Subgroups — #13117571

Order 1664.

528 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#11718891	C₂	1	2	yes	maximal, complemented
3	#11718873	C₂	1	1	yes	maximal, characteristic
4	C₂×#88071	C₂	1	1	yes	maximal, characteristic
5	#11718646	C₂	1	2	yes	maximal, complemented
6	#11718642	C₂	1	2	yes	maximal, complemented
7	#11718642	C₂	1	2	yes	maximal, complemented
8	#11718611	C₂	1	2	yes	maximal
9	C₂²⋊C₄×Dic₁₃	C₂	1	2	yes	maximal
10	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂³×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂³×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×C₂²⋊Q₈	C₄	1	8	yes	complemented
15	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₂×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
20	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
21	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
24	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
25	#88071	C₂²	1	2	yes	—
26	#88071	C₂²	1	2	yes	—
27	#88071	C₂²	1	2	yes	—
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	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₁₃	—	4	1	no	—
34	C₂×Q₈×C₁₃	—	4	1	no	—
35	C₂×Q₈×C₁₃	C₂×C₄	1	4	yes	complemented
36	C₂²×C₄×C₁₃	—	4	1	no	—
37	C₂²×C₄×C₁₃	—	4	1	no	—
38	C₂²×C₄×C₁₃	C₂×C₄	1	4	yes	—
39	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×Dic₁₃	—	2	2	no	—
44	C₂²×Dic₁₃	—	4	1	no	—
45	C₂²×Dic₁₃	—	4	1	no	—
46	C₂²×Dic₁₃	—	4	1	no	—
47	C₂²×Dic₁₃	—	4	1	no	—
48	C₂²×Dic₁₃	Q₈	1	4	yes	complemented
49	C₂²×Dic₁₃	C₂³	1	2	yes	—
50	C₂²×Dic₁₃	C₂³	1	2	yes	—
51	C₂²×Dic₁₃	C₂³	1	2	yes	—
52	C₂²×Dic₁₃	C₂³	1	2	yes	—
53	C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	complemented
54	C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	complemented
55	C₁₃×C₂²⋊C₄	C₂×C₄	1	8	yes	—
56	C₂²⋊Dic₁₃	—	2	4	no	—
57	C₄⋊Dic₁₃	—	4	1	no	—
58	C₄⋊Dic₁₃	—	4	2	no	—
59	C₄⋊Dic₁₃	—	4	1	no	—
60	C₄⋊Dic₁₃	—	4	2	no	—
61	C₄×Dic₁₃	—	4	2	no	—
62	C₄×Dic₁₃	—	4	2	no	—
63	#2329	—	13	1	no	maximal, Sylow
64	C₂³×C₁₃	—	2	1	no	—
65	C₂³×C₁₃	—	2	1	no	—
66	C₂³×C₁₃	—	2	1	no	—
67	C₂³×C₁₃	—	2	1	no	—
68	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
69	C₂³×C₁₃	C₂×Q₈	1	2	yes	—
70	C₂³×C₁₃	#41	1	2	yes	—
71	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
72	Q₈×C₁₃	—	4	4	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	1	no	—
76	C₂×C₄×C₁₃	—	2	1	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	1	no	—
79	C₂×C₄×C₁₃	—	2	1	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	4	4	no	—
82	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
83	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
84	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
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	2	no	—
