Subgroups — #15854468

Order 3328.

604 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13127260	C₂	1	1	yes	maximal, characteristic
2	#13127130	C₂	1	2	yes	maximal, complemented
3	#13114445	C₂	1	1	yes	maximal, characteristic, complemented
4	#13114409	C₂	1	2	yes	maximal, complemented
5	#13114240	C₂	1	1	yes	maximal, characteristic, complemented
6	#13114222	C₂	1	1	yes	maximal, characteristic, complemented
7	#13114063	C₂	1	2	yes	maximal, complemented
8	#13114001	C₂	1	2	yes	maximal, complemented
9	#13113721	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#6106	C₂	1	1	yes	maximal, characteristic, complemented
11	Q₈×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
12	D₄×Q₈×C₁₃	C₂²	1	2	yes	—
13	#11719268	C₂²	1	2	yes	—
14	C₁₃×#500	—	2	1	no	—
15	C₁₃×#500	C₂²	1	1	yes	characteristic
16	C₁₃×#474	—	2	2	no	—
17	C₁₃×#474	C₂²	1	2	yes	—
18	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₄×C₁₃×D₈	—	2	1	no	—
20	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
21	#11718888	—	2	2	no	—
22	#11718886	—	2	2	no	—
23	#11718818	—	2	1	no	—
24	#11718754	C₂²	1	2	yes	—
25	#11718744	C₂²	1	2	yes	—
26	#11718742	C₂²	1	2	yes	—
27	#11718710	—	2	2	no	—
28	#11718706	—	2	2	no	—
29	C₄×#6106	C₂²	1	1	yes	characteristic
30	#11718682	C₂²	1	2	yes	—
31	#11718673	C₂²	1	2	yes	—
32	#11718669	C₂²	1	2	yes	—
33	#11718541	—	2	1	no	—
34	#11718445	—	2	2	no	—
35	C₈×D₅₂	C₂²	1	1	yes	characteristic
36	#11718343	C₂²	1	1	yes	characteristic
37	#11718139	C₂²	1	2	yes	—
38	Q₈⋊#1055	C₂²	1	1	yes	characteristic
39	D₄⋊#1055	C₂²	1	2	yes	—
40	#11718075	C₂²	1	1	yes	characteristic
41	#11718073	C₂²	1	2	yes	—
42	#11718072	C₂²	1	1	yes	characteristic
43	#11718050	C₂²	1	1	yes	characteristic
44	C₈×#1055	C₂²	1	1	yes	characteristic
45	C₂²×Q₈×C₁₃	—	2	2	no	—
46	C₂×C₁₃×D₈	D₄	1	1	yes	characteristic
47	C₁₃×C₄⋊Q₈	—	2	2	no	—
48	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₁₃×C₂²⋊Q₈	—	2	2	no	—
50	C₁₃×C₂²⋊Q₈	—	4	2	no	—
51	C₄×Q₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₄×D₄×C₁₃	—	2	2	no	—
53	C₄×D₄×C₁₃	C₂³	1	2	yes	—
54	C₂×#6106	—	2	2	no	—
55	C₂×#6106	D₄	1	1	yes	characteristic
56	#88192	—	2	2	no	—
57	#88184	—	2	2	no	—
58	#88180	—	2	2	no	—
59	C₂×#6099	—	2	2	no	—
60	C₂×C₈×D₁₃	—	2	1	no	—
61	#88146	—	2	2	no	—
62	#88139	—	4	2	no	—
63	C₄×D₅₂	C₂³	1	1	yes	characteristic
64	#88123	C₂³	1	2	yes	—
