Subgroups — #15756589

Order 3328.

362 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114988	C₂	1	1	yes	maximal, characteristic
2	#13114582	C₂	1	1	yes	maximal, characteristic
3	#13112670	C₂	1	4	yes	maximal
4	C₁₃×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
6	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
7	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
8	C₄³×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊#1055	—	2	1	no	—
10	C₂×C₄⋊#1055	—	2	1	no	—
11	C₂×C₄⋊#1055	C₂²	1	2	yes	—
12	C₂×C₄×#1055	C₂²	1	2	yes	—
13	#11718116	C₄	1	2	yes	—
14	#11718116	C₄	1	2	yes	—
15	C₂²×C₈×C₁₃	—	2	2	no	—
16	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
23	C₂²×#1055	—	2	2	no	—
24	C₂²×#1055	C₂×C₄	1	4	yes	—
25	C₁₃×C₄⋊C₈	—	2	4	no	—
26	C₁₃×C₄⋊C₈	—	4	2	no	—
27	C₁₃×C₄⋊C₈	—	4	2	no	—
28	C₄⋊#1055	—	2	4	no	—
29	C₄⋊#1055	—	4	2	no	—
30	C₄⋊#1055	—	4	2	no	—
31	C₄×#1055	—	2	2	no	—
32	C₄×#1055	—	2	2	no	—
33	#8787	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₁₃	—	4	2	no	—
42	C₂×C₈×C₁₃	—	4	2	no	—
43	C₂×C₈×C₁₃	—	4	2	no	—
44	C₄²×C₁₃	—	2	2	no	—
45	C₄²×C₁₃	—	2	2	no	—
46	C₄²×C₁₃	—	4	1	no	—
47	C₄²×C₁₃	—	4	1	no	—
48	C₄²×C₁₃	D₈	1	2	yes	—
49	C₄²×C₁₃	SD₁₆	1	2	yes	—
50	C₄²×C₁₃	SD₁₆	1	2	yes	—
51	C₄²×C₁₃	Q₁₆	1	2	yes	—
52	C₄²×C₁₃	#41	1	2	yes	—
53	C₄²×C₁₃	#41	1	2	yes	—
54	C₂×#1055	—	2	4	no	—
55	C₂×#1055	—	2	4	no	—
56	C₂×#1055	—	2	4	no	—
57	C₂×#1055	—	4	2	no	—
58	C₂×#1055	—	4	2	no	—
59	C₂×#1055	—	4	2	no	—
60	#1842	—	13	1	no	—
61	#1842	D₁₃	1	1	yes	characteristic
62	#1744	—	13	1	no	—
63	#1277	—	13	4	no	—
64	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃	—	2	1	no	—
66	C₂×C₄×C₁₃	—	2	1	no	—
67	C₂×C₄×C₁₃	—	2	1	no	—
68	C₂×C₄×C₁₃	—	2	1	no	—
69	C₂×C₄×C₁₃	—	2	1	no	—
70	C₂×C₄×C₁₃	—	2	1	no	—
71	C₂×C₄×C₁₃	—	2	1	no	—
72	C₂×C₄×C₁₃	—	2	1	no	—
73	C₂×C₄×C₁₃	—	2	1	no	—
74	C₂×C₄×C₁₃	—	2	1	no	—
75	C₂×C₄×C₁₃	#106	1	2	yes	—
76	C₂×C₄×C₁₃	#106	1	2	yes	—
77	C₂×C₄×C₁₃	#107	1	2	yes	—
78	C₂×C₄×C₁₃	#107	1	2	yes	—
79	C₂×C₄×C₁₃	#117	1	2	yes	—
80	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
81	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
82	C₂×C₄×C₁₃	#125	1	2	yes	—
83	C₂×C₄×C₁₃	#131	1	2	yes	—
84	C₂×C₄×C₁₃	#131	1	2	yes	—
85	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
89	C₈×C₁₃	—	8	2	no	—
90	#1055	—	4	4	no	—
91	#1055	—	8	2	no	—
92	C₂×C₄⋊C₈	—	2	1	no	—
93	C₂×C₄⋊C₈	—	2	1	no	—
94	C₂×C₄⋊C₈	—	13	2	no	—
95	C₂×C₄⋊C₈	—	26	1	no	—
96	C₂×C₄⋊C₈	—	26	1	no	—
97	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
98	C₂×C₄×C₈	—	13	2	no	—
99	C₄³	C₂×D₁₃	1	1	yes	characteristic
100	#336	—	13	2	no	—
101	#336	—	13	2	no	—
102	C₂²×C₁₃	#382	1	1	yes	characteristic
103	C₂²×C₁₃	#424	1	1	yes	characteristic
104	C₂²×C₁₃	#424	1	1	yes	characteristic
105	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
106	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
107	C₂²×C₁₃	#430	1	1	yes	characteristic
108	C₂²×C₁₃	#430	1	1	yes	characteristic
109	C₄×C₁₃	—	2	2	no	—
110	C₄×C₁₃	—	2	2	no	—
111	C₄×C₁₃	—	2	2	no	—
112	C₄×C₁₃	—	2	2	no	—
113	C₄×C₁₃	—	2	2	no	—
114	C₄×C₁₃	#433	1	4	yes	—
115	C₄×C₁₃	#486	1	1	yes	characteristic
116	C₄×C₁₃	#486	1	1	yes	characteristic
117	C₄×C₁₃	#487	1	1	yes	characteristic
118	C₄×C₁₃	#487	1	1	yes	characteristic
119	C₂²×C₈	—	2	2	no	—
120	C₂²×C₈	—	13	4	no	—
121	C₂²×C₈	—	26	2	no	—
122	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
