Subgroups — #15768118

Order 3328.

886 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718762	C₂	1	2	yes	maximal
2	C₂×#11718144	C₂	1	1	yes	maximal, characteristic
3	#13114553	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718115	C₂	1	2	yes	maximal
5	C₂²×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88180	C₂²	1	2	yes	—
10	C₂²×#6099	C₂²	1	2	yes	—
11	C₂×C₁₃×#97	—	2	4	no	—
12	C₂×C₁₃×#97	C₄	1	4	yes	complemented
13	#11718762	—	2	4	no	—
14	#11718762	C₄	1	8	yes	complemented
15	C₂×#88055	—	2	2	no	—
16	C₂×C₈×Dic₁₃	—	2	2	no	—
17	#11718144	C₄	1	4	yes	complemented
18	#11718143	C₄	1	4	yes	complemented
19	#11718115	C₄	1	8	yes	complemented
20	C₂³×C₄×C₁₃	—	2	1	no	—
21	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	—	2	2	no	—
23	C₂²×C₈×C₁₃	—	2	2	no	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₃	D₄	1	2	yes	—
30	C₂²×C₈×C₁₃	D₄	1	4	yes	—
31	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
32	C₂×C₄²×C₁₃	—	2	2	no	—
33	C₂×C₄²×C₁₃	—	2	1	no	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
37	#88180	—	2	2	no	—
38	#88180	—	2	2	no	—
39	#88180	C₂×C₄	1	4	yes	—
40	#88180	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
42	C₂×C₄×Dic₁₃	—	2	2	no	—
43	C₂×#6099	—	2	2	no	—
44	C₂×#6099	—	2	2	no	—
45	C₂×#6099	—	2	2	no	—
46	C₂×#6099	C₂×C₄	1	4	yes	—
47	C₂×#6099	C₂×C₄	1	4	yes	—
48	C₂×#6099	C₂×C₄	1	4	yes	—
49	C₂²×#1055	—	2	2	no	—
50	C₁₃×#97	—	2	8	no	—
51	#88055	—	2	8	no	—
52	C₈×Dic₁₃	—	2	8	no	—
53	#10707	—	13	1	no	maximal, Sylow
54	C₂⁴×C₁₃	#41	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₃	—	2	4	no	—
57	C₂²×C₄×C₁₃	—	4	4	no	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
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	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	#41	1	2	yes	—
68	C₂²×Dic₁₃	—	2	4	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	—	2	4	no	—
72	C₂×C₈×C₁₃	—	2	4	no	—
73	C₂×C₈×C₁₃	C₄²	1	4	yes	—
74	C₂×C₈×C₁₃	C₄²	1	4	yes	—
75	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
76	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
77	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
78	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
79	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
80	C₄²×C₁₃	—	2	2	no	—
81	C₄²×C₁₃	—	2	4	no	—
82	C₄²×C₁₃	C₄²	1	4	yes	—
83	C₂²⋊Dic₁₃	—	2	4	no	—
84	C₂²⋊Dic₁₃	—	2	4	no	—
85	C₄⋊Dic₁₃	—	2	4	no	—
86	C₄⋊Dic₁₃	C₄²	1	8	yes	—
87	C₄×Dic₁₃	—	2	4	no	—
88	C₄×Dic₁₃	—	2	4	no	—
89	#6099	—	2	4	no	—
90	#6099	—	2	4	no	—
91	#6099	—	2	4	no	—
92	#6099	C₄²	1	8	yes	—
93	C₂×#1055	—	2	4	no	—
94	C₂×#1055	—	2	4	no	—
95	C₂×#1055	—	2	4	no	—
96	C₂×#405	—	13	2	no	—
97	C₂²×#97	D₁₃	1	1	yes	characteristic
98	C₂×#341	—	13	1	no	—
99	#1735	—	13	1	no	—
100	C₂×#339	—	13	2	no	—
101	C₂³×C₁₃	—	2	4	no	—
102	C₂³×C₁₃	#117	1	2	yes	—
103	C₂³×C₁₃	#117	1	1	yes	characteristic
104	C₂³×C₁₃	C₄×D₄	1	2	yes	—
105	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
106	C₂³×C₁₃	C₄×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	4	8	no	—
109	C₂×C₄×C₁₃	—	4	4	no	—
