Subgroups — #15871097

Order 3328.

746 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123771	C₂	1	1	yes	maximal, characteristic, complemented
2	#13123548	C₂	1	1	yes	maximal, characteristic
3	#13115814	C₂	1	1	yes	maximal, characteristic, complemented
4	#13115665	C₂	1	1	yes	maximal, characteristic
5	#13115549	C₂	1	4	yes	maximal, complemented
6	#13114974	C₂	1	1	yes	maximal, characteristic, complemented
7	#13114683	C₂	1	4	yes	maximal, complemented
8	C₄×#88055	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#3131	C₂	1	1	yes	maximal, characteristic
10	C₁₃×#533	C₂²	1	1	yes	characteristic
11	C₁₃×#532	C₂²	1	1	yes	characteristic
12	C₂×#88202	C₂²	1	2	yes	—
13	C₁₃×#488	C₄	1	4	yes	complemented
14	C₁₃×#487	C₄	1	2	yes	complemented
15	C₁₃×#486	C₄	1	2	yes	complemented
16	C₁₃×#416	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×#88077	C₂²	1	4	yes	—
19	#11718895	C₂²	1	2	yes	—
20	C₂×#88055	C₂²	1	2	yes	—
21	C₂×#88055	C₂²	1	2	yes	—
22	#11718685	—	2	2	no	—
23	Q₈×#1055	—	2	2	no	—
24	#11718673	—	2	2	no	—
25	D₄×#1055	—	2	2	no	—
26	#11718660	C₂²	1	2	yes	—
27	#11718658	C₂²	1	2	yes	—
28	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
29	#11718585	C₂²	1	1	yes	characteristic
30	#11718585	C₂²	1	1	yes	characteristic
31	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
32	#11718144	C₂²	1	4	yes	—
33	#11718124	C₂²	1	4	yes	—
34	C₂×C₁₃×#41	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
36	C₁₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
37	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
38	C₁₃×#127	C₂×C₄	1	1	yes	characteristic
39	C₁₃×#125	C₂×C₄	1	2	yes	—
40	C₁₃×#124	C₂×C₄	1	2	yes	—
41	C₁₃×C₂²⋊Q₈	—	2	2	no	—
42	C₁₃×#121	—	2	2	no	—
43	C₄×Q₈×C₁₃	—	2	2	no	—
44	C₄×D₄×C₁₃	—	2	2	no	—
45	C₁₃×#117	C₂³	1	2	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	#88202	—	2	8	no	—
48	C₂×C₄×Dic₁₃	—	2	2	no	—
49	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
50	C₂×#6099	—	2	2	no	—
51	C₂×#6099	C₂³	1	2	yes	—
52	C₂×#6099	C₂³	1	2	yes	—
53	C₂²×#1055	—	2	2	no	—
54	C₂²×#1055	C₂³	1	1	yes	characteristic
55	C₂²×#1055	C₂³	1	1	yes	characteristic
56	C₁₃×#103	C₂×C₄	1	8	yes	—
57	C₁₃×#102	C₂×C₄	1	8	yes	—
58	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
60	#88077	—	2	8	no	—
61	C₂²⋊#1055	—	2	4	no	—
62	#88055	—	2	4	no	—
63	#88055	—	2	4	no	—
64	C₄⋊#1055	—	2	2	no	—
65	C₄⋊#1055	—	2	2	no	—
66	C₄⋊#1055	—	2	4	no	—
67	#88043	—	2	2	no	—
68	C₄×#1055	—	2	2	no	—
69	C₄×#1055	D₄	1	2	yes	—
70	C₄×#1055	D₄	1	2	yes	—
71	#21660	—	13	1	no	maximal, Sylow
72	C₁₃×#41	—	2	4	no	—
73	C₁₃×#41	—	2	4	no	—
74	C₂×Q₈×C₁₃	—	2	2	no	—
75	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂×D₄×C₁₃	—	2	2	no	—
77	C₂×D₄×C₁₃	—	2	2	no	—
78	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₁₃	—	2	2	no	—
80	C₂²×C₄×C₁₃	#41	1	2	yes	—
81	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₂²×Dic₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
86	C₁₃×C₄⋊C₄	—	2	2	no	—
87	C₁₃×C₄⋊C₄	—	2	4	no	—
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
90	C₁₃×C₂²⋊C₄	—	2	2	no	—
91	C₁₃×C₂²⋊C₄	—	2	4	no	—
92	C₄²×C₁₃	—	2	2	no	—
93	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₁₃	C₂×D₄	1	2	yes	—
96	C₄×Dic₁₃	—	2	2	no	—
