Subgroups — #13119855

Order 1664.

468 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×#88142	C₂	1	1	yes	maximal, characteristic
3	#11718641	C₂	1	2	yes	maximal
4	#11718640	C₂	1	2	yes	maximal
5	#11718638	C₂	1	1	yes	maximal, characteristic
6	C₄⋊C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic
7	#11718635	C₂	1	2	yes	maximal
8	#11718634	C₂	1	1	yes	maximal, characteristic
9	#11718633	C₂	1	2	yes	maximal
10	C₄×#6101	C₂	1	1	yes	maximal, characteristic
11	#11718589	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₂₆	C₂²	1	2	yes	—
13	C₁₃×#125	—	2	4	no	—
14	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
21	C₂×#6101	C₂²	1	2	yes	—
22	C₂×#6101	C₂²	1	1	yes	characteristic
23	C₂×#6101	C₂²	1	1	yes	characteristic
24	C₂×#6101	C₂²	1	2	yes	—
25	C₂×#6101	C₂²	1	2	yes	—
26	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
28	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
29	#88142	—	2	4	no	—
30	#88071	C₂²	1	2	yes	—
31	#88071	C₂²	1	1	yes	characteristic
32	#88071	C₂²	1	1	yes	characteristic
33	#88071	C₂²	1	2	yes	—
34	#88071	C₂²	1	2	yes	—
35	#88071	C₂²	1	2	yes	—
36	#88071	C₂²	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×Dic₁₃	C₂³	1	2	yes	—
43	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
44	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
45	C₂²×Dic₁₃	C₂³	1	2	yes	—
46	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
47	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
48	C₂×Dic₂₆	—	2	4	no	—
49	C₂×Dic₂₆	—	2	4	no	—
50	C₂×Dic₂₆	—	4	2	no	—
51	C₁₃×C₄⋊C₄	—	2	4	no	—
52	C₁₃×C₄⋊C₄	—	2	4	no	—
53	C₁₃×C₄⋊C₄	—	2	2	no	—
54	C₁₃×C₄⋊C₄	D₄	1	4	yes	—
55	C₄²×C₁₃	—	2	2	no	—
56	C₄⋊Dic₁₃	—	2	2	no	—
57	C₄⋊Dic₁₃	Q₈	1	4	yes	complemented
58	#6101	—	2	4	no	—
59	#6101	—	2	2	no	—
60	#6101	—	2	4	no	—
61	#6101	—	2	2	no	—
62	#6101	—	4	2	no	—
63	C₄×Dic₁₃	—	2	2	no	—
64	C₄×Dic₁₃	—	4	2	no	—
65	C₄×Dic₁₃	Q₈	1	4	yes	complemented
66	#2550	—	13	1	no	maximal, Sylow
67	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₂×C₄×C₁₃	—	2	2	no	—
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	2	1	no	—
71	C₂×C₄×C₁₃	—	2	1	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	1	no	—
74	C₂×C₄×C₁₃	—	2	1	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	1	no	—
77	C₂×C₄×C₁₃	—	2	1	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
80	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
82	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
83	C₂×Dic₁₃	—	2	1	no	—
84	C₂×Dic₁₃	—	2	1	no	—
85	C₂×Dic₁₃	—	2	1	no	—
86	C₂×Dic₁₃	—	2	1	no	—
87	C₂×Dic₁₃	—	2	2	no	—
88	C₂×Dic₁₃	—	2	1	no	—
89	C₂×Dic₁₃	—	2	2	no	—
90	C₂×Dic₁₃	—	2	1	no	—
91	C₂×Dic₁₃	—	2	1	no	—
92	C₂×Dic₁₃	—	2	1	no	—
93	C₂×Dic₁₃	—	2	2	no	—
94	C₂×Dic₁₃	—	2	2	no	—
95	C₂×Dic₁₃	—	2	2	no	—
96	C₂×Dic₁₃	C₂×Q₈	1	2	yes	—
97	C₂×Dic₁₃	C₂×Q₈	1	2	yes	—
98	C₂×Dic₁₃	C₂×Q₈	1	4	yes	—
99	C₂×Dic₁₃	#41	1	2	yes	—
100	C₂×Dic₁₃	#41	1	2	yes	—
101	Dic₂₆	—	4	8	no	—
102	C₂×C₄⋊Q₈	—	13	1	no	—
103	C₂×#125	D₁₃	1	1	yes	characteristic
104	#398	—	13	2	no	—
105	#395	—	13	2	no	—
106	#391	—	13	2	no	—
107	#389	—	13	2	no	—
108	#389	—	13	1	no	—
109	#389	—	13	1	no	—
110	#382	—	13	1	no	—
111	C₄×C₄⋊C₄	—	13	1	no	—
112	C₄×C₄⋊C₄	—	13	1	no	—
113	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
114	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
115	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
116	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
117	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
118	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
119	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
120	C₄×C₁₃	—	2	4	no	—
121	C₄×C₁₃	—	4	1	no	—
122	C₄×C₁₃	—	4	1	no	—
123	C₄×C₁₃	—	4	2	no	—
124	C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
125	Dic₁₃	—	2	2	no	—
126	Dic₁₃	—	2	2	no	—
127	Dic₁₃	—	2	2	no	—
128	Dic₁₃	—	2	2	no	—
129	Dic₁₃	—	2	4	no	—
