Subgroups — #15780314

Order 3328.

484 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	C₂²×#88052	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718777	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718088	C₂	1	2	yes	maximal, complemented
4	#13112946	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1810	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×#88178	—	2	1	no	—
8	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#108	C₂²	1	2	yes	—
10	C₁₃×#407	—	2	1	no	—
11	#11718777	—	2	1	no	—
12	#11718777	—	2	1	no	—
13	#11718777	C₄	1	2	yes	—
14	#11718777	C₄	1	2	yes	—
15	C₂×#88058	C₂²	1	1	yes	characteristic
16	C₂×#88056	—	2	1	no	—
17	C₂×#88052	—	2	4	no	—
18	C₂×#88052	—	2	1	no	—
19	C₂×#88052	C₂²	1	2	yes	—
20	C₂²×#6090	—	2	1	no	—
21	C₁₃×#341	—	2	1	no	—
22	#11718088	—	2	4	no	—
23	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
24	C₂²×Dic₂₆	—	2	1	no	—
25	C₂×C₁₃×M₁₆	—	2	2	no	—
26	C₂²×C₈×C₁₃	—	2	2	no	—
27	C₂²×C₈×C₁₃	—	2	2	no	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊Dic₁₃	—	4	1	no	—
35	C₂×C₄⋊Dic₁₃	—	2	1	no	—
36	#88178	—	2	2	no	—
37	#88178	—	2	2	no	—
38	#88178	—	4	1	no	—
39	#88178	—	4	1	no	—
40	C₂×#6101	—	4	1	no	—
41	C₂×Dic₅₂	—	2	2	no	—
42	C₂×Dic₅₂	—	2	2	no	—
43	C₂×Dic₅₂	—	2	1	no	—
44	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
45	C₁₃×#108	—	2	2	no	—
46	C₁₃×#108	—	2	2	no	—
47	C₁₃×C₂²⋊C₈	—	4	1	no	—
48	C₁₃×C₂²⋊C₈	—	4	1	no	—
49	#88058	—	2	1	no	—
50	#88058	C₂×C₄	1	2	yes	—
51	#88056	—	2	2	no	—
52	#88056	—	2	2	no	—
53	#88052	—	2	4	no	—
54	#88052	—	2	2	no	—
55	#88052	D₄	1	4	yes	—
56	C₂×#6090	—	2	4	no	—
57	C₂×#6090	—	2	2	no	—
58	#13066	—	13	1	no	maximal, Sylow
59	C₂⁴×C₁₃	SD₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₃	—	2	2	no	—
62	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₁₃	—	4	1	no	—
69	C₂²×Dic₁₃	—	4	1	no	—
70	C₂×Dic₂₆	—	2	2	no	—
71	C₂×Dic₂₆	—	4	2	no	—
72	C₂×Dic₂₆	—	4	2	no	—
73	C₂×Dic₂₆	—	4	1	no	—
74	C₁₃×M₁₆	—	4	2	no	—
75	C₁₃×M₁₆	—	4	2	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	2	1	no	—
80	C₂×C₈×C₁₃	—	2	1	no	—
81	C₂×C₈×C₁₃	—	2	1	no	—
82	C₂×C₈×C₁₃	—	2	1	no	—
83	C₂×C₈×C₁₃	—	4	2	no	—
84	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	#41	1	2	yes	—
90	C₂²⋊Dic₁₃	—	4	2	no	—
91	C₂²⋊Dic₁₃	—	4	2	no	—
92	C₄⋊Dic₁₃	—	2	2	no	—
93	C₄⋊Dic₁₃	—	4	1	no	—
94	#6101	—	4	2	no	—
95	#6101	—	4	2	no	—
96	Dic₅₂	—	2	4	no	—
97	Dic₅₂	—	4	1	no	—
98	Dic₅₂	—	4	1	no	—
99	#6090	—	2	4	no	—
100	C₂²×M₃₂	—	13	1	no	—
101	C₂×#467	—	13	1	no	—
102	C₂×#362	—	13	2	no	—
103	#1810	D₁₃	1	1	yes	characteristic
104	#1387	—	13	2	no	—
105	C₂³×C₁₃	—	2	2	no	—
106	C₂³×C₁₃	—	2	2	no	—
107	C₂³×C₁₃	—	4	1	no	—
108	C₂³×C₁₃	#102	1	1	yes	characteristic
109	C₂³×C₁₃	#102	1	1	yes	characteristic
