Subgroups — #15768637

Order 3328.

734 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718144	C₂	1	1	yes	maximal, characteristic
2	#13114553	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718115	C₂	1	2	yes	maximal
4	#13114532	C₂	1	2	yes	maximal
5	C₂²×C₄×C₈×C₁₃	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×C₄×C₈×C₁₃	—	2	4	no	—
12	C₂×C₄×C₈×C₁₃	C₄	1	4	yes	complemented
13	#11718144	C₄	1	4	yes	complemented
14	#11718143	C₄	1	4	yes	complemented
15	#11718142	—	2	4	no	—
16	#11718115	—	2	4	no	—
17	C₂³×C₄×C₁₃	—	2	1	no	—
18	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	—	2	2	no	—
20	C₂²×C₈×C₁₃	—	2	2	no	—
21	C₂²×C₈×C₁₃	—	2	2	no	—
22	C₂²×C₈×C₁₃	—	2	2	no	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
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	2	yes	—
27	C₂²×C₈×C₁₃	D₄	1	2	yes	—
28	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
29	C₂×C₄²×C₁₃	—	2	2	no	—
30	C₂×C₄²×C₁₃	—	2	1	no	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₃	D₄	1	4	yes	—
33	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
34	#88180	—	2	2	no	—
35	#88180	—	2	2	no	—
36	#88180	—	2	2	no	—
37	#88180	—	2	2	no	—
38	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₁₃	—	2	2	no	—
40	C₂×#6099	—	2	2	no	—
41	C₂×#6099	—	2	2	no	—
42	C₂×#6099	—	2	2	no	—
43	C₂×#6099	—	2	2	no	—
44	C₂×#6099	—	2	2	no	—
45	C₂×#6099	C₂×C₄	1	4	yes	—
46	C₂²×#1055	—	2	2	no	—
47	C₄×C₈×C₁₃	—	2	8	no	—
48	#10818	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	#41	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	—	2	8	no	—
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	—	2	4	no	—
53	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	#41	1	2	yes	—
60	C₂²×C₄×C₁₃	#41	1	2	yes	—
61	C₂²×C₄×C₁₃	#41	1	2	yes	—
62	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
63	C₂²×Dic₁₃	—	2	4	no	—
64	C₂×C₈×C₁₃	—	2	2	no	—
65	C₂×C₈×C₁₃	—	2	2	no	—
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	—	2	4	no	—
69	C₂×C₈×C₁₃	—	2	4	no	—
70	C₂×C₈×C₁₃	—	2	4	no	—
71	C₂×C₈×C₁₃	—	2	4	no	—
72	C₂×C₈×C₁₃	C₄²	1	4	yes	—
73	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
74	C₄²×C₁₃	—	2	2	no	—
75	C₄²×C₁₃	—	2	4	no	—
76	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
77	C₂²⋊Dic₁₃	—	4	2	no	—
78	C₂²⋊Dic₁₃	—	4	2	no	—
79	C₄⋊Dic₁₃	—	2	4	no	—
80	C₄⋊Dic₁₃	—	2	4	no	—
81	C₄×Dic₁₃	—	4	2	no	—
82	C₄×Dic₁₃	—	4	2	no	—
83	#6099	—	2	4	no	—
84	#6099	—	2	4	no	—
85	#6099	—	4	2	no	—
86	#6099	—	4	2	no	—
87	C₂×#1055	—	2	4	no	—
88	C₂×#1055	—	4	2	no	—
89	C₂×#1055	—	4	2	no	—
90	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
91	C₂×#341	—	13	1	no	—
92	#1735	—	13	1	no	—
93	C₂×#339	—	13	2	no	—
94	#1729	—	13	2	no	—
95	C₂³×C₁₃	—	2	4	no	—
96	C₂³×C₁₃	#117	1	1	yes	characteristic
97	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
98	C₂³×C₁₃	C₄×Q₈	1	1	yes	characteristic
99	C₂³×C₁₃	#123	1	2	yes	—
