Subgroups — #15846029

Order 3328.

674 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125758	C₂	1	2	yes	maximal
2	C₂×#11718343	C₂	1	1	yes	maximal, characteristic, complemented
3	#13114349	C₂	1	4	yes	maximal, complemented
4	#13114225	C₂	1	4	yes	maximal, complemented
5	#13113584	C₂	1	1	yes	maximal, characteristic, complemented
6	#13113522	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2931	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#131	—	2	1	no	—
9	C₁₃×#532	C₂²	1	1	yes	characteristic
10	C₂×#88123	C₂²	1	2	yes	—
11	C₁₃×#436	—	2	1	no	—
12	C₁₃×#436	C₂²	1	2	yes	—
13	C₈×D₄×C₁₃	—	2	1	no	—
14	C₈×D₄×C₁₃	C₂²	1	2	yes	—
15	C₁₃×#423	C₂²	1	1	yes	characteristic
16	C₁₃×#408	—	2	1	no	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×#88058	—	2	1	no	—
19	C₂×#88058	—	2	1	no	—
20	#11718754	—	2	2	no	—
21	#11718737	—	2	2	no	—
22	#11718701	—	2	4	no	—
23	#11718680	C₂²	1	8	yes	—
24	#11718586	C₂²	1	2	yes	—
25	#11718343	—	2	1	no	—
26	#11718343	—	2	1	no	—
27	#11718343	C₂²	1	2	yes	—
28	#11718343	C₂²	1	2	yes	—
29	#11718078	C₂²	1	8	yes	—
30	#11718050	C₂²	1	2	yes	—
31	#11718049	C₂²	1	2	yes	—
32	C₂×C₁₃×#41	—	2	1	no	—
33	C₂²×Dic₂₆	—	2	2	no	—
34	C₁₃×#131	—	4	4	no	—
35	C₂×C₁₃×M₁₆	—	2	1	no	—
36	C₂×C₁₃×M₁₆	—	2	2	no	—
37	C₂²×C₈×C₁₃	—	2	2	no	—
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
40	C₁₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
41	C₁₃×#127	D₄	1	1	yes	characteristic
42	C₁₃×#124	—	2	1	no	—
43	C₁₃×#121	—	2	1	no	—
44	C₁₃×#121	D₄	1	2	yes	—
45	C₄×D₄×C₁₃	—	2	1	no	—
46	C₄×D₄×C₁₃	C₂³	1	2	yes	—
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×#6105	—	2	8	no	—
49	C₂×C₄⋊Dic₁₃	—	2	2	no	—
50	C₂×C₄⋊Dic₁₃	—	2	2	no	—
51	#88123	—	2	2	no	—
52	#88123	—	2	2	no	—
53	#88123	—	2	2	no	—
54	#88123	C₂³	1	4	yes	—
55	C₁₃×C₄⋊C₈	—	2	1	no	—
56	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
57	C₁₃×C₂²⋊C₈	—	2	2	no	—
58	C₁₃×C₂²⋊C₈	—	2	2	no	—
59	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₂²⋊#1055	—	2	2	no	—
64	#88058	—	2	2	no	—
65	#88058	—	2	2	no	—
66	#88058	—	2	2	no	—
67	#88058	—	2	2	no	—
68	#88050	—	2	8	no	—
69	C₄⋊#1055	C₂³	1	4	yes	—
70	C₄×#1055	—	2	2	no	—
71	#19192	—	13	1	no	maximal, Sylow
72	C₁₃×#41	—	4	2	no	—
73	C₁₃×#41	—	4	2	no	—
74	C₂×Q₈×C₁₃	—	2	1	no	—
75	C₂×D₄×C₁₃	—	2	2	no	—
76	C₂×D₄×C₁₃	—	2	1	no	—
77	C₂×D₄×C₁₃	—	2	1	no	—
78	C₂×D₄×C₁₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₁₃	—	2	2	no	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
83	C₂²×Dic₁₃	—	4	2	no	—
84	C₂×Dic₂₆	—	2	4	no	—
85	C₂×Dic₂₆	—	4	2	no	—
86	C₂×Dic₂₆	—	8	4	no	—
87	C₁₃×M₁₆	—	4	2	no	—
88	C₁₃×M₁₆	—	4	4	no	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	—	4	4	no	—
92	C₂×C₈×C₁₃	SD₁₆	1	4	yes	complemented
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₁₃×C₄⋊C₄	—	2	1	no	—
98	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
99	C₁₃×C₂²⋊C₄	—	2	2	no	—
100	C₁₃×C₂²⋊C₄	—	2	2	no	—
101	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₃	#41	1	1	yes	characteristic
104	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
105	#6105	—	8	8	no	—
106	C₄⋊Dic₁₃	—	2	4	no	—
107	C₄⋊Dic₁₃	—	2	4	no	—
108	C₄⋊Dic₁₃	—	4	2	no	—
109	C₄⋊Dic₁₃	—	4	2	no	—
110	C₂×#1055	—	2	4	no	—
111	#3206	—	13	2	no	—