65	C₁₃×#107	—	2	1	no	—
66	C₁₃×#107	—	2	1	no	—
67	C₁₃×#107	—	2	2	no	—
68	C₁₃×#107	D₄	1	1	yes	characteristic
69	C₁₃×C₄⋊C₈	—	2	1	no	—
70	C₁₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
71	C₁₃×#102	—	2	2	no	—
72	C₁₃×#102	D₄	1	2	yes	—
73	C₄×C₈×C₁₃	—	2	1	no	—
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	#88075	—	2	2	no	—
76	#88075	—	2	2	no	—
77	#88075	—	2	2	no	—
78	#88075	—	2	2	no	—
79	#88075	D₄	1	1	yes	characteristic
80	#88072	—	2	2	no	—
81	C₂²⋊#1055	—	2	2	no	—
82	#88060	—	2	1	no	—
83	#88058	—	2	1	no	—
84	#88058	—	2	1	no	—
85	#88050	—	4	2	no	—
86	#88049	D₄	1	1	yes	characteristic
87	#88048	—	2	2	no	—
88	#88048	D₄	1	1	yes	characteristic
89	#88047	—	2	2	no	—
90	C₄⋊#1055	C₂³	1	1	yes	characteristic
91	C₄⋊#1055	C₂³	1	2	yes	—
92	C₄×#1055	C₂³	1	1	yes	characteristic
93	C₄×#1055	C₂³	1	1	yes	characteristic
94	#19905	—	13	1	no	maximal, Sylow
95	C₂×Q₈×C₁₃	—	2	2	no	—
96	C₂×Q₈×C₁₃	—	4	2	no	—
97	C₂×Q₈×C₁₃	—	4	4	no	—
98	C₂×Q₈×C₁₃	—	8	2	no	—
99	C₂×Q₈×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂×D₄×C₁₃	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₁₃	—	2	2	no	—
102	C₂²×C₄×C₁₃	—	4	2	no	—
103	C₂×C₂²⋊D₁₃	—	4	2	no	—
104	C₂×D₅₂	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×D₁₃	—	2	1	no	—
106	C₂×Dic₂₆	—	2	2	no	—
107	C₁₃×D₈	—	2	2	no	—
108	C₂×C₈×C₁₃	—	2	1	no	—
109	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₁₃×C₄⋊C₄	—	2	1	no	—
112	C₁₃×C₄⋊C₄	—	2	2	no	—
113	C₁₃×C₄⋊C₄	—	2	2	no	—
114	C₁₃×C₄⋊C₄	—	4	2	no	—
115	C₁₃×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
116	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
117	C₁₃×C₂²⋊C₄	—	2	2	no	—
118	C₁₃×C₂²⋊C₄	—	4	2	no	—
119	C₄²×C₁₃	—	2	1	no	—
120	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
121	C₂²⋊Dic₁₃	—	4	2	no	—
122	#6106	—	2	2	no	—
123	#6106	—	8	2	no	—
124	#6103	—	2	1	no	—
125	#6103	—	4	2	no	—
126	C₄⋊Dic₁₃	—	2	2	no	—
127	C₄⋊Dic₁₃	—	2	2	no	—
128	C₄⋊Dic₁₃	C₂×D₄	1	1	yes	characteristic
129	C₄×Dic₁₃	—	4	2	no	—
130	#6099	—	8	2	no	—
131	C₂×#1055	—	2	2	no	—
132	C₂×#1055	—	2	1	no	—
133	C₂×#1055	—	2	1	no	—
134	C₂×#1055	C₂×D₄	1	1	yes	characteristic
135	C₂×#1055	C₂×D₄	1	1	yes	characteristic
136	C₈×D₁₃	—	4	2	no	—
137	Q₈×D₈	D₁₃	1	1	yes	characteristic, complemented
138	#3341	—	13	1	no	—
139	#3329	—	13	2	no	—
140	#1710	—	13	1	no	—
141	#1703	—	13	2	no	—
142	#1676	—	13	1	no	—
