Subgroups — #15962672

Order 3328.

710 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123199	C₂	1	1	yes	maximal, characteristic
2	#13116393	C₂	1	2	yes	maximal
3	#13116079	C₂	1	2	yes	maximal
4	#13115529	C₂	1	4	yes	maximal
5	#13114841	C₂	1	2	yes	maximal
6	#13114841	C₂	1	2	yes	maximal
7	#13114689	C₂	1	1	yes	maximal, characteristic
8	C₄×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
11	C₂×#88124	—	2	1	no	—
12	C₁₃×#391	C₂²	1	2	yes	—
13	C₁₃×#389	—	2	1	no	—
14	C₁₃×#389	—	2	1	no	—
15	C₁₃×#384	C₂²	1	1	yes	characteristic
16	C₄×C₁₃×C₄⋊C₄	—	2	1	no	—
17	C₄×C₁₃×C₄⋊C₄	—	2	1	no	—
18	C₄³×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×#88075	C₂²	1	4	yes	—
20	C₂×#88075	C₂²	1	4	yes	—
21	#11718597	—	2	1	no	—
22	#11718596	C₂²	1	1	yes	characteristic
23	#11718596	C₂²	1	1	yes	characteristic
24	#11718589	—	2	1	no	—
25	#11718589	—	2	1	no	—
26	#11718589	C₂²	1	2	yes	—
27	#11718589	C₂²	1	2	yes	—
28	C₂×C₄⋊#1055	C₂²	1	2	yes	—
29	C₂×#88043	C₂²	1	2	yes	—
30	#11718121	C₂²	1	4	yes	—
31	#11718121	C₂²	1	4	yes	—
32	#11718116	C₂²	1	2	yes	—
33	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
34	C₁₃×C₄⋊Q₈	—	2	4	no	—
35	C₄×Q₈×C₁₃	—	2	4	no	—
36	C₄×Q₈×C₁₃	—	2	4	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
43	C₂×C₄²×C₁₃	—	2	2	no	—
44	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₄⋊Dic₁₃	—	2	1	no	—
52	C₂×C₄⋊Dic₁₃	—	2	1	no	—
53	C₂×C₄⋊Dic₁₃	—	2	1	no	—
54	C₂×C₄⋊Dic₁₃	—	2	1	no	—
55	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
56	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
57	C₂×#6101	—	2	2	no	—
58	C₂×#6101	—	2	2	no	—
59	C₂²×#1055	C₂³	1	4	yes	—
60	#88124	—	4	4	no	—
61	C₁₃×#95	—	2	2	no	—
62	C₁₃×#95	—	2	2	no	—
63	#88075	—	2	8	no	—
64	#88075	—	4	4	no	—
65	#88071	—	2	2	no	—
66	#88071	—	2	2	no	—
67	C₄⋊#1055	—	2	4	no	—
68	#88043	—	4	2	no	—
69	#29132	—	13	1	no	maximal, Sylow
70	C₂×Q₈×C₁₃	—	2	2	no	—
71	C₂×Q₈×C₁₃	—	2	2	no	—
72	C₂×Q₈×C₁₃	—	2	2	no	—
73	C₂×Q₈×C₁₃	—	2	2	no	—
74	C₂×Q₈×C₁₃	—	2	2	no	—
75	C₂×Q₈×C₁₃	#41	1	4	yes	—
76	C₂²×C₄×C₁₃	—	2	2	no	—
77	C₂²×C₄×C₁₃	—	2	2	no	—
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
85	C₂²×Dic₁₃	—	2	2	no	—
86	C₂²×Dic₁₃	—	2	2	no	—
87	C₁₃×C₄⋊C₄	—	2	4	no	—
88	C₁₃×C₄⋊C₄	—	2	4	no	—
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₄⋊C₄	—	2	2	no	—
91	C₁₃×C₄⋊C₄	—	4	4	no	—
92	C₁₃×C₄⋊C₄	—	4	2	no	—
93	C₁₃×C₄⋊C₄	—	4	2	no	—
94	C₄²×C₁₃	—	2	2	no	—
95	C₄²×C₁₃	—	2	2	no	—
96	C₄²×C₁₃	—	2	2	no	—
97	C₄²×C₁₃	—	2	2	no	—
98	C₄²×C₁₃	—	2	2	no	—
99	C₄²×C₁₃	—	2	1	no	—
100	C₄²×C₁₃	—	2	1	no	—
101	C₄²×C₁₃	—	4	4	no	—
102	C₄²×C₁₃	#41	1	4	yes	—
103	C₄⋊Dic₁₃	—	2	4	no	—
104	C₄⋊Dic₁₃	—	2	2	no	—
105	C₄⋊Dic₁₃	—	2	2	no	—
106	C₄⋊Dic₁₃	—	4	2	no	—
107	C₄⋊Dic₁₃	—	4	2	no	—
108	C₄⋊Dic₁₃	—	4	2	no	—
109	C₄⋊Dic₁₃	—	4	2	no	—
110	#6101	—	4	4	no	—
111	#6101	—	4	4	no	—
112	C₂×#1055	—	2	4	no	—
113	C₂×#1055	—	2	4	no	—
114	C₂×#1055	—	2	4	no	—
115	#2859	—	13	1	no	—
116	C₄×C₄⋊Q₈	D₁₃	1	1	yes	characteristic
117	#2099	—	13	2	no	—
118	#2043	—	13	2	no	—
119	#1935	—	13	4	no	—
120	#1803	—	13	2	no	—
121	#1803	—	13	2	no	—
122	#1767	—	13	1	no	—
123	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
124	Q₈×C₁₃	—	2	4	no	—
125	Q₈×C₁₃	—	2	4	no	—
126	Q₈×C₁₃	—	4	2	no	—
