Subgroups — #13123772

Order 1664.

348 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719283	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718685	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718500	C₂	1	1	yes	maximal, characteristic
5	#11718500	C₂	1	1	yes	maximal, characteristic
6	#11718498	C₂	1	1	yes	maximal, characteristic
7	#11718492	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718489	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718489	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718369	C₂	1	1	yes	maximal, characteristic
11	D₁₃×#97	C₂	1	1	yes	maximal, characteristic
12	#11718362	C₂	1	1	yes	maximal, characteristic, complemented
13	#11718362	C₂	1	1	yes	maximal, characteristic, complemented
14	#11718350	C₂	1	1	yes	maximal, characteristic
15	#11718340	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
17	#88200	C₄	1	1	yes	characteristic
18	#88198	C₄	1	1	yes	characteristic
19	C₂×#6094	C₂²	1	1	yes	characteristic
20	C₂×#6094	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
23	#88151	C₄	1	1	yes	characteristic
24	#88150	C₄	1	1	yes	characteristic
25	#88150	C₄	1	1	yes	characteristic
26	#88143	C₄	1	1	yes	characteristic
27	#88143	C₄	1	1	yes	characteristic
28	#88142	C₄	1	1	yes	characteristic
29	#88126	C₂²	1	1	yes	characteristic
30	#88126	C₂²	1	1	yes	characteristic
31	C₄²×D₁₃	C₂²	1	1	yes	characteristic
32	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
33	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
34	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
35	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₁₃×#97	C₂²	1	1	yes	characteristic
39	C₁₃×#97	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
41	#88055	C₂²	1	1	yes	characteristic
42	#88055	C₂²	1	1	yes	characteristic
43	#88055	C₂²	1	1	yes	characteristic
44	#88055	C₂²	1	1	yes	characteristic
45	#88054	C₂²	1	1	yes	characteristic
46	#88054	C₂²	1	1	yes	characteristic
47	#88054	C₂²	1	1	yes	characteristic
48	#88054	C₂²	1	1	yes	characteristic
49	#88054	C₂²	1	1	yes	characteristic
50	#88054	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
52	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
53	C₄⋊#1055	C₂²	1	1	yes	characteristic
54	C₄⋊#1055	C₂²	1	1	yes	characteristic
55	C₄⋊#1055	C₂²	1	1	yes	characteristic
56	#88043	C₂²	1	1	yes	characteristic
57	#88043	C₂²	1	1	yes	characteristic
58	C₄×#1055	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
63	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
64	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
65	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
71	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
72	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
73	C₄²×C₁₃	C₂³	1	1	yes	characteristic
74	C₄²×C₁₃	C₂³	1	1	yes	characteristic
75	C₄²×C₁₃	C₂³	1	1	yes	characteristic
76	#6103	C₂×C₄	1	1	yes	characteristic
77	#6103	C₂×C₄	1	1	yes	characteristic
78	#6103	C₂×C₄	1	1	yes	characteristic
79	#6103	C₂×C₄	1	1	yes	characteristic
80	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
81	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
82	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
83	#6101	C₂×C₄	1	1	yes	characteristic
84	#6101	C₂×C₄	1	1	yes	characteristic
85	#6101	C₂×C₄	1	1	yes	characteristic
