Subgroups — #13124006

Order 1664.

346 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#529	C₂	1	1	yes	maximal, characteristic
2	#11718685	C₂	1	2	yes	maximal, complemented
3	#11718673	C₂	1	2	yes	maximal, complemented
4	#11718672	C₂	1	2	yes	maximal, complemented
5	#11718672	C₂	1	2	yes	maximal, complemented
6	#11718661	C₂	1	2	yes	maximal, complemented
7	#11718660	C₂	1	2	yes	maximal, complemented
8	#11718587	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718585	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#126	C₄	1	2	yes	—
11	C₁₃×#125	C₄	1	1	yes	characteristic
12	C₁₃×#124	C₄	1	1	yes	characteristic
13	C₁₃×#123	C₄	1	2	yes	—
14	C₁₃×C₂²⋊Q₈	C₄	1	1	yes	characteristic
15	C₁₃×#121	C₄	1	1	yes	characteristic
16	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₃	C₂²	1	2	yes	—
19	C₁₃×#117	C₂²	1	2	yes	—
20	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
21	C₂×#6099	C₂²	1	2	yes	—
22	C₂×#6099	C₂²	1	1	yes	characteristic
23	C₂×#6099	C₂²	1	1	yes	characteristic
24	C₂²×#1055	C₂²	1	2	yes	—
25	C₂²⋊#1055	C₂²	1	2	yes	—
26	C₂²⋊#1055	C₂²	1	2	yes	—
27	C₂²⋊#1055	C₂²	1	2	yes	—
28	C₄⋊#1055	C₂²	1	1	yes	characteristic
29	C₄⋊#1055	C₂²	1	2	yes	—
30	C₄⋊#1055	C₂²	1	2	yes	—
31	C₄⋊#1055	C₂²	1	2	yes	—
32	C₄⋊#1055	C₂²	1	1	yes	characteristic
33	C₄⋊#1055	C₂²	1	2	yes	—
34	#88043	C₂²	1	2	yes	—
35	#88043	C₂²	1	2	yes	—
36	C₄×#1055	C₂²	1	2	yes	—
37	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
39	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
46	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
47	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
48	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
49	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
51	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
52	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
53	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
54	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
55	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₁₃	C₂³	1	1	yes	characteristic
58	C₄²×C₁₃	C₂³	1	1	yes	characteristic
59	C₄²×C₁₃	C₂³	1	2	yes	—
60	#6099	—	4	1	no	—
61	#6099	—	4	2	no	—
62	#6099	—	4	1	no	—
63	C₂×#1055	—	4	2	no	—
64	C₂×#1055	C₂³	1	2	yes	—
65	C₂×#1055	C₂³	1	2	yes	—
66	C₂×#1055	C₂³	1	2	yes	—
67	C₂×#1055	C₂³	1	2	yes	—
68	#2995	—	13	1	no	maximal, Sylow
69	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
70	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	Q₈×C₁₃	—	4	1	no	—
72	D₄×C₁₃	—	4	1	no	—
73	D₄×C₁₃	—	4	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	1	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	1	no	—
78	C₂×C₄×C₁₃	—	4	1	no	—
79	C₂×C₄×C₁₃	—	4	1	no	—
80	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
89	#1055	—	2	2	no	—
90	#1055	—	2	2	no	—
91	#1055	—	2	2	no	—
92	#1055	—	2	2	no	—
93	#529	D₁₃	1	1	yes	characteristic
94	#446	—	13	2	no	—
95	#436	—	13	2	no	—
96	#435	—	13	2	no	—
97	#435	—	13	2	no	—
98	#433	—	13	2	no	—
99	#432	—	13	1	no	—
100	#424	—	13	2	no	—
101	#423	—	13	1	no	—
102	C₂²×C₁₃	—	2	2	no	—
103	C₂²×C₁₃	—	2	1	no	—
104	C₂²×C₁₃	—	2	1	no	—
105	C₂²×C₁₃	—	2	1	no	—
106	C₂²×C₁₃	—	2	2	no	—
107	C₂²×C₁₃	—	2	2	no	—
108	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
109	C₄×C₁₃	—	2	2	no	—
110	C₄×C₁₃	—	2	1	no	—
111	C₄×C₁₃	—	2	1	no	—
112	C₄×C₁₃	—	2	1	no	—
113	C₄×C₁₃	—	2	1	no	—
114	C₄×C₁₃	—	2	1	no	—
115	C₄×C₁₃	—	2	2	no	—
116	C₄×C₁₃	—	2	2	no	—
117	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
118	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
119	C₂×M₁₆	—	13	1	no	—
120	C₂×M₁₆	—	13	1	no	—
121	C₂×M₁₆	—	13	2	no	—
122	C₂²×C₈	—	13	2	no	—
123	#126	Dic₁₃	1	2	yes	—
124	#125	Dic₁₃	1	1	yes	characteristic
125	#124	Dic₁₃	1	1	yes	characteristic
126	#123	Dic₁₃	1	2	yes	—
127	C₂²⋊Q₈	Dic₁₃	1	1	yes	characteristic
128	#121	Dic₁₃	1	1	yes	characteristic
