Subgroups — #15863098

Order 3328.

772 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718786	C₂	1	1	yes	maximal, characteristic
2	#13123426	C₂	1	1	yes	maximal, characteristic
3	#13114960	C₂	1	8	yes	maximal, complemented
4	C₂×#11718148	C₂	1	2	yes	maximal
5	C₂×#11718123	C₂	1	2	yes	maximal
6	C₁₃×#2873	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
10	C₂×#88180	C₂²	1	2	yes	—
11	C₂²×#6099	C₂²	1	2	yes	—
12	C₁₃×#405	C₄	1	4	yes	complemented
13	C₁₃×#405	C₂²	1	4	yes	—
14	C₄×C₁₃×M₁₆	C₄	1	4	yes	complemented
15	#11718786	—	2	4	no	—
16	#11718782	—	2	4	no	—
17	C₂×#88059	C₂²	1	4	yes	—
18	#11718767	C₂²	1	4	yes	—
19	#11718148	C₂²	1	8	yes	—
20	#11718123	C₂²	1	8	yes	—
21	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₁₃×M₁₆	D₄	1	4	yes	—
28	C₂×C₁₃×M₁₆	Q₈	1	4	yes	—
29	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
32	C₁₃×#117	—	2	1	no	—
33	C₁₃×#117	—	2	1	no	—
34	C₁₃×#117	C₂×C₄	1	2	yes	—
35	C₁₃×#117	C₂³	1	2	yes	—
36	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
37	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
38	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
40	#88180	—	2	2	no	—
41	#88180	—	2	2	no	—
42	#88180	—	2	2	no	—
43	#88180	C₂³	1	4	yes	—
44	C₂×C₄⋊Dic₁₃	—	2	2	no	—
45	C₂×C₄×Dic₁₃	—	2	2	no	—
46	C₂×#6099	—	2	2	no	—
47	C₂×#6099	—	2	2	no	—
48	C₂×#6099	—	2	2	no	—
49	C₂×#6099	—	2	2	no	—
50	C₂×#6099	—	2	2	no	—
51	C₂×#6099	C₂³	1	4	yes	—
52	C₂²×#1055	—	2	2	no	—
53	C₁₃×#97	C₂×C₄	1	8	yes	complemented
54	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	complemented
55	#88059	—	2	8	no	—
56	#88058	—	2	4	no	—
57	#88057	—	2	4	no	—
58	#20757	—	13	1	no	maximal, Sylow
59	C₂⁴×C₁₃	#41	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	#41	1	2	yes	—
67	C₂²×C₄×C₁₃	#41	1	2	yes	—
68	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₁₃	—	4	2	no	—
72	C₁₃×M₁₆	—	2	4	no	—
73	C₁₃×M₁₆	—	2	2	no	—
74	C₁₃×M₁₆	—	2	2	no	—
75	C₁₃×M₁₆	C₄⋊C₄	1	8	yes	—
76	C₁₃×M₁₆	C₄⋊C₄	1	8	yes	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
81	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
82	C₁₃×C₄⋊C₄	—	2	2	no	—
83	C₁₃×C₄⋊C₄	—	2	2	no	—
84	C₁₃×C₂²⋊C₄	—	2	2	no	—
85	C₁₃×C₂²⋊C₄	—	2	2	no	—
86	C₄²×C₁₃	—	2	2	no	—
87	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
88	C₂²⋊Dic₁₃	—	2	4	no	—
89	C₂²⋊Dic₁₃	—	4	2	no	—
90	C₄⋊Dic₁₃	—	2	4	no	—
91	C₄⋊Dic₁₃	—	4	2	no	—
92	C₄×Dic₁₃	—	2	4	no	—
93	C₄×Dic₁₃	—	4	2	no	—
94	#6099	—	2	4	no	—
95	#6099	—	2	4	no	—
96	#6099	—	4	2	no	—
97	#6099	—	4	2	no	—
98	C₂×#1055	—	2	4	no	—
99	C₂×#1055	—	4	2	no	—
100	C₂×#1055	—	4	2	no	—
101	C₂×#430	—	13	1	no	—
102	#2911	—	13	1	no	—
103	#2873	D₁₃	1	1	yes	characteristic
104	#1833	—	13	8	no	—
105	C₂×#344	—	13	2	no	—
106	C₂×H(3,ℤ/4)	—	13	2	no	—
107	C₂³×C₁₃	—	8	1	no	—
108	C₂³×C₁₃	#117	1	1	yes	characteristic
109	C₂³×C₁₃	#117	1	1	yes	characteristic
110	C₂³×C₁₃	#124	1	2	yes	—
111	C₂³×C₁₃	#125	1	2	yes	—
112	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	8	2	no	—
115	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₁₃	#117	1	2	yes	—
118	C₂×C₄×C₁₃	#117	1	2	yes	—
119	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
120	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#124	1	2	yes	—
122	C₂×C₄×C₁₃	#124	1	2	yes	—
123	C₂×C₄×C₁₃	#124	1	2	yes	—
124	C₂×C₄×C₁₃	#125	1	2	yes	—
