Subgroups — #15756088

Order 3328.

454 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114856	C₂	1	2	yes	maximal
2	C₄×#88058	C₂	1	1	yes	maximal, characteristic, complemented
3	#13113595	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112738	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₃×#41	C₂	1	1	yes	maximal, characteristic
6	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
7	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic, complemented
8	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic, complemented
9	C₈×D₄×C₁₃	—	2	1	no	—
10	C₈×D₄×C₁₃	—	2	1	no	—
11	C₈×D₄×C₁₃	C₄	1	1	yes	characteristic, complemented
12	C₈×D₄×C₁₃	C₄	1	1	yes	characteristic, complemented
13	C₁₃×#431	—	2	1	no	—
14	C₁₃×#431	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	—	2	1	no	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×#88077	—	2	2	no	—
18	C₂×#88058	—	2	1	no	—
19	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
20	#11718585	C₂²	1	2	yes	—
21	#11718143	—	2	1	no	—
22	#11718124	—	2	2	no	—
23	#11718050	—	2	1	no	—
24	#11718050	—	2	1	no	—
25	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	—	2	1	no	—
27	C₂²×C₈×C₁₃	—	2	1	no	—
28	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₄×Q₈×C₁₃	Q₈	1	1	yes	characteristic
32	C₄×D₄×C₁₃	—	2	1	no	—
33	C₄×D₄×C₁₃	—	2	1	no	—
34	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₄×D₄×C₁₃	Q₈	1	1	yes	characteristic
36	C₁₃×#117	—	2	1	no	—
37	C₁₃×#117	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	—	2	1	no	—
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×C₄⋊Dic₁₃	—	2	2	no	—
41	C₂×C₄×Dic₁₃	—	2	2	no	—
42	C₂×#6099	—	2	2	no	—
43	C₁₃×C₄⋊C₈	—	2	1	no	—
44	C₁₃×C₄⋊C₈	—	2	1	no	—
45	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₁₃×C₂²⋊C₈	—	2	1	no	—
48	C₁₃×C₂²⋊C₈	—	2	1	no	—
49	C₁₃×C₂²⋊C₈	—	2	1	no	—
50	C₄×C₈×C₁₃	—	2	1	no	—
51	C₄×C₈×C₁₃	—	2	1	no	—
52	C₄×C₈×C₁₃	—	2	1	no	—
53	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
56	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
57	#88077	—	4	4	no	—
58	#88071	—	2	2	no	—
59	#88058	—	2	4	no	—
60	C₄⋊#1055	—	2	2	no	—
61	C₄⋊#1055	—	2	2	no	—
62	#8667	—	13	1	no	maximal, Sylow
63	C₁₃×#41	—	2	1	no	—
64	C₁₃×#41	—	2	1	no	—
65	C₁₃×#41	D₈	1	2	yes	—
66	C₁₃×#41	Q₁₆	1	2	yes	—
67	C₂×Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
68	C₂×D₄×C₁₃	—	2	1	no	—
69	C₂×D₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	—	2	1	no	—
71	C₂²×C₄×C₁₃	—	2	1	no	—
72	C₂²×C₄×C₁₃	—	2	1	no	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
76	C₂²×Dic₁₃	—	4	2	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	—	2	1	no	—
80	C₂×C₈×C₁₃	—	4	1	no	—
81	C₂×C₈×C₁₃	—	4	2	no	—
82	C₂×C₈×C₁₃	—	4	2	no	—
83	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₁₃×C₄⋊C₄	—	2	1	no	—
88	C₁₃×C₄⋊C₄	—	2	1	no	—
89	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
90	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
91	C₁₃×C₂²⋊C₄	—	2	1	no	—
92	C₁₃×C₂²⋊C₄	—	2	1	no	—
93	C₁₃×C₂²⋊C₄	—	2	1	no	—
94	C₄²×C₁₃	—	2	1	no	—
95	C₄²×C₁₃	—	2	1	no	—
96	C₄²×C₁₃	—	2	1	no	—
97	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
100	C₄²×C₁₃	#41	1	1	yes	characteristic
101	C₄⋊Dic₁₃	—	2	4	no	—
102	C₄⋊Dic₁₃	—	4	2	no	—
103	C₄×Dic₁₃	—	4	2	no	—
104	C₄×Dic₁₃	—	4	2	no	—
105	#6099	—	4	2	no	—
106	#6099	—	4	2	no	—
107	C₂×#1055	—	4	2	no	—
108	C₈×#41	D₁₃	1	1	yes	characteristic
109	#1806	—	13	2	no	—
110	C₄×#107	—	13	1	no	—
111	#1567	—	13	1	no	—
112	#1303	—	13	2	no	—
113	C₂³×C₁₃	—	2	1	no	—
114	C₂³×C₁₃	#121	1	1	yes	characteristic
115	Q₈×C₁₃	—	2	1	no	—
116	Q₈×C₁₃	#107	1	2	yes	complemented