100	C₂³×C₁₃	#126	1	2	yes	—
101	C₂×C₄×C₁₃	—	2	16	no	—
102	C₂×C₄×C₁₃	—	2	8	no	—
103	C₂×C₄×C₁₃	—	2	4	no	—
104	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#117	1	2	yes	—
110	C₂×C₄×C₁₃	#117	1	2	yes	—
111	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	#121	1	2	yes	—
117	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₁₃	#123	1	2	yes	—
119	C₂×C₄×C₁₃	#124	1	2	yes	—
120	C₂×C₄×C₁₃	#125	1	2	yes	—
121	C₂×C₄×C₁₃	#126	1	2	yes	—
122	C₂×Dic₁₃	—	4	4	no	—
123	C₂×Dic₁₃	—	4	8	no	—
124	C₈×C₁₃	—	2	4	no	—
125	C₈×C₁₃	—	2	4	no	—
126	#1055	—	4	4	no	—
127	C₂²×M₁₆	—	13	2	no	—
128	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
129	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
130	C₂×#117	—	13	2	no	—
131	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
132	C₂×C₄×C₈	—	2	4	no	—
133	C₂×C₄×C₈	Dic₁₃	1	4	yes	complemented
134	#341	—	13	4	no	—
135	#340	—	13	4	no	—
136	#339	—	26	4	no	—
137	#336	—	26	4	no	—
138	C₂²×C₁₃	—	2	4	no	—
139	C₂²×C₁₃	—	2	4	no	—
140	C₂²×C₁₃	—	2	2	no	—
141	C₂²×C₁₃	—	2	2	no	—
142	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₁₃	#381	1	1	yes	characteristic
144	C₂²×C₁₃	#383	1	1	yes	characteristic
145	C₂²×C₁₃	#387	1	2	yes	—
146	C₂²×C₁₃	#388	1	2	yes	—
147	C₂²×C₁₃	#443	1	4	yes	—
148	C₄×C₁₃	—	2	8	no	—
149	C₄×C₁₃	#376	1	1	yes	characteristic
150	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
151	C₄×C₁₃	#380	1	1	yes	characteristic
152	C₄×C₁₃	#382	1	1	yes	characteristic
153	C₄×C₁₃	#387	1	2	yes	—
154	C₄×C₁₃	#388	1	2	yes	—
155	Dic₁₃	—	8	4	no	—
156	C₂³×C₄	—	2	1	no	—
157	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
158	C₂×M₁₆	—	13	4	no	—
159	C₂×M₁₆	—	26	2	no	—
160	C₂×M₁₆	—	26	2	no	—
161	C₂×M₁₆	—	26	2	no	—
162	C₂×M₁₆	—	26	2	no	—
163	C₂×M₁₆	—	26	2	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	26	2	no	—
169	C₂²×C₈	Dic₂₆	1	2	yes	—
170	C₂²×C₈	C₄×D₁₃	1	2	yes	—
171	C₂²×C₈	C₄×D₁₃	1	2	yes	—
172	C₂²×C₈	D₅₂	1	2	yes	—
173	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
174	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
175	#117	—	26	2	no	—
176	#117	—	26	2	no	—
177	#117	—	26	2	no	—
178	#117	—	26	2	no	—
179	C₂×C₄⋊C₄	—	13	4	no	—
180	C₂×C₂²⋊C₄	—	26	2	no	—
181	C₂×C₄²	—	2	2	no	—
182	C₂×C₄²	—	2	1	no	—
183	C₂×C₄²	—	26	2	no	—
184	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
185	C₂×C₄²	C₂²⋊D₁₃	1	4	yes	—
186	C₄×C₈	—	2	8	no	—
187	C₂×C₁₃	—	2	1	no	—
188	C₂×C₁₃	—	2	1	no	—
189	C₂×C₁₃	—	2	2	no	—
190	C₂×C₁₃	#1434	1	1	yes	characteristic
191	C₂×C₁₃	#1772	1	2	yes	—
192	C₂×C₁₃	#1937	1	4	yes	—
193	C₂⁴	#6127	1	1	yes	characteristic
194	C₂²×C₄	—	2	8	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	26	4	no	—
198	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
202	C₂²×C₄	#6127	1	1	yes	characteristic
203	C₂²×C₄	#6127	1	2	yes	—
204	C₂²×C₄	#6127	1	2	yes	—
205	C₂²×C₄	#6127	1	2	yes	—
