Subgroups — #15828642

Order 3328.

814 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	#13123803	C₂	1	1	yes	maximal, characteristic
2	C₄×#88202	C₂	1	1	yes	maximal, characteristic
3	#13113139	C₂	1	2	yes	maximal, complemented
4	#13113131	C₂	1	2	yes	maximal, complemented
5	#13113126	C₂	1	1	yes	maximal, characteristic, complemented
6	#13113125	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113120	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113117	C₂	1	2	yes	maximal, complemented
9	M₁₆×#1055	C₂	1	2	yes	maximal, complemented
10	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2965	C₂	1	1	yes	maximal, characteristic
12	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₂×#88202	C₄	1	2	yes	—
14	C₁₃×#446	C₄	1	2	yes	—
15	C₁₃×#436	C₄	1	2	yes	—
16	C₁₃×#435	C₄	1	4	yes	—
17	C₁₃×#431	C₂²	1	2	yes	—
18	C₁₃×#431	C₂²	1	1	yes	characteristic
19	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	#11718685	C₄	1	2	yes	—
22	#11718673	C₄	1	2	yes	—
23	#11718672	C₄	1	4	yes	—
24	#11718658	C₄	1	4	yes	—
25	#11718658	C₄	1	2	yes	—
26	#11718586	C₂²	1	2	yes	—
27	#11718586	C₂²	1	1	yes	characteristic
28	C₄×#6099	C₂²	1	2	yes	—
29	C₄×#6099	C₂²	1	1	yes	characteristic
30	C₄×#6099	C₂²	1	2	yes	—
31	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
32	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
33	C₂×C₄×#1055	C₂²	1	2	yes	—
34	#11718047	C₂²	1	2	yes	—
35	#11718047	C₂²	1	2	yes	—
36	#11718046	C₂²	1	2	yes	—
37	#11718046	C₂²	1	2	yes	—
38	#11718046	C₂²	1	2	yes	—
39	C₈×#1055	C₂²	1	1	yes	characteristic
40	C₈×#1055	C₂²	1	1	yes	characteristic
41	C₈×#1055	C₂²	1	1	yes	characteristic
42	C₈×#1055	C₂²	1	1	yes	characteristic
43	C₈×#1055	C₂²	1	2	yes	—
44	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
47	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
49	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
50	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
51	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
52	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
53	C₁₃×#117	C₂×C₄	1	2	yes	—
54	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
57	#88202	—	2	8	no	—
58	C₂×#6099	C₂×C₄	1	2	yes	—
59	C₂×#6099	C₂×C₄	1	4	yes	—
60	C₂×#6099	C₂×C₄	1	4	yes	—
61	C₂²×#1055	C₂×C₄	1	4	yes	—
62	C₂²×#1055	C₂×C₄	1	2	yes	—
63	C₂²×#1055	C₂×C₄	1	2	yes	—
64	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
65	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
66	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
67	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
68	C₁₃×#97	C₂×C₄	1	4	yes	—
69	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₁₃	C₂³	1	2	yes	—
73	C₂²⋊#1055	C₂×C₄	1	2	yes	—
74	C₂²⋊#1055	C₂×C₄	1	4	yes	—
75	C₄⋊#1055	C₂×C₄	1	4	yes	—
76	C₄⋊#1055	C₂×C₄	1	2	yes	—
77	#88043	C₂×C₄	1	2	yes	—
78	#88043	C₂×C₄	1	4	yes	—
79	#88043	C₂×C₄	1	4	yes	—
80	C₄×#1055	C₂×C₄	1	4	yes	—
81	C₄×#1055	C₂×C₄	1	2	yes	—
82	C₄×#1055	C₂×C₄	1	2	yes	—
83	C₄×#1055	C₂³	1	1	yes	characteristic
84	C₄×#1055	C₂³	1	1	yes	characteristic
85	C₄×#1055	C₂³	1	1	yes	characteristic