117	D₄×C₁₃	—	2	2	no	—
118	D₄×C₁₃	—	2	2	no	—
119	D₄×C₁₃	—	2	1	no	—
120	D₄×C₁₃	#107	1	2	yes	complemented
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	4	1	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	4	1	no	—
131	C₂×C₄×C₁₃	—	4	1	no	—
132	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
135	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
141	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
142	C₂×Dic₁₃	—	4	4	no	—
143	C₂×Dic₁₃	—	8	2	no	—
144	C₂×Dic₁₃	—	8	2	no	—
145	C₈×C₁₃	—	2	1	no	—
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	—	2	1	no	—
148	C₈×C₁₃	—	4	1	no	—
149	C₈×C₁₃	—	4	1	no	—
150	C₈×C₁₃	#104	1	2	yes	complemented
151	#1055	—	8	2	no	—
152	C₄×#41	C₂×D₁₃	1	1	yes	characteristic
153	C₈×Q₈	Dic₁₃	1	1	yes	characteristic, complemented
154	C₈×Q₈	Dic₁₃	1	1	yes	characteristic, complemented
155	C₈×D₄	—	2	1	no	—
156	C₈×D₄	—	2	1	no	—
157	C₈×D₄	Dic₁₃	1	1	yes	characteristic, complemented
158	C₈×D₄	Dic₁₃	1	1	yes	characteristic, complemented
159	#431	—	2	1	no	—
160	#431	C₂×D₁₃	1	1	yes	characteristic
161	C₂×#107	—	26	1	no	—
162	#423	—	13	2	no	—
163	C₂×#104	—	26	2	no	—
164	C₂×C₄×C₈	—	2	1	no	—
165	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
166	C₄×C₄⋊C₄	—	13	2	no	—
167	#340	—	26	1	no	—
168	#339	—	26	2	no	—
169	#335	—	26	1	no	—
170	#335	—	26	1	no	—
171	C₂²×C₁₃	—	2	1	no	—
172	C₂²×C₁₃	—	2	1	no	—
173	C₂²×C₁₃	—	2	2	no	—
174	C₂²×C₁₃	—	4	1	no	—
175	C₂²×C₁₃	—	4	1	no	—
176	C₂²×C₁₃	#380	1	1	yes	characteristic
177	C₂²×C₁₃	#466	1	1	yes	characteristic
178	C₂²×C₁₃	#467	1	1	yes	characteristic
179	C₄×C₁₃	—	2	1	no	—
180	C₄×C₁₃	—	2	2	no	—
181	C₄×C₁₃	—	2	1	no	—
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	—	4	1	no	—
184	C₄×C₁₃	—	4	1	no	—
185	C₄×C₁₃	C₂×#104	1	1	yes	characteristic
186	C₄×C₁₃	#421	1	1	yes	characteristic
187	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
188	C₄×C₁₃	#427	1	1	yes	characteristic
189	C₄×C₁₃	#466	1	1	yes	characteristic
190	C₄×C₁₃	#467	1	1	yes	characteristic
191	Dic₁₃	—	8	4	no	—
192	C₂×#41	D₅₂	1	1	yes	characteristic
193	C₂×M₁₆	—	26	2	no	—
194	C₂²×C₈	—	2	1	no	—
195	C₂²×C₈	—	2	1	no	—
196	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
197	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
198	C₄×Q₈	Dic₂₆	1	1	yes	characteristic
199	C₄×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
200	C₄×D₄	—	2	1	no	—
201	C₄×D₄	—	2	1	no	—
202	C₄×D₄	Dic₂₆	1	1	yes	characteristic
203	C₄×D₄	C₂×Dic₁₃	1	1	yes	characteristic
204	#117	—	2	1	no	—
205	#117	D₅₂	1	1	yes	characteristic
206	C₂×C₄⋊C₄	—	26	2	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	—	26	2	no	—
209	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
210	#107	—	26	4	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	—	26	2	no	—
214	C₄⋊C₈	—	26	2	no	—
215	C₄⋊C₈	C₂×Dic₁₃	1	1	yes	characteristic
216	C₄⋊C₈	C₂×Dic₁₃	1	1	yes	characteristic
217	#104	—	52	4	no	—
218	C₂²⋊C₈	—	2	1	no	—
219	C₂²⋊C₈	—	2	1	no	—
220	C₂²⋊C₈	—	2	1	no	—
221	C₄×C₈	—	2	1	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
225	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
226	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
227	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
228	#95	—	26	2	no	—
229	C₂×C₁₃	—	2	1	no	—
230	C₂×C₁₃	—	2	1	no	—
231	C₂×C₁₃	—	4	2	no	—
232	C₂×C₁₃	#1806	1	1	yes	characteristic
233	C₂×C₁₃	#1808	1	1	yes	characteristic
234	C₂×C₁₃	#1814	1	1	yes	characteristic
235	#41	—	2	1	no	—
236	#41	—	2	1	no	—
237	#41	D₁₀₄	1	2	yes	—
238	#41	Dic₅₂	1	2	yes	—
239	C₂×Q₈	C₄⋊Dic₁₃	1	1	yes	characteristic