92	C₂×Dic₁₃	—	2	2	no	—
93	C₂×Dic₁₃	—	2	2	no	—
94	C₂×Dic₁₃	—	2	2	no	—
95	C₂×Dic₁₃	—	2	2	no	—
96	C₂×Dic₁₃	—	2	2	no	—
97	C₂×Dic₁₃	—	4	4	no	—
98	C₂×Dic₁₃	—	4	4	no	—
99	C₂×Dic₁₃	C₂×Q₈	1	4	yes	—
100	C₂×C₂²⋊Q₈	D₁₃	1	1	yes	characteristic
101	#391	—	13	2	no	—
102	#389	—	13	2	no	—
103	#387	—	13	2	no	—
104	#387	—	13	2	no	—
105	#385	—	13	2	no	—
106	#380	—	13	1	no	—
107	C₄×C₂²⋊C₄	—	13	2	no	—
108	C₂×#95	—	13	1	no	—
109	C₂²×C₁₃	—	2	2	no	—
110	C₂²×C₁₃	—	2	2	no	—
111	C₂²×C₁₃	—	2	2	no	—
112	C₂²×C₁₃	—	2	2	no	—
113	C₂²×C₁₃	—	2	2	no	—
114	C₂²×C₁₃	—	2	2	no	—
115	C₂²×C₁₃	C₄×Q₈	1	4	yes	—
116	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
117	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
118	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
119	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
122	C₂²×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
123	C₄×C₁₃	—	2	2	no	—
124	C₄×C₁₃	—	2	2	no	—
125	C₄×C₁₃	—	2	4	no	—
126	C₄×C₁₃	—	2	4	no	—
127	Dic₁₃	—	2	4	no	—
128	Dic₁₃	—	4	2	no	—
129	Dic₁₃	—	4	2	no	—
130	Dic₁₃	—	4	2	no	—
131	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
132	C₂³×C₄	—	13	1	no	—
133	C₂³×C₄	—	13	1	no	—
134	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
135	C₂²⋊Q₈	Dic₁₃	1	8	yes	complemented
136	C₂×C₄⋊C₄	—	13	1	no	—
137	C₂×C₄⋊C₄	—	13	1	no	—
138	C₂×C₄⋊C₄	—	13	2	no	—
139	C₂×C₄⋊C₄	—	13	2	no	—
140	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
141	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
142	C₂×C₂²⋊C₄	—	13	2	no	—
143	C₂×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
144	C₂×C₄²	—	13	2	no	—
145	C₂×C₄²	—	13	2	no	—
146	#95	—	13	2	no	—
147	#95	—	13	2	no	—
148	#95	—	13	2	no	—
149	#95	—	13	2	no	—
150	#95	—	13	2	no	—
151	#95	—	13	2	no	—
152	#95	—	13	2	no	—
153	C₂×C₁₃	—	2	4	no	—
154	C₂×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
155	C₂×C₁₃	#518	1	1	yes	characteristic
156	C₂×C₁₃	#520	1	1	yes	characteristic
157	C₂×C₁₃	#538	1	1	yes	characteristic
158	C₂×C₁₃	#539	1	1	yes	characteristic
159	C₂×C₁₃	#543	1	1	yes	characteristic
160	C₂×C₁₃	#544	1	1	yes	characteristic
161	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
162	C₂×Q₈	—	4	1	no	—
163	C₂×Q₈	—	4	1	no	—
164	C₂×Q₈	C₂×Dic₁₃	1	4	yes	complemented
165	C₂²×C₄	—	4	1	no	—
166	C₂²×C₄	—	4	1	no	—
167	C₂²×C₄	—	13	2	no	—
168	C₂²×C₄	—	13	4	no	—
169	C₂²×C₄	—	13	2	no	—
170	C₂²×C₄	—	13	2	no	—
171	C₂²×C₄	—	13	2	no	—
172	C₂²×C₄	—	26	2	no	—
173	C₂²×C₄	—	52	1	no	—
174	C₂²×C₄	—	52	1	no	—
175	C₂²×C₄	—	52	1	no	—
176	C₂²×C₄	—	52	1	no	—
177	C₂²×C₄	C₂×Dic₁₃	1	4	yes	—
178	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
179	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
