Subgroups — #15777953

Order 3328.

514 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718350	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718099	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718099	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718094	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112914	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#6094	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88126	C₄	1	1	yes	characteristic
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×#88055	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
16	C₂²×#6090	C₂²	1	1	yes	characteristic
17	C₂²×#6090	C₂²	1	1	yes	characteristic
18	C₂×#88043	C₄	1	1	yes	characteristic
19	#11718350	—	2	2	no	—
20	#11718350	—	2	2	no	—
21	#11718099	C₄	1	4	yes	complemented
22	#11718099	C₄	1	4	yes	complemented
23	#11718094	—	2	2	no	—
24	C₁₆×Dic₁₃	—	2	2	no	—
25	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×#6103	C₈	1	2	yes	—
30	C₂×#6101	C₈	1	2	yes	—
31	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
33	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
34	C₂×#6094	—	2	2	no	—
35	C₂×#6094	—	2	2	no	—
36	C₂×#6094	C₈	1	4	yes	—
37	C₂×C₈×D₁₃	—	2	1	no	—
38	C₂×C₈×D₁₃	—	2	1	no	—
39	C₂×C₈×D₁₃	—	2	1	no	—
40	C₂×C₈×D₁₃	—	2	1	no	—
41	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
43	#88126	—	2	2	no	—
44	#88126	—	2	2	no	—
45	C₂×C₁₃×C₁₆	—	2	1	no	—
46	C₂×C₁₃×C₁₆	—	2	1	no	—
47	C₂×C₁₃×C₁₆	—	2	1	no	—
48	C₂×C₁₃×C₁₆	—	2	1	no	—
49	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
50	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₁₃	—	2	1	no	—
52	C₄×C₈×C₁₃	—	2	1	no	—
53	#88055	—	2	2	no	—
54	C₈×Dic₁₃	D₄	1	2	yes	—
55	C₈×Dic₁₃	D₄	1	2	yes	—
56	C₂×#6090	—	2	1	no	—
57	C₂×#6090	—	2	1	no	—
58	C₂×#6090	—	2	1	no	—
59	C₂×#6090	—	2	1	no	—
60	C₂×#6090	C₂×C₄	1	2	yes	—
61	C₂×#6090	C₂×C₄	1	2	yes	—
62	#88043	—	2	2	no	—
63	#12600	—	13	1	no	maximal, Sylow
64	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
68	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
69	C₂×C₄×D₁₃	—	2	1	no	—
70	C₂×C₄×D₁₃	—	2	1	no	—
71	C₂×C₄×D₁₃	—	2	1	no	—
72	C₂×C₄×D₁₃	—	2	1	no	—
73	C₂×C₄×D₁₃	C₄²	1	2	yes	—
74	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
75	C₂×C₈×C₁₃	—	2	1	no	—
76	C₂×C₈×C₁₃	—	2	1	no	—
77	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
84	C₄²×C₁₃	—	2	1	no	—
85	C₄²×C₁₃	—	2	1	no	—
86	#6103	—	2	4	no	—
87	#6101	—	2	4	no	—
88	C₄×Dic₁₃	C₂²⋊C₄	1	2	yes	—
89	C₄×Dic₁₃	M₁₆	1	2	yes	—
90	C₂×#1055	—	2	1	no	—
91	C₂×#1055	—	2	1	no	—
92	C₂×#1055	C₄²	1	2	yes	—
93	C₂×#1055	C₂²⋊C₄	1	2	yes	—
94	C₂×#1055	C₂×C₈	1	2	yes	—
95	C₂×#1055	M₁₆	1	2	yes	—
96	#6094	—	2	8	no	—
97	C₈×D₁₃	—	2	2	no	—
98	C₈×D₁₃	—	2	2	no	—
99	C₈×D₁₃	—	2	2	no	—
100	C₈×D₁₃	—	2	2	no	—
101	C₁₃×C₁₆	—	2	2	no	—
102	C₁₃×C₁₆	—	2	2	no	—
103	#6090	—	2	2	no	—
104	#6090	—	2	2	no	—
105	C₂×#405	—	13	1	no	—
106	C₂×C₂²⋊C₁₆	—	13	1	no	—
107	C₂×C₂²⋊C₁₆	—	13	1	no	—
108	C₂×#346	—	13	1	no	—
109	C₂×C₄×C₁₆	—	13	1	no	—
110	#1374	—	13	1	no	—
111	#1374	D₁₃	1	1	yes	characteristic, complemented
112	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
113	C₂²×D₁₃	—	2	1	no	—
114	C₂²×D₁₃	—	2	1	no	—
115	C₂²×D₁₃	—	2	2	no	—
