Subgroups — #15761959

Order 3328.

478 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114953	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718138	C₂	1	2	yes	maximal
4	C₂×#11718139	C₂	1	2	yes	maximal
5	C₁₃×#2862	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₁₃×#391	—	2	2	no	—
9	C₁₃×#391	C₄	1	4	yes	—
10	C₁₃×#382	—	2	1	no	—
11	C₁₃×#382	—	2	1	no	—
12	C₄³×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊#1055	C₂²	1	2	yes	—
14	C₂×C₄⋊#1055	C₂²	1	2	yes	—
15	C₂×C₄×#1055	—	2	1	no	—
16	#11718139	—	2	4	no	—
17	#11718138	—	2	4	no	—
18	#11718116	—	2	1	no	—
19	#11718116	—	2	2	no	—
20	C₂²×Q₈×C₁₃	—	2	1	no	—
21	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
22	C₁₃×C₄⋊Q₈	—	2	2	no	—
23	C₁₃×C₄⋊Q₈	—	2	2	no	—
24	C₁₃×C₄⋊Q₈	—	2	2	no	—
25	C₁₃×C₄⋊Q₈	—	2	2	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
28	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
29	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	D₄	1	2	yes	—
33	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²×#1055	—	2	2	no	—
36	C₂²×#1055	—	2	2	no	—
37	C₁₃×#95	—	2	4	no	—
38	C₁₃×#95	—	2	4	no	—
39	C₄⋊#1055	—	2	2	no	—
40	C₄⋊#1055	—	2	2	no	—
41	C₄⋊#1055	—	2	2	no	—
42	C₄⋊#1055	D₄	1	4	yes	—
43	C₄×#1055	—	4	2	no	—
44	#9693	—	13	1	no	maximal, Sylow
45	C₂×Q₈×C₁₃	—	2	4	no	—
46	C₂×Q₈×C₁₃	—	4	4	no	—
47	C₂×Q₈×C₁₃	—	4	1	no	—
48	C₂×Q₈×C₁₃	—	4	1	no	—
49	C₂×Q₈×C₁₃	—	4	4	no	—
50	C₂²×C₄×C₁₃	—	2	2	no	—
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₁₃×C₄⋊C₄	—	2	4	no	—
58	C₁₃×C₄⋊C₄	—	4	2	no	—
59	C₁₃×C₄⋊C₄	—	4	2	no	—
60	C₁₃×C₄⋊C₄	—	4	1	no	—
61	C₁₃×C₄⋊C₄	—	4	1	no	—
62	C₄²×C₁₃	—	2	4	no	—
63	C₄²×C₁₃	—	2	2	no	—
64	C₄²×C₁₃	—	4	2	no	—
65	C₄²×C₁₃	SD₁₆	1	2	yes	—
66	C₄²×C₁₃	Q₁₆	1	2	yes	—
67	C₄²×C₁₃	C₂×D₄	1	2	yes	—
68	C₄²×C₁₃	#41	1	2	yes	—
69	C₂×#1055	—	2	4	no	—
70	C₂×#1055	—	4	2	no	—
71	C₂×#1055	—	4	2	no	—
72	C₂×#1055	—	4	2	no	—
73	C₂×#1055	—	4	4	no	—
74	#2862	D₁₃	1	1	yes	characteristic
75	#1831	—	13	1	no	—
76	C₄×C₄⋊C₈	—	13	1	no	—
77	C₂×#333	—	13	2	no	—
78	C₂×#332	—	13	2	no	—
79	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	Q₈×C₁₃	—	8	4	no	—
81	Q₈×C₁₃	—	8	2	no	—
82	Q₈×C₁₃	—	8	2	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	4	2	no	—
90	C₂×C₄×C₁₃	—	4	2	no	—
91	C₂×C₄×C₁₃	—	4	4	no	—
92	C₂×C₄×C₁₃	—	4	4	no	—
93	C₂×C₄×C₁₃	#103	1	4	yes	—
94	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₁₃	#121	1	2	yes	—
96	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
97	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
100	C₂×C₄×C₁₃	#135	1	4	yes	—
101	C₂×C₄×C₁₃	#137	1	2	yes	—
102	#1055	—	4	4	no	—
103	#1055	—	8	2	no	—
104	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
105	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
106	C₂×C₄⋊C₈	—	13	2	no	—
107	C₂×C₄⋊C₈	—	13	2	no	—
108	C₂×C₄×C₈	—	26	1	no	—
109	#391	—	2	2	no	—
110	#391	Dic₁₃	1	4	yes	—
111	#382	—	2	1	no	—
112	#382	—	2	1	no	—
113	C₄³	C₂×D₁₃	1	1	yes	characteristic
114	#336	—	26	2	no	—
115	#336	—	26	1	no	—
116	#333	—	26	4	no	—
117	#332	—	26	4	no	—
118	C₂²×C₁₃	#390	1	1	yes	characteristic
119	C₂²×C₁₃	#413	1	2	yes	—
120	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
121	C₂²×C₁₃	#416	1	2	yes	—
122	C₂²×C₁₃	#419	1	1	yes	characteristic
123	C₄×C₁₃	—	2	4	no	—
124	C₄×C₁₃	—	2	2	no	—
125	C₄×C₁₃	—	4	2	no	—
126	C₄×C₁₃	—	8	2	no	—
127	C₄×C₁₃	—	8	2	no	—
