Subgroups — #13123841

Order 1664.

370 conjugacy classes of subgroups.

The subgroup lattice has 246 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	1	yes	maximal, characteristic, complemented
3	Q₈×#1055	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718673	C₂	1	2	yes	maximal, complemented
5	#11718673	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718672	C₂	1	2	yes	maximal, complemented
7	D₄×#1055	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718661	C₂	1	2	yes	maximal, complemented
9	#11718658	C₂	1	2	yes	maximal, complemented
10	#11718586	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#6099	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#126	C₄	1	2	yes	—
13	C₁₃×#125	C₄	1	1	yes	characteristic
14	C₁₃×#124	C₄	1	1	yes	characteristic
15	C₁₃×#123	C₄	1	2	yes	—
16	C₁₃×C₂²⋊Q₈	C₄	1	1	yes	characteristic
17	C₁₃×#121	C₄	1	1	yes	characteristic
18	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₁₃	C₂²	1	2	yes	—
21	C₁₃×#117	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
23	C₂×#6099	C₂²	1	2	yes	—
24	C₂×#6099	C₂²	1	1	yes	characteristic
25	C₂×#6099	C₂²	1	1	yes	characteristic
26	C₂²×#1055	C₂²	1	2	yes	—
27	C₂²⋊#1055	C₂²	1	2	yes	—
28	C₂²⋊#1055	C₂²	1	2	yes	—
29	C₂²⋊#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	1	yes	characteristic
34	#88043	C₂²	1	1	yes	characteristic
35	#88043	C₂²	1	2	yes	—
36	#88043	C₂²	1	1	yes	characteristic
37	C₄×#1055	C₂²	1	2	yes	—
38	C₄×#1055	C₂²	1	2	yes	—
39	C₄×#1055	C₂²	1	1	yes	characteristic
40	C₄×#1055	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
52	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
53	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
54	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
55	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
56	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
57	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
58	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
59	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
60	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
61	C₄²×C₁₃	C₂³	1	1	yes	characteristic
62	C₄²×C₁₃	C₂³	1	1	yes	characteristic
63	C₄²×C₁₃	C₂³	1	2	yes	—
64	#6099	—	2	2	no	—
65	#6099	—	2	4	no	—
66	#6099	—	2	2	no	—
67	C₂×#1055	—	2	4	no	—
68	C₂×#1055	C₂³	1	2	yes	—
69	C₂×#1055	C₂³	1	1	yes	characteristic
70	C₂×#1055	C₂³	1	1	yes	characteristic
71	C₂×#1055	C₂³	1	2	yes	—
72	C₂×#1055	C₂³	1	1	yes	characteristic
73	C₂×#1055	C₂³	1	1	yes	characteristic
74	#2969	—	13	1	no	maximal, Sylow
75	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	Q₈×C₁₃	—	2	2	no	—
78	D₄×C₁₃	—	2	2	no	—
79	D₄×C₁₃	—	4	2	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	4	1	no	—
86	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
95	#1055	—	2	2	no	—
96	#1055	—	2	1	no	—
97	#1055	—	2	1	no	—
98	#1055	—	2	2	no	—
99	#1055	#41	1	2	yes	—
100	#1055	#41	1	2	yes	—
101	#529	D₁₃	1	1	yes	characteristic
102	#446	—	13	1	no	—
103	C₈×Q₈	—	13	1	no	—
104	#436	—	13	1	no	—
105	#436	—	13	2	no	—
106	#435	—	13	2	no	—
107	C₈×D₄	—	13	1	no	—
108	#433	—	13	2	no	—
109	#431	—	13	1	no	—
110	#405	—	13	2	no	—
111	C₄×M₁₆	—	13	1	no	—
112	C₂²×C₁₃	—	2	2	no	—
113	C₂²×C₁₃	—	2	1	no	—
114	C₂²×C₁₃	—	2	1	no	—
115	C₂²×C₁₃	—	2	2	no	—
116	C₂²×C₁₃	—	2	1	no	—
117	C₂²×C₁₃	—	2	2	no	—
118	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
119	C₄×C₁₃	—	2	2	no	—
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	—	2	1	no	—
122	C₄×C₁₃	—	2	2	no	—
123	C₄×C₁₃	—	2	1	no	—
124	C₄×C₁₃	—	2	1	no	—
125	C₄×C₁₃	—	2	2	no	—
126	C₄×C₁₃	#131	1	2	yes	—
127	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
128	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
129	C₂×M₁₆	—	13	1	no	—
130	C₂×M₁₆	—	13	1	no	—
131	C₂×M₁₆	—	13	2	no	—
132	C₂²×C₈	—	13	2	no	—
133	#126	Dic₁₃	1	2	yes	—
134	#125	Dic₁₃	1	1	yes	characteristic
135	#124	Dic₁₃	1	1	yes	characteristic