86	C₄×#1055	C₂³	1	1	yes	characteristic
87	C₄×#1055	C₂³	1	2	yes	—
88	C₄×#1055	C₂³	1	2	yes	—
89	#17335	—	13	1	no	maximal, Sylow
90	C₁₃×#41	—	2	4	no	—
91	C₂×Q₈×C₁₃	C₄²	1	1	yes	characteristic
92	C₂×D₄×C₁₃	C₄²	1	2	yes	—
93	C₂×D₄×C₁₃	C₄²	1	1	yes	characteristic
94	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
100	C₁₃×M₁₆	—	2	8	no	—
101	C₂×C₈×C₁₃	—	2	4	no	—
102	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
103	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
104	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
105	C₁₃×C₄⋊C₄	C₄²	1	2	yes	—
106	C₁₃×C₄⋊C₄	C₄²	1	1	yes	characteristic
107	C₁₃×C₄⋊C₄	C₄²	1	2	yes	—
108	C₁₃×C₄⋊C₄	C₄²	1	1	yes	characteristic
109	C₁₃×C₂²⋊C₄	C₄²	1	2	yes	—
110	C₁₃×C₂²⋊C₄	C₄²	1	2	yes	—
111	C₁₃×C₂²⋊C₄	C₄²	1	2	yes	—
112	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
114	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
116	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
118	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
119	#6099	—	2	4	no	—
120	#6099	—	2	8	no	—
121	#6099	—	2	8	no	—
122	C₂×#1055	—	2	8	no	—
123	C₂×#1055	—	2	4	no	—
124	C₂×#1055	M₁₆	1	8	yes	complemented
125	C₂×#1055	C₂²×C₄	1	2	yes	—
126	C₂×#1055	C₂²×C₄	1	4	yes	—
127	C₂×#1055	C₂²×C₄	1	4	yes	—
128	C₂×#1055	C₂²×C₄	1	2	yes	—
129	C₂×#1055	C₂²×C₄	1	2	yes	—
130	C₂×#1055	C₂²×C₄	1	2	yes	—
131	#2965	—	13	1	no	—
132	#2965	D₁₃	1	1	yes	characteristic
133	C₄×#131	—	13	1	no	—
134	#1455	—	13	2	no	—
135	#1453	—	13	2	no	—
136	#1451	—	13	1	no	—
137	#1451	—	13	1	no	—
138	#1450	—	13	1	no	—
139	C₈×M₁₆	—	13	2	no	—
140	C₈×M₁₆	—	13	2	no	—
141	C₂×C₈²	—	13	1	no	—
142	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
143	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
144	Q₈×C₁₃	—	2	2	no	—
145	D₄×C₁₃	—	2	4	no	—
146	D₄×C₁₃	—	2	2	no	—
147	C₂×C₄×C₁₃	—	2	4	no	—
148	C₂×C₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	2	4	no	—
150	C₂×C₄×C₁₃	—	2	2	no	—
151	C₂×C₄×C₁₃	—	2	4	no	—
152	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
154	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
156	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
157	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
159	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
161	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
162	C₈×C₁₃	—	2	2	no	—
163	C₈×C₁₃	—	2	4	no	—
164	C₈×C₁₃	#131	1	4	yes	complemented
165	#1055	—	2	4	no	—
166	#1055	—	2	2	no	—
167	#1055	—	2	4	no	—
168	#1055	C₂×M₁₆	1	4	yes	—
169	#1055	C₂×M₁₆	1	4	yes	—
170	#1055	#131	1	4	yes	complemented
171	C₂×#131	—	13	2	no	—
172	C₄×#41	C₂×D₁₃	1	1	yes	characteristic
173	#446	—	13	2	no	—
174	#446	Dic₁₃	1	2	yes	—
175	#436	—	13	2	no	—
176	#436	Dic₁₃	1	2	yes	—
177	#435	—	13	4	no	—
178	#435	Dic₁₃	1	4	yes	—
179	#431	—	13	2	no	—
180	#431	—	13	1	no	—
181	#431	C₂×D₁₃	1	2	yes	—
182	#431	C₂×D₁₃	1	1	yes	characteristic
183	#405	—	13	2	no	—
184	#405	—	13	4	no	—
185	C₄×M₁₆	—	13	1	no	—
186	C₄×M₁₆	—	13	2	no	—
187	C₄×M₁₆	—	13	2	no	—
188	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
189	C₂×C₄×C₈	—	13	1	no	—
190	C₂×C₄×C₈	—	13	2	no	—
