Subgroups — #15777971

Order 3328.

482 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123258	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718107	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718100	C₂	1	1	yes	maximal, characteristic
4	M₃₂×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718094	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112925	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88161	C₄	1	1	yes	characteristic
9	C₂×#88155	C₂²	1	1	yes	characteristic
10	#11719227	C₄	1	1	yes	characteristic
11	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₁₃×#405	C₂²	1	1	yes	characteristic
14	M₁₆×Dic₁₃	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
16	C₂×#88052	C₂²	1	1	yes	characteristic
17	C₂²×#6090	C₂²	1	1	yes	characteristic
18	#11718658	C₄	1	1	yes	characteristic
19	#11718369	—	2	1	no	—
20	#11718368	—	2	1	no	—
21	#11718350	—	2	1	no	—
22	C₄×C₈×D₁₃	—	2	1	no	—
23	#11718107	C₄	1	4	yes	complemented
24	#11718100	C₄	1	4	yes	complemented
25	#11718094	—	2	1	no	—
26	#11718094	—	2	1	no	—
27	#11718094	—	2	1	no	—
28	C₁₆×Dic₁₃	—	2	1	no	—
29	C₂×#6127	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×#6099	C₂×C₄	1	1	yes	characteristic
35	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
36	#88161	—	2	2	no	—
37	#88161	C₈	1	4	yes	—
38	#88155	—	2	1	no	—
39	#88155	—	2	1	no	—
40	#88155	C₂×C₄	1	2	yes	—
41	#88155	C₂×C₄	1	2	yes	—
42	C₂×#6094	—	2	1	no	—
43	C₂×#6094	—	2	1	no	—
44	C₂×C₈×D₁₃	—	2	1	no	—
45	C₂×C₈×D₁₃	—	2	1	no	—
46	#88147	C₈	1	2	yes	—
47	#88141	C₈	1	2	yes	—
48	#88135	—	2	2	no	—
49	#88126	—	2	1	no	—
50	C₄²×D₁₃	—	2	1	no	—
51	C₁₃×M₃₂	—	2	1	no	—
52	C₁₃×M₃₂	C₂×C₄	1	2	yes	complemented
53	C₂×C₁₃×C₁₆	—	2	1	no	—
54	C₂×C₁₃×C₁₆	—	2	1	no	—
55	C₂×C₁₃×C₁₆	—	2	1	no	—
56	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	complemented
57	C₁₃×#97	—	2	1	no	—
58	C₄×C₈×C₁₃	—	2	1	no	—
59	#88055	—	2	1	no	—
60	C₈×Dic₁₃	—	2	1	no	—
61	C₈×Dic₁₃	D₄	1	1	yes	characteristic
62	C₈×Dic₁₃	D₄	1	1	yes	characteristic
63	C₈×Dic₁₃	D₄	1	1	yes	characteristic
64	C₈×Dic₁₃	D₄	1	1	yes	characteristic
65	#88052	—	2	1	no	—
66	#88052	C₂×C₄	1	2	yes	complemented
67	C₂×#6090	—	2	1	no	—
68	C₂×#6090	—	2	1	no	—
69	C₂×#6090	—	2	1	no	—
70	C₂×#6090	C₂×C₄	1	2	yes	complemented
71	#88043	—	2	1	no	—
72	C₄×#1055	—	2	1	no	—
73	#12604	—	13	1	no	maximal, Sylow
74	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₂²⋊D₁₃	—	2	1	no	—
76	C₂²×Dic₁₃	M₁₆	1	2	yes	—
77	#6127	—	2	1	no	—
78	#6127	—	2	1	no	—
79	#6127	C₄²	1	2	yes	—
80	#6127	C₂×C₈	1	2	yes	—
81	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₁₃	—	2	1	no	—
83	C₂×C₄×D₁₃	—	2	2	no	—
84	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
