Subgroups — #15866286

Order 3328.

982 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718767	C₂	1	1	yes	maximal, characteristic
2	C₂²×#88058	C₂	1	1	yes	maximal, characteristic
3	#13117456	C₂	1	2	yes	maximal
4	#13115579	C₂	1	4	yes	maximal, complemented
5	#13115569	C₂	1	4	yes	maximal, complemented
6	#13114556	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×C₄⋊Dic₁₃	C₂²	1	2	yes	—
13	C₁₃×#405	—	2	4	no	—
14	C₂×C₁₃×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
16	#11718767	C₄	1	8	yes	complemented
17	C₂×#88058	—	2	4	no	—
18	C₂×#88058	C₂²	1	2	yes	—
19	C₂×#88058	C₂²	1	2	yes	—
20	C₂×#88058	C₂²	1	2	yes	—
21	C₂×#88057	C₂²	1	2	yes	—
22	#11718638	—	2	2	no	—
23	#11718637	—	2	2	no	—
24	#11718607	—	2	4	no	—
25	#11718589	C₂²	1	8	yes	—
26	#11718143	C₂²	1	8	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂³×Dic₁₃	—	2	2	no	—
29	C₂×C₁₃×M₁₆	—	2	2	no	—
30	C₂×C₁₃×M₁₆	—	2	2	no	—
31	C₂×C₁₃×M₁₆	Q₈	1	4	yes	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₃	D₄	1	4	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	—	2	2	no	—
40	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
41	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
42	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
44	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
45	#88180	—	2	4	no	—
46	#88180	C₂×C₄	1	8	yes	—
47	C₂×C₄⋊Dic₁₃	—	2	4	no	—
48	C₂×C₄⋊Dic₁₃	—	2	4	no	—
49	C₂×C₄⋊Dic₁₃	—	2	2	no	—
50	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
51	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
52	C₂×#6101	—	2	8	no	—
53	C₂×C₄×Dic₁₃	—	2	2	no	—
54	C₁₃×#97	—	2	4	no	—
55	C₄×C₈×C₁₃	—	2	4	no	—
56	#88071	—	2	8	no	—
57	#88058	—	2	4	no	—
58	#88058	—	2	4	no	—
59	#88058	C₂×C₄	1	8	yes	complemented
60	#88057	C₂×C₄	1	8	yes	complemented
61	#21156	—	13	1	no	maximal, Sylow
62	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₁₃	#41	1	2	yes	—
71	C₂²×C₄×C₁₃	#41	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₁₃	—	2	4	no	—
74	C₂²×Dic₁₃	—	2	4	no	—
75	C₂²×Dic₁₃	—	4	4	no	—
76	C₂²×Dic₁₃	—	8	4	no	—
77	C₁₃×M₁₆	—	2	8	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	C₄⋊C₄	1	16	yes	complemented
81	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
86	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
87	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₂²⋊C₄	—	2	4	no	—
91	C₄²×C₁₃	—	2	2	no	—
92	C₄²×C₁₃	—	2	2	no	—
93	C₂²⋊Dic₁₃	—	2	8	no	—
94	C₄⋊Dic₁₃	—	2	4	no	—
95	C₄⋊Dic₁₃	—	2	4	no	—
96	C₄⋊Dic₁₃	—	2	4	no	—
97	C₄⋊Dic₁₃	—	4	4	no	—
98	C₄⋊Dic₁₃	C₂²×C₄	1	8	yes	—
99	#6101	—	8	8	no	—
100	C₄×Dic₁₃	—	2	8	no	—
101	C₂×#427	—	13	1	no	—
102	C₂²×#107	—	13	1	no	—
103	C₂×#405	D₁₃	1	1	yes	characteristic
104	#2317	—	13	2	no	—
105	#1944	—	13	4	no	—
106	#1942	—	13	4	no	—
107	#1736	—	13	2	no	—
108	C₂³×C₁₃	—	2	2	no	—
109	C₂³×C₁₃	—	2	2	no	—
110	C₂³×C₁₃	C₄×D₄	1	2	yes	—
111	C₂³×C₁₃	#121	1	2	yes	—
112	C₂³×C₁₃	C₄⋊Q₈	1	2	yes	—
113	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	—	2	4	no	—
118	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
119	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
121	C₂×C₄×C₁₃	#121	1	2	yes	—
122	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
