Subgroups — #15861536

Order 3328.

886 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718685	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×#1055	C₂	1	1	yes	maximal, characteristic
3	#13114711	C₂	1	2	yes	maximal
4	#13114705	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₄×#1055	C₂	1	1	yes	maximal, characteristic
6	C₄⋊C₄×#1055	C₂	1	2	yes	maximal
7	#13114693	C₂	1	2	yes	maximal
8	#13114689	C₂	1	1	yes	maximal, characteristic
9	#13114523	C₂	1	2	yes	maximal
10	C₄²×#1055	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#2295	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₁₃×#391	C₄	1	2	yes	—
14	C₁₃×#389	C₄	1	2	yes	—
15	C₁₃×#387	C₄	1	4	yes	—
16	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₁₃×#376	C₂²	1	2	yes	—
19	C₄³×C₁₃	C₂²	1	1	yes	characteristic
20	#11718685	C₄	1	8	yes	complemented
21	Q₈×#1055	C₄	1	8	yes	complemented
22	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊#1055	C₂²	1	2	yes	—
25	C₂×C₄⋊#1055	C₂²	1	2	yes	—
26	C₂×#88043	C₂²	1	2	yes	—
27	C₂×#88043	C₂²	1	2	yes	—
28	C₂×#88043	C₂²	1	2	yes	—
29	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
30	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
31	C₂×C₄×#1055	C₂²	1	2	yes	—
32	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
33	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
34	C₂×C₄×#1055	C₂²	1	2	yes	—
35	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
36	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
37	#11718116	C₂²	1	2	yes	—
38	#11718116	C₂²	1	2	yes	—
39	#11718116	C₂²	1	2	yes	—
40	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
42	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
43	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
44	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
51	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
53	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₂²×#1055	C₂³	1	1	yes	characteristic
55	C₂²×#1055	C₂³	1	1	yes	characteristic
56	C₂²×#1055	C₂³	1	2	yes	—
57	C₂²×#1055	C₂³	1	2	yes	—
58	C₂²×#1055	C₂³	1	1	yes	characteristic
59	C₂²×#1055	C₂³	1	1	yes	characteristic
60	C₁₃×#95	C₂×C₄	1	2	yes	—
61	C₁₃×#95	C₂×C₄	1	4	yes	—
62	C₁₃×#95	C₂×C₄	1	4	yes	—
63	C₁₃×#95	C₂×C₄	1	4	yes	—
64	C₄⋊#1055	C₂×C₄	1	4	yes	—
65	C₄⋊#1055	C₂×C₄	1	8	yes	—
66	C₄⋊#1055	C₂×C₄	1	8	yes	—
67	C₄⋊#1055	C₂×C₄	1	4	yes	—
68	#88043	—	2	4	no	—
69	#88043	—	2	4	no	—
70	#88043	C₂×C₄	1	8	yes	—
71	C₄×#1055	—	2	2	no	—
72	C₄×#1055	—	2	2	no	—
73	C₄×#1055	—	2	4	no	—
74	C₄×#1055	—	2	2	no	—
75	C₄×#1055	C₂×C₄	1	4	yes	—
76	C₄×#1055	C₂×C₄	1	8	yes	—
77	C₄×#1055	C₂×C₄	1	4	yes	—
78	C₄×#1055	Q₈	1	4	yes	complemented
79	#20516	—	13	1	no	maximal, Sylow
80	C₂×Q₈×C₁₃	—	2	4	no	—
81	C₂×Q₈×C₁₃	C₄²	1	4	yes	—
82	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
83	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₁₃×C₄⋊C₄	—	2	8	no	—
91	C₁₃×C₄⋊C₄	—	2	4	no	—
92	C₁₃×C₄⋊C₄	C₄²	1	8	yes	—
93	C₁₃×C₄⋊C₄	C₄²	1	4	yes	—
94	C₄²×C₁₃	—	2	4	no	—
95	C₄²×C₁₃	—	2	4	no	—
96	C₄²×C₁₃	—	2	1	no	—
97	C₄²×C₁₃	—	2	1	no	—
98	C₄²×C₁₃	—	2	2	no	—
99	C₄²×C₁₃	—	2	2	no	—
100	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
101	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
102	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
103	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
104	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
105	C₄²×C₁₃	#41	1	2	yes	—
106	C₂×#1055	—	2	1	no	—
