Subgroups — #13117922

Order 1664.

496 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88142	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#389	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	#11718635	C₂	1	2	yes	maximal
6	#11718634	C₂	1	1	yes	maximal, characteristic
7	#11718633	C₂	1	2	yes	maximal
8	C₄×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
9	C₄×#6101	C₂	1	1	yes	maximal, characteristic
10	#11718310	C₂	1	2	yes	maximal
11	#11718307	C₂	1	2	yes	maximal
12	C₂²×Dic₂₆	C₂²	1	2	yes	—
13	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
19	C₂×#6101	C₂²	1	1	yes	characteristic
20	C₂×#6101	C₂²	1	2	yes	—
21	C₂×#6101	C₂²	1	2	yes	—
22	C₂×#6101	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
24	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
26	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
27	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
28	#88142	C₄	1	8	yes	complemented
29	C₁₃×#95	C₂²	1	2	yes	—
30	#88071	C₂²	1	1	yes	characteristic
31	#88071	C₂²	1	2	yes	—
32	#88071	C₂²	1	2	yes	—
33	#88071	C₂²	1	2	yes	—
34	#88071	C₂²	1	1	yes	characteristic
35	#88071	C₂²	1	2	yes	—
36	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
42	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
43	C₂²×Dic₁₃	C₂³	1	2	yes	—
44	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
45	C₂²×Dic₁₃	C₂³	1	2	yes	—
46	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
47	C₂×Dic₂₆	—	4	2	no	—
48	C₂×Dic₂₆	—	4	2	no	—
49	C₂×Dic₂₆	C₂×C₄	1	8	yes	—
50	C₁₃×C₄⋊C₄	—	2	2	no	—
51	C₁₃×C₄⋊C₄	—	4	2	no	—
52	C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
53	C₄²×C₁₃	—	2	2	no	—
54	C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
55	C₄⋊Dic₁₃	Q₈	1	4	yes	complemented
56	#6101	—	2	2	no	—
57	#6101	—	4	2	no	—
58	#6101	C₂×C₄	1	8	yes	—
59	#6101	Q₈	1	4	yes	complemented
60	C₄×Dic₁₃	—	2	2	no	—
61	C₄×Dic₁₃	—	2	4	no	—
62	C₄×Dic₁₃	—	4	2	no	—
63	C₄×Dic₁₃	—	4	2	no	—
64	C₄×Dic₁₃	C₂×C₄	1	4	yes	—
65	#2369	—	13	1	no	maximal, Sylow
66	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
67	C₂×C₄×C₁₃	—	2	1	no	—
68	C₂×C₄×C₁₃	—	2	1	no	—
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	2	1	no	—
73	C₂×C₄×C₁₃	—	2	1	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	1	no	—
77	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
78	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
80	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
81	C₂×C₄×C₁₃	#41	1	2	yes	—
82	C₂×Dic₁₃	—	2	1	no	—
83	C₂×Dic₁₃	—	2	2	no	—
84	C₂×Dic₁₃	—	2	1	no	—
85	C₂×Dic₁₃	—	2	2	no	—
86	C₂×Dic₁₃	—	2	2	no	—
87	C₂×Dic₁₃	—	2	2	no	—
88	C₂×Dic₁₃	—	2	1	no	—
89	C₂×Dic₁₃	—	2	1	no	—
90	C₂×Dic₁₃	—	2	1	no	—
91	C₂×Dic₁₃	—	2	1	no	—
92	C₂×Dic₁₃	—	2	1	no	—
93	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
94	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
95	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
96	C₂×Dic₁₃	C₂×Q₈	1	2	yes	—
97	C₂×Dic₁₃	C₂×Q₈	1	4	yes	—
98	C₂×Dic₁₃	C₂×Q₈	1	2	yes	—
99	C₂×Dic₁₃	#41	1	2	yes	—
100	Dic₂₆	—	4	8	no	—
101	C₂×C₄⋊Q₈	—	13	1	no	—
102	#391	—	13	2	no	—
103	#391	—	13	2	no	—
104	#389	—	13	2	no	—
105	#389	—	13	1	no	—
106	#389	D₁₃	1	1	yes	characteristic
107	C₄×C₄⋊C₄	—	13	1	no	—
108	C₄×C₄⋊C₄	—	13	1	no	—
109	C₄×C₄⋊C₄	—	13	1	no	—
110	C₄×C₄⋊C₄	—	13	1	no	—
111	C₄×C₄⋊C₄	—	13	2	no	—
112	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
114	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
115	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
116	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
117	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
119	C₄×C₁₃	—	2	4	no	—
120	C₄×C₁₃	—	2	2	no	—
121	C₄×C₁₃	—	4	2	no	—
122	C₄×C₁₃	—	4	1	no	—
123	C₄×C₁₃	C₄×Q₈	1	4	yes	—
124	Dic₁₃	—	2	2	no	—
125	Dic₁₃	—	2	4	no	—
126	Dic₁₃	—	2	2	no	—
