Subgroups — #15860938

Order 3328.

1018 conjugacy classes of subgroups.

The subgroup lattice has 302 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	C₂×#11718762	C₂	1	2	yes	maximal
4	#13114720	C₂	1	4	yes	maximal, complemented
5	#13114715	C₂	1	4	yes	maximal, complemented
6	#13114546	C₂	1	2	yes	maximal
7	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
11	C₂×#88180	C₂²	1	2	yes	—
12	C₂²×#6099	C₂²	1	2	yes	—
13	C₁₃×#405	C₄	1	8	yes	complemented
14	C₂×C₁₃×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
16	#11718786	C₄	1	8	yes	complemented
17	#11718782	C₄	1	8	yes	complemented
18	C₂×#88059	C₂²	1	4	yes	—
19	#11718767	C₂²	1	4	yes	—
20	#11718762	—	2	2	no	—
21	#11718762	—	2	2	no	—
22	#11718762	C₂²	1	4	yes	—
23	#11718762	C₂²	1	4	yes	—
24	C₂×#88055	—	2	2	no	—
25	C₂×C₈×Dic₁₃	—	2	2	no	—
26	#11718115	C₂²	1	8	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
29	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
30	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₁₃	D₄	1	2	yes	—
33	C₂²×C₈×C₁₃	D₄	1	2	yes	—
34	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₁₃×#117	—	2	2	no	—
39	C₁₃×#117	C₂×C₄	1	4	yes	—
40	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
41	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
42	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
43	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
44	#88180	—	2	2	no	—
45	#88180	—	2	2	no	—
46	#88180	C₂×C₄	1	4	yes	—
47	#88180	C₂³	1	4	yes	—
48	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
49	C₂×C₄×Dic₁₃	—	2	2	no	—
50	C₂×#6099	—	2	2	no	—
51	C₂×#6099	—	2	4	no	—
52	C₂×#6099	C₂×C₄	1	8	yes	—
53	C₂×#6099	C₂³	1	4	yes	—
54	C₂²×#1055	—	2	2	no	—
55	C₁₃×#97	C₂×C₄	1	8	yes	complemented
56	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	complemented
57	#88059	C₂×C₄	1	16	yes	complemented
58	#88058	C₂×C₄	1	8	yes	—
59	#88057	C₂×C₄	1	8	yes	—
60	#88055	—	2	4	no	—
61	#88055	—	2	4	no	—
62	C₈×Dic₁₃	—	2	4	no	—
63	C₈×Dic₁₃	—	2	4	no	—
64	#20444	—	13	1	no	maximal, Sylow
65	C₂⁴×C₁₃	#41	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	#41	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₂²×Dic₁₃	—	2	4	no	—
76	C₁₃×M₁₆	C₄²	1	16	yes	—
77	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
78	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
88	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
89	C₁₃×C₄⋊C₄	—	2	4	no	—
90	C₁₃×C₂²⋊C₄	—	2	4	no	—
91	C₄²×C₁₃	—	2	2	no	—
92	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
93	C₂²⋊Dic₁₃	—	2	4	no	—
94	C₂²⋊Dic₁₃	—	2	4	no	—
95	C₄⋊Dic₁₃	—	2	4	no	—
96	C₄⋊Dic₁₃	C₂²×C₄	1	8	yes	—
97	C₄×Dic₁₃	—	2	4	no	—
98	C₄×Dic₁₃	—	2	4	no	—
99	#6099	—	2	8	no	—
100	#6099	—	2	4	no	—
101	#6099	C₂²×C₄	1	8	yes	—
102	C₂×#1055	—	2	8	no	—
103	C₂×#1055	—	2	4	no	—
104	C₂×#430	—	13	1	no	—
105	#2911	—	13	1	no	—
106	C₂×#405	—	13	2	no	—
107	C₂×#405	D₁₃	1	1	yes	characteristic
108	#1773	—	13	4	no	—
109	#1772	—	13	4	no	—
110	#1733	—	13	2	no	—
111	C₂³×C₁₃	—	4	2	no	—
112	C₂³×C₁₃	#117	1	2	yes	—
113	C₂³×C₁₃	C₄×D₄	1	2	yes	—
114	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
115	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	—	4	4	no	—
118	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
119	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
120	C₂×C₄×C₁₃	#117	1	2	yes	—
121	C₂×C₄×C₁₃	C₄×D₄	1	4	yes	—
122	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
