Subgroups — #15861005

Order 3328.

1126 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88057	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718762	C₂	1	1	yes	maximal, characteristic
3	#13117192	C₂	1	2	yes	maximal
4	#13114735	C₂	1	4	yes	maximal, complemented
5	#106×Dic₁₃	C₂	1	4	yes	maximal, complemented
6	#13114557	C₂	1	2	yes	maximal
7	C₂×C₁₃×#427	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#117	C₂²	1	2	yes	—
10	C₂×#88180	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
12	C₂²×#6099	C₂²	1	1	yes	characteristic
13	C₁₃×#427	C₄	1	8	yes	complemented
14	C₂×C₁₃×#107	C₂²	1	2	yes	—
15	C₂×C₁₃×#106	C₂²	1	2	yes	—
16	C₂×#88057	C₄	1	8	yes	complemented
17	C₂×#88057	C₂²	1	2	yes	—
18	C₂×#88057	C₂²	1	2	yes	—
19	#11718762	C₄	1	8	yes	complemented
20	C₂×#88055	C₂²	1	2	yes	—
21	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
22	C₄⋊C₄×Dic₁₃	C₂²	1	8	yes	—
23	C₂²⋊C₄×Dic₁₃	—	2	4	no	—
24	C₄×C₄⋊Dic₁₃	—	2	4	no	—
25	#11718121	C₂²	1	8	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₂³×Dic₁₃	—	2	2	no	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₁₃×#117	—	2	4	no	—
34	C₁₃×#117	C₂×C₄	1	8	yes	—
35	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
36	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
37	C₂×C₄²×C₁₃	—	2	2	no	—
38	C₂×C₂²⋊Dic₁₃	D₄	1	2	yes	—
39	#88180	C₂×C₄	1	4	yes	—
40	#88180	Q₈	1	4	yes	—
41	C₂×C₄⋊Dic₁₃	—	2	2	no	—
42	C₂×C₄⋊Dic₁₃	—	2	4	no	—
43	C₂×C₄⋊Dic₁₃	C₂×C₄	1	8	yes	—
44	C₂×C₄⋊Dic₁₃	D₄	1	2	yes	—
45	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
46	C₂×C₄×Dic₁₃	—	2	8	no	—
47	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
48	C₂×#6099	—	2	2	no	—
49	C₂×#6099	—	2	2	no	—
50	C₂×#6099	C₂×C₄	1	4	yes	—
51	C₂²×#1055	C₂³	1	2	yes	—
52	C₁₃×#107	C₂×C₄	1	8	yes	complemented
53	C₁₃×#106	C₂×C₄	1	8	yes	complemented
54	#88071	—	2	8	no	—
55	#88057	C₂×C₄	1	8	yes	—
56	#88057	C₂×C₄	1	8	yes	—
57	#88055	C₂×C₄	1	8	yes	complemented
58	C₈×Dic₁₃	C₂×C₄	1	8	yes	complemented
59	#20454	—	13	1	no	maximal, Sylow
60	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	—	2	4	no	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₁₃	#41	1	2	yes	—
68	C₂²×C₄×C₁₃	#41	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₁₃	—	2	4	no	—
71	C₂²×Dic₁₃	—	2	8	no	—
72	C₂²×Dic₁₃	—	8	4	no	—
73	C₂²×Dic₁₃	C₂×D₄	1	4	yes	—
74	C₂×C₈×C₁₃	C₄²	1	16	yes	complemented
75	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
81	C₁₃×C₄⋊C₄	—	2	4	no	—
82	C₁₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
83	C₁₃×C₂²⋊C₄	—	2	8	no	—
84	C₄²×C₁₃	—	2	8	no	—
85	C₂²⋊Dic₁₃	C₄⋊C₄	1	8	yes	—
86	C₄⋊Dic₁₃	—	2	8	no	—
87	C₄⋊Dic₁₃	—	2	4	no	—
88	C₄⋊Dic₁₃	C₄⋊C₄	1	8	yes	—
89	C₄⋊Dic₁₃	C₂²×C₄	1	8	yes	—
90	C₄×Dic₁₃	—	8	8	no	—
91	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
92	C₄×Dic₁₃	C₂×Q₈	1	4	yes	—
93	#6099	—	2	8	no	—
94	C₂×#1055	—	2	2	no	—
95	C₂×#1055	—	2	2	no	—
96	C₂×#1055	C₂²×C₄	1	4	yes	—
97	C₂×#427	D₁₃	1	1	yes	characteristic
98	C₂²×#106	—	13	1	no	—
99	C₂×#405	—	13	1	no	—
100	#2274	—	13	2	no	—
101	#1776	—	13	4	no	—
102	C₄×#106	—	13	4	no	—
103	#1736	—	13	2	no	—
104	C₂³×C₁₃	—	2	4	no	—
105	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂³×C₁₃	C₄×D₄	1	2	yes	—
107	C₂³×C₁₃	C₄×D₄	1	2	yes	—
108	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	—	2	8	no	—
110	C₂×C₄×C₁₃	—	4	4	no	—
111	C₂×C₄×C₁₃	—	4	4	no	—
112	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