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	C₄⋊Dic₁₃	1	1	yes	characteristic
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	52	2	no	—
246	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
247	C₂²×C₄	#6127	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
249	M₁₆	—	52	2	no	—
250	M₁₆	—	52	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	52	2	no	—
258	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
259	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
261	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	26	4	no	—
265	C₄⋊C₄	—	52	2	no	—
266	C₄⋊C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
267	C₄⋊C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	2	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	52	2	no	—
275	C₄²	—	52	2	no	—
276	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
277	C₄²	#6127	1	1	yes	characteristic
278	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
279	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
280	C₁₃	#8667	1	1	yes	characteristic, complemented, Sylow
281	C₂³	—	2	1	no	—
282	C₂³	#88184	1	1	yes	characteristic
283	Q₈	—	2	1	no	—
284	Q₈	#88058	1	2	yes	complemented
285	D₄	—	2	2	no	—
286	D₄	—	2	2	no	—
287	D₄	—	2	1	no	—
288	D₄	#88058	1	2	yes	complemented
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	52	4	no	—
301	C₂×C₄	—	104	2	no	—
302	C₂×C₄	—	104	2	no	—
303	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
304	C₂×C₄	#88158	1	1	yes	characteristic
305	C₂×C₄	#88158	1	1	yes	characteristic
306	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
307	C₂×C₄	#88178	1	1	yes	characteristic
308	C₂×C₄	#88178	1	1	yes	characteristic
309	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
310	C₂×C₄	#88180	1	1	yes	characteristic
311	C₂×C₄	#88184	1	1	yes	characteristic
312	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	4	1	no	—
317	C₈	—	4	1	no	—
318	C₈	—	104	2	no	—
319	C₈	#88077	1	2	yes	complemented
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	#11718776	1	1	yes	characteristic
326	C₂²	#11718777	1	1	yes	characteristic
327	C₂²	#11718873	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	2	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	104	4	no	—
335	C₄	C₂×#88058	1	1	yes	characteristic
336	C₄	#11718767	1	1	yes	characteristic
337	C₄	#11718776	1	1	yes	characteristic
338	C₄	#11718777	1	1	yes	characteristic
339	C₄	C₂×#88077	1	1	yes	characteristic
340	C₄	#11718897	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	4	2	no	—
344	C₂	#13114856	1	1	yes	characteristic
345	C₂	#13114865	1	1	yes	characteristic
346	C₂	#13114890	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88184
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₈×C₁₃×#41	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂×C₁₃×#41	D₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₈×#41	D₁₃
hypocenter	C₁₃	#8667
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. C₄×#88058	C₂	semidirect
3. #13113595	C₂	semidirect
4. #13112738	C₂	semidirect
7. C₈×Q₈×C₁₃	C₄	semidirect
8. C₈×Q₈×C₁₃	C₄	semidirect
11. C₈×D₄×C₁₃	C₄	semidirect
12. C₈×D₄×C₁₃	C₄	semidirect
116. Q₈×C₁₃	#107	semidirect
120. D₄×C₁₃	#107	semidirect
150. C₈×C₁₃	#104	semidirect
153. C₈×Q₈	Dic₁₃	semidirect
154. C₈×Q₈	Dic₁₃	semidirect
157. C₈×D₄	Dic₁₃	semidirect
158. C₈×D₄	Dic₁₃	semidirect
280. C₁₃	#8667	semidirect
284. Q₈	#88058	semidirect
288. D₄	#88058	semidirect
319. C₈	#88077	semidirect

Maximal subgroups

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

subgroup	index
#13114856	2
C₄×#88058	2
#13113595	2
#13112738	2
C₈×C₁₃×#41	2
#8667	13