125	C₂×C₄×C₁₃	#125	1	2	yes	—
126	C₂×C₄×C₁₃	#125	1	2	yes	—
127	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
133	C₂×Dic₁₃	—	4	4	no	—
134	C₂×Dic₁₃	—	8	4	no	—
135	C₈×C₁₃	—	2	4	no	—
136	C₈×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
137	#1055	—	4	4	no	—
138	C₂²×M₁₆	—	13	2	no	—
139	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
140	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
141	C₂×#117	—	13	2	no	—
142	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
143	#430	—	26	4	no	—
144	C₂×#108	—	13	4	no	—
145	#428	—	26	4	no	—
146	#427	—	13	4	no	—
147	#405	Dic₁₃	1	4	yes	complemented
148	#405	C₂×D₁₃	1	4	yes	—
149	C₄×M₁₆	Dic₁₃	1	4	yes	complemented
150	#344	—	13	8	no	—
151	H(3,ℤ/4)	—	13	8	no	—
152	C₂²×C₁₃	—	4	2	no	—
153	C₂²×C₁₃	—	4	1	no	—
154	C₂²×C₁₃	—	4	1	no	—
155	C₂²×C₁₃	—	4	2	no	—
156	C₂²×C₁₃	—	4	1	no	—
157	C₂²×C₁₃	#382	1	2	yes	—
158	C₂²×C₁₃	#382	1	2	yes	—
159	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
160	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
161	C₂²×C₁₃	C₂×#125	1	1	yes	characteristic
162	C₄×C₁₃	—	4	4	no	—
163	C₄×C₁₃	#382	1	2	yes	—
164	C₄×C₁₃	#382	1	2	yes	—
165	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
166	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
167	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
168	C₄×C₁₃	C₂×#125	1	1	yes	characteristic
169	Dic₁₃	—	8	4	no	—
170	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	13	4	no	—
174	C₂×M₁₆	—	26	2	no	—
175	C₂×M₁₆	—	26	2	no	—
176	C₂×M₁₆	—	26	2	no	—
177	C₂×M₁₆	—	26	2	no	—
178	C₂×M₁₆	—	26	2	no	—
179	C₂×M₁₆	Dic₂₆	1	4	yes	—
180	C₂×M₁₆	D₅₂	1	4	yes	—
181	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
182	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
183	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
184	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
185	C₂²×C₈	—	26	2	no	—
186	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
187	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
188	#117	—	2	1	no	—
189	#117	—	2	1	no	—
190	#117	—	13	4	no	—
191	#117	—	26	2	no	—
192	#117	—	26	2	no	—
193	#117	—	26	2	no	—
194	#117	C₂×Dic₁₃	1	2	yes	—
195	#117	C₂²×D₁₃	1	2	yes	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	26	2	no	—
198	C₂×C₂²⋊C₄	—	2	1	no	—
199	C₂×C₂²⋊C₄	—	26	2	no	—
200	C₂×C₄²	—	26	2	no	—
201	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
202	#108	—	26	8	no	—
203	#107	—	26	4	no	—
204	#106	—	26	4	no	—
205	#97	C₂×Dic₁₃	1	8	yes	complemented
206	C₄×C₈	C₂×Dic₁₃	1	8	yes	complemented
207	C₂×C₁₃	—	2	1	no	—
208	C₂×C₁₃	—	2	1	no	—
209	C₂×C₁₃	—	2	1	no	—
210	C₂×C₁₃	—	2	2	no	—
211	C₂×C₁₃	—	2	2	no	—
212	C₂×C₁₃	C₂×#382	1	1	yes	characteristic
213	C₂⁴	#6127	1	1	yes	characteristic
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	52	2	no	—
216	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
217	C₂²×C₄	C₂×D₅₂	1	2	yes	—
218	C₂²×C₄	#6127	1	1	yes	characteristic
219	C₂²×C₄	#6127	1	1	yes	characteristic
220	C₂²×C₄	#6127	1	1	yes	characteristic
221	C₂²×C₄	#6127	1	2	yes	—
222	C₂²×C₄	#6127	1	2	yes	—
223	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
224	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
225	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
226	M₁₆	—	2	4	no	—