180	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
181	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
182	C₄⋊C₄	—	52	1	no	—
183	C₄⋊C₄	—	52	1	no	—
184	C₄⋊C₄	—	52	2	no	—
185	C₄⋊C₄	—	52	2	no	—
186	C₄⋊C₄	C₂×Dic₁₃	1	4	yes	complemented
187	C₄⋊C₄	C₂×Dic₁₃	1	8	yes	complemented
188	C₂²⋊C₄	—	26	4	no	—
189	C₂²⋊C₄	C₂×Dic₁₃	1	8	yes	—
190	C₄²	—	52	2	no	—
191	C₄²	—	52	2	no	—
192	C₁₃	#2329	1	1	yes	characteristic, complemented, Sylow
193	C₂³	—	2	1	no	—
194	C₂³	—	2	1	no	—
195	C₂³	—	2	1	no	—
196	C₂³	—	2	1	no	—
197	C₂³	Q₈×D₁₃	1	2	yes	—
198	C₂³	#6131	1	2	yes	—
199	C₂³	C₂²×Dic₁₃	1	2	yes	—
200	C₂³	C₂³×D₁₃	1	1	yes	characteristic
201	Q₈	—	4	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	13	4	no	—
212	C₂×C₄	—	26	2	no	—
213	C₂×C₄	—	26	2	no	—
214	C₂×C₄	—	26	2	no	—
215	C₂×C₄	—	26	2	no	—
216	C₂×C₄	—	26	2	no	—
217	C₂×C₄	—	26	2	no	—
218	C₂×C₄	—	26	2	no	—
219	C₂×C₄	—	26	2	no	—
220	C₂×C₄	—	26	2	no	—
221	C₂×C₄	—	26	2	no	—
222	C₂×C₄	—	26	2	no	—
223	C₂×C₄	—	52	4	no	—
224	C₂×C₄	—	52	4	no	—
225	C₂×C₄	C₂²×Dic₁₃	1	4	yes	—
226	C₂×C₄	C₂²×Dic₁₃	1	4	yes	—
227	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
228	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	Q₈×Dic₁₃	1	4	yes	—
236	C₂²	#88251	1	1	yes	characteristic
237	C₂²	#88251	1	1	yes	characteristic
238	C₂²	C₂×Q₈×D₁₃	1	1	yes	characteristic
239	C₂²	C₂×#6131	1	1	yes	characteristic
240	C₂²	#88256	1	1	yes	characteristic
241	C₂²	#88257	1	1	yes	characteristic
242	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	2	4	no	—
246	C₄	—	2	4	no	—
247	C₄	—	26	4	no	—
248	C₄	—	52	2	no	—
249	C₄	—	52	2	no	—
250	C₄	—	52	2	no	—
251	C₂	—	2	4	no	—
252	C₂	#11719328	1	1	yes	characteristic
253	C₂	#11719339	1	1	yes	characteristic
254	C₂	#11719349	1	1	yes	characteristic
255	C₂	#11719350	1	1	yes	characteristic
256	C₂	#11719503	1	1	yes	characteristic
257	C₂	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
258	C₂	#11719532	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂³×C₄
Fitting	C₂×C₁₃×C₂²⋊Q₈	C₂
socle, cosocle	C₂³×C₁₃	C₂⁴
prime radical	C₂⁴×C₁₃	C₂³
hypercenter	C₂×C₂²⋊Q₈	D₁₃
hypocenter	C₁₃	#2329
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #11718891	C₂	semidirect
5. #11718646	C₂	semidirect
6. #11718642	C₂	semidirect
7. #11718642	C₂	semidirect
14. C₁₃×C₂²⋊Q₈	C₄	semidirect
35. C₂×Q₈×C₁₃	C₂×C₄	semidirect
48. C₂²×Dic₁₃	Q₈	semidirect
53. C₁₃×C₄⋊C₄	C₂×C₄	semidirect
54. C₁₃×C₄⋊C₄	C₂×C₄	semidirect
135. C₂²⋊Q₈	Dic₁₃	semidirect
164. C₂×Q₈	C₂×Dic₁₃	semidirect
186. C₄⋊C₄	C₂×Dic₁₃	semidirect
187. C₄⋊C₄	C₂×Dic₁₃	semidirect
192. C₁₃	#2329	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×C₂²⋊Q₈	2
#11718891	2
#11718873	2
C₂×#88071	2
#11718646	2
#11718642	2
#11718611	2
C₂²⋊C₄×Dic₁₃	2
#2329	13