Subgroups — #15753985

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	#13123547	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×#1055	C₂	1	1	yes	maximal, characteristic, complemented
3	#13113483	C₂	1	1	yes	maximal, characteristic
4	#13113330	C₂	1	1	yes	maximal, characteristic
5	#13113120	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112731	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1303	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88161	—	2	1	no	—
9	#11719182	C₂²	1	1	yes	characteristic
10	C₁₃×#431	C₂²	1	1	yes	characteristic
11	C₁₃×#423	C₂²	1	1	yes	characteristic
12	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	#11718790	—	2	1	no	—
14	#11718586	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
16	C₂×C₄×#1055	—	2	1	no	—
17	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
18	#11718499	—	2	1	no	—
19	#11718499	C₄	1	2	yes	complemented
20	#11718495	—	2	1	no	—
21	#11718495	C₄	1	2	yes	complemented
22	#11718116	—	2	1	no	—
23	#11718078	C₄	1	2	yes	complemented
24	#11718077	C₄	1	2	yes	complemented
25	#11718046	—	2	1	no	—
26	C₈×#1055	—	2	1	no	—
27	#11718043	C₄	1	4	yes	—
28	C₂×#6127	—	2	1	no	—
29	C₂×C₁₃×M₁₆	—	2	1	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
31	C₂×C₄²×C₁₃	—	2	1	no	—
32	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
33	#88184	—	2	1	no	—
34	#88184	C₂×C₄	1	2	yes	—
35	C₂²×#1055	—	2	1	no	—
36	C₂²×#1055	D₄	1	1	yes	characteristic
37	C₂²×#1055	D₄	1	1	yes	characteristic
38	#88161	—	4	2	no	—
39	#88161	—	4	2	no	—
40	C₂×#6094	—	2	2	no	—
41	C₂×C₈×D₁₃	—	2	2	no	—
42	#88129	—	2	1	no	—
43	#88128	C₂×C₄	1	1	yes	characteristic
44	C₄×D₅₂	—	2	1	no	—
45	C₄×D₅₂	C₂×C₄	1	2	yes	—
46	#88123	C₂×C₄	1	1	yes	characteristic
47	C₁₃×C₄⋊C₈	—	2	1	no	—
48	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
49	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₁₃×C₂²⋊C₈	—	2	1	no	—
51	C₄×C₈×C₁₃	—	2	1	no	—
52	C₁₃×#95	C₂×C₄	1	2	yes	—
53	C₂²⋊#1055	C₂×C₄	1	2	yes	—
54	#88060	—	2	2	no	—
55	#88060	—	2	2	no	—
56	C₄⋊#1055	—	2	1	no	—
57	C₄⋊#1055	—	2	1	no	—
58	C₄⋊#1055	C₂×C₄	1	2	yes	—
59	C₄×#1055	—	2	1	no	—
60	C₄×#1055	—	4	2	no	—
61	C₄×#1055	C₂×C₄	1	1	yes	characteristic
62	C₄×#1055	C₂×C₄	1	1	yes	characteristic
63	C₄×#1055	D₄	1	1	yes	characteristic
64	C₄×#1055	D₄	1	1	yes	characteristic
65	#8251	—	13	1	no	maximal, Sylow
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
70	C₂×C₂²⋊D₁₃	—	2	2	no	—
71	#6127	—	4	1	no	—
72	#6127	—	4	1	no	—
73	#6127	—	4	1	no	—
74	#6127	—	4	1	no	—
75	C₂×D₅₂	—	2	1	no	—
76	C₂×D₅₂	—	2	1	no	—
77	C₂×D₅₂	C₄²	1	2	yes	—
78	C₂×C₄×D₁₃	—	2	2	no	—
79	C₂×Dic₂₆	—	2	1	no	—
80	C₁₃×M₁₆	—	4	1	no	—
81	C₁₃×M₁₆	—	4	1	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₁₃×C₄⋊C₄	—	2	4	no	—
85	C₄²×C₁₃	—	4	1	no	—
86	C₄²×C₁₃	—	4	1	no	—
87	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₃	#41	1	1	yes	characteristic
91	#6103	—	2	2	no	—
92	#6103	—	2	2	no	—
93	C₄⋊Dic₁₃	—	2	1	no	—
94	C₄⋊Dic₁₃	C₄²	1	2	yes	—
95	C₂×#1055	—	2	1	no	—
96	C₂×#1055	—	2	1	no	—
97	C₂×#1055	—	2	2	no	—
98	C₂×#1055	—	2	2	no	—
99	C₂×#1055	—	4	1	no	—
100	C₂×#1055	—	4	1	no	—
101	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×#1055	D₈	1	2	yes	—
106	C₂×#1055	SD₁₆	1	2	yes	—
107	#6094	—	4	4	no	—
108	C₈×D₁₃	—	4	4	no	—
109	#2931	—	13	1	no	—