127	Q₈×C₁₃	—	4	2	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	1	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	2	1	no	—
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	2	4	no	—
135	C₂×C₄×C₁₃	—	2	1	no	—
136	C₂×C₄×C₁₃	—	2	1	no	—
137	C₂×C₄×C₁₃	—	2	1	no	—
138	C₂×C₄×C₁₃	—	2	1	no	—
139	C₂×C₄×C₁₃	—	2	1	no	—
140	C₂×C₄×C₁₃	—	2	1	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	—	4	2	no	—
145	C₂×C₄×C₁₃	—	4	2	no	—
146	C₂×C₄×C₁₃	#124	1	2	yes	—
147	C₂×C₄×C₁₃	#124	1	2	yes	—
148	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
149	C₂×C₄×C₁₃	#137	1	2	yes	—
150	C₂×C₄×C₁₃	#137	1	2	yes	—
151	C₂×C₄×C₁₃	#137	1	2	yes	—
152	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
153	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
154	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
156	C₂×Dic₁₃	—	4	2	no	—
157	C₂×Dic₁₃	—	4	2	no	—
158	C₂×Dic₁₃	—	4	2	no	—
159	C₂×Dic₁₃	—	4	2	no	—
160	C₂×Dic₁₃	—	4	2	no	—
161	C₂×Dic₁₃	—	4	2	no	—
162	#1055	—	4	4	no	—
163	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
164	C₂×#125	—	26	1	no	—
165	C₂×C₄×Q₈	C₂×D₁₃	1	2	yes	—
166	C₂×C₄⋊C₈	—	13	2	no	—
167	C₂×#103	—	13	4	no	—
168	C₂×#103	—	13	4	no	—
169	C₂×#97	—	13	2	no	—
170	#391	C₂×D₁₃	1	2	yes	—
171	#389	—	2	1	no	—
172	#389	—	2	1	no	—
173	#389	—	13	2	no	—
174	#389	—	13	2	no	—
175	#389	—	26	1	no	—
176	#389	—	26	1	no	—
177	#384	—	13	1	no	—
178	#384	—	13	1	no	—
179	#384	C₂×D₁₃	1	1	yes	characteristic
180	#382	—	26	1	no	—
181	C₄×C₄⋊C₄	—	2	1	no	—
182	C₄×C₄⋊C₄	—	2	1	no	—
183	C₄³	C₂×D₁₃	1	1	yes	characteristic
184	#340	—	13	4	no	—
185	#340	—	13	4	no	—
186	#336	—	13	2	no	—
187	C₂²×C₁₃	#525	1	1	yes	characteristic
188	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
189	C₂²×C₁₃	#537	1	1	yes	characteristic
190	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
191	C₂²×C₁₃	C₂×#137	1	1	yes	characteristic
192	C₂²×C₁₃	C₂×#137	1	1	yes	characteristic
193	C₂²×C₁₃	C₂×#137	1	1	yes	characteristic
194	C₄×C₁₃	—	2	2	no	—
195	C₄×C₁₃	—	2	2	no	—
196	C₄×C₁₃	—	2	2	no	—
197	C₄×C₁₃	—	4	4	no	—
198	C₄×C₁₃	—	4	1	no	—
199	C₄×C₁₃	—	4	1	no	—
200	C₄×C₁₃	—	8	2	no	—
201	C₄×C₁₃	#491	1	4	yes	—
202	C₄×C₁₃	#552	1	2	yes	—
203	C₄×C₁₃	#553	1	2	yes	—
204	Dic₁₃	—	8	2	no	—
205	Dic₁₃	—	8	2	no	—
206	C₂²×Q₈	C₂²×D₁₃	1	2	yes	—
207	C₂²×C₈	—	13	4	no	—
208	C₄⋊Q₈	—	2	4	no	—
209	#125	—	52	4	no	—
210	C₄×Q₈	—	2	4	no	—
211	C₄×Q₈	—	2	4	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	13	2	no	—
218	C₂×C₄⋊C₄	—	13	2	no	—
219	C₂×C₄⋊C₄	—	26	2	no	—
220	C₂×C₄⋊C₄	—	26	2	no	—
221	C₂×C₄⋊C₄	—	26	1	no	—
222	C₂×C₄⋊C₄	—	26	1	no	—
223	C₂×C₄⋊C₄	—	26	1	no	—
224	C₂×C₄⋊C₄	—	26	1	no	—
225	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
226	C₂×C₄²	—	2	2	no	—
227	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
228	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
229	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
230	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
231	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
233	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
234	C₄⋊C₈	—	26	4	no	—
235	#103	—	26	8	no	—
236	#103	—	52	4	no	—
237	#97	—	52	2	no	—
238	#95	—	2	2	no	—
239	#95	—	2	2	no	—
240	#95	—	26	2	no	—
241	#95	—	26	2	no	—
242	C₂×C₁₃	#2562	1	1	yes	characteristic
243	C₂×C₁₃	#3112	1	1	yes	characteristic