110	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#117	1	2	yes	—
120	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
128	C₂×Dic₁₃	—	2	8	no	—
129	C₂×Dic₁₃	—	4	8	no	—
130	C₈×C₁₃	—	2	4	no	—
131	C₈×C₁₃	#95	1	8	yes	—
132	#1055	—	2	8	no	—
133	C₂²×M₁₆	—	13	2	no	—
134	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
135	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
136	C₂×#117	—	13	2	no	—
137	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
138	#405	—	13	8	no	—
139	#405	—	26	4	no	—
140	C₂×#97	—	2	4	no	—
141	C₂×#97	—	26	2	no	—
142	C₂×#97	Dic₁₃	1	4	yes	complemented
143	C₂×C₄×C₈	—	26	2	no	—
144	#341	—	13	4	no	—
145	#340	—	13	4	no	—
146	#339	—	13	8	no	—
147	C₂²×C₁₃	—	2	4	no	—
148	C₂²×C₁₃	—	2	4	no	—
149	C₂²×C₁₃	—	2	2	no	—
150	C₂²×C₁₃	—	2	2	no	—
151	C₂²×C₁₃	#376	1	1	yes	characteristic
152	C₂²×C₁₃	C₄×C₂²⋊C₄	1	2	yes	—
153	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
154	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
155	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
156	C₂²×C₁₃	#444	1	4	yes	—
157	C₄×C₁₃	—	4	4	no	—
158	C₄×C₁₃	C₄³	1	1	yes	characteristic
159	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
160	C₄×C₁₃	C₄×C₂²⋊C₄	1	2	yes	—
161	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
162	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
163	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
164	Dic₁₃	—	4	8	no	—
165	C₂³×C₄	—	2	1	no	—
166	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
167	C₂×M₁₆	—	13	4	no	—
168	C₂×M₁₆	—	13	4	no	—
169	C₂×M₁₆	—	13	4	no	—
170	C₂×M₁₆	—	26	2	no	—
171	C₂×M₁₆	—	26	2	no	—
172	C₂×M₁₆	—	26	2	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	26	2	no	—
176	C₂²×C₈	Dic₂₆	1	2	yes	—
177	C₂²×C₈	C₄×D₁₃	1	2	yes	—
178	C₂²×C₈	C₄×D₁₃	1	2	yes	—
179	C₂²×C₈	C₄×D₁₃	1	4	yes	—
180	C₂²×C₈	D₅₂	1	2	yes	—
181	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
182	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
183	C₂²×C₈	C₂²⋊D₁₃	1	4	yes	—
184	#117	—	13	4	no	—
185	#117	—	13	4	no	—
186	#117	—	26	2	no	—
187	#117	—	26	2	no	—
188	C₂×C₄⋊C₄	—	13	4	no	—
189	C₂×C₂²⋊C₄	—	26	2	no	—
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	26	2	no	—
193	C₂×C₄²	C₄×D₁₃	1	4	yes	—
194	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
195	#97	—	2	8	no	—
196	#97	—	26	8	no	—
197	C₄×C₈	—	26	8	no	—
198	C₂×C₁₃	—	2	1	no	—
199	C₂×C₁₃	—	2	1	no	—
200	C₂×C₁₃	—	2	2	no	—
201	C₂×C₁₃	C₄×#95	1	1	yes	characteristic
202	C₂×C₁₃	#1765	1	4	yes	—
203	C₂×C₁₃	#1773	1	2	yes	—
204	C₂⁴	#6127	1	1	yes	characteristic
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	4	4	no	—
208	C₂²×C₄	—	26	4	no	—
209	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
211	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
213	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
215	C₂²×C₄	#6127	1	1	yes	characteristic
216	C₂²×C₄	#6127	1	2	yes	—
217	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
219	M₁₆	—	13	8	no	—
220	M₁₆	—	26	4	no	—
221	M₁₆	—	26	4	no	—
222	M₁₆	—	26	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	26	4	no	—