206	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
208	M₁₆	—	26	4	no	—
209	M₁₆	—	26	4	no	—
210	M₁₆	—	52	2	no	—
211	M₁₆	—	52	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	26	4	no	—
221	C₂×C₈	—	52	2	no	—
222	C₂×C₈	—	52	2	no	—
223	C₂×C₈	C₄×Dic₁₃	1	4	yes	—
224	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
225	C₄⋊C₄	—	26	4	no	—
226	C₄⋊C₄	—	26	4	no	—
227	C₂²⋊C₄	—	52	2	no	—
228	C₂²⋊C₄	—	52	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	4	no	—
231	C₄²	—	52	2	no	—
232	C₄²	—	52	2	no	—
233	C₄²	C₂²⋊Dic₁₃	1	4	yes	—
234	C₁₃	#10818	1	1	yes	characteristic, complemented, Sylow
235	C₂³	—	2	4	no	—
236	C₂³	C₄×Dic₂₆	1	1	yes	characteristic
237	C₂³	C₄×D₅₂	1	1	yes	characteristic
238	C₂³	#88130	1	2	yes	—
239	C₂³	#88180	1	1	yes	characteristic
240	C₂³	#88183	1	2	yes	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	16	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	52	4	no	—
245	C₂×C₄	—	52	8	no	—
246	C₂×C₄	C₄×Dic₂₆	1	1	yes	characteristic
247	C₂×C₄	#88124	1	2	yes	—
248	C₂×C₄	#88126	1	2	yes	—
249	C₂×C₄	#88126	1	2	yes	—
250	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
251	C₂×C₄	#88129	1	2	yes	—
252	C₂×C₄	#88130	1	2	yes	—
253	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
254	C₂×C₄	#88178	1	2	yes	—
255	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
256	C₂×C₄	#88180	1	1	yes	characteristic
257	C₂×C₄	#88180	1	1	yes	characteristic
258	C₂×C₄	#88180	1	1	yes	characteristic
259	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
260	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
261	C₂×C₄	#88183	1	2	yes	—
262	C₂×C₄	#88184	1	2	yes	—
263	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
264	C₈	—	2	4	no	—
265	C₈	—	2	4	no	—
266	C₈	—	52	4	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	#11718361	1	4	yes	—
272	C₂²	#11718594	1	2	yes	—
273	C₂²	C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
274	C₂²	#11718598	1	1	yes	characteristic
275	C₂²	#11718601	1	2	yes	—
276	C₂²	#11718872	1	1	yes	characteristic
277	C₄	—	2	8	no	—
278	C₄	—	104	4	no	—
279	C₄	#11718593	1	1	yes	characteristic
280	C₄	#11718594	1	2	yes	—
281	C₄	#11718597	1	1	yes	characteristic
282	C₄	#11718601	1	2	yes	—
283	C₄	C₄×C₂²⋊Dic₁₃	1	1	yes	characteristic
284	C₄	#11718873	1	1	yes	characteristic
285	C₂	—	2	1	no	—
286	C₂	—	2	2	no	—
287	C₂	—	2	1	no	—
288	C₂	#13113063	1	1	yes	characteristic
289	C₂	#13114715	1	2	yes	—
290	C₂	#13115540	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₄×D₄
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	#10818
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
12. C₂×C₄×C₈×C₁₃	C₄	semidirect
13. #11718144	C₄	semidirect
14. #11718143	C₄	semidirect
133. C₂×C₄×C₈	Dic₁₃	semidirect
234. C₁₃	#10818	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718144	2
#13114553	2
C₂×#11718115	2
#13114532	2
C₂²×C₄×C₈×C₁₃	2
#10818	13