85	C₁₃×M₁₆	—	2	1	no	—
86	C₁₃×M₁₆	C₂×C₈	1	2	yes	—
87	C₂×C₈×C₁₃	—	2	1	no	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
93	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
94	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
95	C₁₃×C₂²⋊C₄	—	2	1	no	—
96	C₄²×C₁₃	—	2	1	no	—
97	#6103	—	2	2	no	—
98	#6101	—	2	2	no	—
99	C₄×Dic₁₃	—	2	2	no	—
100	C₄×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
101	C₄×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
102	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
103	#6099	—	2	1	no	—
104	#6099	C₄²	1	2	yes	—
105	C₂×#1055	—	2	1	no	—
106	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×#1055	C₂×C₈	1	2	yes	—
109	C₂×#1055	M₁₆	1	2	yes	—
110	#6094	—	2	2	no	—
111	#6094	—	2	2	no	—
112	#6094	—	2	4	no	—
113	C₈×D₁₃	—	2	2	no	—
114	C₈×D₁₃	—	2	2	no	—
115	C₈×D₁₃	—	2	4	no	—
116	C₁₃×C₁₆	—	2	2	no	—
117	C₁₃×C₁₆	—	2	2	no	—
118	#6090	—	2	2	no	—
119	#6090	—	2	2	no	—
120	C₄×#131	—	13	1	no	—
121	C₂×#350	—	13	1	no	—
122	C₂×#350	—	13	1	no	—
123	C₄×M₃₂	—	13	1	no	—
124	C₂×#346	—	13	1	no	—
125	#1379	—	13	1	no	—
126	#1379	D₁₃	1	1	yes	characteristic, complemented
127	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
128	C₂²×D₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
136	C₂²⋊D₁₃	—	2	2	no	—
137	C₂²⋊D₁₃	—	2	2	no	—
138	C₂×Dic₁₃	—	2	1	no	—
139	C₂×Dic₁₃	—	4	2	no	—
140	C₂×Dic₁₃	C₂²⋊C₈	1	2	yes	—
141	C₂×Dic₁₃	C₂²⋊C₈	1	2	yes	—
142	D₅₂	—	2	1	no	—
143	D₅₂	C₄×C₈	1	2	yes	—
144	C₄×D₁₃	—	2	2	no	—
145	C₄×D₁₃	—	2	2	no	—
146	C₄×D₁₃	—	4	4	no	—
147	Dic₂₆	—	2	1	no	—
148	Dic₂₆	C₄×C₈	1	2	yes	—
149	C₈×C₁₃	—	2	2	no	—
150	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
151	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
152	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
153	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
154	#1055	—	2	2	no	—
155	#1055	C₂²⋊C₈	1	2	yes	—
156	#1055	C₂²⋊C₈	1	2	yes	—
157	C₂×#131	—	13	1	no	—
158	C₂×#131	—	13	1	no	—
159	C₄×#41	—	13	1	no	—
160	C₂×M₃₂	—	13	1	no	—
161	C₂×M₃₂	C₂×D₁₃	1	1	yes	characteristic
162	C₂²×C₁₆	—	13	1	no	—
163	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
164	#405	—	13	1	no	—
165	#405	—	26	1	no	—
166	#405	—	26	1	no	—
167	#405	C₂×D₁₃	1	1	yes	characteristic
168	C₄×M₁₆	—	13	1	no	—
169	C₄×M₁₆	—	26	1	no	—
170	C₂×C₄×C₈	—	13	1	no	—
171	C₂×C₄×C₈	—	26	1	no	—
172	#350	—	13	4	no	—
173	#350	—	13	4	no	—
174	#346	—	26	1	no	—
175	#346	—	26	1	no	—
176	#346	—	26	1	no	—
177	C₄×C₁₆	—	26	1	no	—
178	C₂²×C₁₃	—	2	2	no	—
179	C₂²×C₁₃	C₄×M₁₆	1	1	yes	characteristic
180	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
181	C₂²×C₁₃	#431	1	1	yes	characteristic