127	Dic₁₃	—	2	4	no	—
128	Dic₁₃	—	4	1	no	—
129	Dic₁₃	C₄×Q₈	1	4	yes	—
130	C₂²×Q₈	—	13	2	no	—
131	C₄⋊Q₈	—	13	8	no	—
132	C₂×C₄⋊C₄	—	13	2	no	—
133	C₂×C₄⋊C₄	—	13	1	no	—
134	C₂×C₄⋊C₄	—	13	1	no	—
135	C₂×C₄⋊C₄	—	13	1	no	—
136	C₂×C₄⋊C₄	—	13	1	no	—
137	C₂×C₄⋊C₄	—	13	2	no	—
138	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
139	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
140	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
141	C₂×C₄²	—	13	1	no	—
142	C₂×C₄²	—	13	2	no	—
143	C₂×C₄²	—	13	1	no	—
144	C₂×C₄²	—	13	2	no	—
145	C₂×C₄²	—	13	2	no	—
146	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
147	#95	—	13	2	no	—
148	#95	—	13	2	no	—
149	#95	—	13	2	no	—
150	#95	—	13	1	no	—
151	#95	—	13	2	no	—
152	#95	—	13	1	no	—
153	#95	C₂×D₁₃	1	2	yes	—
154	C₂×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
155	C₂×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
156	C₂×C₁₃	#518	1	1	yes	characteristic
157	C₂×C₁₃	#554	1	1	yes	characteristic
158	C₂×C₁₃	#554	1	1	yes	characteristic
159	C₂×C₁₃	#555	1	1	yes	characteristic
160	C₂×C₁₃	Q₈²	1	1	yes	characteristic
161	C₂×Q₈	—	13	8	no	—
162	C₂×Q₈	—	52	2	no	—
163	C₂×Q₈	—	52	2	no	—
164	C₂²×C₄	—	13	1	no	—
165	C₂²×C₄	—	13	1	no	—
166	C₂²×C₄	—	13	1	no	—
167	C₂²×C₄	—	13	2	no	—
168	C₂²×C₄	—	13	2	no	—
169	C₂²×C₄	—	13	1	no	—
170	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
171	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
172	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
173	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
174	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	13	8	no	—
178	C₄⋊C₄	—	13	4	no	—
179	C₄⋊C₄	—	13	4	no	—
180	C₄⋊C₄	—	13	4	no	—
181	C₄⋊C₄	—	26	2	no	—
182	C₄⋊C₄	—	52	2	no	—
183	C₄⋊C₄	C₄×D₁₃	1	4	yes	—
184	C₄²	—	2	2	no	—
185	C₄²	—	13	4	no	—
186	C₄²	—	26	4	no	—
187	C₄²	—	26	2	no	—
188	C₄²	—	52	2	no	—
189	C₄²	—	52	2	no	—
190	C₁₃	#2369	1	1	yes	characteristic, complemented, Sylow
191	C₂³	C₂³×D₁₃	1	1	yes	characteristic
192	Q₈	—	52	8	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	13	2	no	—
204	C₂×C₄	—	13	2	no	—
205	C₂×C₄	—	13	2	no	—
206	C₂×C₄	—	13	2	no	—
207	C₂×C₄	—	13	4	no	—
208	C₂×C₄	—	13	2	no	—
209	C₂×C₄	—	13	4	no	—
210	C₂×C₄	—	26	1	no	—
211	C₂×C₄	—	26	2	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	2	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	C₂×C₄×D₁₃	1	4	yes	—
222	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
223	C₂×C₄	#6129	1	2	yes	—
224	C₂×C₄	Q₈×D₁₃	1	2	yes	—
225	C₂×C₄	Q₈×D₁₃	1	2	yes	—
226	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
227	C₂²	C₂×#6129	1	1	yes	characteristic
228	C₂²	#88251	1	1	yes	characteristic
229	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
230	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
231	C₂²	#41×D₁₃	1	1	yes	characteristic
232	C₂²	#88256	1	1	yes	characteristic
233	C₄	—	2	4	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	13	4	no	—
238	C₄	—	26	4	no	—
239	C₄	—	26	2	no	—
240	C₄	—	26	2	no	—
241	C₄	—	26	4	no	—
242	C₄	—	52	1	no	—
243	C₄	#88141	1	4	yes	—
244	C₂	C₂×#88141	1	1	yes	characteristic
245	C₂	#11719250	1	1	yes	characteristic
246	C₂	C₄×Q₈×D₁₃	1	1	yes	characteristic
247	C₂	#11719321	1	1	yes	characteristic
248	C₂	#11719340	1	1	yes	characteristic
249	C₂	#11719395	1	1	yes	characteristic
250	C₂	#11719427	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#389	C₂
socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
hypercenter	#389	D₁₃
hypocenter	C₁₃	#2369
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
28. #88142	C₄	semidirect
55. C₄⋊Dic₁₃	Q₈	semidirect
59. #6101	Q₈	semidirect
190. C₁₃	#2369	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88142	2
C₁₃×#389	2
C₄⋊C₄×Dic₁₃	2
#11718635	2
#11718634	2
#11718633	2
C₄×C₄⋊Dic₁₃	2
C₄×#6101	2
#11718310	2
#11718307	2
#2369	13