191	C₂×C₄×C₈	—	13	1	no	—
192	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
193	#322	—	13	2	no	—
194	#322	—	13	2	no	—
195	#322	—	13	2	no	—
196	#322	—	13	2	no	—
197	#322	—	13	2	no	—
198	C₈²	—	13	1	no	—
199	C₈²	—	13	1	no	—
200	C₈²	—	13	1	no	—
201	C₈²	—	13	1	no	—
202	C₈²	—	13	2	no	—
203	C₂²×C₁₃	—	2	1	no	—
204	C₂²×C₁₃	—	2	1	no	—
205	C₂²×C₁₃	—	2	2	no	—
206	C₂²×C₁₃	—	2	2	no	—
207	C₂²×C₁₃	—	2	2	no	—
208	C₂²×C₁₃	C₄×M₁₆	1	2	yes	—
209	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
210	C₄×C₁₃	—	2	2	no	—
211	C₄×C₁₃	—	2	1	no	—
212	C₄×C₁₃	—	2	2	no	—
213	C₄×C₁₃	—	2	1	no	—
214	C₄×C₁₃	—	2	2	no	—
215	C₄×C₁₃	C₄×M₁₆	1	2	yes	—
216	C₄×C₁₃	C₂²×M₁₆	1	2	yes	—
217	C₄×C₁₃	C₂×#131	1	2	yes	—
218	C₄×C₁₃	C₂×#131	1	2	yes	—
219	C₂×#41	C₄×D₁₃	1	1	yes	characteristic
220	#131	—	26	8	no	—
221	C₂×M₁₆	—	13	4	no	—
222	C₂×M₁₆	—	13	2	no	—
223	C₂×M₁₆	—	13	4	no	—
224	C₂×M₁₆	C₂×Dic₁₃	1	4	yes	—
225	C₂²×C₈	—	13	2	no	—
226	C₂²×C₈	—	13	4	no	—
227	C₂²×C₈	—	13	2	no	—
228	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
229	C₄×Q₈	C₄×D₁₃	1	1	yes	characteristic
230	C₄×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
231	C₄×D₄	C₄×D₁₃	1	2	yes	—
232	C₄×D₄	C₄×D₁₃	1	1	yes	characteristic
233	C₄×D₄	C₂×Dic₁₃	1	2	yes	—
234	C₄×D₄	C₂×Dic₁₃	1	1	yes	characteristic
235	#117	C₄×D₁₃	1	2	yes	—
236	#117	C₄×D₁₃	1	1	yes	characteristic
237	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
239	C₄⋊C₈	—	13	4	no	—
240	C₄⋊C₈	—	13	2	no	—
241	C₄⋊C₈	C₂×Dic₁₃	1	4	yes	—
242	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
243	C₂²⋊C₈	—	13	2	no	—
244	C₂²⋊C₈	—	13	4	no	—
245	C₂²⋊C₈	C₂×Dic₁₃	1	2	yes	—
246	C₂²⋊C₈	C₂×Dic₁₃	1	4	yes	—
247	#97	—	13	4	no	—
248	#97	—	13	2	no	—
249	#97	—	13	4	no	—
250	#97	C₂×Dic₁₃	1	4	yes	—
251	C₄×C₈	—	13	2	no	—
252	C₄×C₈	—	13	4	no	—
253	C₄×C₈	—	13	2	no	—
254	C₄×C₈	—	13	1	no	—
255	C₄×C₈	—	13	1	no	—
256	C₄×C₈	—	13	1	no	—
257	C₄×C₈	—	13	1	no	—
258	C₄×C₈	—	13	2	no	—
259	C₄×C₈	—	13	2	no	—
260	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
261	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
262	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₁₃	1	2	yes	—
264	C₂×C₁₃	—	2	2	no	—
265	C₂×C₁₃	—	4	2	no	—
266	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
267	C₂×C₁₃	C₄×#131	1	1	yes	characteristic
268	C₂×C₁₃	C₄×#131	1	1	yes	characteristic
269	#41	—	2	4	no	—
270	C₂×Q₈	C₄×Dic₁₃	1	1	yes	characteristic
271	C₂×D₄	C₄×Dic₁₃	1	2	yes	—
272	C₂×D₄	C₄×Dic₁₃	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
275	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
276	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
277	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
278	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
279	M₁₆	—	2	8	no	—
280	M₁₆	—	26	8	no	—
281	M₁₆	—	26	8	no	—
282	M₁₆	—	26	4	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	—	13	8	no	—
285	C₂×C₈	—	13	2	no	—
286	C₂×C₈	—	13	2	no	—
287	C₂×C₈	—	13	2	no	—
288	C₂×C₈	—	13	4	no	—
289	C₂×C₈	—	13	2	no	—
290	C₂×C₈	—	13	4	no	—
291	C₂×C₈	—	26	8	no	—
292	C₂×C₈	—	26	4	no	—
293	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