110	C₈×C₄⋊C₄	—	13	1	no	—
111	#1540	—	13	1	no	—
112	#1502	—	13	1	no	—
113	#1450	—	13	1	no	—
114	#1303	D₁₃	1	1	yes	characteristic, complemented
115	#1301	—	13	1	no	—
116	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	4	no	—
120	C₂×C₄×C₁₃	—	4	1	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
133	C₂²⋊D₁₃	—	4	2	no	—
134	C₂²⋊D₁₃	—	4	2	no	—
135	C₂×Dic₁₃	—	2	2	no	—
136	D₅₂	—	4	2	no	—
137	D₅₂	—	4	1	no	—
138	D₅₂	—	4	1	no	—
139	D₅₂	—	4	2	no	—
140	C₄×D₁₃	—	4	2	no	—
141	C₄×D₁₃	—	4	2	no	—
142	Dic₂₆	—	4	1	no	—
143	Dic₂₆	—	4	1	no	—
144	C₈×C₁₃	—	4	2	no	—
145	C₈×C₁₃	—	4	2	no	—
146	#1055	—	2	1	no	—
147	#1055	—	2	1	no	—
148	#1055	—	4	2	no	—
149	#1055	—	4	2	no	—
150	#1055	#102	1	2	yes	—
151	#1055	#102	1	2	yes	—
152	C₂×#131	—	26	1	no	—
153	#532	—	13	1	no	—
154	#436	—	13	2	no	—
155	#436	—	26	1	no	—
156	C₈×D₄	—	13	2	no	—
157	C₈×D₄	—	26	1	no	—
158	#431	—	13	1	no	—
159	#431	C₂×D₁₃	1	1	yes	characteristic
160	#423	C₂×D₁₃	1	1	yes	characteristic
161	C₂×C₄⋊C₈	—	13	1	no	—
162	#408	—	26	1	no	—
163	C₂×C₄×C₈	—	13	1	no	—
164	C₂×C₄×C₈	—	26	1	no	—
165	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
166	#336	—	26	1	no	—
167	#332	—	13	2	no	—
168	#331	—	13	2	no	—
169	#327	—	13	4	no	—
170	#322	—	26	1	no	—
171	C₈²	—	26	1	no	—
172	C₂²×C₁₃	—	2	1	no	—
173	C₂²×C₁₃	—	2	1	no	—
174	C₂²×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
175	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
176	C₂²×C₁₃	C₄×SD₁₆	1	1	yes	characteristic
177	C₂×D₁₃	—	4	2	no	—
178	C₂×D₁₃	—	4	2	no	—
179	C₂×D₁₃	—	4	2	no	—
180	C₄×C₁₃	—	2	1	no	—
181	C₄×C₁₃	—	2	1	no	—
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	—	4	4	no	—
184	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
185	C₄×C₁₃	#416	1	1	yes	characteristic
186	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
187	C₄×C₁₃	C₄×SD₁₆	1	1	yes	characteristic
188	C₄×C₁₃	#443	1	2	yes	—
189	Dic₁₃	—	4	2	no	—
190	C₂×#41	—	26	1	no	—
191	#131	—	52	2	no	—
192	#131	—	52	2	no	—
193	C₂×M₁₆	—	2	1	no	—
194	C₂×M₁₆	—	26	2	no	—
195	C₂²×C₈	—	13	1	no	—
196	C₂²×C₈	—	13	1	no	—
197	C₂²×C₈	—	26	2	no	—
198	C₂²×C₈	—	26	1	no	—
199	C₄⋊Q₈	—	13	1	no	—
200	#127	—	13	1	no	—
201	#124	—	26	1	no	—
202	#121	—	13	2	no	—
203	#121	—	26	1	no	—
204	C₄×D₄	—	13	2	no	—
205	C₄×D₄	—	26	1	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	13	2	no	—
211	C₄⋊C₈	—	26	1	no	—
212	C₄⋊C₈	—	26	1	no	—
213	C₄⋊C₈	C₄×D₁₃	1	2	yes	—
214	C₄⋊C₈	C₄×D₁₃	1	2	yes	—
215	C₂²⋊C₈	—	2	1	no	—
216	C₂²⋊C₈	—	13	2	no	—
217	C₂²⋊C₈	—	26	2	no	—
218	C₂²⋊C₈	—	26	2	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	13	1	no	—
221	C₄×C₈	—	13	1	no	—
222	C₄×C₈	—	13	1	no	—
223	C₄×C₈	—	13	1	no	—
224	C₄×C₈	—	26	1	no	—
225	C₄×C₈	—	52	2	no	—
226	#95	C₄×D₁₃	1	2	yes	—
227	C₂×C₁₃	—	2	1	no	—
228	C₂×C₁₃	—	2	1	no	—
229	C₂×C₁₃	#1757	1	1	yes	characteristic
230	C₂×C₁₃	C₄×#102	1	1	yes	characteristic
231	C₂×C₁₃	#1764	1	1	yes	characteristic
232	D₁₃	—	8	2	no	—
233	#41	—	52	1	no	—
234	#41	—	52	1	no	—
235	#41	—	52	1	no	—
236	#41	—	52	1	no	—
237	C₂×Q₈	—	26	1	no	—
238	C₂×D₄	—	13	2	no	—
239	C₂×D₄	—	26	2	no	—
240	C₂×D₄	—	26	1	no	—
241	C₂×D₄	—	26	1	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	26	2	no	—
244	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
245	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
