Subgroups — #13123996

Order 1664.

364 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×#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	1	yes	maximal, characteristic, complemented
5	#11718672	C₂	1	2	yes	maximal, complemented
6	D₄×#1055	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×#1055	C₂	1	2	yes	maximal, complemented
8	#11718661	C₂	1	2	yes	maximal, complemented
9	#11718660	C₂	1	2	yes	maximal, complemented
10	#11718586	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄⋊#1055	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	2	yes	—
20	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
21	C₁₃×#117	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
23	C₂×#6099	C₂²	1	2	yes	—
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	C₂²⋊#1055	C₂²	1	1	yes	characteristic
29	C₂²⋊#1055	C₂²	1	1	yes	characteristic
30	C₄⋊#1055	C₂²	1	1	yes	characteristic
31	C₄⋊#1055	C₂²	1	2	yes	—
32	C₄⋊#1055	C₂²	1	2	yes	—
33	C₄⋊#1055	C₂²	1	2	yes	—
34	C₄⋊#1055	C₂²	1	1	yes	characteristic
35	C₄⋊#1055	C₂²	1	1	yes	characteristic
36	C₄⋊#1055	C₂²	1	1	yes	characteristic
37	#88043	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	2	yes	—
43	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
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	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
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	—	4	2	no	—
65	C₂×#1055	—	4	2	no	—
66	C₂×#1055	—	4	2	no	—
67	C₂×#1055	C₂³	1	1	yes	characteristic
68	C₂×#1055	C₂³	1	1	yes	characteristic
69	C₂×#1055	C₂³	1	2	yes	—
70	C₂×#1055	C₂³	1	1	yes	characteristic
71	C₂×#1055	C₂³	1	2	yes	—
72	C₂×#1055	C₂³	1	1	yes	characteristic
73	#2994	—	13	1	no	maximal, Sylow
74	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	Q₈×C₁₃	—	2	2	no	—
77	D₄×C₁₃	—	2	4	no	—
78	D₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	2	1	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
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	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
94	#1055	—	2	1	no	—
95	#1055	—	2	2	no	—
96	#1055	—	2	2	no	—
97	#1055	—	2	1	no	—
98	#1055	—	2	1	no	—
99	#1055	—	2	1	no	—
100	#529	D₁₃	1	1	yes	characteristic
101	#446	—	13	1	no	—
102	C₈×Q₈	—	13	1	no	—
103	#436	—	13	1	no	—
104	#435	—	13	2	no	—
105	C₈×D₄	—	13	1	no	—
106	C₈×D₄	—	13	2	no	—
107	#433	—	13	2	no	—
108	#431	—	13	1	no	—
109	#424	—	13	2	no	—
110	C₂×C₄⋊C₈	—	13	1	no	—
111	C₂²×C₁₃	—	2	1	no	—
112	C₂²×C₁₃	—	2	1	no	—
113	C₂²×C₁₃	—	2	2	no	—
114	C₂²×C₁₃	—	2	2	no	—
115	C₂²×C₁₃	—	2	2	no	—
116	C₂²×C₁₃	#131	1	2	yes	—
117	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₄×C₁₃	—	2	1	no	—
119	C₄×C₁₃	—	2	2	no	—
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	—	2	1	no	—
122	C₄×C₁₃	—	2	1	no	—
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	—	2	2	no	—
125	C₄×C₁₃	#131	1	2	yes	—
126	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
127	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
128	C₂×M₁₆	—	13	2	no	—
129	C₂²×C₈	—	13	2	no	—
130	C₂²×C₈	—	13	2	no	—
131	#126	Dic₁₃	1	2	yes	—
132	#125	Dic₁₃	1	1	yes	characteristic
133	#124	Dic₁₃	1	1	yes	characteristic
134	#123	Dic₁₃	1	2	yes	—
135	C₂²⋊Q₈	Dic₁₃	1	1	yes	characteristic
136	#121	Dic₁₃	1	1	yes	characteristic
137	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
138	C₄×D₄	C₂×D₁₃	1	2	yes	—
139	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
140	#117	C₂×D₁₃	1	2	yes	—
141	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
142	C₄⋊C₈	—	13	2	no	—
143	C₄⋊C₈	—	13	1	no	—
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	C₄⋊C₈	—	13	1	no	—
149	C₂²⋊C₈	—	13	1	no	—
150	C₂²⋊C₈	—	13	1	no	—
151	C₂²⋊C₈	—	13	2	no	—
152	C₂²⋊C₈	—	13	2	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	C₂×C₁₃	—	2	2	no	—
158	C₂×C₁₃	—	4	2	no	—
159	C₂×C₁₃	#520	1	1	yes	characteristic
160	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
161	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
162	C₂×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
163	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
164	C₂×D₄	C₂×Dic₁₃	1	1	yes	characteristic
165	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
166	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
167	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
169	M₁₆	—	52	2	no	—
170	C₂×C₈	—	13	1	no	—
171	C₂×C₈	—	13	1	no	—
172	C₂×C₈	—	13	1	no	—
173	C₂×C₈	—	13	2	no	—
174	C₂×C₈	—	13	1	no	—
175	C₂×C₈	—	13	2	no	—
176	C₂×C₈	—	52	2	no	—
177	C₂×C₈	—	52	2	no	—
178	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
179	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
180	C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
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	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
185	C₂²⋊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	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
190	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
191	C₄²	C₂²×D₁₃	1	1	yes	characteristic
192	C₄²	C₂²×D₁₃	1	2	yes	—
193	C₄²	C₂²×D₁₃	1	1	yes	characteristic
194	C₁₃	#2994	1	1	yes	characteristic, complemented, Sylow
195	C₂³	C₂²×Dic₁₃	1	2	yes	—
196	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
197	Q₈	—	2	2	no	—
198	D₄	—	2	4	no	—
199	D₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
207	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
208	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
209	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
210	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
211	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
212	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
213	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
215	C₈	—	26	1	no	—
216	C₈	—	26	2	no	—
217	C₈	—	26	2	no	—
218	C₈	—	26	1	no	—
219	C₈	—	26	1	no	—
220	C₈	—	26	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	#88202	1	2	yes	—
227	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	2	no	—
235	C₄	#88202	1	2	yes	—
236	C₄	#88256	1	1	yes	characteristic
237	C₄	#88257	1	1	yes	characteristic
238	C₂	—	2	2	no	—
239	C₂	—	4	2	no	—
240	C₂	C₂×#88202	1	1	yes	characteristic
241	C₂	C₂×#88202	1	1	yes	characteristic
242	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	#2994
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. #11718672	C₂	semidirect
6. D₄×#1055	C₂	semidirect
7. D₄×#1055	C₂	semidirect
8. #11718661	C₂	semidirect
9. #11718660	C₂	semidirect
10. #11718586	C₂	semidirect
11. C₂×C₄⋊#1055	C₂	semidirect
194. C₁₃	#2994	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
#11718660	2
#11718586	2
C₂×C₄⋊#1055	2
#2994	13