Subgroups — #15771008

Order 3328.

682 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88058	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718763	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718146	C₂	1	2	yes	maximal
4	#13114556	C₂	1	2	yes	maximal
5	C₂×C₁₃×#341	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88180	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
10	C₂²×#6099	C₂²	1	1	yes	characteristic
11	C₂×#88058	—	2	2	no	—
12	C₂×#88058	—	2	4	no	—
13	#11718763	—	2	2	no	—
14	#11718763	C₄	1	4	yes	complemented
15	C₂×#88054	—	2	2	no	—
16	C₁₃×#341	—	2	2	no	—
17	#11718146	—	2	4	no	—
18	#11718143	—	2	4	no	—
19	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂³×Dic₁₃	—	2	2	no	—
21	C₂×C₁₃×M₁₆	—	2	2	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	2	no	—
26	C₂×C₁₃×M₁₆	Q₈	1	4	yes	—
27	C₂²×C₈×C₁₃	—	2	2	no	—
28	C₂²×C₈×C₁₃	—	2	2	no	—
29	C₂²×C₈×C₁₃	—	2	2	no	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₃	D₄	1	2	yes	—
32	C₂²×C₈×C₁₃	D₄	1	2	yes	—
33	C₂×C₂²⋊Dic₁₃	—	2	1	no	—
34	#88180	—	2	1	no	—
35	#88180	—	2	1	no	—
36	#88180	C₂×C₄	1	2	yes	—
37	#88180	Q₈	1	2	yes	—
38	C₂×C₄⋊Dic₁₃	—	2	2	no	—
39	C₂×C₄⋊Dic₁₃	—	2	2	no	—
40	C₂×C₄⋊Dic₁₃	—	2	2	no	—
41	C₂×C₄⋊Dic₁₃	—	2	2	no	—
42	C₂×C₄⋊Dic₁₃	—	2	2	no	—
43	C₂×C₄⋊Dic₁₃	D₄	1	4	yes	—
44	C₂×C₄⋊Dic₁₃	D₄	1	2	yes	—
45	C₂×C₄×Dic₁₃	—	2	1	no	—
46	C₂×#6099	—	2	1	no	—
47	C₂×#6099	—	2	2	no	—
48	C₂×#6099	—	2	2	no	—
49	C₂×#6099	C₂×C₄	1	2	yes	—
50	C₂²×#1055	—	2	1	no	—
51	#88058	—	2	8	no	—
52	#88054	—	2	8	no	—
53	#11239	—	13	1	no	maximal, Sylow
54	C₂⁴×C₁₃	#41	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	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	#41	1	2	yes	—
63	C₂²×C₄×C₁₃	#41	1	2	yes	—
64	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
65	C₂²×Dic₁₃	—	2	2	no	—
66	C₂²×Dic₁₃	—	2	4	no	—
67	C₂²×Dic₁₃	—	4	2	no	—
68	C₂²×Dic₁₃	—	4	2	no	—
69	C₂²×Dic₁₃	—	8	4	no	—
70	C₁₃×M₁₆	—	2	4	no	—
71	C₁₃×M₁₆	—	2	4	no	—
72	C₁₃×M₁₆	—	4	2	no	—
73	C₁₃×M₁₆	—	4	2	no	—
74	C₂×C₈×C₁₃	—	2	4	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	—	2	4	no	—
79	C₂×C₈×C₁₃	—	4	2	no	—
80	C₂×C₈×C₁₃	—	4	2	no	—
81	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
82	C₂²⋊Dic₁₃	—	2	2	no	—
83	C₂²⋊Dic₁₃	—	2	2	no	—
84	C₄⋊Dic₁₃	—	2	2	no	—
85	C₄⋊Dic₁₃	—	2	4	no	—
86	C₄⋊Dic₁₃	—	2	4	no	—
87	C₄⋊Dic₁₃	—	4	2	no	—
88	C₄⋊Dic₁₃	—	4	2	no	—
89	C₄⋊Dic₁₃	C₄⋊C₄	1	4	yes	—
90	C₄×Dic₁₃	—	2	2	no	—
91	C₄×Dic₁₃	—	2	2	no	—
92	#6099	—	2	4	no	—
93	#6099	—	4	2	no	—
94	C₂×#1055	—	2	2	no	—
95	C₂×#1055	—	4	1	no	—
96	C₂×#1055	—	4	1	no	—
97	C₂²×#107	—	13	1	no	—
98	C₂×#424	—	13	1	no	—
99	C₂×#343	—	13	2	no	—
100	C₂×#341	D₁₃	1	1	yes	characteristic
101	#1736	—	13	2	no	—
102	C₂³×C₁₃	—	2	2	no	—
103	C₂³×C₁₃	—	2	2	no	—
104	C₂³×C₁₃	#117	1	1	yes	characteristic
105	C₂³×C₁₃	#121	1	1	yes	characteristic
106	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂³×C₁₃	#123	1	2	yes	—
108	C₂³×C₁₃	#125	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
