Subgroups — #15881062

Order 3328.

982 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₅₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718541	C₂	1	2	yes	maximal, complemented
3	#13117787	C₂	1	2	yes	maximal
4	#13115597	C₂	1	2	yes	maximal, complemented
5	#13115032	C₂	1	4	yes	maximal, complemented
6	#13114862	C₂	1	2	yes	maximal, complemented
7	C₄×#88058	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₁₃	C₂²	1	2	yes	—
10	C₂×#88146	C₂²	1	4	yes	—
11	C₂×C₄×D₅₂	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#107	C₂²	1	2	yes	—
13	C₂×C₁₃×#107	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×#384	C₂²	1	2	yes	—
16	C₂×#88060	C₂²	1	2	yes	—
17	C₂×#88058	C₂²	1	1	yes	characteristic
18	C₂×#88058	C₂²	1	1	yes	characteristic
19	#11718651	—	2	4	no	—
20	#11718649	—	2	2	no	—
21	#11718648	C₂²	1	4	yes	—
22	C₄⋊C₄×Dic₁₃	—	2	2	no	—
23	C₂×#88048	C₂²	1	4	yes	—
24	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
25	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
26	#11718541	—	2	8	no	—
27	C₈×D₅₂	C₄	1	8	yes	complemented
28	#11718143	C₂²	1	2	yes	—
29	#11718121	C₂²	1	4	yes	—
30	C₂²×D₅₂	D₄	1	1	yes	characteristic
31	C₂²×C₄×D₁₃	C₂³	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
36	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×#6103	—	2	4	no	—
39	C₂×#6103	—	2	4	no	—
40	C₂×#6103	D₄	1	2	yes	—
41	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
42	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
43	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
44	C₂×C₄×Dic₁₃	—	2	2	no	—
45	C₂×C₄×Dic₁₃	—	2	4	no	—
46	C₂²×#1055	C₂³	1	2	yes	—
47	C₂×C₈×D₁₃	—	2	4	no	—
48	C₂×C₈×D₁₃	—	2	4	no	—
49	C₂×C₈×D₁₃	C₂×C₄	1	8	yes	—
50	#88146	—	2	8	no	—
51	#88146	—	4	4	no	—
52	C₄×D₅₂	C₂×C₄	1	4	yes	—
53	C₄×D₅₂	Q₈	1	4	yes	—
54	C₁₃×#107	—	2	4	no	—
55	C₁₃×#107	—	4	2	no	—
56	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
57	#88071	—	2	2	no	—
58	#88071	—	2	4	no	—
59	#88060	C₂×C₄	1	8	yes	—
60	#88058	—	2	2	no	—
61	#88058	D₄	1	4	yes	complemented
62	#88048	—	2	8	no	—
63	C₄⋊#1055	C₂×C₄	1	4	yes	—
64	#22658	—	13	1	no	maximal, Sylow
65	C₂³×D₁₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₃	—	2	4	no	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₁₃	—	2	2	no	—
71	C₂²×Dic₁₃	—	2	2	no	—
72	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
73	C₂×D₅₂	—	2	2	no	—
74	C₂×D₅₂	—	2	2	no	—
75	C₂×D₅₂	C₄⋊C₄	1	4	yes	—
76	C₂×C₄×D₁₃	—	2	2	no	—
77	C₂×C₄×D₁₃	—	2	2	no	—
78	C₂×C₄×D₁₃	—	2	2	no	—
79	C₂×C₄×D₁₃	—	2	2	no	—
80	C₂×C₄×D₁₃	C₂²×C₄	1	4	yes	—
81	C₂×C₄×D₁₃	C₂×Q₈	1	4	yes	—
82	C₂×C₈×C₁₃	—	2	1	no	—
83	C₂×C₈×C₁₃	—	2	1	no	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₃	#41	1	2	yes	—
88	C₁₃×C₄⋊C₄	—	2	4	no	—
89	C₁₃×C₄⋊C₄	—	2	4	no	—
90	C₁₃×C₄⋊C₄	—	4	4	no	—
91	C₁₃×C₄⋊C₄	—	8	2	no	—