97	C₄×Dic₁₃	—	4	4	no	—
98	C₄×Dic₁₃	#41	1	4	yes	—
99	#6099	—	2	4	no	—
100	#6099	—	2	4	no	—
101	#6099	—	4	4	no	—
102	C₂×#1055	—	2	2	no	—
103	C₂×#1055	—	2	2	no	—
104	C₂×#1055	—	2	2	no	—
105	C₂×#1055	—	4	4	no	—
106	C₂×#1055	C₂×D₄	1	2	yes	—
107	C₂×#1055	C₂×D₄	1	1	yes	characteristic
108	C₂×#1055	C₂×D₄	1	1	yes	characteristic
109	C₂×#1055	#41	1	4	yes	—
110	#3131	D₁₃	1	1	yes	characteristic
111	#2960	—	13	1	no	—
112	#2931	—	13	1	no	—
113	#1994	—	13	1	no	—
114	#1962	—	13	1	no	—
115	#1938	—	13	4	no	—
116	#1837	—	13	1	no	—
117	#1765	—	13	4	no	—
118	C₄×#97	—	13	1	no	—
119	C₂³×C₁₃	—	2	2	no	—
120	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
121	Q₈×C₁₃	—	2	4	no	—
122	Q₈×C₁₃	—	4	4	no	—
123	D₄×C₁₃	—	2	4	no	—
124	D₄×C₁₃	—	4	4	no	—
125	D₄×C₁₃	—	4	4	no	—
126	C₂×C₄×C₁₃	—	2	4	no	—
127	C₂×C₄×C₁₃	—	2	4	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
136	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
137	C₂×Dic₁₃	—	2	4	no	—
138	C₂×Dic₁₃	—	4	4	no	—
139	C₈×C₁₃	—	2	4	no	—
140	#1055	—	2	2	no	—
141	#1055	—	4	2	no	—
142	#1055	—	4	2	no	—
143	#1055	#124	1	2	yes	—
144	#1055	#124	1	2	yes	—
145	C₂×#131	—	13	2	no	—
146	#533	C₂×D₁₃	1	1	yes	characteristic
147	#532	C₂×D₁₃	1	1	yes	characteristic
148	#488	Dic₁₃	1	4	yes	complemented
149	#487	Dic₁₃	1	2	yes	complemented
150	#486	Dic₁₃	1	2	yes	complemented
151	#446	—	26	2	no	—
152	C₈×Q₈	—	26	2	no	—
153	#436	—	26	2	no	—
154	C₈×D₄	—	26	2	no	—
155	#424	—	13	2	no	—
156	#423	—	13	1	no	—
157	#423	—	13	1	no	—
158	C₂×#104	—	13	4	no	—
159	#416	C₂×D₁₃	1	4	yes	—
160	#408	—	13	2	no	—
161	#405	—	13	2	no	—
162	C₂×#97	—	13	2	no	—
163	C₂×#97	—	13	2	no	—
164	C₂×C₄×C₈	—	13	1	no	—
165	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
166	C₄³	—	13	1	no	—
167	#341	—	13	4	no	—
168	#339	—	13	4	no	—
169	C₂²×C₁₃	—	2	2	no	—
170	C₂²×C₁₃	—	4	4	no	—
171	C₂²×C₁₃	—	4	2	no	—
172	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂²×C₁₃	#529	1	2	yes	—
174	C₄×C₁₃	—	2	2	no	—
175	C₄×C₁₃	—	4	4	no	—
176	C₄×C₁₃	—	4	2	no	—
177	C₄×C₁₃	#444	1	4	yes	—
178	C₄×C₁₃	C₄×#41	1	2	yes	—
179	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
180	C₄×C₁₃	#532	1	1	yes	characteristic
181	Dic₁₃	—	4	4	no	—
182	C₂×#41	C₂²×D₁₃	1	2	yes	—
183	#131	—	26	8	no	—
184	C₂×M₁₆	—	13	2	no	—
185	C₂×M₁₆	—	13	2	no	—
186	C₂×M₁₆	—	26	2	no	—
187	C₂²×C₈	—	13	1	no	—
188	C₂²×C₈	—	13	1	no	—
189	C₂²×C₈	—	26	2	no	—
190	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
191	C₄⋊Q₈	C₂×Dic₁₃	1	1	yes	characteristic
192	C₄⋊Q₈	C₂×Dic₁₃	1	2	yes	—
193	#127	C₂×Dic₁₃	1	1	yes	characteristic
194	#125	C₂×Dic₁₃	1	2	yes	—
195	#124	C₂×Dic₁₃	1	2	yes	—
196	C₂²⋊Q₈	—	2	2	no	—
197	#121	—	2	2	no	—
198	C₄×Q₈	—	2	2	no	—
199	C₄×D₄	—	2	2	no	—
200	#117	C₂²×D₁₃	1	2	yes	—
201	C₂×C₄²	—	13	2	no	—
202	C₂×C₄²	—	26	2	no	—
203	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
204	C₄⋊C₈	—	26	4	no	—
205	C₄⋊C₈	—	26	2	no	—
206	C₄⋊C₈	—	26	2	no	—
207	#104	—	26	8	no	—
208	#103	C₂×Dic₁₃	1	8	yes	—
209	#102	C₂×Dic₁₃	1	8	yes	—
210	C₂²⋊C₈	—	26	4	no	—
211	#97	—	26	2	no	—
212	#97	—	26	4	no	—
213	#97	—	26	4	no	—
214	C₄×C₈	—	13	2	no	—
215	C₄×C₈	—	13	2	no	—
216	C₄×C₈	—	26	2	no	—
217	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
