Subgroups — #15754109

Order 3328.

438 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114689	C₂	1	1	yes	maximal, characteristic
2	C₄²×#1055	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718072	C₂	1	2	yes	maximal
4	#13112670	C₂	1	2	yes	maximal
5	C₁₃×#1767	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₁₃×#97	—	2	1	no	—
8	C₄³×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊#1055	C₂²	1	2	yes	—
10	C₂×#88043	—	2	1	no	—
11	C₂×C₄×#1055	C₄	1	2	yes	—
12	C₂×C₄×#1055	C₄	1	2	yes	—
13	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
14	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
15	C₁₃×#336	—	2	1	no	—
16	#11718116	—	2	1	no	—
17	#11718072	C₄	1	8	yes	complemented
18	C₂²×C₈×C₁₃	—	2	2	no	—
19	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂²×#1055	—	2	2	no	—
25	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
26	C₂²×#1055	C₂×C₄	1	2	yes	—
27	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
28	C₁₃×C₄⋊C₈	—	2	2	no	—
29	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
30	C₁₃×#97	—	4	2	no	—
31	C₄⋊#1055	—	2	2	no	—
32	C₄⋊#1055	C₂×C₄	1	4	yes	—
33	#88043	—	4	2	no	—
34	C₄×#1055	—	2	2	no	—
35	C₄×#1055	—	2	4	no	—
36	C₄×#1055	—	2	4	no	—
37	C₄×#1055	C₂×C₄	1	4	yes	—
38	#8273	—	13	1	no	maximal, Sylow
39	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₁₃	—	2	4	no	—
45	C₂×C₈×C₁₃	—	4	2	no	—
46	C₂×C₈×C₁₃	—	4	2	no	—
47	C₄²×C₁₃	—	2	4	no	—
48	C₄²×C₁₃	—	2	2	no	—
49	C₄²×C₁₃	—	2	2	no	—
50	C₄²×C₁₃	—	2	2	no	—
51	C₄²×C₁₃	SD₁₆	1	2	yes	—
52	C₄²×C₁₃	SD₁₆	1	2	yes	—
53	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
54	C₄²×C₁₃	#41	1	2	yes	—
55	C₂×#1055	—	2	2	no	—
56	C₂×#1055	—	2	2	no	—
57	C₂×#1055	—	2	2	no	—
58	C₂×#1055	—	2	4	no	—
59	C₂×#1055	—	4	2	no	—
60	C₂×#1055	—	4	2	no	—
61	C₂×#1055	C₄²	1	2	yes	—
62	C₂×#1055	C₄²	1	2	yes	—
63	C₂×#1055	M₁₆	1	4	yes	—
64	C₂×#1055	M₁₆	1	4	yes	—
65	#1767	—	13	1	no	—
66	#1767	D₁₃	1	1	yes	characteristic
67	C₄²×C₈	—	13	1	no	—
68	C₂×#334	—	13	2	no	—
69	#1277	—	13	2	no	—
70	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	#106	1	4	yes	—
77	C₂×C₄×C₁₃	#108	1	4	yes	—
78	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
82	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
83	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
84	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₃	#135	1	2	yes	—
87	C₈×C₁₃	—	4	4	no	—
88	#1055	—	2	2	no	—
89	#1055	—	2	4	no	—
90	#1055	—	4	4	no	—
91	#1055	#97	1	4	yes	complemented
92	C₂×C₄⋊C₈	—	13	2	no	—
93	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
94	C₂×#97	—	2	1	no	—
95	C₂×#97	—	26	1	no	—
96	C₂×C₄×C₈	—	13	1	no	—
97	C₂×C₄×C₈	—	13	2	no	—
98	C₂×C₄×C₈	—	13	2	no	—
99	C₂×C₄×C₈	—	13	1	no	—
100	C₄³	C₂×D₁₃	1	1	yes	characteristic
101	#336	—	2	1	no	—
102	#336	—	26	1	no	—
103	#334	—	13	8	no	—
104	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₁₃	#423	1	2	yes	—
106	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
107	C₂²×C₁₃	#427	1	1	yes	characteristic
108	C₂²×C₁₃	C₂×#108	1	2	yes	—
109	C₄×C₁₃	—	2	4	no	—
110	C₄×C₁₃	—	2	4	no	—
111	C₄×C₁₃	—	2	2	no	—
112	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