92	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
93	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
94	#6103	—	4	8	no	—
95	#6103	—	8	4	no	—
96	#6103	C₄⋊C₄	1	8	yes	—
97	C₄⋊Dic₁₃	—	2	2	no	—
98	C₄⋊Dic₁₃	—	2	2	no	—
99	C₄⋊Dic₁₃	—	2	2	no	—
100	C₄⋊Dic₁₃	C₄⋊C₄	1	4	yes	—
101	C₄⋊Dic₁₃	C₂×D₄	1	4	yes	—
102	C₄×Dic₁₃	—	4	8	no	—
103	C₄×Dic₁₃	—	4	4	no	—
104	C₂×#1055	—	2	2	no	—
105	C₂×#1055	—	2	2	no	—
106	C₂×#1055	C₂²×C₄	1	4	yes	—
107	C₈×D₁₃	—	2	16	no	—
108	C₂×C₈×D₄	—	13	1	no	—
109	C₂×#427	—	13	2	no	—
110	#2354	—	13	2	no	—
111	#1948	—	13	2	no	—
112	#1854	—	13	4	no	—
113	#1845	D₁₃	1	1	yes	characteristic, complemented
114	#1808	—	13	2	no	—
115	C₄×#107	—	13	1	no	—
116	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
117	C₂²×D₁₃	—	2	2	no	—
118	C₂²×D₁₃	—	2	2	no	—
119	C₂²×D₁₃	—	2	2	no	—
120	C₂²×D₁₃	—	2	2	no	—
121	C₂²×D₁₃	—	2	2	no	—
122	C₂²×D₁₃	—	2	2	no	—
123	C₂²×D₁₃	C₂×C₄⋊C₄	1	4	yes	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	4	4	no	—
127	C₂×C₄×C₁₃	—	4	4	no	—
128	C₂×C₄×C₁₃	—	4	4	no	—
129	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₁₃	#135	1	2	yes	—
132	C₂×C₄×C₁₃	#135	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	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
139	C₂×Dic₁₃	—	2	2	no	—
140	C₂×Dic₁₃	—	2	2	no	—
141	C₂×Dic₁₃	—	2	4	no	—
142	C₂×Dic₁₃	—	4	2	no	—
143	C₂×Dic₁₃	—	4	2	no	—
144	C₂×Dic₁₃	—	4	2	no	—
145	C₂×Dic₁₃	—	4	2	no	—
146	C₂×Dic₁₃	—	4	2	no	—
147	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
148	D₅₂	—	2	8	no	—
149	C₄×D₁₃	—	2	4	no	—
150	C₄×D₁₃	—	2	4	no	—
151	C₄×D₁₃	—	2	4	no	—
152	C₄×D₁₃	—	2	4	no	—
153	C₈×C₁₃	—	2	2	no	—
154	C₈×C₁₃	C₄×D₄	1	4	yes	—
155	#1055	—	2	4	no	—
156	C₂³×C₈	—	13	2	no	—
157	C₂×C₄×D₄	—	13	1	no	—
158	C₂×#117	—	13	4	no	—
159	C₈×D₄	—	13	8	no	—
160	#427	—	26	8	no	—
161	C₂×#107	—	13	1	no	—
162	C₂×#107	—	13	1	no	—
163	C₂×#107	C₂×D₁₃	1	2	yes	—
164	C₂×#107	C₂×D₁₃	1	2	yes	—
165	C₂×#106	—	13	4	no	—
166	C₂×C₄⋊C₈	—	13	1	no	—
167	C₂×C₂²⋊C₈	—	13	2	no	—
168	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
169	#390	—	26	2	no	—
170	#388	—	26	4	no	—
171	#384	C₂×D₁₃	1	2	yes	—
172	#380	—	13	4	no	—
173	C₄×C₄⋊C₄	—	13	2	no	—
174	C₄×C₄⋊C₄	—	26	2	no	—
175	#340	—	13	2	no	—
176	#340	—	13	4	no	—
177	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
178	C₂²×C₁₃	D₄²	1	1	yes	characteristic
179	C₂²×C₁₃	#547	1	1	yes	characteristic
180	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
181	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
182	C₂²×C₁₃	#576	1	1	yes	characteristic
183	C₂²×C₁₃	#578	1	1	yes	characteristic
184	C₂×D₁₃	—	2	4	no	—
185	C₂×D₁₃	—	2	4	no	—
186	C₂×D₁₃	—	2	4	no	—
187	C₂×D₁₃	—	4	2	no	—
188	C₂×D₁₃	—	4	2	no	—
189	C₂×D₁₃	—	4	2	no	—
190	C₂×D₁₃	—	4	2	no	—
191	C₄×C₁₃	—	2	2	no	—
192	C₄×C₁₃	—	8	4	no	—
193	C₄×C₁₃	#427	1	4	yes	complemented