113	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#121	1	2	yes	—
115	C₂×C₄×C₁₃	#121	1	2	yes	—
116	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
122	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#125	1	2	yes	—
124	C₂×C₄×C₁₃	#126	1	2	yes	—
125	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
126	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
127	C₂×Dic₁₃	—	2	4	no	—
128	C₂×Dic₁₃	—	4	2	no	—
129	C₂×Dic₁₃	—	4	2	no	—
130	C₂×Dic₁₃	—	4	4	no	—
131	C₂×Dic₁₃	—	8	4	no	—
132	C₂×Dic₁₃	—	8	4	no	—
133	C₂×Dic₁₃	—	8	4	no	—
134	C₈×C₁₃	—	2	4	no	—
135	C₈×C₁₃	—	4	4	no	—
136	#1055	—	4	2	no	—
137	C₂²×M₁₆	—	13	1	no	—
138	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
139	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
140	C₂×#117	—	13	1	no	—
141	C₂²×C₄⋊C₄	—	13	2	no	—
142	C₂×#107	—	26	4	no	—
143	C₂×#107	—	26	2	no	—
144	#424	—	13	4	no	—
145	#424	—	26	2	no	—
146	C₂×C₄⋊C₈	—	26	2	no	—
147	#343	—	26	4	no	—
148	#341	—	2	2	no	—
149	#340	—	26	4	no	—
150	C₂²×C₁₃	—	2	2	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	2	4	no	—
153	C₂²×C₁₃	—	2	2	no	—
154	C₂²×C₁₃	—	2	2	no	—
155	C₂²×C₁₃	#380	1	1	yes	characteristic
156	C₂²×C₁₃	#387	1	2	yes	—
157	C₂²×C₁₃	#394	1	1	yes	characteristic
158	C₂²×C₁₃	#398	1	1	yes	characteristic
159	C₂²×C₁₃	#398	1	2	yes	—
160	C₂²×C₁₃	#472	1	2	yes	—
161	C₂²×C₁₃	#473	1	2	yes	—
162	C₄×C₁₃	#386	1	1	yes	characteristic
163	C₄×C₁₃	#389	1	2	yes	—
164	C₄×C₁₃	#391	1	1	yes	characteristic
165	C₄×C₁₃	#395	1	1	yes	characteristic
166	C₄×C₁₃	#396	1	2	yes	—
167	C₄×C₁₃	#400	1	1	yes	characteristic
168	Dic₁₃	—	4	4	no	—
169	Dic₁₃	—	8	4	no	—
170	C₂³×C₄	—	26	2	no	—
171	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	13	2	no	—
178	C₂×M₁₆	—	26	1	no	—
179	C₂×M₁₆	—	26	2	no	—
180	C₂×M₁₆	—	26	2	no	—
181	C₂×M₁₆	Dic₂₆	1	4	yes	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	26	1	no	—
186	C₂²×C₈	C₄×D₁₃	1	2	yes	—
187	C₂²×C₈	D₅₂	1	2	yes	—
188	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
189	#117	—	13	2	no	—
190	#117	—	13	2	no	—
191	#117	—	26	1	no	—
192	#117	—	26	1	no	—
193	C₂×C₄⋊C₄	—	13	2	no	—
194	C₂×C₄⋊C₄	—	13	4	no	—
195	C₂×C₄⋊C₄	—	26	2	no	—
196	C₂×C₄⋊C₄	—	26	2	no	—
197	C₂×C₄⋊C₄	—	26	2	no	—
198	C₂×C₄⋊C₄	—	26	2	no	—
199	C₂×C₄⋊C₄	—	26	2	no	—
200	C₂×C₂²⋊C₄	—	26	1	no	—
201	C₂×C₄²	—	26	1	no	—
202	#107	—	26	8	no	—
203	C₄⋊C₈	—	26	8	no	—
204	C₂×C₁₃	—	2	2	no	—
205	C₂×C₁₃	—	2	2	no	—
206	C₂×C₁₃	#1441	1	1	yes	characteristic
207	C₂×C₁₃	#1816	1	2	yes	—
208	C₂×C₁₃	#2045	1	1	yes	characteristic
209	C₂×C₁₃	#2047	1	1	yes	characteristic
210	C₂×C₁₃	#2089	1	2	yes	—
211	C₂⁴	#6131	1	1	yes	characteristic
212	C₂²×C₄	—	26	2	no	—
213	C₂²×C₄	—	26	4	no	—
214	C₂²×C₄	—	52	2	no	—
215	C₂²×C₄	—	52	2	no	—
216	C₂²×C₄	—	104	4	no	—
217	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
218	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
220	C₂²×C₄	#6127	1	2	yes	—
221	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