107	C₂×#1055	—	2	1	no	—
108	C₂×#1055	—	2	1	no	—
109	C₂×#1055	—	2	1	no	—
110	C₂×#1055	—	2	2	no	—
111	C₂×#1055	—	2	2	no	—
112	C₂×#1055	—	2	2	no	—
113	C₂×#1055	—	2	2	no	—
114	C₂×#1055	C₂²×C₄	1	2	yes	—
115	C₂×#1055	C₂²×C₄	1	2	yes	—
116	C₂×#1055	C₂²×C₄	1	4	yes	—
117	C₂×#1055	C₂²×C₄	1	4	yes	—
118	C₂×#1055	C₂²×C₄	1	2	yes	—
119	C₂×#1055	C₂²×C₄	1	2	yes	—
120	C₂×#1055	C₂×Q₈	1	2	yes	—
121	C₂×#1055	C₂×Q₈	1	2	yes	—
122	C₂×#1055	#41	1	2	yes	—
123	C₂×#1055	#41	1	2	yes	—
124	C₂×#446	—	13	1	no	—
125	C₂×C₈×Q₈	—	13	1	no	—
126	#2295	D₁₃	1	1	yes	characteristic
127	#1771	—	13	2	no	—
128	#1770	—	13	1	no	—
129	C₈×C₄⋊C₄	—	13	2	no	—
130	C₈×C₄⋊C₄	—	13	1	no	—
131	#1768	—	13	2	no	—
132	#1767	—	13	1	no	—
133	#1726	—	13	2	no	—
134	C₄²×C₈	—	13	1	no	—
135	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
136	Q₈×C₁₃	—	2	8	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	2	2	no	—
140	C₂×C₄×C₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	—	2	2	no	—
142	C₂×C₄×C₁₃	—	2	4	no	—
143	C₂×C₄×C₁₃	—	2	4	no	—
144	C₂×C₄×C₁₃	—	2	4	no	—
145	C₂×C₄×C₁₃	—	2	4	no	—
146	C₂×C₄×C₁₃	—	2	4	no	—
147	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
148	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
149	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
150	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
151	C₂×C₄×C₁₃	C₄×Q₈	1	4	yes	—
152	C₂×C₄×C₁₃	#131	1	4	yes	—
153	C₂×C₄×C₁₃	#131	1	4	yes	—
154	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
160	#1055	—	2	2	no	—
161	#1055	—	2	2	no	—
162	#1055	—	2	4	no	—
163	#1055	—	2	4	no	—
164	#1055	C₄×Q₈	1	4	yes	—
165	#1055	C₄×Q₈	1	4	yes	—
166	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
167	#446	—	13	8	no	—
168	C₈×Q₈	—	13	8	no	—
169	C₂×C₄⋊C₈	—	13	1	no	—
170	C₂×C₄⋊C₈	—	13	2	no	—
171	C₂×C₄⋊C₈	—	13	1	no	—
172	C₂×C₄⋊C₈	—	13	2	no	—
173	C₂×#97	—	13	2	no	—
174	C₂×#97	—	13	2	no	—
175	C₂×#97	—	13	2	no	—
176	C₂×C₄×C₈	—	13	1	no	—
177	C₂×C₄×C₈	—	13	2	no	—
178	C₂×C₄×C₈	—	13	1	no	—
179	C₂×C₄×C₈	—	13	1	no	—
180	C₂×C₄×C₈	—	13	1	no	—
181	C₂×C₄×C₈	—	13	1	no	—
182	C₂×C₄×C₈	—	13	1	no	—
183	C₂×C₄×C₈	—	13	2	no	—
184	#391	Dic₁₃	1	2	yes	—
185	#389	Dic₁₃	1	2	yes	—
186	#387	Dic₁₃	1	4	yes	—
187	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
188	C₄×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
189	#376	C₂×D₁₃	1	2	yes	—
190	C₄³	C₂×D₁₃	1	1	yes	characteristic
191	#336	—	13	2	no	—
192	#336	—	13	2	no	—
193	#336	—	13	2	no	—
194	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
195	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
196	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
197	C₂²×C₁₃	C₂×#131	1	2	yes	—
198	C₂²×C₁₃	C₂×#131	1	2	yes	—
199	C₄×C₁₃	—	2	4	no	—
200	C₄×C₁₃	—	2	2	no	—
201	C₄×C₁₃	—	2	4	no	—
202	C₄×C₁₃	—	2	4	no	—
203	C₄×C₁₃	—	2	4	no	—
204	C₄×C₁₃	—	4	4	no	—
205	C₄×C₁₃	#405	1	4	yes	—
206	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
207	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
208	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
209	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
210	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
211	C₂²×C₈	—	13	1	no	—
212	C₂²×C₈	—	13	1	no	—
213	C₂²×C₈	—	13	1	no	—
214	C₂²×C₈	—	13	1	no	—
215	C₂²×C₈	—	13	2	no	—
216	C₂²×C₈	—	13	2	no	—
217	C₄×Q₈	C₄×D₁₃	1	4	yes	—
218	C₄×Q₈	C₄×D₁₃	1	4	yes	—
219	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
220	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