130	Dic₁₃	—	4	2	no	—
131	C₂²×Q₈	—	13	2	no	—
132	C₄⋊Q₈	—	26	4	no	—
133	#125	—	2	4	no	—
134	C₂×C₄⋊C₄	—	13	1	no	—
135	C₂×C₄⋊C₄	—	13	2	no	—
136	C₂×C₄⋊C₄	—	13	1	no	—
137	C₂×C₄⋊C₄	—	13	1	no	—
138	C₂×C₄⋊C₄	—	13	1	no	—
139	C₂×C₄⋊C₄	—	13	2	no	—
140	C₂×C₄⋊C₄	—	13	2	no	—
141	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
142	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
143	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
144	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
145	C₂×C₄²	—	13	1	no	—
146	C₂×C₄²	—	13	1	no	—
147	C₂×C₄²	—	13	2	no	—
148	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
149	#95	—	13	1	no	—
150	#95	—	13	1	no	—
151	#95	—	13	2	no	—
152	#95	—	13	2	no	—
153	#95	—	13	2	no	—
154	#95	—	13	2	no	—
155	#95	—	13	2	no	—
156	C₂×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
157	C₂×C₁₃	#533	1	1	yes	characteristic
158	C₂×C₁₃	#544	1	1	yes	characteristic
159	C₂×C₁₃	#546	1	1	yes	characteristic
160	C₂×C₁₃	#548	1	1	yes	characteristic
161	C₂×C₁₃	#554	1	1	yes	characteristic
162	C₂×C₁₃	#557	1	1	yes	characteristic
163	C₂×Q₈	—	26	4	no	—
164	C₂×Q₈	—	26	4	no	—
165	C₂×Q₈	—	52	2	no	—
166	C₂²×C₄	—	13	1	no	—
167	C₂²×C₄	—	13	1	no	—
168	C₂²×C₄	—	13	2	no	—
169	C₂²×C₄	—	13	1	no	—
170	C₂²×C₄	—	13	1	no	—
171	C₂²×C₄	—	13	2	no	—
172	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
173	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
174	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
175	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
176	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	13	4	no	—
181	C₄⋊C₄	—	26	2	no	—
182	C₄⋊C₄	—	26	4	no	—
183	C₄⋊C₄	—	26	2	no	—
184	C₄⋊C₄	—	26	4	no	—
185	C₄⋊C₄	—	26	2	no	—
186	C₄⋊C₄	—	52	2	no	—
187	C₄⋊C₄	C₂²⋊D₁₃	1	4	yes	—
188	C₄²	—	2	2	no	—
189	C₄²	—	13	4	no	—
190	C₄²	—	26	2	no	—
191	C₄²	—	52	2	no	—
192	C₁₃	#2550	1	1	yes	characteristic, complemented, Sylow
193	C₂³	C₂³×D₁₃	1	1	yes	characteristic
194	Q₈	—	52	8	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	13	2	no	—
207	C₂×C₄	—	13	2	no	—
208	C₂×C₄	—	13	4	no	—
209	C₂×C₄	—	13	2	no	—
210	C₂×C₄	—	13	2	no	—
211	C₂×C₄	—	26	1	no	—
212	C₂×C₄	—	26	1	no	—
213	C₂×C₄	—	26	2	no	—
214	C₂×C₄	—	26	1	no	—
215	C₂×C₄	—	26	2	no	—
216	C₂×C₄	—	26	1	no	—
217	C₂×C₄	—	26	1	no	—
218	C₂×C₄	—	26	1	no	—
219	C₂×C₄	—	26	1	no	—
220	C₂×C₄	—	26	1	no	—
221	C₂×C₄	—	26	2	no	—
222	C₂×C₄	—	26	2	no	—
223	C₂×C₄	—	26	2	no	—
224	C₂×C₄	Q₈×D₁₃	1	2	yes	—
225	C₂×C₄	Q₈×D₁₃	1	2	yes	—
226	C₂×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
227	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
228	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
229	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
230	C₂²	#41×D₁₃	1	1	yes	characteristic
231	C₂²	#41×D₁₃	1	1	yes	characteristic
232	C₂²	#88256	1	1	yes	characteristic
233	C₂²	#88257	1	1	yes	characteristic
234	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
235	C₄	—	2	4	no	—
236	C₄	—	4	1	no	—
237	C₄	—	4	1	no	—
238	C₄	—	4	2	no	—
239	C₄	—	26	2	no	—
240	C₄	—	26	2	no	—
241	C₄	—	26	2	no	—
242	C₄	—	26	2	no	—
243	C₄	—	26	4	no	—
244	C₄	—	52	2	no	—
245	C₄	#88198	1	4	yes	—
246	C₂	#11719390	1	1	yes	characteristic
247	C₂	#11719393	1	1	yes	characteristic
248	C₂	#11719394	1	1	yes	characteristic
249	C₂	#11719395	1	1	yes	characteristic
250	C₂	C₂×#88198	1	1	yes	characteristic
251	C₂	#11719534	1	1	yes	characteristic
252	C₂	#11719536	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂×C₁₃×#125	C₂
hypercenter	C₂×#125	D₁₃
hypocenter	C₁₃	#2550
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
57. C₄⋊Dic₁₃	Q₈	semidirect
65. C₄×Dic₁₃	Q₈	semidirect
192. C₁₃	#2550	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#125	2
C₂×#88142	2
#11718641	2
#11718640	2
#11718638	2
C₄⋊C₄×Dic₁₃	2
#11718635	2
#11718634	2
#11718633	2
C₄×#6101	2
#11718589	2
#2550	13