294	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
295	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
296	C₄⋊C₄	C₄×Dic₁₃	1	2	yes	—
297	C₄⋊C₄	C₄×Dic₁₃	1	1	yes	characteristic
298	C₄⋊C₄	C₄×Dic₁₃	1	2	yes	—
299	C₄⋊C₄	C₄×Dic₁₃	1	1	yes	characteristic
300	C₂²⋊C₄	C₄×Dic₁₃	1	2	yes	—
301	C₂²⋊C₄	C₄×Dic₁₃	1	2	yes	—
302	C₂²⋊C₄	C₄×Dic₁₃	1	2	yes	—
303	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
304	C₄²	C₂×C₄×D₁₃	1	2	yes	—
305	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
306	C₄²	C₂×C₄×D₁₃	1	2	yes	—
307	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
308	C₄²	C₂²×Dic₁₃	1	2	yes	—
309	C₄²	C₂³×D₁₃	1	1	yes	characteristic
310	C₁₃	#17335	1	1	yes	characteristic, complemented, Sylow
311	C₂³	C₂×C₄×Dic₁₃	1	2	yes	—
312	C₂³	C₂×C₄×Dic₁₃	1	1	yes	characteristic
313	Q₈	—	2	2	no	—
314	D₄	—	2	4	no	—
315	D₄	—	2	2	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	M₁₆×D₁₃	1	4	yes	—
322	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
323	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
324	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
325	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
326	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
327	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
328	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
329	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
330	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
331	C₈	—	2	2	no	—
332	C₈	—	2	4	no	—
333	C₈	—	13	4	no	—
334	C₈	—	13	4	no	—
335	C₈	—	13	4	no	—
336	C₈	—	26	4	no	—
337	C₈	—	26	2	no	—
338	C₈	—	26	4	no	—
339	C₈	#88202	1	4	yes	complemented
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	M₁₆×Dic₁₃	1	2	yes	—
346	C₂²	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
347	C₄	—	2	2	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	2	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	2	no	—
352	C₄	M₁₆×Dic₁₃	1	2	yes	—
353	C₄	C₂×M₁₆×D₁₃	1	2	yes	—
354	C₄	D₁₃×#131	1	2	yes	—
355	C₄	C₂×#88202	1	2	yes	—
356	C₂	—	2	2	no	—
357	C₂	—	4	2	no	—
358	C₂	C₂×M₁₆×Dic₁₃	1	1	yes	characteristic
359	C₂	#131×Dic₁₃	1	1	yes	characteristic
360	C₂	C₄×#88202	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂²×C₄²
Fitting	C₁₃×#2965	C₂
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#2965	D₁₃
hypocenter	C₁₃	#17335
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ #2965

Complements

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

N	H ≅ G/N	type
3. #13113139	C₂	semidirect
4. #13113131	C₂	semidirect
5. #13113126	C₂	semidirect
6. #13113125	C₂	semidirect
7. #13113120	C₂	semidirect
8. #13113117	C₂	semidirect
9. M₁₆×#1055	C₂	semidirect
10. C₂×C₈×#1055	C₂	semidirect
124. C₂×#1055	M₁₆	semidirect
164. C₈×C₁₃	#131	semidirect
170. #1055	#131	semidirect
310. C₁₃	#17335	semidirect
339. C₈	#88202	semidirect

Maximal subgroups

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

subgroup	index
#13123803	2
C₄×#88202	2
#13113139	2
#13113131	2
#13113126	2
#13113125	2
#13113120	2
#13113117	2
M₁₆×#1055	2
C₂×C₈×#1055	2
C₁₃×#2965	2
#17335	13