112	C₂×#500	—	13	1	no	—
113	#2931	D₁₃	1	1	yes	characteristic
114	#1693	—	13	4	no	—
115	#1673	—	13	4	no	—
116	#1564	—	13	1	no	—
117	#1549	—	13	2	no	—
118	C₂³×C₁₃	—	2	2	no	—
119	C₂³×C₁₃	#121	1	1	yes	characteristic
120	Q₈×C₁₃	—	4	2	no	—
121	D₄×C₁₃	—	2	4	no	—
122	D₄×C₁₃	—	4	4	no	—
123	D₄×C₁₃	—	4	2	no	—
124	D₄×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
134	C₂×C₄×C₁₃	#137	1	2	yes	—
135	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
138	C₂×Dic₁₃	—	4	4	no	—
139	C₂×Dic₁₃	—	8	4	no	—
140	Dic₂₆	—	4	8	no	—
141	Dic₂₆	—	8	4	no	—
142	C₈×C₁₃	—	2	1	no	—
143	C₈×C₁₃	—	2	1	no	—
144	C₈×C₁₃	—	4	2	no	—
145	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
146	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
147	#1055	—	4	4	no	—
148	C₂×#131	—	2	1	no	—
149	#532	C₂×D₁₃	1	1	yes	characteristic
150	C₂×C₄⋊Q₈	—	13	2	no	—
151	#500	—	13	2	no	—
152	#500	—	13	2	no	—
153	#500	—	26	1	no	—
154	#500	—	26	1	no	—
155	#492	—	26	2	no	—
156	#487	—	26	2	no	—
157	#461	—	13	8	no	—
158	#448	—	26	4	no	—
159	#436	—	2	1	no	—
160	#436	C₂×D₁₃	1	2	yes	—
161	C₈×D₄	—	2	1	no	—
162	C₈×D₄	C₂×D₁₃	1	2	yes	—
163	#431	—	13	2	no	—
164	C₂×#107	—	26	1	no	—
165	C₂×#107	—	26	1	no	—
166	#423	C₂×D₁₃	1	1	yes	characteristic
167	#408	—	2	1	no	—
168	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
169	#335	—	13	2	no	—
170	#334	—	13	2	no	—
171	#332	—	13	8	no	—
172	C₂²×C₁₃	—	2	1	no	—
173	C₂²×C₁₃	—	2	1	no	—
174	C₂²×C₁₃	—	4	2	no	—
175	C₂²×C₁₃	—	4	2	no	—
176	C₂²×C₁₃	—	4	2	no	—
177	C₂²×C₁₃	#461	1	2	yes	—
178	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
179	C₄×C₁₃	—	2	1	no	—
180	C₄×C₁₃	—	2	1	no	—
181	C₄×C₁₃	—	4	2	no	—
182	C₄×C₁₃	#461	1	2	yes	—
183	C₄×C₁₃	#473	1	2	yes	—
184	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
185	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
186	C₄×C₁₃	#576	1	2	yes	—
187	Dic₁₃	—	8	4	no	—
188	C₂×#41	—	2	1	no	—
189	C₂²×Q₈	—	26	2	no	—
190	C₂×SD₁₆	—	26	8	no	—
191	#131	—	4	4	no	—
192	C₂×M₁₆	—	2	1	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
196	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
197	C₄⋊Q₈	—	13	4	no	—
198	C₄⋊Q₈	—	26	2	no	—
199	C₄⋊Q₈	—	26	2	no	—
200	C₄⋊Q₈	—	26	2	no	—
201	C₄⋊Q₈	D₅₂	1	1	yes	characteristic
202	#127	D₅₂	1	1	yes	characteristic
203	#124	—	2	1	no	—
204	#121	—	2	1	no	—
205	#121	D₅₂	1	2	yes	—
206	C₄×D₄	—	2	1	no	—
207	C₄×D₄	C₂²×D₁₃	1	2	yes	—
208	C₂×C₄⋊C₄	—	26	2	no	—
209	C₂×C₄⋊C₄	—	26	2	no	—
210	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
211	#107	—	26	2	no	—
212	#107	—	26	2	no	—
213	#107	—	26	2	no	—
214	#107	—	26	2	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	13	4	no	—
217	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
218	#103	—	26	8	no	—
219	C₂²⋊C₈	—	2	2	no	—
220	C₂²⋊C₈	—	2	2	no	—
221	C₂²⋊C₈	—	26	2	no	—
222	C₄×C₈	—	26	2	no	—
223	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
224	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
225	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
226	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
227	C₂×C₁₃	—	2	2	no	—
228	C₂×C₁₃	—	8	2	no	—
229	C₂×C₁₃	C₂×#461	1	1	yes	characteristic
230	C₂×C₁₃	#3058	1	1	yes	characteristic
231	C₂×C₁₃	C₂×#473	1	1	yes	characteristic