129	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
130	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
131	C₄×D₄	C₂×D₁₃	1	2	yes	—
132	#117	C₂×D₁₃	1	2	yes	—
133	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
134	C₄⋊C₈	—	13	2	no	—
135	C₄⋊C₈	—	13	2	no	—
136	C₄⋊C₈	—	13	1	no	—
137	C₄⋊C₈	—	13	2	no	—
138	C₄⋊C₈	—	13	2	no	—
139	C₄⋊C₈	—	13	1	no	—
140	C₂²⋊C₈	—	13	2	no	—
141	C₂²⋊C₈	—	13	2	no	—
142	C₂²⋊C₈	—	13	2	no	—
143	#97	—	13	2	no	—
144	#97	—	13	2	no	—
145	C₄×C₈	—	13	2	no	—
146	C₂×C₁₃	—	4	1	no	—
147	C₂×C₁₃	—	4	2	no	—
148	C₂×C₁₃	#520	1	1	yes	characteristic
149	C₂×C₁₃	#568	1	1	yes	characteristic
150	C₂×C₁₃	#568	1	1	yes	characteristic
151	C₂×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
152	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
153	C₂×D₄	C₂×Dic₁₃	1	1	yes	characteristic
154	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
155	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
156	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
157	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
158	M₁₆	—	52	1	no	—
159	M₁₆	—	52	2	no	—
160	M₁₆	—	52	1	no	—
161	C₂×C₈	—	13	2	no	—
162	C₂×C₈	—	13	2	no	—
163	C₂×C₈	—	13	2	no	—
164	C₂×C₈	—	13	2	no	—
165	C₂×C₈	—	52	2	no	—
166	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
167	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
168	C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
169	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
170	C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
171	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
172	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
173	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
174	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
175	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
176	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
177	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
178	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
179	C₄²	C₂²×D₁₃	1	2	yes	—
180	C₄²	C₂²×D₁₃	1	1	yes	characteristic
181	C₄²	C₂²×D₁₃	1	1	yes	characteristic
182	C₁₃	#2995	1	1	yes	characteristic, complemented, Sylow
183	C₂³	C₂²×Dic₁₃	1	2	yes	—
184	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
185	Q₈	—	4	1	no	—
186	D₄	—	4	1	no	—
187	D₄	—	4	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
195	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
196	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
197	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
198	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
199	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
200	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
201	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
202	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
203	C₈	—	26	2	no	—
204	C₈	—	26	2	no	—
205	C₈	—	26	2	no	—
206	C₈	—	26	2	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	1	no	—
213	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	2	1	no	—
222	C₄	#88256	1	1	yes	characteristic
223	C₄	#88257	1	1	yes	characteristic
224	C₂	—	4	1	no	—
225	C₂	—	4	2	no	—
226	C₂	#11719524	1	1	yes	characteristic
227	C₂	#11719524	1	1	yes	characteristic
228	C₂	#11719532	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#529	C₂
prime radical	C₁₃×#121	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#529	D₁₃
hypocenter	C₁₃	#2995
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
2. #11718685	C₂	semidirect
3. #11718673	C₂	semidirect
4. #11718672	C₂	semidirect
5. #11718672	C₂	semidirect
6. #11718661	C₂	semidirect
7. #11718660	C₂	semidirect
8. #11718587	C₂	semidirect
9. #11718585	C₂	semidirect
182. C₁₃	#2995	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#529	2
#11718685	2
#11718673	2
#11718672	2
#11718661	2
#11718660	2
#11718587	2
#11718585	2
#2995	13