244	C₂×C₁₃	#3113	1	1	yes	characteristic
245	C₂×C₁₃	C₂×#491	1	1	yes	characteristic
246	C₂×C₁₃	#3141	1	1	yes	characteristic
247	C₂×C₁₃	#3254	1	1	yes	characteristic
248	C₂×C₁₃	#3255	1	1	yes	characteristic
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	#6127	1	4	yes	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	26	2	no	—
258	C₂²×C₄	—	26	2	no	—
259	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
260	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
261	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
262	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
263	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
265	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
266	C₂×C₈	—	26	4	no	—
267	C₂×C₈	—	26	4	no	—
268	C₂×C₈	—	26	4	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	4	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	26	4	no	—
277	C₄⋊C₄	—	26	2	no	—
278	C₄⋊C₄	—	26	2	no	—
279	C₄⋊C₄	—	52	4	no	—
280	C₄⋊C₄	—	52	2	no	—
281	C₄⋊C₄	—	52	2	no	—
282	C₄⋊C₄	—	52	2	no	—
283	C₄⋊C₄	—	52	2	no	—
284	C₄⋊C₄	—	52	4	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	1	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	4	4	no	—
293	C₄²	#6129	1	4	yes	—
294	C₁₃	#29132	1	1	yes	characteristic, complemented, Sylow
295	C₂³	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
296	Q₈	—	2	4	no	—
297	Q₈	—	2	4	no	—
298	Q₈	—	4	2	no	—
299	Q₈	—	4	2	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	52	2	no	—
319	C₂×C₄	—	52	2	no	—
320	C₂×C₄	—	52	2	no	—
321	C₂×C₄	—	52	2	no	—
322	C₂×C₄	—	52	2	no	—
323	C₂×C₄	—	52	2	no	—
324	C₂×C₄	#88190	1	2	yes	—
325	C₂×C₄	#88190	1	2	yes	—
326	C₂×C₄	#88196	1	2	yes	—
327	C₂×C₄	#88196	1	2	yes	—
328	C₂×C₄	#88203	1	2	yes	—
329	C₂×C₄	#88205	1	2	yes	—
330	C₂×C₄	C₂×#6127	1	2	yes	—
331	C₂×C₄	C₂×#6129	1	2	yes	—
332	C₂×C₄	#88256	1	1	yes	characteristic
333	C₂×C₄	#88257	1	1	yes	characteristic
334	C₈	—	52	4	no	—
335	C₂²	#11719496	1	1	yes	characteristic
336	C₂²	C₂×#88190	1	1	yes	characteristic
337	C₂²	C₂×#88196	1	1	yes	characteristic
338	C₂²	C₂×#88196	1	1	yes	characteristic
339	C₂²	C₂×#88203	1	1	yes	characteristic
340	C₂²	C₂×#88205	1	1	yes	characteristic
341	C₂²	#11719538	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	2	2	no	—
344	C₄	—	2	2	no	—
345	C₄	—	4	4	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	8	2	no	—
349	C₄	—	104	2	no	—
350	C₄	—	104	2	no	—
351	C₄	#11718743	1	4	yes	—
352	C₄	#11719246	1	2	yes	—
353	C₄	#11719270	1	2	yes	—
354	C₂	#13120002	1	1	yes	characteristic
355	C₂	#13124958	1	1	yes	characteristic
356	C₂	#13124966	1	1	yes	characteristic
357	C₂	C₂×#11718743	1	1	yes	characteristic
358	C₂	#13125229	1	1	yes	characteristic
359	C₂	#13126219	1	1	yes	characteristic
360	C₂	#13126227	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₁₃
commutator, cosocle	C₂²×C₄×C₁₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
hypercenter	C₄×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#29132
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
294. C₁₃	#29132	semidirect

Maximal subgroups

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

subgroup	index
#13123199	2
#13116393	2
#13116079	2
#13115529	2
#13114841	2
#13114689	2
C₄×C₁₃×C₄⋊Q₈	2
#29132	13