232	#41	—	4	2	no	—
233	#41	—	4	2	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	26	4	no	—
236	C₂×Q₈	—	52	2	no	—
237	C₂×Q₈	—	104	4	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	C₂×D₅₂	1	2	yes	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	52	2	no	—
244	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
245	C₂²×C₄	#6127	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
247	SD₁₆	—	104	8	no	—
248	M₁₆	—	4	2	no	—
249	M₁₆	—	4	4	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	4	4	no	—
253	C₂×C₈	—	26	4	no	—
254	C₂×C₈	#6105	1	4	yes	complemented
255	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
256	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
257	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
258	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	26	4	no	—
261	C₄⋊C₄	—	26	4	no	—
262	C₄⋊C₄	—	52	2	no	—
263	C₄⋊C₄	—	52	2	no	—
264	C₄⋊C₄	C₂×D₅₂	1	2	yes	—
265	C₂²⋊C₄	—	2	2	no	—
266	C₂²⋊C₄	—	2	2	no	—
267	C₄²	C₂×D₅₂	1	1	yes	characteristic
268	C₄²	#6127	1	1	yes	characteristic
269	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
270	C₄²	C₂³×D₁₃	1	1	yes	characteristic
271	C₁₃	#19192	1	1	yes	characteristic, complemented, Sylow
272	C₂³	—	2	2	no	—
273	C₂³	#88184	1	1	yes	characteristic
274	Q₈	—	4	2	no	—
275	Q₈	—	52	8	no	—
276	Q₈	—	104	4	no	—
277	D₄	—	2	4	no	—
278	D₄	—	4	4	no	—
279	D₄	—	4	2	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	52	4	no	—
287	C₂×C₄	—	104	4	no	—
288	C₂×C₄	#88184	1	1	yes	characteristic
289	C₂×C₄	#88184	1	1	yes	characteristic
290	C₂×C₄	#88184	1	1	yes	characteristic
291	C₂×C₄	C₂×#6105	1	2	yes	—
292	C₂×C₄	#88205	1	2	yes	—
293	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
294	C₂×C₄	C₂×#6127	1	1	yes	characteristic
295	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	4	2	no	—
299	C₈	—	52	4	no	—
300	C₈	C₂×#6105	1	2	yes	—
301	C₈	C₂×#6105	1	2	yes	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	4	2	no	—
307	C₂²	#11718680	1	2	yes	—
308	C₂²	C₂×#88184	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	4	2	no	—
312	C₄	—	104	4	no	—
313	C₄	#11718680	1	2	yes	—
314	C₄	#11718804	1	2	yes	—
315	C₄	#11719457	1	2	yes	—
316	C₄	C₂²×#6105	1	1	yes	characteristic
317	C₄	C₂×#88205	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	8	2	no	—
320	C₂	C₂×#11718680	1	1	yes	characteristic
321	C₂	#13124484	1	1	yes	characteristic
322	C₂	C₂×#11718804	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88184
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#2931	C₂
socle	C₂²×C₁₃	C₂×#121
prime radical	C₁₃×#532	C₂²
hypercenter	#2931	D₁₃
hypocenter	C₁₃	#19192
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

Complements

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

N	H ≅ G/N	type
2. C₂×#11718343	C₂	semidirect
3. #13114349	C₂	semidirect
4. #13114225	C₂	semidirect
5. #13113584	C₂	semidirect
6. #13113522	C₂	semidirect
92. C₂×C₈×C₁₃	SD₁₆	semidirect
254. C₂×C₈	#6105	semidirect
271. C₁₃	#19192	semidirect

Maximal subgroups

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

subgroup	index
#13125758	2
C₂×#11718343	2
#13114349	2
#13114225	2
#13113584	2
#13113522	2
C₁₃×#2931	2
#19192	13