124	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
125	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
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×Dic₁₃	—	2	8	no	—
132	C₂×Dic₁₃	—	4	8	no	—
133	C₈×C₁₃	C₂×C₄²	1	8	yes	—
134	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
135	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
136	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
137	#1055	—	2	8	no	—
138	C₂²×M₁₆	—	13	2	no	—
139	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
140	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
141	C₂×#117	—	13	2	no	—
142	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
143	#430	—	13	8	no	—
144	C₂×#108	—	13	4	no	—
145	#428	—	13	8	no	—
146	#427	—	13	4	no	—
147	#405	—	13	4	no	—
148	#405	—	13	4	no	—
149	#405	—	26	2	no	—
150	#405	—	26	2	no	—
151	#405	Dic₁₃	1	8	yes	complemented
152	C₂×#97	—	26	2	no	—
153	C₂×#97	C₂×D₁₃	1	2	yes	—
154	C₂×C₄×C₈	—	26	2	no	—
155	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
156	#339	—	13	8	no	—
157	C₂²×C₁₃	—	2	4	no	—
158	C₂²×C₁₃	—	2	2	no	—
159	C₂²×C₁₃	—	4	2	no	—
160	C₂²×C₁₃	—	4	2	no	—
161	C₂²×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
162	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
163	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
164	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
165	C₄×C₁₃	—	4	4	no	—
166	C₄×C₁₃	C₄×C₄⋊C₄	1	4	yes	—
167	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
168	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
169	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
170	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
171	Dic₁₃	—	4	8	no	—
172	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
173	C₂×M₁₆	—	13	8	no	—
174	C₂×M₁₆	—	13	4	no	—
175	C₂×M₁₆	—	26	2	no	—
176	C₂×M₁₆	—	26	4	no	—
177	C₂×M₁₆	C₄×D₁₃	1	4	yes	—
178	C₂×M₁₆	C₄×D₁₃	1	4	yes	—
179	C₂×M₁₆	C₂×Dic₁₃	1	4	yes	—
180	C₂²×C₈	—	26	2	no	—
181	C₂²×C₈	Dic₂₆	1	4	yes	—
182	C₂²×C₈	D₅₂	1	2	yes	—
183	C₂²×C₈	D₅₂	1	2	yes	—
184	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
185	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
187	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
188	#117	—	2	2	no	—
189	#117	—	13	4	no	—
190	#117	—	13	4	no	—
191	#117	—	26	2	no	—
192	#117	—	26	2	no	—
193	#117	C₂×Dic₁₃	1	4	yes	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	13	4	no	—
196	C₂×C₂²⋊C₄	—	2	1	no	—
197	C₂×C₂²⋊C₄	—	26	2	no	—
198	C₂×C₄²	—	26	2	no	—
199	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
200	#108	—	13	16	no	—
201	#107	—	13	8	no	—
202	#106	—	13	8	no	—
203	#97	—	26	4	no	—
204	#97	—	26	4	no	—
205	#97	C₂×Dic₁₃	1	8	yes	complemented
206	C₄×C₈	—	26	4	no	—
207	C₄×C₈	—	26	4	no	—
208	C₄×C₈	C₂×Dic₁₃	1	8	yes	complemented
209	C₂×C₁₃	—	2	1	no	—
210	C₂×C₁₃	—	2	1	no	—
211	C₂×C₁₃	—	2	4	no	—
212	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
213	C₂×C₁₃	#2955	1	2	yes	—
214	C₂⁴	#6127	1	1	yes	characteristic
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	26	4	no	—
217	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
218	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
219	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
220	C₂²×C₄	C₂×D₅₂	1	2	yes	—
221	C₂²×C₄	#6127	1	1	yes	characteristic
222	C₂²×C₄	#6127	1	2	yes	—
223	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
