Subgroups — #13119413

Order 1664.

388 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#391	C₂	1	1	yes	maximal, characteristic
2	#11718891	C₂	1	1	yes	maximal, characteristic
3	#11718891	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	#11718634	C₂	1	1	yes	maximal, characteristic
6	#11718596	C₂	1	1	yes	maximal, characteristic
7	C₄×#6101	C₂	1	1	yes	maximal, characteristic
8	#11718321	C₂	1	2	yes	maximal
9	#11718320	C₂	1	2	yes	maximal
10	#11718316	C₂	1	2	yes	maximal
11	#11718314	C₂	1	2	yes	maximal
12	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
19	C₂×#6101	C₂²	1	1	yes	characteristic
20	C₂×#6101	C₂²	1	2	yes	—
21	C₂×#6101	C₂²	1	1	yes	characteristic
22	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
25	C₁₃×#95	C₂²	1	2	yes	—
26	C₁₃×#95	C₂²	1	2	yes	—
27	#88071	C₂²	1	1	yes	characteristic
28	#88071	C₂²	1	2	yes	—
29	#88071	C₂²	1	2	yes	—
30	#88071	C₂²	1	2	yes	—
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	C₂×Q₈×C₁₃	—	2	2	no	—
36	C₂×Q₈×C₁₃	—	2	2	no	—
37	C₂×Q₈×C₁₃	—	2	2	no	—
38	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
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	2	yes	—
47	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
48	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×C₄⋊C₄	—	2	2	no	—
50	C₄²×C₁₃	—	2	2	no	—
51	C₄⋊Dic₁₃	—	2	2	no	—
52	C₄⋊Dic₁₃	—	2	2	no	—
53	C₄⋊Dic₁₃	—	2	4	no	—
54	C₄⋊Dic₁₃	—	4	2	no	—
55	#6101	—	2	2	no	—
56	#6101	—	4	2	no	—
57	#6101	Q₈	1	4	yes	complemented
58	C₄×Dic₁₃	—	2	2	no	—
59	C₄×Dic₁₃	—	2	2	no	—
60	C₄×Dic₁₃	—	4	2	no	—
61	#2511	—	13	1	no	maximal, Sylow
62	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
63	Q₈×C₁₃	—	4	4	no	—
64	C₂×C₄×C₁₃	—	2	1	no	—
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	2	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	2	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
75	C₂×C₄×C₁₃	#41	1	2	yes	—
76	C₂×C₄×C₁₃	#41	1	2	yes	—
77	C₂×C₄×C₁₃	#41	1	2	yes	—
78	C₂×C₄×C₁₃	#41	1	2	yes	—
79	C₂×Dic₁₃	—	2	1	no	—
80	C₂×Dic₁₃	—	2	1	no	—
81	C₂×Dic₁₃	—	2	1	no	—
82	C₂×Dic₁₃	—	2	1	no	—
83	C₂×Dic₁₃	—	2	1	no	—
84	C₂×Dic₁₃	—	2	1	no	—
85	C₂×Dic₁₃	—	2	1	no	—
86	C₂×Dic₁₃	—	2	2	no	—
87	C₂×Dic₁₃	—	2	2	no	—
88	C₂×Dic₁₃	—	2	2	no	—
89	C₂×Dic₁₃	—	2	2	no	—
90	C₂×Dic₁₃	—	2	2	no	—
91	C₂×Dic₁₃	—	2	1	no	—
92	C₂×Dic₁₃	—	2	2	no	—
93	C₂×Dic₁₃	—	2	1	no	—
94	C₂×Dic₁₃	—	2	1	no	—
95	C₂×Dic₁₃	C₂×Q₈	1	2	yes	—
96	C₂×Dic₁₃	C₂×Q₈	1	2	yes	—
97	#400	—	13	2	no	—
98	#400	—	13	2	no	—
99	#391	—	13	1	no	—
100	#391	—	13	1	no	—
101	#391	D₁₃	1	1	yes	characteristic
102	#389	—	13	1	no	—
103	#387	—	13	2	no	—
104	#387	—	13	2	no	—
105	#384	—	13	1	no	—
106	C₄×C₄⋊C₄	—	13	1	no	—
107	C₄×C₄⋊C₄	—	13	1	no	—
108	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
109	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
110	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
111	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
112	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
113	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
115	C₄×C₁₃	—	2	2	no	—
116	C₄×C₁₃	—	2	2	no	—
117	C₄×C₁₃	—	4	2	no	—
118	C₄×C₁₃	—	4	2	no	—
119	C₄×C₁₃	#126	1	4	yes	—
120	Dic₁₃	—	2	2	no	—
121	Dic₁₃	—	2	2	no	—
122	Dic₁₃	—	4	1	no	—
123	Dic₁₃	—	4	1	no	—
124	Dic₁₃	—	4	2	no	—
125	Dic₁₃	—	4	2	no	—
126	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
127	C₂×C₄⋊C₄	—	13	1	no	—
128	C₂×C₄⋊C₄	—	13	1	no	—