218	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
219	C₂×C₁₃	—	2	2	no	—
220	C₂×C₁₃	—	8	2	no	—
221	C₂×C₁₃	C₄×#124	1	1	yes	characteristic
222	C₂×C₁₃	C₂×#444	1	2	yes	—
223	#41	—	2	4	no	—
224	#41	—	2	4	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	C₂²×Dic₁₃	1	2	yes	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂²×Dic₁₃	1	2	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	26	2	no	—
232	C₂²×C₄	#6129	1	2	yes	—
233	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
234	M₁₆	—	26	4	no	—
235	M₁₆	—	26	4	no	—
236	M₁₆	—	52	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	13	1	no	—
240	C₂×C₈	—	13	1	no	—
241	C₂×C₈	—	13	4	no	—
242	C₂×C₈	—	13	2	no	—
243	C₂×C₈	—	26	2	no	—
244	C₂×C₈	—	26	2	no	—
245	C₂×C₈	—	26	2	no	—
246	C₂×C₈	—	52	4	no	—
247	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	C₂²×Dic₁₃	1	4	yes	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	13	4	no	—
256	C₄²	—	26	2	no	—
257	C₄²	—	52	4	no	—
258	C₄²	D₄×D₁₃	1	2	yes	—
259	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
260	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
261	C₁₃	#21660	1	1	yes	characteristic, complemented, Sylow
262	C₂³	—	2	2	no	—
263	C₂³	C₂×#6129	1	1	yes	characteristic
264	Q₈	—	2	4	no	—
265	Q₈	—	4	4	no	—
266	D₄	—	2	4	no	—
267	D₄	—	4	4	no	—
268	D₄	—	4	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	26	4	no	—
275	C₂×C₄	—	52	4	no	—
276	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
277	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
278	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
279	C₂×C₄	#41×D₁₃	1	2	yes	—
280	C₂×C₄	#41×D₁₃	1	2	yes	—
281	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
282	C₈	—	2	4	no	—
283	C₈	—	13	2	no	—
284	C₈	—	13	2	no	—
285	C₈	—	26	2	no	—
286	C₈	—	52	2	no	—
287	C₈	—	52	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	4	no	—
290	C₂²	—	4	2	no	—
291	C₂²	#11719385	1	2	yes	—
292	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
293	C₄	—	2	2	no	—
294	C₄	—	4	4	no	—
295	C₄	—	4	2	no	—
296	C₄	—	52	4	no	—
297	C₄	#11718858	1	4	yes	—
298	C₄	#11719373	1	1	yes	characteristic
299	C₄	D₁₃×#124	1	1	yes	characteristic
300	C₄	#41×Dic₁₃	1	2	yes	—
301	C₂	—	2	2	no	—
302	C₂	—	8	2	no	—
303	C₂	#124×Dic₁₃	1	1	yes	characteristic
304	C₂	C₂×#11718858	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#3131	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₁₃×#532	C₂²
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#3131	D₁₃
hypocenter	C₁₃	#21660
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3131

Complements

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

N	H ≅ G/N	type
1. #13123771	C₂	semidirect
3. #13115814	C₂	semidirect
5. #13115549	C₂	semidirect
6. #13114974	C₂	semidirect
7. #13114683	C₂	semidirect
8. C₄×#88055	C₂	semidirect
13. C₁₃×#488	C₄	semidirect
14. C₁₃×#487	C₄	semidirect
15. C₁₃×#486	C₄	semidirect
148. #488	Dic₁₃	semidirect
149. #487	Dic₁₃	semidirect
150. #486	Dic₁₃	semidirect
261. C₁₃	#21660	semidirect

Maximal subgroups

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

subgroup	index
#13123771	2
#13123548	2
#13115814	2
#13115665	2
#13115549	2
#13114974	2
#13114683	2
C₄×#88055	2
C₁₃×#3131	2
#21660	13