224	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
225	M₁₆	—	13	8	no	—
226	M₁₆	—	26	8	no	—
227	M₁₆	—	26	4	no	—
228	M₁₆	C₄×Dic₁₃	1	16	yes	—
229	C₂×C₈	—	26	8	no	—
230	C₂×C₈	—	26	4	no	—
231	C₂×C₈	C₄⋊Dic₁₃	1	8	yes	—
232	C₂×C₈	C₄⋊Dic₁₃	1	8	yes	—
233	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
234	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
235	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
236	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
237	C₂×C₈	C₂×D₅₂	1	2	yes	—
238	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
239	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
240	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
241	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
242	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	13	8	no	—
245	C₄⋊C₄	—	26	4	no	—
246	C₂²⋊C₄	—	2	4	no	—
247	C₂²⋊C₄	—	26	4	no	—
248	C₂²⋊C₄	—	26	4	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	26	4	no	—
251	C₄²	—	26	4	no	—
252	C₄²	C₂²×Dic₁₃	1	4	yes	—
253	C₁₃	#20444	1	1	yes	characteristic, complemented, Sylow
254	C₂³	—	4	2	no	—
255	C₂³	C₄×Dic₂₆	1	2	yes	—
256	C₂³	C₄×D₅₂	1	2	yes	—
257	C₂³	#88180	1	2	yes	—
258	C₂³	C₂×#6127	1	1	yes	characteristic
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	26	8	no	—
262	C₂×C₄	—	52	8	no	—
263	C₂×C₄	C₄×Dic₂₆	1	4	yes	—
264	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
265	C₂×C₄	C₄×D₅₂	1	4	yes	—
266	C₂×C₄	C₄×D₅₂	1	2	yes	—
267	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
268	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
269	C₂×C₄	#88180	1	2	yes	—
270	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
272	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
273	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
274	C₂×C₄	C₂×#6127	1	1	yes	characteristic
275	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
276	C₈	—	26	8	no	—
277	C₈	C₂×C₄×Dic₁₃	1	8	yes	—
278	C₈	C₂×C₄⋊Dic₁₃	1	4	yes	—
279	C₈	C₂×C₄⋊Dic₁₃	1	2	yes	—
280	C₈	C₂×C₄⋊Dic₁₃	1	2	yes	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	C₄×C₄⋊Dic₁₃	1	4	yes	—
286	C₂²	C₂×C₄×Dic₂₆	1	1	yes	characteristic
287	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
288	C₂²	C₂×#88180	1	1	yes	characteristic
289	C₄	—	4	4	no	—
290	C₄	—	52	8	no	—
291	C₄	C₄×C₄⋊Dic₁₃	1	4	yes	—
292	C₄	C₂×C₄×Dic₂₆	1	1	yes	characteristic
293	C₄	C₂×C₄×D₅₂	1	1	yes	characteristic
294	C₄	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
295	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
296	C₂	—	2	1	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	4	no	—
299	C₂	C₂×C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
300	C₂	#13123740	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₅₂
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#405	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#405	D₁₃
hypocenter	C₁₃	#20444
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
4. #13114720	C₂	semidirect
5. #13114715	C₂	semidirect
13. C₁₃×#405	C₄	semidirect
16. #11718786	C₄	semidirect
17. #11718782	C₄	semidirect
55. C₁₃×#97	C₂×C₄	semidirect
56. C₄×C₈×C₁₃	C₂×C₄	semidirect
57. #88059	C₂×C₄	semidirect
151. #405	Dic₁₃	semidirect
205. #97	C₂×Dic₁₃	semidirect
208. C₄×C₈	C₂×Dic₁₃	semidirect
253. C₁₃	#20444	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
C₂×#11718762	2
#13114720	2
#13114715	2
#13114546	2
C₂×C₁₃×#405	2
#20444	13