110	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#121	1	2	yes	—
122	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#135	1	2	yes	—
124	C₂×Dic₁₃	—	4	2	no	—
125	C₂×Dic₁₃	—	4	2	no	—
126	C₂×Dic₁₃	—	8	2	no	—
127	C₂×Dic₁₃	—	8	2	no	—
128	Dic₂₆	—	4	2	no	—
129	Dic₂₆	—	4	2	no	—
130	Dic₂₆	—	8	2	no	—
131	C₈×C₁₃	—	2	1	no	—
132	C₈×C₁₃	—	2	1	no	—
133	C₈×C₁₃	—	8	2	no	—
134	C₈×C₁₃	C₄×D₄	1	2	yes	—
135	C₈×C₁₃	#120	1	1	yes	characteristic
136	C₈×C₁₃	#123	1	1	yes	characteristic
137	C₂²×Q₁₆	—	13	1	no	—
138	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
139	C₂×C₂²⋊Q₈	—	26	1	no	—
140	C₂×M₃₂	—	13	2	no	—
141	C₂×M₃₂	—	26	4	no	—
142	C₂×M₃₂	—	26	1	no	—
143	C₂²×C₁₆	—	26	1	no	—
144	#467	—	13	2	no	—
145	#467	—	13	2	no	—
146	#467	—	26	1	no	—
147	#467	—	26	1	no	—
148	C₂×#108	C₂×D₁₃	1	2	yes	—
149	C₂×#107	—	13	1	no	—
150	C₂×#103	—	26	1	no	—
151	#407	—	2	1	no	—
152	#362	—	26	4	no	—
153	#341	—	2	1	no	—
154	C₂²×C₁₃	—	2	1	no	—
155	C₂²×C₁₃	—	2	1	no	—
156	C₂²×C₁₃	—	2	1	no	—
157	C₂²×C₁₃	—	2	1	no	—
158	C₂²×C₁₃	—	2	1	no	—
159	C₂²×C₁₃	—	2	1	no	—
160	C₂²×C₁₃	—	2	1	no	—
161	C₂²×C₁₃	—	4	4	no	—
162	C₂²×C₁₃	#362	1	2	yes	complemented
163	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
164	C₂²×C₁₃	#465	1	2	yes	—
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	#386	1	1	yes	characteristic
168	C₄×C₁₃	#416	1	1	yes	characteristic
169	C₄×C₁₃	#466	1	2	yes	—
170	Dic₁₃	—	8	2	no	—
171	Dic₁₃	—	8	2	no	—
172	C₂²×Q₈	—	26	1	no	—
173	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
174	C₂×Q₁₆	—	13	2	no	—
175	C₂×Q₁₆	—	26	1	no	—
176	C₂×Q₁₆	—	26	2	no	—
177	C₂×Q₁₆	—	26	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
182	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
183	C₂²×C₈	D₅₂	1	1	yes	characteristic
184	C₂²×C₈	D₅₂	1	1	yes	characteristic
185	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
187	C₂²⋊Q₈	—	26	2	no	—
188	C₂²⋊Q₈	—	26	2	no	—
189	C₂²⋊Q₈	—	52	1	no	—
190	C₂²⋊Q₈	—	52	1	no	—
191	C₂×C₄⋊C₄	—	26	1	no	—
192	C₂×C₄⋊C₄	—	52	1	no	—
193	C₂×C₂²⋊C₄	—	52	1	no	—
194	M₃₂	—	13	4	no	—
195	M₃₂	—	26	4	no	—
196	M₃₂	—	26	2	no	—
197	C₂×C₁₆	—	26	4	no	—
198	C₂×C₁₆	—	26	2	no	—
199	#108	—	2	2	no	—
200	#108	—	2	2	no	—
201	#107	—	13	2	no	—
202	#107	—	26	1	no	—
203	#103	—	26	2	no	—
204	#103	—	26	2	no	—
205	C₂²⋊C₈	—	4	1	no	—
206	C₂²⋊C₈	—	4	1	no	—
207	C₂×C₁₃	—	2	2	no	—
208	C₂×C₁₃	—	2	2	no	—
209	C₂×C₁₃	—	4	1	no	—
210	C₂×C₁₃	#1893	1	1	yes	characteristic
211	C₂×C₁₃	C₂×#362	1	1	yes	characteristic
212	C₂×C₁₃	#2148	1	1	yes	characteristic
213	C₂⁴	#6106	1	1	yes	characteristic
214	C₂×Q₈	—	26	2	no	—
215	C₂×Q₈	—	52	1	no	—
216	C₂×Q₈	—	52	2	no	—
217	C₂×Q₈	—	52	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	52	1	no	—
221	C₂²×C₄	—	52	1	no	—
222	C₂²×C₄	#6103	1	1	yes	characteristic