116	C₂²×D₁₃	—	2	2	no	—
117	C₂²×D₁₃	C₄×C₈	1	2	yes	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
127	C₂×Dic₁₃	—	2	1	no	—
128	C₂×Dic₁₃	—	2	1	no	—
129	C₂×Dic₁₃	—	2	2	no	—
130	C₂×Dic₁₃	—	2	2	no	—
131	C₂×Dic₁₃	C₄×C₈	1	2	yes	—
132	C₂×Dic₁₃	C₂²⋊C₈	1	4	yes	—
133	C₄×D₁₃	—	2	2	no	—
134	C₄×D₁₃	—	2	2	no	—
135	C₄×D₁₃	—	2	2	no	—
136	C₄×D₁₃	—	2	2	no	—
137	C₈×C₁₃	—	2	2	no	—
138	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
139	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
140	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
141	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
142	#1055	—	2	2	no	—
143	#1055	C₂²⋊C₈	1	4	yes	—
144	C₂²×M₁₆	—	13	1	no	—
145	C₂³×C₈	—	13	1	no	—
146	C₂×#117	—	13	1	no	—
147	C₂²×C₁₆	—	13	1	no	—
148	C₂²×C₁₆	—	13	1	no	—
149	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
150	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
151	#405	—	26	2	no	—
152	#405	—	26	2	no	—
153	C₂×#97	—	13	1	no	—
154	C₂×#97	—	13	1	no	—
155	C₂×C₄×C₈	—	13	1	no	—
156	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
157	C₂²⋊C₁₆	—	13	4	no	—
158	C₂²⋊C₁₆	—	13	4	no	—
159	#346	—	26	2	no	—
160	C₄×C₁₆	—	26	2	no	—
161	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
162	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
163	C₂²×C₁₃	#431	1	1	yes	characteristic
164	C₂²×C₁₃	#504	1	2	yes	—
165	C₂²×C₁₃	#504	1	2	yes	—
166	C₂×D₁₃	—	2	2	no	—
167	C₂×D₁₃	—	2	2	no	—
168	C₂×D₁₃	—	2	2	no	—
169	C₂×D₁₃	—	4	2	no	—
170	C₂×D₁₃	—	4	2	no	—
171	C₄×C₁₃	—	4	2	no	—
172	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
173	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
174	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
175	C₄×C₁₃	#431	1	1	yes	characteristic
176	Dic₁₃	—	2	4	no	—
177	Dic₁₃	—	2	2	no	—
178	C₂³×C₄	—	13	1	no	—
179	C₂×M₁₆	—	13	4	no	—
180	C₂×M₁₆	—	26	2	no	—
181	C₂×M₁₆	—	26	2	no	—
182	C₂²×C₈	—	13	1	no	—
183	C₂²×C₈	—	13	1	no	—
184	C₂²×C₈	—	13	2	no	—
185	C₂²×C₈	—	13	2	no	—
186	C₂²×C₈	—	26	1	no	—
187	C₂²×C₈	—	26	1	no	—
188	C₂²×C₈	—	26	1	no	—
189	C₂²×C₈	—	26	1	no	—
190	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
191	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
192	#117	—	26	2	no	—
193	#117	—	26	2	no	—
194	C₂×C₄⋊C₄	—	13	2	no	—
195	C₂×C₂²⋊C₄	—	13	2	no	—
196	C₂×C₄²	—	13	1	no	—
197	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
198	C₂×C₁₆	—	2	1	no	—
199	C₂×C₁₆	—	2	1	no	—
200	C₂×C₁₆	—	2	1	no	—
201	C₂×C₁₆	—	2	1	no	—
202	C₂×C₁₆	—	13	2	no	—
203	C₂×C₁₆	—	13	2	no	—
204	C₂×C₁₆	—	26	1	no	—
205	C₂×C₁₆	—	26	1	no	—
206	C₂×C₁₆	—	26	1	no	—
207	C₂×C₁₆	—	26	1	no	—
208	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
209	C₂×C₁₆	C₄×D₁₃	1	2	yes	—
210	#97	—	26	2	no	—
211	#97	—	26	2	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	13	2	no	—
215	C₄×C₈	—	13	2	no	—
216	C₂×C₁₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
217	C₂×C₁₃	#2108	1	1	yes	characteristic
218	C₂×C₁₃	#2108	1	1	yes	characteristic
219	C₂×C₁₃	#2113	1	2	yes	—
220	C₂×C₁₃	#2164	1	2	yes	—
221	D₁₃	—	4	2	no	—
222	C₂⁴	—	13	1	no	—
223	C₂²×C₄	—	13	2	no	—
224	C₂²×C₄	—	13	1	no	—
225	C₂²×C₄	—	13	2	no	—
226	C₂²×C₄	—	13	2	no	—
227	C₂²×C₄	—	26	1	no	—
228	C₂²×C₄	—	26	1	no	—