182	C₂×D₁₃	—	2	2	no	—
183	C₂×D₁₃	—	4	2	no	—
184	C₄×C₁₃	—	4	2	no	—
185	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
186	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
187	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
188	C₄×C₁₃	#431	1	1	yes	characteristic
189	Dic₁₃	—	2	2	no	—
190	Dic₁₃	—	2	2	no	—
191	Dic₁₃	—	2	2	no	—
192	C₂×#41	—	13	1	no	—
193	#131	—	13	4	no	—
194	#131	—	13	2	no	—
195	#131	—	13	2	no	—
196	#131	—	26	2	no	—
197	#131	—	26	1	no	—
198	#131	—	26	1	no	—
199	C₂×M₁₆	—	13	1	no	—
200	C₂×M₁₆	—	26	1	no	—
201	C₂×M₁₆	—	26	1	no	—
202	C₂×M₁₆	C₄×D₁₃	1	1	yes	characteristic
203	C₂²×C₈	—	13	1	no	—
204	C₂²×C₈	—	26	1	no	—
205	C₂²×C₈	—	26	1	no	—
206	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
207	C₄×Q₈	—	13	2	no	—
208	C₄×D₄	—	13	2	no	—
209	C₄×D₄	—	26	2	no	—
210	#117	—	26	1	no	—
211	#117	C₄×D₁₃	1	1	yes	characteristic
212	C₂×C₄²	—	13	1	no	—
213	C₂×C₄²	—	26	1	no	—
214	M₃₂	—	2	1	no	—
215	M₃₂	—	13	2	no	—
216	M₃₂	—	26	1	no	—
217	M₃₂	C₄×D₁₃	1	2	yes	complemented
218	C₂×C₁₆	—	2	1	no	—
219	C₂×C₁₆	—	2	1	no	—
220	C₂×C₁₆	—	2	1	no	—
221	C₂×C₁₆	—	13	2	no	—
222	C₂×C₁₆	—	26	1	no	—
223	C₂×C₁₆	—	26	1	no	—
224	C₂×C₁₆	—	26	1	no	—
225	C₂×C₁₆	C₄×D₁₃	1	2	yes	complemented
226	#97	—	2	1	no	—
227	#97	—	26	1	no	—
228	#97	—	26	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	13	1	no	—
231	C₄×C₈	—	13	1	no	—
232	C₄×C₈	—	13	1	no	—
233	C₄×C₈	—	13	1	no	—
234	C₄×C₈	—	26	1	no	—
235	C₄×C₈	—	26	1	no	—
236	C₂×C₁₃	—	2	1	no	—
237	C₂×C₁₃	—	2	1	no	—
238	C₂×C₁₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
239	C₂×C₁₃	#2117	1	2	yes	—
240	D₁₃	—	4	2	no	—
241	#41	—	13	2	no	—
242	#41	—	13	2	no	—
243	#41	—	26	1	no	—
244	#41	—	26	1	no	—
245	C₂×Q₈	—	13	1	no	—
246	C₂×D₄	—	13	1	no	—
247	C₂×D₄	—	26	1	no	—
248	C₂²×C₄	—	13	2	no	—
249	C₂²×C₄	—	26	2	no	—
250	C₂²×C₄	—	26	1	no	—
251	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
252	M₁₆	—	2	1	no	—
253	M₁₆	—	13	2	no	—
254	M₁₆	—	26	4	no	—
255	M₁₆	—	26	1	no	—
256	M₁₆	—	26	2	no	—
257	M₁₆	—	26	2	no	—
258	M₁₆	C₈×D₁₃	1	2	yes	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	13	1	no	—
261	C₂×C₈	—	13	1	no	—
262	C₂×C₈	—	13	2	no	—
263	C₂×C₈	—	13	2	no	—
264	C₂×C₈	—	26	4	no	—
265	C₂×C₈	—	26	1	no	—
266	C₂×C₈	—	26	2	no	—
267	C₂×C₈	—	26	2	no	—
268	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
270	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
271	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
272	C₂×C₈	#6129	1	1	yes	characteristic
273	C₂×C₈	#6131	1	1	yes	characteristic
274	C₄⋊C₄	—	26	2	no	—
275	C₄⋊C₄	C₈×D₁₃	1	2	yes	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	26	2	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	13	1	no	—