223	C₂²×C₄	#6103	1	1	yes	characteristic
224	C₂²×C₄	#6104	1	1	yes	characteristic
225	C₂²×C₄	#6104	1	1	yes	characteristic
226	C₂²×C₄	#6106	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
228	Q₁₆	—	26	4	no	—
229	Q₁₆	—	52	1	no	—
230	Q₁₆	—	52	1	no	—
231	M₁₆	—	4	2	no	—
232	M₁₆	—	4	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	#6103	1	1	yes	characteristic
242	C₂×C₈	#6103	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
244	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
245	C₂×C₈	D₄×D₁₃	1	2	yes	—
246	C₂×C₈	#6129	1	2	yes	—
247	C₄⋊C₄	—	26	2	no	—
248	C₄⋊C₄	—	52	1	no	—
249	C₄⋊C₄	—	52	2	no	—
250	C₄⋊C₄	—	52	2	no	—
251	C₂²⋊C₄	—	52	2	no	—
252	C₂²⋊C₄	—	52	2	no	—
253	C₁₆	—	26	4	no	—
254	C₁₃	#13066	1	1	yes	characteristic, complemented, Sylow
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	4	1	no	—
258	C₂³	#88049	1	1	yes	characteristic
259	C₂³	#88049	1	1	yes	characteristic
260	C₂³	C₂×#6106	1	1	yes	characteristic
261	Q₈	—	52	2	no	—
262	Q₈	—	52	2	no	—
263	Q₈	—	104	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	52	2	no	—
272	C₂×C₄	—	52	2	no	—
273	C₂×C₄	—	104	2	no	—
274	C₂×C₄	—	104	2	no	—
275	C₂×C₄	#88049	1	1	yes	characteristic
276	C₂×C₄	#88049	1	1	yes	characteristic
277	C₂×C₄	C₂×#6103	1	1	yes	characteristic
278	C₂×C₄	C₂×#6104	1	1	yes	characteristic
279	C₂×C₄	#88193	1	2	yes	—
280	C₂×C₄	#88204	1	2	yes	—
281	C₈	—	2	1	no	—
282	C₈	—	2	1	no	—
283	C₈	—	8	2	no	—
284	C₈	#88135	1	2	yes	—
285	C₈	#88136	1	1	yes	characteristic
286	C₈	#88140	1	1	yes	characteristic
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	4	4	no	—
295	C₂²	#11718088	1	2	yes	complemented
296	C₂²	C₂×#88049	1	1	yes	characteristic
297	C₂²	#11718709	1	2	yes	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	104	2	no	—
301	C₄	—	104	2	no	—
302	C₄	#11718619	1	1	yes	characteristic
303	C₄	#11718663	1	1	yes	characteristic
304	C₄	#11718699	1	2	yes	—
305	C₂	—	2	2	no	—
306	C₂	—	2	2	no	—
307	C₂	—	4	1	no	—
308	C₂	#13115262	1	1	yes	characteristic
309	C₂	C₂×#11718088	1	1	yes	characteristic
310	C₂	#13116583	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88049
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1810	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂⁴×C₁₃	SD₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1810	D₁₃
hypocenter	C₁₃	#13066
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

Complements

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

N	H ≅ G/N	type
3. C₂×#11718088	C₂	semidirect
4. #13112946	C₂	semidirect
162. C₂²×C₁₃	#362	semidirect
254. C₁₃	#13066	semidirect
295. C₂²	#11718088	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88052	2
C₂×#11718777	2
C₂×#11718088	2
#13112946	2
C₁₃×#1810	2
#13066	13