143	#1673	—	13	1	no	—
144	#1646	—	13	2	no	—
145	#1637	—	13	2	no	—
146	#1589	—	13	1	no	—
147	C₈×SD₁₆	—	13	1	no	—
148	C₂³×C₁₃	—	2	2	no	—
149	C₂²×D₁₃	—	2	1	no	—
150	Q₈×C₁₃	—	2	1	no	—
151	Q₈×C₁₃	—	4	2	no	—
152	Q₈×C₁₃	—	4	2	no	—
153	Q₈×C₁₃	—	8	4	no	—
154	Q₈×C₁₃	#135	1	2	yes	—
155	D₄×C₁₃	—	2	2	no	—
156	D₄×C₁₃	—	2	2	no	—
157	C₂×C₄×C₁₃	—	2	1	no	—
158	C₂×C₄×C₁₃	—	2	2	no	—
159	C₂×C₄×C₁₃	—	2	1	no	—
160	C₂×C₄×C₁₃	—	4	2	no	—
161	C₂×C₄×C₁₃	—	4	2	no	—
162	C₂×C₄×C₁₃	—	4	2	no	—
163	C₂×C₄×C₁₃	—	8	4	no	—
164	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
167	C₂²⋊D₁₃	—	16	2	no	—
168	C₂×Dic₁₃	—	2	1	no	—
169	C₂×Dic₁₃	—	4	2	no	—
170	D₅₂	—	2	1	no	—
171	D₅₂	—	2	1	no	—
172	C₄×D₁₃	—	4	1	no	—
173	C₄×D₁₃	—	4	1	no	—
174	Dic₂₆	—	8	2	no	—
175	C₈×C₁₃	—	2	1	no	—
176	C₈×C₁₃	—	4	1	no	—
177	C₈×C₁₃	#137	1	2	yes	—
178	#1055	—	2	1	no	—
179	#1055	—	2	1	no	—
180	#1055	—	4	1	no	—
181	#1055	—	4	1	no	—
182	#1055	—	4	2	no	—
183	#555	—	13	2	no	—
184	D₄×Q₈	C₂×D₁₃	1	2	yes	—
185	#500	—	2	1	no	—
186	#500	—	13	1	no	—
187	#500	C₂×D₁₃	1	1	yes	characteristic
188	#487	—	13	2	no	—
189	#487	—	13	2	no	—
190	#485	—	26	2	no	—
191	#482	—	26	2	no	—
192	#477	—	13	2	no	—
193	#476	—	13	2	no	—
194	#474	—	2	2	no	—
195	#474	C₂×D₁₃	1	2	yes	—
196	#468	—	26	2	no	—
197	#466	—	26	1	no	—
198	#461	—	13	2	no	—
199	#448	—	26	2	no	—
200	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
201	C₄×SD₁₆	—	13	1	no	—
202	C₄×D₈	—	2	1	no	—
203	C₄×D₈	C₂×D₁₃	1	1	yes	characteristic
204	#436	—	13	2	no	—
205	C₈×D₄	—	13	1	no	—
206	#427	—	26	1	no	—
207	#419	—	26	2	no	—
208	#335	—	13	1	no	—
209	#334	—	13	1	no	—
210	#333	—	13	2	no	—
211	#332	—	13	2	no	—
212	Q₈⋊C₈	—	13	1	no	—
213	D₄⋊C₈	—	13	2	no	—
214	D₄⋊C₈	—	13	1	no	—
215	C₈²	—	13	1	no	—
216	C₂²×C₁₃	—	4	2	no	—
217	C₂²×C₁₃	—	4	2	no	—
218	C₂²×C₁₃	—	4	2	no	—
219	C₂²×C₁₃	D₄²	1	1	yes	characteristic
220	C₂×D₁₃	—	4	1	no	—
221	C₂×D₁₃	—	4	1	no	—
222	C₂×D₁₃	—	4	1	no	—
223	C₄×C₁₃	—	2	2	no	—
224	C₄×C₁₃	—	2	1	no	—
225	C₄×C₁₃	—	4	2	no	—
226	C₄×C₁₃	—	4	1	no	—
227	C₄×C₁₃	—	8	2	no	—
228	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
229	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
230	C₄×C₁₃	#576	1	1	yes	characteristic
231	C₄×C₁₃	#578	1	1	yes	characteristic
232	Dic₁₃	—	4	1	no	—
233	Dic₁₃	—	8	2	no	—
234	C₂²×Q₈	—	2	2	no	—
235	C₂×SD₁₆	—	13	1	no	—
236	C₂×SD₁₆	—	26	2	no	—