227	M₁₆	—	2	2	no	—
228	M₁₆	—	2	2	no	—
229	M₁₆	—	26	4	no	—
230	M₁₆	—	26	4	no	—
231	M₁₆	—	52	2	no	—
232	M₁₆	—	52	2	no	—
233	M₁₆	C₄⋊Dic₁₃	1	8	yes	—
234	M₁₆	C₄⋊Dic₁₃	1	8	yes	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	26	4	no	—
237	C₂×C₈	—	52	2	no	—
238	C₂×C₈	—	52	2	no	—
239	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
240	C₂×C₈	C₂×D₅₂	1	4	yes	—
241	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
242	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	26	4	no	—
246	C₄⋊C₄	—	52	2	no	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₂²⋊C₄	—	26	4	no	—
250	C₂²⋊C₄	—	52	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	26	4	no	—
253	C₄²	—	52	2	no	—
254	C₄²	C₂²×Dic₁₃	1	4	yes	—
255	C₁₃	#20757	1	1	yes	characteristic, complemented, Sylow
256	C₂³	—	8	1	no	—
257	C₂³	#88124	1	2	yes	—
258	C₂³	#88129	1	2	yes	—
259	C₂³	#88180	1	1	yes	characteristic
260	C₂³	#88180	1	1	yes	characteristic
261	C₂³	C₂×#6127	1	1	yes	characteristic
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	8	2	no	—
264	C₂×C₄	—	52	4	no	—
265	C₂×C₄	—	104	4	no	—
266	C₂×C₄	#88124	1	2	yes	—
267	C₂×C₄	#88124	1	2	yes	—
268	C₂×C₄	#88124	1	2	yes	—
269	C₂×C₄	#88129	1	2	yes	—
270	C₂×C₄	#88129	1	2	yes	—
271	C₂×C₄	#88129	1	2	yes	—
272	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
273	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
274	C₂×C₄	#88180	1	2	yes	—
275	C₂×C₄	#88180	1	2	yes	—
276	C₂×C₄	#88180	1	1	yes	characteristic
277	C₂×C₄	#88180	1	1	yes	characteristic
278	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
279	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
280	C₂×C₄	C₂×#6127	1	1	yes	characteristic
281	C₂×C₄	C₂×#6127	1	1	yes	characteristic
282	C₂×C₄	C₂×#6127	1	1	yes	characteristic
283	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
284	C₈	—	2	4	no	—
285	C₈	—	52	4	no	—
286	C₈	C₂×C₄⋊Dic₁₃	1	8	yes	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	#11718597	1	2	yes	—
293	C₂²	#11718597	1	2	yes	—
294	C₂²	C₂×#88124	1	1	yes	characteristic
295	C₂²	C₂×#88129	1	1	yes	characteristic
296	C₂²	C₂×#88180	1	1	yes	characteristic
297	C₄	—	4	4	no	—
298	C₄	—	104	4	no	—
299	C₄	#11718597	1	2	yes	—
300	C₄	#11718597	1	2	yes	—
301	C₄	C₂×#88124	1	1	yes	characteristic
302	C₄	C₂×#88129	1	1	yes	characteristic
303	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
304	C₄	C₂×#88180	1	1	yes	characteristic
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	C₂×#11718597	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#88180
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#2873	C₂
socle	C₂×C₁₃	C₂×#382
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#2873	D₁₃
hypocenter	C₁₃	#20757
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×#117 ◁ #2873

Complements

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

N	H ≅ G/N	type
3. #13114960	C₂	semidirect
12. C₁₃×#405	C₄	semidirect
14. C₄×C₁₃×M₁₆	C₄	semidirect
53. C₁₃×#97	C₂×C₄	semidirect
54. C₄×C₈×C₁₃	C₂×C₄	semidirect
147. #405	Dic₁₃	semidirect
149. C₄×M₁₆	Dic₁₃	semidirect
205. #97	C₂×Dic₁₃	semidirect
206. C₄×C₈	C₂×Dic₁₃	semidirect
255. C₁₃	#20757	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718786	2
#13123426	2
#13114960	2
C₂×#11718148	2
C₂×#11718123	2
C₁₃×#2873	2
#20757	13