129	C₂×C₄⋊C₄	—	13	1	no	—
130	C₂×C₄⋊C₄	—	13	2	no	—
131	C₂×C₄⋊C₄	—	13	1	no	—
132	C₂×C₄⋊C₄	—	13	2	no	—
133	C₂×C₄⋊C₄	—	13	2	no	—
134	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
135	C₂×C₄²	—	13	1	no	—
136	C₂×C₄²	—	13	1	no	—
137	C₂×C₄²	—	13	2	no	—
138	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
139	#95	—	13	1	no	—
140	#95	—	13	1	no	—
141	#95	—	13	2	no	—
142	#95	—	13	2	no	—
143	#95	—	13	2	no	—
144	#95	—	13	2	no	—
145	#95	—	13	2	no	—
146	#95	—	13	2	no	—
147	#95	C₂×D₁₃	1	2	yes	—
148	#95	C₂×D₁₃	1	2	yes	—
149	C₂×C₁₃	C₂×#126	1	1	yes	characteristic
150	C₂×C₁₃	#533	1	1	yes	characteristic
151	C₂×C₁₃	#542	1	1	yes	characteristic
152	C₂×C₁₃	#552	1	1	yes	characteristic
153	C₂×C₁₃	#554	1	1	yes	characteristic
154	C₂×C₁₃	#557	1	1	yes	characteristic
155	C₂×C₁₃	#559	1	1	yes	characteristic
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	—	2	2	no	—
159	C₂²×C₄	—	13	1	no	—
160	C₂²×C₄	—	13	1	no	—
161	C₂²×C₄	—	13	1	no	—
162	C₂²×C₄	—	13	1	no	—
163	C₂²×C₄	—	13	2	no	—
164	C₂²×C₄	—	13	2	no	—
165	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
166	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
167	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	13	4	no	—
172	C₄⋊C₄	—	26	2	no	—
173	C₄⋊C₄	—	26	2	no	—
174	C₄⋊C₄	—	26	2	no	—
175	C₄⋊C₄	—	26	4	no	—
176	C₄⋊C₄	—	52	2	no	—
177	C₄⋊C₄	—	52	2	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	26	2	no	—
180	C₄²	—	26	2	no	—
181	C₄²	—	52	2	no	—
182	C₁₃	#2511	1	1	yes	characteristic, complemented, Sylow
183	C₂³	C₂³×D₁₃	1	1	yes	characteristic
184	Q₈	—	4	4	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	13	2	no	—
196	C₂×C₄	—	13	2	no	—
197	C₂×C₄	—	26	1	no	—
198	C₂×C₄	—	26	1	no	—
199	C₂×C₄	—	26	1	no	—
200	C₂×C₄	—	26	1	no	—
201	C₂×C₄	—	26	2	no	—
202	C₂×C₄	—	26	2	no	—
203	C₂×C₄	—	26	1	no	—
204	C₂×C₄	—	26	1	no	—
205	C₂×C₄	—	26	2	no	—
206	C₂×C₄	—	26	2	no	—
207	C₂×C₄	—	26	1	no	—
208	C₂×C₄	—	26	2	no	—
209	C₂×C₄	—	26	2	no	—
210	C₂×C₄	—	26	1	no	—
211	C₂×C₄	—	26	1	no	—
212	C₂×C₄	—	26	1	no	—
213	C₂×C₄	#6127	1	2	yes	—
214	C₂×C₄	#6129	1	2	yes	—
215	C₂×C₄	#6129	1	2	yes	—
216	C₂×C₄	Q₈×D₁₃	1	2	yes	—
217	C₂×C₄	#6131	1	2	yes	—
218	C₂²	C₂×#6127	1	1	yes	characteristic
219	C₂²	C₂×#6129	1	1	yes	characteristic
220	C₂²	C₂×#6129	1	1	yes	characteristic
221	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
222	C₂²	C₂×#6131	1	1	yes	characteristic
223	C₂²	#88256	1	1	yes	characteristic
224	C₂²	#88257	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	2	no	—
229	C₄	—	26	2	no	—
230	C₄	—	26	2	no	—
231	C₄	—	52	2	no	—
232	C₄	—	52	2	no	—
233	C₄	—	52	1	no	—
234	C₄	—	52	1	no	—
235	C₄	#88133	1	4	yes	—
236	C₂	C₂×#88133	1	1	yes	characteristic
237	C₂	#11719272	1	1	yes	characteristic
238	C₂	#11719290	1	1	yes	characteristic
239	C₂	#11719340	1	1	yes	characteristic
240	C₂	#11719347	1	1	yes	characteristic
241	C₂	#11719387	1	1	yes	characteristic
242	C₂	#11719433	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₁₃×#391	C₂
hypercenter	#391	D₁₃
hypocenter	C₁₃	#2511
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ #391

Complements

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

N	H ≅ G/N	type
57. #6101	Q₈	semidirect
182. C₁₃	#2511	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#391	2
#11718891	2
C₄⋊C₄×Dic₁₃	2
#11718634	2
#11718596	2
C₄×#6101	2
#11718321	2
#11718320	2
#11718316	2
#11718314	2
#2511	13