246	C₂²×C₄	#6129	1	1	yes	characteristic
247	M₁₆	—	4	1	no	—
248	M₁₆	—	4	1	no	—
249	M₁₆	—	52	4	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	13	1	no	—
253	C₂×C₈	—	13	1	no	—
254	C₂×C₈	—	13	2	no	—
255	C₂×C₈	—	13	2	no	—
256	C₂×C₈	—	13	1	no	—
257	C₂×C₈	—	13	1	no	—
258	C₂×C₈	—	26	1	no	—
259	C₂×C₈	—	26	1	no	—
260	C₂×C₈	—	26	2	no	—
261	C₂×C₈	—	26	2	no	—
262	C₂×C₈	—	52	4	no	—
263	C₂×C₈	—	52	1	no	—
264	C₂×C₈	—	52	1	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	13	2	no	—
267	C₄⋊C₄	—	26	1	no	—
268	C₂²⋊C₄	—	26	2	no	—
269	C₂²⋊C₄	—	26	2	no	—
270	C₄²	—	4	1	no	—
271	C₄²	—	4	1	no	—
272	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
273	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
274	C₄²	D₄×D₁₃	1	1	yes	characteristic
275	C₄²	#6131	1	1	yes	characteristic
276	C₁₃	#8251	1	1	yes	characteristic, complemented, Sylow
277	C₂³	—	26	2	no	—
278	C₂³	#88135	1	1	yes	characteristic
279	Q₈	—	52	1	no	—
280	Q₈	—	52	1	no	—
281	D₄	—	52	2	no	—
282	D₄	—	52	2	no	—
283	D₄	—	52	2	no	—
284	D₄	—	52	2	no	—
285	D₄	—	52	1	no	—
286	D₄	—	52	1	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	26	2	no	—
293	C₂×C₄	—	52	2	no	—
294	C₂×C₄	—	52	2	no	—
295	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
296	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
297	C₂×C₄	#88135	1	1	yes	characteristic
298	C₂×C₄	#88146	1	1	yes	characteristic
299	C₂×C₄	#88147	1	1	yes	characteristic
300	C₂×C₄	#88147	1	1	yes	characteristic
301	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
302	C₂×C₄	SD₁₆×D₁₃	1	1	yes	characteristic
303	C₂×C₄	#88170	1	1	yes	characteristic
304	C₂×C₄	#88173	1	1	yes	characteristic
305	C₈	—	4	2	no	—
306	C₈	—	4	2	no	—
307	C₈	—	13	2	no	—
308	C₈	—	13	2	no	—
309	C₈	—	26	1	no	—
310	C₈	—	26	1	no	—
311	C₈	—	52	2	no	—
312	C₈	—	52	2	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	52	2	no	—
316	C₂²	—	52	2	no	—
317	C₂²	—	52	2	no	—
318	C₂²	#11718327	1	1	yes	characteristic
319	C₂²	#11718417	1	1	yes	characteristic
320	C₂²	#11718449	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	4	no	—
325	C₄	—	52	2	no	—
326	C₄	D₁₃×#102	1	1	yes	characteristic
327	C₄	#11718465	1	1	yes	characteristic
328	C₄	#11718485	1	2	yes	—
329	C₄	#11718513	1	1	yes	characteristic
330	C₄	#11718533	1	1	yes	characteristic
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	104	2	no	—
334	C₂	#13114635	1	1	yes	characteristic
335	C₂	#13114650	1	1	yes	characteristic
336	C₂	#13114670	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1303	C₂
socle	C₂²×C₁₃	C₄×C₂²⋊C₄
prime radical	#11719182	C₂²
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1303	D₁₃
hypocenter	C₁₃	#8251
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1303

Complements

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

N	H ≅ G/N	type
2. C₄⋊C₄×#1055	C₂	semidirect
5. #13113120	C₂	semidirect
6. #13112731	C₂	semidirect
7. C₁₃×#1303	C₂	semidirect
19. #11718499	C₄	semidirect
21. #11718495	C₄	semidirect
23. #11718078	C₄	semidirect
24. #11718077	C₄	semidirect
114. #1303	D₁₃	semidirect
276. C₁₃	#8251	semidirect

Maximal subgroups

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

subgroup	index
#13123547	2
C₄⋊C₄×#1055	2
#13113483	2
#13113330	2
#13113120	2
#13112731	2
C₁₃×#1303	2
#8251	13