194	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
195	C₄×C₁₃	#520	1	1	yes	characteristic
196	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
197	C₄×C₁₃	#554	1	1	yes	characteristic
198	Dic₁₃	—	2	4	no	—
199	Dic₁₃	—	4	4	no	—
200	Dic₁₃	—	8	2	no	—
201	C₂²×D₄	—	13	1	no	—
202	C₂³×C₄	—	13	2	no	—
203	C₂²×C₈	—	13	2	no	—
204	C₂²×C₈	—	13	8	no	—
205	C₂²×C₈	—	26	4	no	—
206	C₂²×C₈	—	26	4	no	—
207	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
208	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
209	C₄×D₄	—	13	4	no	—
210	C₄×D₄	—	13	4	no	—
211	#117	—	26	8	no	—
212	#117	—	52	4	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	13	1	no	—
216	C₂×C₄⋊C₄	—	13	2	no	—
217	C₂×C₄⋊C₄	—	13	2	no	—
218	C₂×C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
219	C₂×C₂²⋊C₄	—	13	2	no	—
220	C₂×C₂²⋊C₄	—	26	4	no	—
221	C₂×C₂²⋊C₄	—	26	4	no	—
222	C₂×C₄²	—	26	4	no	—
223	C₂×C₄²	—	26	2	no	—
224	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
225	#107	—	2	4	no	—
226	#107	—	4	2	no	—
227	#107	—	13	4	no	—
228	#107	—	26	2	no	—
229	#106	—	26	8	no	—
230	C₄⋊C₈	—	13	4	no	—
231	C₂²⋊C₈	—	13	8	no	—
232	C₄×C₈	C₄×D₁₃	1	4	yes	—
233	#95	—	26	4	no	—
234	#95	—	26	2	no	—
235	C₂×C₁₃	D₄×C₄⋊C₄	1	1	yes	characteristic
236	C₂×C₁₃	C₂×#427	1	1	yes	characteristic
237	C₂×C₁₃	#2913	1	1	yes	characteristic
238	C₂×C₁₃	#3283	1	1	yes	characteristic
239	C₂×C₁₃	#3287	1	1	yes	characteristic
240	C₂×C₁₃	#3341	1	1	yes	characteristic
241	C₂×C₁₃	#3342	1	1	yes	characteristic
242	D₁₃	—	4	4	no	—
243	C₂⁴	—	13	2	no	—
244	C₂×D₄	—	13	4	no	—
245	C₂×D₄	—	26	2	no	—
246	C₂×D₄	—	26	2	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	13	2	no	—
249	C₂²×C₄	—	13	4	no	—
250	C₂²×C₄	—	13	4	no	—
251	C₂²×C₄	—	26	2	no	—
252	C₂²×C₄	—	26	2	no	—
253	C₂²×C₄	—	26	2	no	—
254	C₂²×C₄	—	26	2	no	—
255	C₂²×C₄	—	26	2	no	—
256	C₂²×C₄	—	26	2	no	—
257	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
258	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
259	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	13	4	no	—
263	C₂×C₈	—	26	2	no	—
264	C₂×C₈	—	26	2	no	—
265	C₂×C₈	—	26	16	no	—
266	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
267	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
268	C₂×C₈	D₄×D₁₃	1	2	yes	—
269	C₂×C₈	#6131	1	2	yes	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	4	4	no	—
273	C₄⋊C₄	—	8	2	no	—
274	C₄⋊C₄	—	13	4	no	—
275	C₄⋊C₄	—	13	4	no	—
276	C₄⋊C₄	—	26	2	no	—
277	C₄⋊C₄	—	26	2	no	—
278	C₄⋊C₄	—	26	2	no	—
279	C₂²⋊C₄	—	13	8	no	—
280	C₂²⋊C₄	—	52	8	no	—
281	C₂²⋊C₄	—	104	4	no	—
282	C₄²	—	52	8	no	—
283	C₄²	—	52	4	no	—
284	C₄²	C₂×C₄×D₁₃	1	2	yes	—
285	C₄²	Q₈×D₁₃	1	2	yes	—
286	C₁₃	#22658	1	1	yes	characteristic, complemented, Sylow
287	C₂³	—	13	4	no	—
288	C₂³	—	26	2	no	—
289	C₂³	—	26	2	no	—
290	C₂³	—	26	2	no	—