136	#123	Dic₁₃	1	2	yes	—
137	C₂²⋊Q₈	Dic₁₃	1	1	yes	characteristic
138	#121	Dic₁₃	1	1	yes	characteristic
139	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
140	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
141	C₄×D₄	C₂×D₁₃	1	2	yes	—
142	#117	C₂×D₁₃	1	2	yes	—
143	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
144	C₄⋊C₈	—	13	1	no	—
145	C₄⋊C₈	—	13	1	no	—
146	C₄⋊C₈	—	13	2	no	—
147	C₄⋊C₈	—	13	2	no	—
148	C₂²⋊C₈	—	13	2	no	—
149	C₂²⋊C₈	—	13	2	no	—
150	C₂²⋊C₈	—	13	2	no	—
151	#97	—	13	1	no	—
152	#97	—	13	1	no	—
153	#97	—	13	2	no	—
154	C₄×C₈	—	13	1	no	—
155	C₄×C₈	—	13	1	no	—
156	C₄×C₈	—	13	2	no	—
157	C₄×C₈	—	13	2	no	—
158	C₂×C₁₃	—	4	1	no	—
159	C₂×C₁₃	—	4	2	no	—
160	C₂×C₁₃	C₄×#41	1	1	yes	characteristic
161	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
162	C₂×C₁₃	#568	1	1	yes	characteristic
163	C₂×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
164	C₂×D₄	C₂×Dic₁₃	1	1	yes	characteristic
165	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
166	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
167	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
170	M₁₆	—	26	2	no	—
171	M₁₆	—	26	2	no	—
172	M₁₆	—	26	4	no	—
173	C₂×C₈	—	13	1	no	—
174	C₂×C₈	—	13	1	no	—
175	C₂×C₈	—	13	2	no	—
176	C₂×C₈	—	13	1	no	—
177	C₂×C₈	—	13	1	no	—
178	C₂×C₈	—	13	2	no	—
179	C₂×C₈	—	26	4	no	—
180	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
181	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
182	C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
183	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
184	C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
185	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
186	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
187	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
188	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
189	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
190	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
191	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
192	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
193	C₄²	C₂²×D₁₃	1	1	yes	characteristic
194	C₄²	C₂²×D₁₃	1	1	yes	characteristic
195	C₄²	C₂²×D₁₃	1	2	yes	—
196	C₁₃	#2969	1	1	yes	characteristic, complemented, Sylow
197	C₂³	C₂²×Dic₁₃	1	2	yes	—
198	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
199	Q₈	—	2	2	no	—
200	D₄	—	2	2	no	—
201	D₄	—	4	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
209	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
210	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
211	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
212	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
213	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
214	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
215	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
217	C₈	—	13	2	no	—
218	C₈	—	13	2	no	—
219	C₈	—	26	2	no	—
220	C₈	—	26	1	no	—
221	C₈	—	26	1	no	—
222	C₈	—	26	2	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	1	no	—
229	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	1	no	—
237	C₄	#88202	1	2	yes	—
238	C₄	#41×D₁₃	1	1	yes	characteristic
239	C₄	#41×D₁₃	1	1	yes	characteristic
240	C₂	—	4	1	no	—
241	C₂	—	4	2	no	—
242	C₂	C₂×#88202	1	1	yes	characteristic
243	C₂	#11719524	1	1	yes	characteristic
244	C₂	#41×Dic₁₃	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	#2969
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. Q₈×#1055	C₂	semidirect
4. #11718673	C₂	semidirect
5. #11718673	C₂	semidirect
6. #11718672	C₂	semidirect
7. D₄×#1055	C₂	semidirect
8. #11718661	C₂	semidirect
9. #11718658	C₂	semidirect
10. #11718586	C₂	semidirect
11. C₄×#6099	C₂	semidirect
196. C₁₃	#2969	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
Q₈×#1055	2
#11718673	2
#11718672	2
D₄×#1055	2
#11718661	2
#11718658	2
#11718586	2
C₄×#6099	2
#2969	13