86	#6101	C₂×C₄	1	1	yes	characteristic
87	#6101	C₂×C₄	1	1	yes	characteristic
88	#6101	C₂×C₄	1	1	yes	characteristic
89	#6101	C₂×C₄	1	1	yes	characteristic
90	#6101	C₂×C₄	1	1	yes	characteristic
91	#6101	C₂×C₄	1	1	yes	characteristic
92	#6101	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
96	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
97	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
98	C₂×#1055	C₂³	1	1	yes	characteristic
99	C₂×#1055	C₂³	1	1	yes	characteristic
100	C₂×#1055	C₂³	1	1	yes	characteristic
101	C₂×#1055	C₂³	1	1	yes	characteristic
102	#6094	—	2	2	no	—
103	#6094	—	2	2	no	—
104	C₈×D₁₃	—	2	2	no	—
105	C₈×D₁₃	Q₈	1	4	yes	complemented
106	#2961	—	13	1	no	maximal, Sylow
107	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
108	Q₈×C₁₃	—	4	1	no	—
109	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
123	C₄×D₁₃	—	4	1	no	—
124	C₄×D₁₃	—	4	1	no	—
125	C₄×D₁₃	C₂×Q₈	1	2	yes	—
126	C₄×D₁₃	#41	1	2	yes	—
127	Dic₂₆	—	4	1	no	—
128	Dic₂₆	—	4	1	no	—
129	Dic₂₆	—	4	1	no	—
130	C₈×C₁₃	—	2	1	no	—
131	C₈×C₁₃	—	2	1	no	—
132	C₈×C₁₃	—	2	1	no	—
133	C₈×C₁₃	C₂×Q₈	1	2	yes	—
134	#1055	—	2	1	no	—
135	#1055	—	2	1	no	—
136	#1055	—	2	1	no	—
137	#1055	C₂×Q₈	1	2	yes	—
138	#533	—	13	1	no	—
139	#446	—	13	1	no	—
140	#446	—	13	1	no	—
141	#446	—	13	1	no	—
142	#446	—	13	1	no	—
143	#446	—	13	1	no	—
144	#446	—	13	1	no	—
145	#446	—	13	1	no	—
146	#446	D₁₃	1	1	yes	characteristic, complemented
147	#424	—	13	1	no	—
148	#424	—	13	1	no	—
149	#423	—	13	1	no	—
150	#405	—	13	1	no	—
151	#405	—	13	1	no	—
152	C₂×#97	—	13	1	no	—
153	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
154	C₂×D₁₃	—	2	1	no	—
155	C₂×D₁₃	—	2	1	no	—
156	C₂×D₁₃	C₄×Q₈	1	2	yes	—
157	C₄×C₁₃	—	2	1	no	—
158	C₄×C₁₃	—	2	1	no	—
159	C₄×C₁₃	—	2	1	no	—
160	C₄×C₁₃	—	2	1	no	—
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
164	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
165	Dic₁₃	—	2	1	no	—
166	Dic₁₃	—	2	1	no	—
167	Dic₁₃	—	2	1	no	—
168	Dic₁₃	—	2	1	no	—
169	Dic₁₃	—	2	1	no	—
170	Dic₁₃	—	2	1	no	—
171	Dic₁₃	C₄×Q₈	1	2	yes	—
172	C₂×M₁₆	—	13	1	no	—
173	C₂×M₁₆	—	13	1	no	—
174	C₂²×C₈	—	13	1	no	—
175	C₂²×C₈	—	13	1	no	—
176	C₄⋊Q₈	—	13	1	no	—
177	C₄⋊Q₈	—	13	1	no	—
178	#125	—	13	1	no	—
179	#125	—	13	1	no	—
180	C₂²⋊Q₈	—	13	1	no	—
181	C₂²⋊Q₈	—	13	1	no	—
182	C₂²⋊Q₈	—	13	1	no	—
183	C₂²⋊Q₈	—	13	1	no	—
184	C₄×Q₈	—	13	1	no	—
185	C₄×Q₈	—	13	1	no	—
186	C₄×Q₈	—	13	1	no	—
187	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
188	#117	—	13	1	no	—
189	#117	—	13	1	no	—
190	C₂×C₄²	—	13	1	no	—
191	C₄⋊C₈	—	13	1	no	—
192	C₄⋊C₈	—	13	1	no	—
193	C₄⋊C₈	—	13	1	no	—
194	C₄⋊C₈	—	13	1	no	—
195	C₄⋊C₈	—	13	1	no	—
196	C₄⋊C₈	—	13	1	no	—
197	C₄⋊C₈	—	13	1	no	—
198	C₄⋊C₈	—	13	1	no	—
199	C₄⋊C₈	—	13	1	no	—
200	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
201	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
202	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
203	#97	—	13	1	no	—
204	#97	—	13	1	no	—
205	#97	—	13	1	no	—
206	#97	—	13	1	no	—
207	#97	—	13	1	no	—
208	#97	—	13	1	no	—
209	#97	C₂×D₁₃	1	1	yes	characteristic
210	#97	C₂×D₁₃	1	1	yes	characteristic
211	C₄×C₈	—	13	1	no	—
212	C₄×C₈	—	13	1	no	—
213	C₄×C₈	—	13	1	no	—
214	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
215	C₂×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
216	C₂×C₁₃	#568	1	1	yes	characteristic