222	C₂²×C₄	D₄×D₁₃	1	2	yes	—
223	C₂²×C₄	#6129	1	1	yes	characteristic
224	C₂²×C₄	#6129	1	2	yes	—
225	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
227	M₁₆	—	2	4	no	—
228	M₁₆	—	2	4	no	—
229	M₁₆	—	4	2	no	—
230	M₁₆	—	4	2	no	—
231	M₁₆	—	26	4	no	—
232	M₁₆	—	52	2	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	26	2	no	—
241	C₂×C₈	—	52	1	no	—
242	C₂×C₈	—	52	1	no	—
243	C₂×C₈	#6103	1	4	yes	—
244	C₄⋊C₄	—	13	4	no	—
245	C₄⋊C₄	—	26	2	no	—
246	C₄⋊C₄	—	26	4	no	—
247	C₄⋊C₄	—	26	4	no	—
248	C₄⋊C₄	—	52	2	no	—
249	C₄⋊C₄	—	52	2	no	—
250	C₂²⋊C₄	—	26	2	no	—
251	C₂²⋊C₄	—	26	2	no	—
252	C₄²	—	26	2	no	—
253	C₄²	—	26	2	no	—
254	C₁₃	#11239	1	1	yes	characteristic, complemented, Sylow
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	#88144	1	2	yes	—
258	C₂³	#88146	1	1	yes	characteristic
259	C₂³	#88148	1	2	yes	—
260	C₂³	#88149	1	1	yes	characteristic
261	C₂³	#88200	1	1	yes	characteristic
262	C₂×C₄	—	26	4	no	—
263	C₂×C₄	—	52	2	no	—
264	C₂×C₄	—	52	2	no	—
265	C₂×C₄	—	52	4	no	—
266	C₂×C₄	—	104	4	no	—
267	C₂×C₄	—	104	4	no	—
268	C₂×C₄	—	104	4	no	—
269	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
270	C₂×C₄	#88123	1	1	yes	characteristic
271	C₂×C₄	#88129	1	1	yes	characteristic
272	C₂×C₄	#88132	1	2	yes	—
273	C₂×C₄	#88132	1	2	yes	—
274	C₂×C₄	#88133	1	2	yes	—
275	C₂×C₄	#88135	1	2	yes	—
276	C₂×C₄	#88136	1	1	yes	characteristic
277	C₂×C₄	#88138	1	2	yes	—
278	C₂×C₄	#88140	1	1	yes	characteristic
279	C₂×C₄	#88142	1	2	yes	—
280	C₂×C₄	#88143	1	2	yes	—
281	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
282	C₂×C₄	#88151	1	1	yes	characteristic
283	C₂×C₄	#88178	1	2	yes	—
284	C₂×C₄	C₂×#6103	1	1	yes	characteristic
285	C₂×C₄	#88192	1	1	yes	characteristic
286	C₂×C₄	#88193	1	2	yes	—
287	C₈	—	2	4	no	—
288	C₈	—	4	4	no	—
289	C₈	—	52	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	#11718316	1	2	yes	—
296	C₂²	#11718322	1	1	yes	characteristic
297	C₂²	#11718333	1	1	yes	characteristic
298	C₂²	#11718557	1	2	yes	—
299	C₂²	#11718558	1	2	yes	—
300	C₂²	#11718645	1	2	yes	—
301	C₂²	#11718648	1	1	yes	characteristic
302	C₄	—	52	4	no	—
303	C₄	—	104	4	no	—
304	C₄	#11718308	1	2	yes	—
305	C₄	#11718317	1	1	yes	characteristic
306	C₄	#11718320	1	1	yes	characteristic
307	C₄	#11718590	1	1	yes	characteristic
308	C₄	#11718609	1	2	yes	—
309	C₄	#11718619	1	1	yes	characteristic
310	C₂	—	2	2	no	—
311	C₂	—	2	2	no	—
312	C₂	#13113090	1	1	yes	characteristic
313	C₂	#13114897	1	2	yes	—
314	C₂	#13116093	1	1	yes	characteristic
315	C₂	#13116105	1	1	yes	characteristic
316	C₂	#13116343	1	2	yes	—

	subgroup	G/H
center	C₂³	#88149
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×#341	C₂
socle	C₂³×C₁₃	#121
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×#341	D₁₃
hypocenter	C₁₃	#11239
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
14. #11718763	C₄	semidirect
254. C₁₃	#11239	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88058	2
C₂×#11718763	2
C₂×#11718146	2
#13114556	2
C₂×C₁₃×#341	2
#11239	13