123	C₂×C₄×C₁₃	#125	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	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
131	C₂×Dic₁₃	—	2	8	no	—
132	C₂×Dic₁₃	—	4	4	no	—
133	C₂×Dic₁₃	—	4	4	no	—
134	C₂×Dic₁₃	—	4	4	no	—
135	C₂×Dic₁₃	—	4	8	no	—
136	C₂×Dic₁₃	—	8	8	no	—
137	C₈×C₁₃	—	2	4	no	—
138	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
139	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
140	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
141	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
142	C₂×#117	—	13	2	no	—
143	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
144	C₂²×C₄⋊C₄	—	13	2	no	—
145	#427	—	13	8	no	—
146	C₂×#107	—	13	2	no	—
147	C₂×#107	—	13	2	no	—
148	C₂×#107	—	13	2	no	—
149	C₂×#107	—	26	4	no	—
150	C₂×#106	—	13	2	no	—
151	#405	—	2	4	no	—
152	C₂×#97	C₂×D₁₃	1	2	yes	—
153	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
154	#389	—	13	8	no	—
155	#385	—	26	4	no	—
156	#384	—	26	2	no	—
157	#382	—	26	2	no	—
158	#340	—	13	8	no	—
159	C₂²×C₁₃	—	2	4	no	—
160	C₂²×C₁₃	—	2	4	no	—
161	C₂²×C₁₃	—	2	2	no	—
162	C₂²×C₁₃	—	2	2	no	—
163	C₂²×C₁₃	#389	1	4	yes	—
164	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
166	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
167	C₂²×C₁₃	#576	1	2	yes	—
168	C₂²×C₁₃	#578	1	2	yes	—
169	C₄×C₁₃	—	4	2	no	—
170	C₄×C₁₃	—	4	2	no	—
171	C₄×C₁₃	#389	1	4	yes	—
172	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
173	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
174	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
175	C₄×C₁₃	C₂×#125	1	1	yes	characteristic
176	Dic₁₃	—	4	8	no	—
177	Dic₁₃	—	8	4	no	—
178	C₂³×C₄	—	26	2	no	—
179	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	Dic₂₆	1	4	yes	—
183	C₂²×C₈	Dic₂₆	1	4	yes	—
184	C₂²×C₈	C₄×D₁₃	1	4	yes	—
185	C₂²×C₈	C₂²⋊D₁₃	1	4	yes	—
186	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
187	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	#117	—	13	8	no	—
192	#117	—	26	4	no	—
193	C₂×C₄⋊C₄	—	13	4	no	—
194	C₂×C₄⋊C₄	—	13	4	no	—
195	C₂×C₄⋊C₄	—	26	4	no	—
196	C₂×C₄⋊C₄	—	26	4	no	—
197	C₂×C₄⋊C₄	—	26	2	no	—
198	C₂×C₄⋊C₄	—	26	8	no	—
199	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
200	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
201	C₂×C₂²⋊C₄	—	26	2	no	—
202	C₂×C₂²⋊C₄	D₅₂	1	2	yes	—
203	C₂×C₄²	—	26	2	no	—
204	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
205	#107	—	13	8	no	—
206	#107	—	26	4	no	—
207	#107	—	26	4	no	—
208	#106	—	13	8	no	—
209	#97	—	2	4	no	—
210	C₄×C₈	—	2	4	no	—
211	#95	—	26	8	no	—
212	C₂×C₁₃	—	2	4	no	—
213	C₂×C₁₃	C₂×#389	1	1	yes	characteristic
214	C₂×C₁₃	#2951	1	2	yes	—
215	C₂×C₁₃	#3058	1	1	yes	characteristic
216	C₂×C₁₃	#3059	1	1	yes	characteristic
217	C₂×C₁₃	#3164	1	2	yes	—
218	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
219	C₂²×C₄	—	26	4	no	—
220	C₂²×C₄	—	26	4	no	—
221	C₂²×C₄	—	52	4	no	—
222	C₂²×C₄	—	104	4	no	—
223	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
224	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
225	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
226	C₂²×C₄	C₂×D₅₂	1	2	yes	—
227	C₂²×C₄	C₂×D₅₂	1	2	yes	—
228	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