237	C₂×D₈	C₂²⋊D₁₃	1	1	yes	characteristic
238	C₂×M₁₆	—	26	2	no	—
239	C₂²×C₈	—	26	1	no	—
240	C₄⋊Q₈	—	2	2	no	—
241	C₄⋊Q₈	—	13	2	no	—
242	C₄⋊Q₈	C₂²×D₁₃	1	2	yes	—
243	#124	—	52	2	no	—
244	C₂²⋊Q₈	—	2	2	no	—
245	C₂²⋊Q₈	—	4	2	no	—
246	#121	—	26	2	no	—
247	#121	—	26	2	no	—
248	C₄×Q₈	C₂²×D₁₃	1	1	yes	characteristic
249	C₄×D₄	—	2	2	no	—
250	C₄×D₄	—	13	1	no	—
251	C₄×D₄	C₂²×D₁₃	1	2	yes	—
252	#117	—	26	2	no	—
253	#117	—	26	2	no	—
254	#107	—	2	1	no	—
255	#107	—	2	1	no	—
256	#107	—	2	2	no	—
257	#107	—	26	1	no	—
258	#107	—	26	1	no	—
259	#107	—	26	2	no	—
260	#107	C₂²⋊D₁₃	1	1	yes	characteristic
261	#106	—	13	1	no	—
262	#106	—	26	2	no	—
263	C₄⋊C₈	—	2	1	no	—
264	C₄⋊C₈	—	13	2	no	—
265	C₄⋊C₈	—	13	1	no	—
266	C₄⋊C₈	C₂²×D₁₃	1	1	yes	characteristic
267	#103	—	13	1	no	—
268	#103	—	26	2	no	—
269	#103	—	26	2	no	—
270	#103	—	26	2	no	—
271	#103	—	26	2	no	—
272	#103	—	52	2	no	—
273	#102	—	2	2	no	—
274	#102	—	13	1	no	—
275	#102	—	26	2	no	—
276	#102	C₂²⋊D₁₃	1	2	yes	—
277	C₂²⋊C₈	—	26	2	no	—
278	C₂²⋊C₈	—	26	1	no	—
279	C₄×C₈	—	2	1	no	—
280	C₄×C₈	—	13	1	no	—
281	C₄×C₈	—	13	1	no	—
282	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
283	C₂×C₁₃	—	8	2	no	—
284	C₂×C₁₃	#3276	1	1	yes	characteristic
285	C₂×C₁₃	#3280	1	1	yes	characteristic
286	C₂×C₁₃	#3290	1	1	yes	characteristic
287	D₁₃	—	8	1	no	—
288	C₂×Q₈	—	2	2	no	—
289	C₂×Q₈	—	4	2	no	—
290	C₂×Q₈	—	4	4	no	—
291	C₂×Q₈	—	8	2	no	—
292	C₂×Q₈	—	26	2	no	—
293	C₂×Q₈	D₄×D₁₃	1	1	yes	characteristic
294	C₂×D₄	—	13	1	no	—
295	C₂×D₄	—	52	2	no	—
296	C₂×D₄	C₂×C₂²⋊D₁₃	1	2	yes	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	4	2	no	—
299	C₂²×C₄	—	26	1	no	—
300	SD₁₆	—	26	2	no	—
301	SD₁₆	—	104	2	no	—
302	D₈	—	2	2	no	—
303	M₁₆	—	104	2	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	13	1	no	—
306	C₂×C₈	—	13	1	no	—
307	C₂×C₈	—	26	2	no	—
308	C₂×C₈	—	26	1	no	—
309	C₂×C₈	—	26	1	no	—
310	C₂×C₈	—	52	2	no	—
311	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
312	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	4	2	no	—
317	C₄⋊C₄	—	13	1	no	—
318	C₄⋊C₄	—	26	2	no	—
319	C₄⋊C₄	—	26	2	no	—
320	C₄⋊C₄	D₄×D₁₃	1	1	yes	characteristic
321	C₄⋊C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
322	C₂²⋊C₄	—	2	2	no	—
323	C₂²⋊C₄	—	4	2	no	—
324	C₂²⋊C₄	—	26	1	no	—
325	C₂²⋊C₄	—	52	2	no	—
326	C₂²⋊C₄	—	52	2	no	—
327	C₄²	—	2	1	no	—
328	C₄²	—	52	2	no	—