291	C₂³	—	26	2	no	—
292	C₂³	—	26	2	no	—
293	C₂³	—	26	2	no	—
294	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
295	D₄	—	26	8	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	13	4	no	—
302	C₂×C₄	—	26	2	no	—
303	C₂×C₄	—	26	2	no	—
304	C₂×C₄	—	26	4	no	—
305	C₂×C₄	—	26	4	no	—
306	C₂×C₄	—	26	4	no	—
307	C₂×C₄	—	26	4	no	—
308	C₂×C₄	—	26	4	no	—
309	C₂×C₄	—	52	2	no	—
310	C₂×C₄	—	52	2	no	—
311	C₂×C₄	—	52	2	no	—
312	C₂×C₄	—	52	2	no	—
313	C₂×C₄	—	52	2	no	—
314	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
315	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
316	C₂×C₄	#88166	1	2	yes	—
317	C₂×C₄	#88173	1	2	yes	—
318	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
319	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
320	C₂×C₄	#88251	1	1	yes	characteristic
321	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
322	C₂×C₄	C₂×#6131	1	1	yes	characteristic
323	C₂×C₄	#88254	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	26	4	no	—
326	C₈	#88147	1	4	yes	—
327	C₂²	—	26	4	no	—
328	C₂²	—	26	4	no	—
329	C₂²	—	26	4	no	—
330	C₂²	—	52	2	no	—
331	C₂²	—	52	2	no	—
332	C₂²	—	52	2	no	—
333	C₂²	—	52	2	no	—
334	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
335	C₂²	#11719422	1	1	yes	characteristic
336	C₂²	#11719431	1	1	yes	characteristic
337	C₂²	C₂×#88166	1	1	yes	characteristic
338	C₂²	#11719462	1	1	yes	characteristic
339	C₂²	C₂×#88173	1	1	yes	characteristic
340	C₂²	#11719471	1	1	yes	characteristic
341	C₄	—	2	2	no	—
342	C₄	—	8	4	no	—
343	C₄	—	26	4	no	—
344	C₄	—	52	4	no	—
345	C₄	—	104	2	no	—
346	C₄	#11718541	1	4	yes	complemented
347	C₄	C₂×#88147	1	1	yes	characteristic
348	C₄	#11719209	1	1	yes	characteristic
349	C₄	#11719432	1	1	yes	characteristic
350	C₄	#11719435	1	1	yes	characteristic
351	C₂	—	52	4	no	—
352	C₂	#13117736	1	1	yes	characteristic
353	C₂	C₂×#11718541	1	1	yes	characteristic
354	C₂	#13123432	1	1	yes	characteristic
355	C₂	#13126556	1	1	yes	characteristic
356	C₂	#13126594	1	1	yes	characteristic
357	C₂	#13127258	1	1	yes	characteristic
358	C₂	#13127266	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#1845	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂²×D₅₂	D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#1845	D₁₃
hypocenter	C₁₃	#22658
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
2. C₂×#11718541	C₂	semidirect
4. #13115597	C₂	semidirect
5. #13115032	C₂	semidirect
6. #13114862	C₂	semidirect
7. C₄×#88058	C₂	semidirect
8. C₁₃×#1845	C₂	semidirect
27. C₈×D₅₂	C₄	semidirect
61. #88058	D₄	semidirect
113. #1845	D₁₃	semidirect
193. C₄×C₁₃	#427	semidirect
286. C₁₃	#22658	semidirect
346. C₄	#11718541	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₅₂	2
C₂×#11718541	2
#13117787	2
#13115597	2
#13115032	2
#13114862	2
C₄×#88058	2
C₁₃×#1845	2
#22658	13