217	C₂×C₁₃	#568	1	1	yes	characteristic
218	D₁₃	—	2	2	no	—
219	C₂×Q₈	—	13	1	no	—
220	C₂×Q₈	—	13	1	no	—
221	C₂×Q₈	—	13	1	no	—
222	C₂×Q₈	C₄×D₁₃	1	1	yes	characteristic
223	C₂²×C₄	—	13	1	no	—
224	C₂²×C₄	—	13	1	no	—
225	C₂²×C₄	—	13	1	no	—
226	M₁₆	—	26	2	no	—
227	M₁₆	—	26	2	no	—
228	C₂×C₈	—	13	1	no	—
229	C₂×C₈	—	13	4	no	—
230	C₂×C₈	—	13	1	no	—
231	C₂×C₈	—	13	1	no	—
232	C₂×C₈	—	13	1	no	—
233	C₂×C₈	—	26	2	no	—
234	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
235	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
236	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
237	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
238	C₄⋊C₄	—	13	1	no	—
239	C₄⋊C₄	—	13	1	no	—
240	C₄⋊C₄	—	13	1	no	—
241	C₄⋊C₄	—	13	1	no	—
242	C₄⋊C₄	—	13	1	no	—
243	C₄⋊C₄	—	13	1	no	—
244	C₄⋊C₄	—	13	1	no	—
245	C₄⋊C₄	—	13	1	no	—
246	C₄⋊C₄	—	13	1	no	—
247	C₄⋊C₄	—	13	1	no	—
248	C₄⋊C₄	—	13	1	no	—
249	C₄⋊C₄	—	13	1	no	—
250	C₄⋊C₄	—	13	1	no	—
251	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
252	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
253	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
254	C₂²⋊C₄	—	13	1	no	—
255	C₂²⋊C₄	—	13	1	no	—
256	C₂²⋊C₄	—	13	1	no	—
257	C₂²⋊C₄	—	13	1	no	—
258	C₄²	—	13	1	no	—
259	C₄²	—	13	1	no	—
260	C₄²	—	13	1	no	—
261	C₄²	—	13	1	no	—
262	C₄²	—	13	1	no	—
263	C₄²	C₂²×D₁₃	1	1	yes	characteristic
264	C₄²	C₂²×D₁₃	1	1	yes	characteristic
265	C₄²	C₂²×D₁₃	1	1	yes	characteristic
266	C₁₃	#2961	1	1	yes	characteristic, complemented, Sylow
267	C₂³	—	13	1	no	—
268	Q₈	—	4	1	no	—
269	Q₈	—	52	1	no	—
270	Q₈	—	52	1	no	—
271	Q₈	—	52	1	no	—
272	C₂×C₄	—	13	1	no	—
273	C₂×C₄	—	13	2	no	—
274	C₂×C₄	—	13	2	no	—
275	C₂×C₄	—	13	1	no	—
276	C₂×C₄	—	13	1	no	—
277	C₂×C₄	—	13	1	no	—
278	C₂×C₄	—	13	1	no	—
279	C₂×C₄	—	13	1	no	—
280	C₂×C₄	—	13	1	no	—
281	C₂×C₄	—	52	1	no	—
282	C₂×C₄	—	52	1	no	—
283	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
285	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
286	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	13	2	no	—
294	C₈	—	26	1	no	—
295	C₈	—	26	1	no	—
296	C₈	—	26	1	no	—
297	C₈	Q₈×D₁₃	1	2	yes	—
298	C₂²	—	13	2	no	—
299	C₂²	—	26	1	no	—
300	C₂²	—	26	1	no	—
301	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	13	2	no	—
309	C₄	—	26	1	no	—
310	C₄	—	26	1	no	—
311	C₄	—	26	1	no	—
312	C₄	—	26	1	no	—
313	C₄	—	26	1	no	—
314	C₄	—	26	1	no	—
315	C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
316	C₄	#41×D₁₃	1	1	yes	characteristic
317	C₂	—	26	2	no	—
318	C₂	C₄×Q₈×D₁₃	1	1	yes	characteristic
319	C₂	#11719450	1	1	yes	characteristic
320	C₂	#11719455	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#446	C₂
prime radical	C₂²×D₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#446	D₁₃
hypocenter	C₁₃	#2961
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. C₁₃×#446	C₂	semidirect
3. #11718685	C₂	semidirect
7. #11718492	C₂	semidirect
8. #11718489	C₂	semidirect
9. #11718489	C₂	semidirect
12. #11718362	C₂	semidirect
13. #11718362	C₂	semidirect
15. #11718340	C₂	semidirect
105. C₈×D₁₃	Q₈	semidirect
146. #446	D₁₃	semidirect
266. C₁₃	#2961	semidirect

Maximal subgroups

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

subgroup	index
#11719283	2
C₁₃×#446	2
#11718685	2
#11718500	2
#11718498	2
#11718492	2
#11718489	2
#11718369	2
D₁₃×#97	2
#11718362	2
#11718350	2
#11718340	2
#2961	13