329	C₄²	C₂³×D₁₃	1	1	yes	characteristic
330	C₁₃	#19905	1	1	yes	characteristic, complemented, Sylow
331	C₂³	—	2	2	no	—
332	C₂³	—	26	1	no	—
333	Q₈	—	2	1	no	—
334	Q₈	—	4	2	no	—
335	Q₈	—	4	2	no	—
336	Q₈	—	8	4	no	—
337	Q₈	—	104	2	no	—
338	Q₈	#88166	1	2	yes	—
339	D₄	—	2	2	no	—
340	D₄	—	2	2	no	—
341	D₄	—	26	1	no	—
342	D₄	—	26	1	no	—
343	D₄	—	208	2	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	8	4	no	—
351	C₂×C₄	—	26	1	no	—
352	C₂×C₄	—	52	1	no	—
353	C₂×C₄	—	52	1	no	—
354	C₂×C₄	—	52	2	no	—
355	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
356	C₂×C₄	#88251	1	1	yes	characteristic
357	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	4	1	no	—
360	C₈	—	26	1	no	—
361	C₈	—	26	1	no	—
362	C₈	—	52	1	no	—
363	C₈	—	52	2	no	—
364	C₈	—	52	1	no	—
365	C₈	#88196	1	2	yes	—
366	C₂²	—	4	2	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	4	2	no	—
369	C₂²	—	52	1	no	—
370	C₂²	—	52	1	no	—
371	C₂²	—	52	1	no	—
372	C₂²	D₄×C₂²⋊D₁₃	1	1	yes	characteristic
373	C₄	—	2	2	no	—
374	C₄	—	2	1	no	—
375	C₄	—	4	2	no	—
376	C₄	—	4	1	no	—
377	C₄	—	8	2	no	—
378	C₄	—	52	1	no	—
379	C₄	—	104	2	no	—
380	C₄	C₂×#88166	1	1	yes	characteristic
381	C₄	#11719462	1	1	yes	characteristic
382	C₄	C₂×#88196	1	1	yes	characteristic
383	C₄	#11719543	1	1	yes	characteristic
384	C₂	—	8	2	no	—
385	C₂	—	104	1	no	—
386	C₂	#13126482	1	1	yes	characteristic
387	C₂	#13126521	1	1	yes	characteristic
388	C₂	#13126622	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₁₃
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	Q₈×C₁₃×D₈	C₂
socle	C₂²×C₁₃	D₄²
prime radical	#13127260	C₂
hypercenter	Q₈×D₈	D₁₃
hypocenter	C₁₃	#19905
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈×D₈

Complements

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

N	H ≅ G/N	type
2. #13127130	C₂	semidirect
3. #13114445	C₂	semidirect
4. #13114409	C₂	semidirect
5. #13114240	C₂	semidirect
6. #13114222	C₂	semidirect
7. #13114063	C₂	semidirect
8. #13114001	C₂	semidirect
9. #13113721	C₂	semidirect
10. C₈×#6106	C₂	semidirect
11. Q₈×C₁₃×D₈	C₂	semidirect
137. Q₈×D₈	D₁₃	semidirect
330. C₁₃	#19905	semidirect

Maximal subgroups

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

subgroup	index
#13127260	2
#13127130	2
#13114445	2
#13114409	2
#13114240	2
#13114222	2
#13114063	2
#13114001	2
#13113721	2
C₈×#6106	2
Q₈×C₁₃×D₈	2
#19905	13