128	C₄×C₁₃	#461	1	2	yes	—
129	C₄×C₁₃	#462	1	2	yes	—
130	C₄×C₁₃	#488	1	2	yes	—
131	C₄×C₁₃	#495	1	2	yes	—
132	C₂²×Q₈	—	2	1	no	—
133	C₂²×Q₈	C₂×Dic₁₃	1	2	yes	—
134	C₂²×C₈	—	26	2	no	—
135	C₂²×C₈	—	26	2	no	—
136	C₄⋊Q₈	—	2	2	no	—
137	C₄⋊Q₈	—	2	2	no	—
138	C₄⋊Q₈	—	2	2	no	—
139	C₄⋊Q₈	—	2	2	no	—
140	C₂×C₄⋊C₄	—	2	1	no	—
141	C₂×C₄⋊C₄	—	2	1	no	—
142	C₂×C₄⋊C₄	—	2	1	no	—
143	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
144	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
145	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
146	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
147	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
148	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
149	C₄⋊C₈	—	13	4	no	—
150	C₄⋊C₈	—	26	2	no	—
151	C₄⋊C₈	—	26	2	no	—
152	C₄⋊C₈	—	26	2	no	—
153	C₄×C₈	—	52	2	no	—
154	#95	—	2	4	no	—
155	#95	—	2	4	no	—
156	C₂×C₁₃	#1542	1	2	yes	—
157	C₂×C₁₃	#1554	1	2	yes	—
158	C₂×C₁₃	#1958	1	1	yes	characteristic
159	C₂×C₁₃	#1967	1	1	yes	characteristic
160	C₂×C₁₃	#1981	1	1	yes	characteristic
161	C₂×Q₈	—	2	4	no	—
162	C₂×Q₈	—	4	4	no	—
163	C₂×Q₈	—	4	1	no	—
164	C₂×Q₈	—	4	1	no	—
165	C₂×Q₈	—	4	4	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
169	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
170	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
173	C₂×C₈	—	26	4	no	—
174	C₂×C₈	—	52	2	no	—
175	C₂×C₈	—	52	2	no	—
176	C₂×C₈	—	52	2	no	—
177	C₂×C₈	—	52	4	no	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	4	1	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	4	no	—
185	C₄²	—	4	2	no	—
186	C₄²	#6106	1	2	yes	—
187	C₄²	#6107	1	2	yes	—
188	C₄²	D₄×D₁₃	1	2	yes	—
189	C₄²	#6129	1	2	yes	—
190	C₁₃	#9693	1	1	yes	characteristic, complemented, Sylow
191	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
192	Q₈	—	8	4	no	—
193	Q₈	—	8	2	no	—
194	Q₈	—	8	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	#88075	1	4	yes	—
206	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
207	C₂×C₄	#88190	1	1	yes	characteristic
208	C₂×C₄	#88192	1	2	yes	—
209	C₂×C₄	#88194	1	1	yes	characteristic
210	C₂×C₄	C₂×#6106	1	1	yes	characteristic
211	C₂×C₄	#88196	1	2	yes	—
212	C₂×C₄	C₂×#6107	1	1	yes	characteristic
213	C₂×C₄	#88204	1	4	yes	—
214	C₈	—	52	4	no	—
215	C₈	—	104	2	no	—
216	C₂²	#11718882	1	1	yes	characteristic
217	C₂²	C₂×#88075	1	1	yes	characteristic
218	C₂²	#11718886	1	1	yes	characteristic
219	C₂²	#11718894	1	2	yes	—
220	C₂²	#11718898	1	2	yes	—
221	C₄	—	2	2	no	—
222	C₄	—	2	4	no	—
223	C₄	—	4	2	no	—
224	C₄	—	8	2	no	—
225	C₄	—	8	2	no	—
226	C₄	#11718717	1	2	yes	—
227	C₄	#11718720	1	2	yes	—
228	C₄	#11718744	1	2	yes	—
229	C₄	#11718751	1	2	yes	—
230	C₂	#13113493	1	2	yes	—
231	C₂	#13113546	1	2	yes	—
232	C₂	#13115648	1	1	yes	characteristic
233	C₂	#13115686	1	1	yes	characteristic
234	C₂	#13115760	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#2862	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#2862	D₁₃
hypocenter	C₁₃	#9693
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

Complements

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

N	H ≅ G/N	type
190. C₁₃	#9693	semidirect

Maximal subgroups

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

subgroup	index
#13114953	2
C₄×C₄⋊#1055	2
C₂×#11718138	2
C₂×#11718139	2
C₁₃×#2862	2
#9693	13