229	C₂²×C₄	—	26	1	no	—
230	C₂²×C₄	—	26	1	no	—
231	C₂²×C₄	C₈×D₁₃	1	2	yes	—
232	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
233	M₁₆	—	26	8	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	13	2	no	—
237	C₂×C₈	—	13	2	no	—
238	C₂×C₈	—	13	2	no	—
239	C₂×C₈	—	13	2	no	—
240	C₂×C₈	—	26	1	no	—
241	C₂×C₈	—	26	1	no	—
242	C₂×C₈	—	26	2	no	—
243	C₂×C₈	—	26	2	no	—
244	C₂×C₈	—	26	2	no	—
245	C₂×C₈	—	26	2	no	—
246	C₂×C₈	C₈×D₁₃	1	2	yes	—
247	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
248	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
249	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
250	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
251	C₂×C₈	#6129	1	1	yes	characteristic
252	C₂×C₈	#6131	1	1	yes	characteristic
253	C₄⋊C₄	—	26	4	no	—
254	C₂²⋊C₄	—	26	4	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	13	2	no	—
258	C₄²	—	13	2	no	—
259	C₁₆	—	2	2	no	—
260	C₁₆	—	2	2	no	—
261	C₁₆	—	26	2	no	—
262	C₁₆	—	26	2	no	—
263	C₁₃	#12600	1	1	yes	characteristic, complemented, Sylow
264	C₂³	—	13	2	no	—
265	C₂³	—	26	1	no	—
266	C₂³	—	26	2	no	—
267	C₂³	—	26	2	no	—
268	C₂³	—	26	1	no	—
269	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	13	2	no	—
274	C₂×C₄	—	13	4	no	—
275	C₂×C₄	—	26	1	no	—
276	C₂×C₄	—	26	2	no	—
277	C₂×C₄	—	26	2	no	—
278	C₂×C₄	—	26	1	no	—
279	C₂×C₄	—	26	2	no	—
280	C₂×C₄	—	26	2	no	—
281	C₂×C₄	—	26	2	no	—
282	C₂×C₄	—	26	2	no	—
283	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
284	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
285	C₂×C₄	#88146	1	1	yes	characteristic
286	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
287	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
288	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	13	4	no	—
291	C₈	—	26	2	no	—
292	C₈	#88135	1	1	yes	characteristic
293	C₈	#88135	1	1	yes	characteristic
294	C₈	#88147	1	1	yes	characteristic
295	C₈	#88147	1	1	yes	characteristic
296	C₂²	—	26	2	no	—
297	C₂²	—	26	2	no	—
298	C₂²	—	26	2	no	—
299	C₂²	—	52	2	no	—
300	C₂²	—	52	2	no	—
301	C₂²	C₄×C₈×D₁₃	1	1	yes	characteristic
302	C₂²	D₁₃×C₂²⋊C₈	1	1	yes	characteristic
303	C₂²	#11718494	1	1	yes	characteristic
304	C₂²	#11718562	1	2	yes	—
305	C₂²	#11718568	1	2	yes	—
306	C₄	—	4	2	no	—
307	C₄	—	26	4	no	—
308	C₄	—	26	2	no	—
309	C₄	#11718327	1	1	yes	characteristic
310	C₄	#11718368	1	1	yes	characteristic
311	C₄	D₁₃×C₂²⋊C₈	1	1	yes	characteristic
312	C₄	#11718494	1	1	yes	characteristic
313	C₂	—	52	2	no	—
314	C₂	#13114593	1	1	yes	characteristic
315	C₂	#13116431	1	1	yes	characteristic
316	C₂	#13116434	1	1	yes	characteristic
317	C₂	#13116448	1	2	yes	—
318	C₂	#13116642	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₁₃×#1374	C₂
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1374	D₁₃
hypocenter	C₁₃	#12600
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
4. C₂×#11718094	C₂	semidirect
5. C₂×C₁₆×Dic₁₃	C₂	semidirect
6. #13112914	C₂	semidirect
7. C₁₃×#1374	C₂	semidirect
21. #11718099	C₄	semidirect
22. #11718099	C₄	semidirect
111. #1374	D₁₃	semidirect
263. C₁₃	#12600	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718350	2
C₂×#11718099	2
C₂×#11718094	2
C₂×C₁₆×Dic₁₃	2
#13112914	2
C₁₃×#1374	2
#12600	13