113	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
114	C₄×C₁₃	#432	1	2	yes	—
115	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
116	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	—	13	1	no	—
119	C₂²×C₈	—	13	1	no	—
120	C₂²×C₈	—	13	2	no	—
121	C₂²×C₈	—	26	2	no	—
122	C₂×C₄²	Dic₂₆	1	2	yes	—
123	C₂×C₄²	C₄×D₁₃	1	2	yes	—
124	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
125	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
126	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	—	13	4	no	—
129	C₄⋊C₈	—	26	2	no	—
130	C₄⋊C₈	C₄×D₁₃	1	4	yes	—
131	#97	—	4	2	no	—
132	#97	—	52	2	no	—
133	C₄×C₈	—	13	4	no	—
134	C₄×C₈	—	26	4	no	—
135	C₄×C₈	—	26	4	no	—
136	C₄×C₈	—	26	2	no	—
137	C₂×C₁₃	#1566	1	2	yes	—
138	C₂×C₁₃	#1568	1	2	yes	—
139	C₂×C₁₃	#1767	1	1	yes	characteristic
140	C₂×C₁₃	C₄×#106	1	1	yes	characteristic
141	C₂×C₁₃	C₄×#108	1	1	yes	characteristic
142	C₂²×C₄	#6101	1	2	yes	—
143	C₂²×C₄	#6101	1	2	yes	—
144	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	4	2	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	13	2	no	—
151	C₂×C₈	—	13	4	no	—
152	C₂×C₈	—	13	2	no	—
153	C₂×C₈	—	13	4	no	—
154	C₂×C₈	—	26	2	no	—
155	C₂×C₈	—	26	2	no	—
156	C₂×C₈	—	26	2	no	—
157	C₂×C₈	—	26	4	no	—
158	C₂×C₈	—	52	2	no	—
159	C₂×C₈	—	52	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	4	no	—
164	C₄²	#6105	1	2	yes	—
165	C₄²	#6106	1	2	yes	—
166	C₄²	C₂×C₄×D₁₃	1	2	yes	—
167	C₄²	#6127	1	2	yes	—
168	C₁₃	#8273	1	1	yes	characteristic, complemented, Sylow
169	C₂³	C₂×#6101	1	1	yes	characteristic
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	#88048	1	4	yes	—
176	C₂×C₄	#88062	1	4	yes	—
177	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
178	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
179	C₂×C₄	#88126	1	1	yes	characteristic
180	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
181	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
182	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
183	C₂×C₄	C₂×#6105	1	1	yes	characteristic
184	C₂×C₄	C₂×#6106	1	1	yes	characteristic
185	C₂×C₄	#88204	1	2	yes	—
186	C₈	—	4	4	no	—
187	C₈	—	13	4	no	—
188	C₈	—	26	4	no	—
189	C₈	—	26	2	no	—
190	C₈	—	52	4	no	—
191	C₂²	C₄×#6101	1	1	yes	characteristic
192	C₂²	C₂×#88048	1	1	yes	characteristic
193	C₂²	#11718659	1	1	yes	characteristic
194	C₂²	#11718779	1	2	yes	—
195	C₂²	C₂×#88062	1	2	yes	—
196	C₄	—	2	2	no	—
197	C₄	—	2	4	no	—
198	C₄	—	2	4	no	—
199	C₄	D₁₃×#97	1	1	yes	characteristic
200	C₄	#11718366	1	2	yes	—
201	C₄	#11718368	1	1	yes	characteristic
202	C₄	C₄×#6105	1	2	yes	—
203	C₄	C₄×#6106	1	2	yes	—
204	C₂	#13113591	1	2	yes	—
205	C₂	#13113597	1	2	yes	—
206	C₂	#13114690	1	1	yes	characteristic
207	C₂	C₄×#88048	1	1	yes	characteristic
208	C₂	C₄×#88062	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1767	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1767	D₁₃
hypocenter	C₁₃	#8273
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
17. #11718072	C₄	semidirect
91. #1055	#97	semidirect
168. C₁₃	#8273	semidirect

Maximal subgroups

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

subgroup	index
#13114689	2
C₄²×#1055	2
C₂×#11718072	2
#13112670	2
C₁₃×#1767	2
#8273	13