280	C₄²	—	13	1	no	—
281	C₄²	—	13	2	no	—
282	C₄²	—	26	2	no	—
283	C₁₆	—	2	2	no	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	26	2	no	—
286	C₁₆	—	26	2	no	—
287	C₁₃	#12604	1	1	yes	characteristic, complemented, Sylow
288	C₂³	—	26	1	no	—
289	C₂³	M₁₆×D₁₃	1	1	yes	characteristic
290	Q₈	—	13	2	no	—
291	Q₈	—	26	1	no	—
292	D₄	—	13	2	no	—
293	D₄	—	26	2	no	—
294	D₄	—	26	2	no	—
295	D₄	—	26	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	13	2	no	—
298	C₂×C₄	—	13	2	no	—
299	C₂×C₄	—	26	1	no	—
300	C₂×C₄	—	26	2	no	—
301	C₂×C₄	—	26	2	no	—
302	C₂×C₄	—	52	4	no	—
303	C₂×C₄	—	52	2	no	—
304	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
305	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
306	C₂×C₄	#88146	1	1	yes	characteristic
307	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
309	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	13	2	no	—
312	C₈	—	13	2	no	—
313	C₈	—	26	2	no	—
314	C₈	#88135	1	1	yes	characteristic
315	C₈	#88135	1	1	yes	characteristic
316	C₈	#88147	1	1	yes	characteristic
317	C₈	#88147	1	1	yes	characteristic
318	C₂²	—	2	2	no	—
319	C₂²	—	26	2	no	—
320	C₂²	—	52	2	no	—
321	C₂²	#11718368	1	1	yes	characteristic
322	C₂²	D₁₃×C₂²⋊C₈	1	1	yes	characteristic
323	C₂²	#11718494	1	1	yes	characteristic
324	C₄	—	4	2	no	—
325	C₄	—	26	2	no	—
326	C₄	—	26	2	no	—
327	C₄	—	26	2	no	—
328	C₄	#11718327	1	1	yes	characteristic
329	C₄	C₄×C₈×D₁₃	1	1	yes	characteristic
330	C₄	D₁₃×C₂²⋊C₈	1	1	yes	characteristic
331	C₄	#11718494	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	2	1	no	—
334	C₂	—	52	2	no	—
335	C₂	#13114593	1	1	yes	characteristic
336	C₂	#13116462	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₁₃
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1379	C₂
socle	C₂²×C₁₃	C₂×C₂²⋊C₈
prime radical	C₂×#6127	C₂×C₄
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1379	D₁₃
hypocenter	C₁₃	#12604
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1379

Complements

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

N	H ≅ G/N	type
4. M₃₂×Dic₁₃	C₂	semidirect
5. C₂×#11718094	C₂	semidirect
6. #13112925	C₂	semidirect
7. C₁₃×#1379	C₂	semidirect
23. #11718107	C₄	semidirect
24. #11718100	C₄	semidirect
52. C₁₃×M₃₂	C₂×C₄	semidirect
56. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
66. #88052	C₂×C₄	semidirect
70. C₂×#6090	C₂×C₄	semidirect
126. #1379	D₁₃	semidirect
217. M₃₂	C₄×D₁₃	semidirect
225. C₂×C₁₆	C₄×D₁₃	semidirect
287. C₁₃	#12604	semidirect

Maximal subgroups

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

subgroup	index
#13123258	2
C₂×#11718107	2
C₂×#11718100	2
M₃₂×Dic₁₃	2
C₂×#11718094	2
#13112925	2
C₁₃×#1379	2
#12604	13