229	C₂²×C₄	#6127	1	2	yes	—
230	C₂²×C₄	#6127	1	2	yes	—
231	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
232	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
233	M₁₆	—	2	8	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	#6101	1	16	yes	complemented
237	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
238	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
239	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
240	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
241	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
242	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
243	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	13	8	no	—
247	C₄⋊C₄	—	26	4	no	—
248	C₄⋊C₄	—	26	4	no	—
249	C₄⋊C₄	—	26	4	no	—
250	C₄⋊C₄	—	52	4	no	—
251	C₄⋊C₄	—	104	8	no	—
252	C₂²⋊C₄	—	2	4	no	—
253	C₂²⋊C₄	—	26	8	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	26	8	no	—
257	C₁₃	#21156	1	1	yes	characteristic, complemented, Sylow
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	#88123	1	2	yes	—
261	C₂³	C₄×D₅₂	1	2	yes	—
262	C₂³	#88184	1	2	yes	—
263	C₂³	C₂²×D₅₂	1	1	yes	characteristic
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	26	8	no	—
269	C₂×C₄	—	52	4	no	—
270	C₂×C₄	—	52	4	no	—
271	C₂×C₄	—	52	8	no	—
272	C₂×C₄	—	52	4	no	—
273	C₂×C₄	—	104	8	no	—
274	C₂×C₄	C₄×Dic₂₆	1	4	yes	—
275	C₂×C₄	#88123	1	2	yes	—
276	C₂×C₄	#88124	1	4	yes	—
277	C₂×C₄	C₄×D₅₂	1	2	yes	—
278	C₂×C₄	C₂×#6101	1	4	yes	—
279	C₂×C₄	#88178	1	4	yes	—
280	C₂×C₄	#88184	1	2	yes	—
281	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
282	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
284	C₂×C₄	C₂×#6127	1	1	yes	characteristic
285	C₂×C₄	C₂×#6127	1	1	yes	characteristic
286	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
287	C₈	—	2	4	no	—
288	C₈	C₂×#6101	1	4	yes	—
289	C₈	C₂×#6101	1	4	yes	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	4	no	—
294	C₂²	#11718589	1	4	yes	—
295	C₂²	C₂×#88123	1	1	yes	characteristic
296	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
297	C₂²	#11719457	1	2	yes	—
298	C₂²	#11719458	1	2	yes	—
299	C₂²	C₂×#88184	1	1	yes	characteristic
300	C₄	—	4	2	no	—
301	C₄	—	4	2	no	—
302	C₄	—	52	8	no	—
303	C₄	—	104	4	no	—
304	C₄	#11718589	1	4	yes	—
305	C₄	C₂×C₄×Dic₂₆	1	1	yes	characteristic
306	C₄	C₂×#88124	1	1	yes	characteristic
307	C₄	C₂²×#6101	1	1	yes	characteristic
308	C₄	C₂×#88178	1	1	yes	characteristic
309	C₂	—	2	4	no	—
310	C₂	C₂×#11718589	1	1	yes	characteristic
311	C₂	#13123713	1	2	yes	—
312	C₂	#13124484	1	1	yes	characteristic
313	C₂	#13124491	1	1	yes	characteristic
314	C₂	#13125375	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#405	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#405	D₁₃
hypocenter	C₁₃	#21156
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
4. #13115579	C₂	semidirect
5. #13115569	C₂	semidirect
16. #11718767	C₄	semidirect
59. #88058	C₂×C₄	semidirect
60. #88057	C₂×C₄	semidirect
80. C₂×C₈×C₁₃	C₄⋊C₄	semidirect
236. C₂×C₈	#6101	semidirect
257. C₁₃	#21156	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718767	2
C₂²×#88058	2
#13117456	2
#13115579	2
#13115569	2
#13114556	2
C₂×C₁₃×#405	2
#21156	13