123	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
124	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
125	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
126	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
127	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
128	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
129	C₄⋊C₈	—	2	4	no	—
130	C₄⋊C₈	—	4	2	no	—
131	C₄⋊C₈	—	4	2	no	—
132	C₄⋊C₈	—	26	4	no	—
133	C₄⋊C₈	—	52	2	no	—
134	C₄⋊C₈	—	52	2	no	—
135	C₄×C₈	—	26	2	no	—
136	C₄×C₈	—	26	2	no	—
137	C₂×C₁₃	#1570	1	1	yes	characteristic
138	C₂×C₁₃	#1570	1	1	yes	characteristic
139	C₂×C₁₃	#1571	1	1	yes	characteristic
140	C₂×C₁₃	#1571	1	1	yes	characteristic
141	C₂×C₁₃	#1832	1	1	yes	characteristic
142	C₂×C₁₃	#1834	1	1	yes	characteristic
143	C₂×C₁₃	#1838	1	1	yes	characteristic
144	C₂²×C₄	#6101	1	1	yes	characteristic
145	C₂²×C₄	#6101	1	1	yes	characteristic
146	C₂²×C₄	#6101	1	1	yes	characteristic
147	C₂²×C₄	#6101	1	1	yes	characteristic
148	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
150	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	26	4	no	—
155	C₂×C₈	—	26	4	no	—
156	C₂×C₈	—	26	4	no	—
157	C₂×C₈	—	52	2	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	—	4	1	no	—
163	C₄²	—	4	1	no	—
164	C₄²	#6104	1	2	yes	—
165	C₄²	#6105	1	2	yes	—
166	C₄²	#6106	1	2	yes	—
167	C₄²	#6107	1	2	yes	—
168	C₄²	#6129	1	2	yes	—
169	C₄²	#6131	1	2	yes	—
170	C₁₃	#8787	1	1	yes	characteristic, complemented, Sylow
171	C₂³	C₂×#6101	1	1	yes	characteristic
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	#88047	1	2	yes	—
183	C₂×C₄	#88047	1	2	yes	—
184	C₂×C₄	#88048	1	2	yes	—
185	C₂×C₄	#88048	1	2	yes	—
186	C₂×C₄	#88144	1	2	yes	—
187	C₂×C₄	#88146	1	2	yes	—
188	C₂×C₄	#88155	1	2	yes	—
189	C₂×C₄	#88161	1	2	yes	—
190	C₂×C₄	C₂×#6104	1	1	yes	characteristic
191	C₂×C₄	C₂×#6105	1	1	yes	characteristic
192	C₂×C₄	#88190	1	1	yes	characteristic
193	C₂×C₄	C₂×#6106	1	1	yes	characteristic
194	C₂×C₄	C₂×#6107	1	1	yes	characteristic
195	C₂×C₄	#88201	1	1	yes	characteristic
196	C₈	—	8	2	no	—
197	C₈	—	52	4	no	—
198	C₈	—	104	2	no	—
199	C₂²	C₂×#88047	1	1	yes	characteristic
200	C₂²	C₂×#88048	1	1	yes	characteristic
201	C₂²	#11718638	1	1	yes	characteristic
202	C₂²	#11718763	1	1	yes	characteristic
203	C₂²	#11718780	1	1	yes	characteristic
204	C₂²	#11718785	1	1	yes	characteristic
205	C₂²	#11718785	1	1	yes	characteristic
206	C₄	—	2	2	no	—
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	2	2	no	—
210	C₄	—	2	2	no	—
211	C₄	#11718501	1	4	yes	—
212	C₄	#11718731	1	1	yes	characteristic
213	C₄	#11718742	1	1	yes	characteristic
214	C₄	#11718749	1	1	yes	characteristic
215	C₄	#11718754	1	1	yes	characteristic
216	C₂	#13113606	1	1	yes	characteristic
217	C₂	#13113608	1	1	yes	characteristic
218	C₂	#13113610	1	1	yes	characteristic
219	C₂	#13113612	1	1	yes	characteristic
220	C₂	#13114955	1	1	yes	characteristic
221	C₂	#13114963	1	1	yes	characteristic
222	C₂	#13114976	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1842	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1842	D₁₃
hypocenter	C₁₃	#8787
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
170. C₁₃	#8787	semidirect

Maximal subgroups

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

subgroup	index
#13114988	2
#13114582	2
#13112670	2
C₁₃×#1842	2
#8787	13