228	C₂×C₈	—	26	4	no	—
229	C₂×C₈	—	26	4	no	—
230	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
231	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
232	C₂×C₈	#6101	1	8	yes	—
233	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
234	C₂×C₈	#6103	1	4	yes	—
235	C₂×C₈	#6103	1	4	yes	—
236	C₂×C₈	C₂²⋊Dic₁₃	1	4	yes	—
237	C₄⋊C₄	—	13	8	no	—
238	C₄⋊C₄	—	26	4	no	—
239	C₂²⋊C₄	—	26	4	no	—
240	C₂²⋊C₄	—	26	4	no	—
241	C₄²	—	2	4	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	26	4	no	—
244	C₄²	—	26	4	no	—
245	C₄²	C₄×Dic₁₃	1	4	yes	—
246	C₁₃	#10707	1	1	yes	characteristic, complemented, Sylow
247	C₂³	—	2	4	no	—
248	C₂³	C₄×Dic₂₆	1	1	yes	characteristic
249	C₂³	#88126	1	2	yes	—
250	C₂³	C₄×D₅₂	1	1	yes	characteristic
251	C₂³	#88180	1	1	yes	characteristic
252	C₂³	C₄×C₂²⋊D₁₃	1	2	yes	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	4	8	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	26	8	no	—
257	C₂×C₄	—	52	8	no	—
258	C₂×C₄	C₄×Dic₂₆	1	1	yes	characteristic
259	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
260	C₂×C₄	C₄²×D₁₃	1	2	yes	—
261	C₂×C₄	C₄²×D₁₃	1	2	yes	—
262	C₂×C₄	#88126	1	2	yes	—
263	C₂×C₄	C₄×D₅₂	1	2	yes	—
264	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
266	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
268	C₂×C₄	C₂×#6101	1	2	yes	—
269	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
270	C₂×C₄	#88180	1	1	yes	characteristic
271	C₂×C₄	C₂×#6103	1	2	yes	—
272	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
273	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
274	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
275	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	—	26	8	no	—
278	C₈	#88071	1	8	yes	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	#11718378	1	4	yes	—
284	C₂²	C₄×#6101	1	2	yes	—
285	C₂²	#11718593	1	1	yes	characteristic
286	C₂²	C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
287	C₂²	C₄×#6103	1	2	yes	—
288	C₂²	C₄×C₂²⋊Dic₁₃	1	1	yes	characteristic
289	C₄	—	4	4	no	—
290	C₄	—	52	8	no	—
291	C₄	C₄²×Dic₁₃	1	1	yes	characteristic
292	C₄	C₄×#6101	1	2	yes	—
293	C₄	C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
294	C₄	C₄×#6103	1	2	yes	—
295	C₄	C₂×#88071	1	1	yes	characteristic
296	C₄	C₄×C₂²⋊Dic₁₃	1	1	yes	characteristic
297	C₂	—	2	1	no	—
298	C₂	—	2	2	no	—
299	C₂	—	2	1	no	—
300	C₂	C₄×#88071	1	1	yes	characteristic
301	C₂	#13114680	1	4	yes	—
302	C₂	#13114720	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₄×C₁₃	C₄³
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂²×C₁₃×#97	C₂
socle	C₂³×C₁₃	C₄×D₄
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂²×#97	D₁₃
hypocenter	C₁₃	#10707
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
12. C₂×C₁₃×#97	C₄	semidirect
14. #11718762	C₄	semidirect
17. #11718144	C₄	semidirect
18. #11718143	C₄	semidirect
19. #11718115	C₄	semidirect
142. C₂×#97	Dic₁₃	semidirect
246. C₁₃	#10707	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718762	2
C₂×#11718144	2
#13114553	2
C₂×#11718115	2
C₂²×C₁₃×#97	2
#10707	13