113	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₁₃	#117	1	4	yes	—
115	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
118	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
119	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
125	C₂×Dic₁₃	—	2	8	no	—
126	C₂×Dic₁₃	—	2	8	no	—
127	C₂×Dic₁₃	—	4	8	no	—
128	C₂×Dic₁₃	—	8	8	no	—
129	C₂×Dic₁₃	C₂×C₄⋊C₄	1	8	yes	—
130	C₈×C₁₃	C₂×C₄²	1	4	yes	—
131	C₈×C₁₃	C₂×C₄²	1	4	yes	—
132	#1055	—	2	4	no	—
133	C₂²×M₁₆	—	13	1	no	—
134	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
135	C₂×#117	—	13	1	no	—
136	C₂×#117	C₂×D₁₃	1	2	yes	—
137	C₂²×C₄⋊C₄	—	13	2	no	—
138	#427	Dic₁₃	1	8	yes	complemented
139	C₂×#107	C₂×D₁₃	1	2	yes	—
140	C₂×#106	—	13	8	no	—
141	C₂×#106	—	13	2	no	—
142	C₂×#106	—	13	2	no	—
143	C₂×#106	C₂×D₁₃	1	2	yes	—
144	#405	—	13	8	no	—
145	C₂×#97	—	13	2	no	—
146	C₂×C₄×C₈	—	13	2	no	—
147	C₄×C₄⋊C₄	—	13	8	no	—
148	C₄×C₄⋊C₄	—	26	4	no	—
149	C₄×C₂²⋊C₄	—	26	4	no	—
150	#340	—	13	8	no	—
151	C₂²×C₁₃	—	2	4	no	—
152	C₂²×C₁₃	—	2	4	no	—
153	C₂²×C₁₃	—	2	4	no	—
154	C₂²×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
155	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
156	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
157	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
158	C₂²×C₁₃	#577	1	4	yes	—
159	C₄×C₁₃	—	4	8	no	—
160	C₄×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
161	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
162	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
163	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
164	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
165	Dic₁₃	—	2	8	no	—
166	Dic₁₃	—	4	8	no	—
167	C₂³×C₄	—	26	2	no	—
168	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
169	C₂×M₁₆	—	13	4	no	—
170	C₂×M₁₆	—	26	2	no	—
171	C₂×M₁₆	—	26	2	no	—
172	C₂²×C₈	—	13	2	no	—
173	C₂²×C₈	C₄×D₁₃	1	4	yes	—
174	C₂²×C₈	C₄×D₁₃	1	4	yes	—
175	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
176	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
177	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
178	#117	—	2	4	no	—
179	#117	—	13	4	no	—
180	#117	—	13	4	no	—
181	#117	C₂×Dic₁₃	1	8	yes	—
182	C₂×C₄⋊C₄	—	13	2	no	—
183	C₂×C₄⋊C₄	—	13	8	no	—
184	C₂×C₄⋊C₄	—	13	4	no	—
185	C₂×C₄⋊C₄	—	26	2	no	—
186	C₂×C₄⋊C₄	—	26	4	no	—
187	C₂×C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
188	C₂×C₂²⋊C₄	—	2	2	no	—
189	C₂×C₂²⋊C₄	—	13	2	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	13	2	no	—
192	C₂×C₄²	—	26	8	no	—
193	#107	C₂×Dic₁₃	1	8	yes	complemented
194	#106	—	13	8	no	—
195	#106	—	13	8	no	—
196	#106	C₂×Dic₁₃	1	8	yes	complemented
197	#97	—	13	8	no	—
198	C₄×C₈	—	13	8	no	—
199	#95	—	26	8	no	—
200	C₂×C₁₃	—	2	4	no	—
201	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
202	C₂×C₁₃	#2912	1	2	yes	—
203	C₂×C₁₃	#2949	1	2	yes	—
204	C₂×C₁₃	#2952	1	2	yes	—
205	C₂⁴	D₄×D₁₃	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	13	4	no	—
208	C₂²×C₄	—	26	4	no	—
209	C₂²×C₄	—	26	8	no	—
210	C₂²×C₄	—	104	4	no	—
211	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
212	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
213	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
214	C₂²×C₄	#6129	1	2	yes	—
215	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
216	C₂²×C₄	#6131	1	2	yes	—
217	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
