17	#11718047	C₄	1	2	yes	—
18	C₈×#1055	—	2	2	no	—
19	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
20	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
23	C₂²×#1055	—	2	1	no	—
24	C₂²×#1055	Q₈	1	2	yes	—
25	C₁₃×#108	C₈	1	8	yes	complemented
26	C₁₃×#97	—	2	2	no	—
27	C₄×C₈×C₁₃	—	2	2	no	—
28	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²⋊#1055	—	2	2	no	—
33	C₄⋊#1055	C₂×C₄	1	4	yes	—
34	C₄×#1055	—	2	2	no	—
35	C₄×#1055	—	2	1	no	—
36	C₄×#1055	C₂×C₄	1	2	yes	—
37	C₄×#1055	C₂×C₄	1	2	yes	—
38	C₄×#1055	D₄	1	2	yes	—
39	#8294	—	13	1	no	maximal, Sylow
40	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
43	C₁₃×M₁₆	—	2	4	no	—
44	C₁₃×M₁₆	C₂×C₈	1	8	yes	—
45	C₂×C₈×C₁₃	—	2	4	no	—
46	C₂×C₈×C₁₃	—	2	2	no	—
47	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
52	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₄²×C₁₃	#41	1	1	yes	characteristic
56	C₂×#1055	—	2	2	no	—
57	C₂×#1055	—	2	4	no	—
58	C₂×#1055	—	4	2	no	—
59	C₂×#1055	—	4	2	no	—
60	C₂×#1055	C₄⋊C₄	1	4	yes	—
61	C₄×#108	D₁₃	1	1	yes	characteristic
62	#1564	—	13	1	no	—
63	C₈×M₁₆	—	13	2	no	—
64	C₂×#322	—	13	1	no	—
65	#1278	—	13	2	no	—
66	C₂³×C₁₃	C₄×Q₈	1	1	yes	characteristic
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	2	no	—
69	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
76	C₂×C₄×C₁₃	#131	1	2	yes	—
77	C₈×C₁₃	—	2	1	no	—
78	C₈×C₁₃	—	2	1	no	—
79	C₈×C₁₃	—	2	8	no	—
80	C₈×C₁₃	C₄×C₈	1	2	yes	—
81	C₈×C₁₃	C₄×C₈	1	2	yes	—
82	#1055	—	2	4	no	—
83	#1055	—	4	2	no	—
84	#1055	—	4	4	no	—
85	#431	—	13	2	no	—
86	C₂×#108	Dic₁₃	1	2	yes	—
87	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
88	C₂×C₄×C₈	—	13	1	no	—
89	C₂×C₄×C₈	—	13	1	no	—
90	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
91	#341	Dic₁₃	1	2	yes	—
92	#335	—	13	2	no	—
93	#334	—	13	2	no	—
94	#322	—	13	2	no	—
95	#322	—	13	2	no	—
96	#322	—	26	2	no	—
97	C₈²	—	26	2	no	—
98	C₂²×C₁₃	—	2	1	no	—
99	C₂²×C₁₃	—	2	1	no	—
100	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₁₃	C₈×Q₈	1	2	yes	—
102	C₄×C₁₃	—	2	1	no	—
103	C₄×C₁₃	—	2	1	no	—
104	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
105	C₄×C₁₃	#405	1	1	yes	characteristic
106	C₄×C₁₃	#430	1	2	yes	—
107	C₄×C₁₃	C₈×D₄	1	2	yes	—
108	C₄×C₁₃	#444	1	2	yes	—
109	C₂×M₁₆	C₂×Dic₁₃	1	4	yes	—
110	C₂²×C₈	—	13	2	no	—
111	C₂²×C₈	—	26	1	no	—
112	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
113	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
114	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
115	#108	#1055	1	8	yes	complemented
116	C₄⋊C₈	—	13	4	no	—
117	C₂²⋊C₈	—	26	2	no	—
118	#97	—	2	2	no	—
119	C₄×C₈	—	2	2	no	—
120	C₄×C₈	—	13	2	no	—
121	C₄×C₈	—	13	2	no	—
122	C₄×C₈	—	13	2	no	—
123	C₄×C₈	—	26	2	no	—
124	C₄×C₈	—	26	1	no	—
125	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
126	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
127	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
128	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
129	C₂×C₁₃	—	2	2	no	—
130	C₂×C₁₃	C₈×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₁₃	#1773	1	1	yes	characteristic
132	C₂×C₁₃	#1778	1	1	yes	characteristic
133	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
134	C₂²×C₄	#6131	1	1	yes	characteristic
135	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
136	M₁₆	—	2	4	no	—
137	M₁₆	C₂×#1055	1	8	yes	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	13	4	no	—
141	C₂×C₈	—	26	2	no	—
142	C₂×C₈	—	26	4	no	—
143	C₂×C₈	—	52	2	no	—
144	C₂×C₈	—	52	2	no	—
145	C₂×C₈	C₂×#1055	1	4	yes	—
146	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
147	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
148	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
149	C₂×C₈	C₄×Dic₁₃	1	1	yes	characteristic
150	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
151	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
152	C₄²	D₄×D₁₃	1	1	yes	characteristic
153	C₄²	#6129	1	1	yes	characteristic
154	C₁₃	#8294	1	1	yes	characteristic, complemented, Sylow
155	C₂³	Q₈×Dic₁₃	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	#88141	1	1	yes	characteristic
159	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
160	C₂×C₄	#88146	1	1	yes	characteristic
161	C₂×C₄	#88147	1	1	yes	characteristic
162	C₂×C₄	C₂²×#1055	1	2	yes	—
163	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
164	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
165	C₂×C₄	#88202	1	2	yes	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	8	no	—
169	C₈	—	26	4	no	—
170	C₈	—	52	2	no	—
171	C₈	—	52	4	no	—
172	C₈	C₄×#1055	1	2	yes	—
173	C₈	C₄×#1055	1	2	yes	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
177	C₂²	Q₈×#1055	1	2	yes	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	#11718554	1	2	yes	—
181	C₄	#11718555	1	2	yes	—
182	C₄	C₂×C₄×#1055	1	1	yes	characteristic
183	C₄	#11718658	1	1	yes	characteristic
184	C₄	D₄×#1055	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	C₄⋊C₄×#1055	1	1	yes	characteristic
187	C₂	#13114723	1	1	yes	characteristic
188	C₂	#13114744	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₁₃
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×#108	C₂
socle	C₂²×C₁₃	C₄×C₄⋊C₄
prime radical	C₂³×C₁₃	C₄×Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×#108	D₁₃
hypocenter	C₁₃	#8294
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
25. C₁₃×#108	C₈	semidirect
115. #108	#1055	semidirect
154. C₁₃	#8294	semidirect

Maximal subgroups

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

subgroup	index
#13113584	2
M₁₆×#1055	2
C₂×#11718047	2
#13112674	2
C₄×C₁₃×#108	2
#8294	13