218	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
219	M₁₆	—	26	8	no	—
220	C₂×C₈	—	13	4	no	—
221	C₂×C₈	—	26	2	no	—
222	C₂×C₈	—	26	2	no	—
223	C₂×C₈	C₄×Dic₁₃	1	16	yes	complemented
224	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
225	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
226	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
227	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
228	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
229	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	13	8	no	—
232	C₄⋊C₄	—	13	8	no	—
233	C₄⋊C₄	—	26	8	no	—
234	C₄⋊C₄	—	26	4	no	—
235	C₄⋊C₄	C₂²×Dic₁₃	1	8	yes	—
236	C₂²⋊C₄	—	2	8	no	—
237	C₂²⋊C₄	—	13	8	no	—
238	C₄²	—	2	8	no	—
239	C₄²	—	13	4	no	—
240	C₄²	—	13	4	no	—
241	C₄²	—	104	8	no	—
242	C₁₃	#20454	1	1	yes	characteristic, complemented, Sylow
243	C₂³	—	2	4	no	—
244	C₂³	C₄⋊C₄×D₁₃	1	2	yes	—
245	C₂³	#88147	1	2	yes	—
246	C₂³	D₄×Dic₁₃	1	2	yes	—
247	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
248	C₂×C₄	—	2	8	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	13	8	no	—
252	C₂×C₄	—	26	8	no	—
253	C₂×C₄	—	26	8	no	—
254	C₂×C₄	—	52	8	no	—
255	C₂×C₄	—	104	8	no	—
256	C₂×C₄	#88141	1	4	yes	—
257	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
258	C₂×C₄	#88146	1	4	yes	—
259	C₂×C₄	#88147	1	2	yes	—
260	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
261	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
262	C₂×C₄	Q₈×Dic₁₃	1	4	yes	—
263	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
264	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
265	C₂×C₄	C₂×#6129	1	1	yes	characteristic
266	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
267	C₂×C₄	C₂×#6131	1	1	yes	characteristic
268	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
269	C₈	—	26	4	no	—
270	C₈	C₂×C₄×Dic₁₃	1	4	yes	—
271	C₈	C₂×C₄×Dic₁₃	1	4	yes	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	C₄⋊C₄×Dic₁₃	1	4	yes	—
276	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
277	C₂²	C₂×#88147	1	1	yes	characteristic
278	C₂²	#11719475	1	4	yes	—
279	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
280	C₄	—	4	8	no	—
281	C₄	—	26	8	no	—
282	C₄	—	52	8	no	—
283	C₄	C₄⋊C₄×Dic₁₃	1	4	yes	—
284	C₄	C₂×#88141	1	1	yes	characteristic
285	C₄	C₂×#88146	1	1	yes	characteristic
286	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
287	C₄	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
288	C₂	—	2	4	no	—
289	C₂	C₂×C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
290	C₂	#13123428	1	2	yes	—
291	C₂	#13123694	1	2	yes	—
292	C₂	#13123720	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#427	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#427	D₁₃
hypocenter	C₁₃	#20454
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
4. #13114735	C₂	semidirect
5. #106×Dic₁₃	C₂	semidirect
13. C₁₃×#427	C₄	semidirect
16. C₂×#88057	C₄	semidirect
19. #11718762	C₄	semidirect
52. C₁₃×#107	C₂×C₄	semidirect
53. C₁₃×#106	C₂×C₄	semidirect
57. #88055	C₂×C₄	semidirect
58. C₈×Dic₁₃	C₂×C₄	semidirect
74. C₂×C₈×C₁₃	C₄²	semidirect
138. #427	Dic₁₃	semidirect
193. #107	C₂×Dic₁₃	semidirect
196. #106	C₂×Dic₁₃	semidirect
223. C₂×C₈	C₄×Dic₁₃	semidirect
242. C₁₃	#20454	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88057	2
C₂×#11718762	2
#13117192	2
#13114735	2
#106×Dic₁₃	2
#13114557	2
C₂×C₁₃×#427	2
#20454	13