221	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
222	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
223	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
224	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
225	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
226	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
227	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
228	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
229	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
230	C₄⋊C₈	—	13	4	no	—
231	C₄⋊C₈	—	13	8	no	—
232	C₄⋊C₈	—	13	4	no	—
233	C₄⋊C₈	—	13	8	no	—
234	#97	—	13	8	no	—
235	#97	—	26	4	no	—
236	#97	—	26	4	no	—
237	C₄×C₈	—	13	4	no	—
238	C₄×C₈	—	13	8	no	—
239	C₄×C₈	—	13	4	no	—
240	C₄×C₈	—	13	4	no	—
241	C₄×C₈	—	26	2	no	—
242	C₄×C₈	—	26	4	no	—
243	C₄×C₈	—	26	2	no	—
244	C₄×C₈	—	26	2	no	—
245	#95	C₂×Dic₁₃	1	4	yes	—
246	#95	C₂×Dic₁₃	1	4	yes	—
247	#95	C₂×Dic₁₃	1	2	yes	—
248	#95	C₂×Dic₁₃	1	4	yes	—
249	C₂×C₁₃	C₄²×Q₈	1	1	yes	characteristic
250	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
251	C₂×C₁₃	C₄×#131	1	1	yes	characteristic
252	C₂×C₁₃	#2960	1	2	yes	—
253	C₂×C₁₃	#2966	1	2	yes	—
254	C₂×Q₈	—	2	4	no	—
255	C₂×Q₈	C₄×Dic₁₃	1	4	yes	—
256	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
257	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
258	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
259	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
260	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
261	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
262	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
263	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
264	C₂×C₈	—	13	2	no	—
265	C₂×C₈	—	13	2	no	—
266	C₂×C₈	—	13	2	no	—
267	C₂×C₈	—	13	2	no	—
268	C₂×C₈	—	13	2	no	—
269	C₂×C₈	—	13	2	no	—
270	C₂×C₈	—	13	2	no	—
271	C₂×C₈	—	13	4	no	—
272	C₂×C₈	—	13	2	no	—
273	C₂×C₈	—	13	4	no	—
274	C₂×C₈	—	26	1	no	—
275	C₂×C₈	—	26	1	no	—
276	C₂×C₈	—	26	2	no	—
277	C₂×C₈	—	26	2	no	—
278	C₂×C₈	—	26	1	no	—
279	C₂×C₈	—	26	1	no	—
280	C₂×C₈	—	26	2	no	—
281	C₂×C₈	—	26	2	no	—
282	C₄⋊C₄	—	2	8	no	—
283	C₄⋊C₄	—	2	4	no	—
284	C₄⋊C₄	C₄×Dic₁₃	1	8	yes	—
285	C₄⋊C₄	C₄×Dic₁₃	1	4	yes	—
286	C₄²	—	2	4	no	—
287	C₄²	—	2	4	no	—
288	C₄²	—	2	1	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	2	no	—
292	C₄²	C₂×C₄×D₁₃	1	4	yes	—
293	C₄²	C₂×C₄×D₁₃	1	2	yes	—
294	C₄²	C₂×C₄×D₁₃	1	2	yes	—
295	C₄²	C₂×C₄×D₁₃	1	4	yes	—
296	C₄²	Q₈×D₁₃	1	2	yes	—
297	C₄²	#6131	1	2	yes	—
298	C₁₃	#20516	1	1	yes	characteristic, complemented, Sylow
299	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
300	Q₈	—	2	8	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	4	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
312	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
313	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
314	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
315	C₂×C₄	Q₈×Dic₁₃	1	4	yes	—
316	C₂×C₄	#88202	1	4	yes	—
317	C₂×C₄	#88202	1	4	yes	—
318	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
319	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
320	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
321	C₂×C₄	C₂×#6131	1	1	yes	characteristic
322	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
323	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
324	C₈	—	13	4	no	—
325	C₈	—	13	4	no	—
326	C₈	—	26	2	no	—
327	C₈	—	26	2	no	—
328	C₈	—	26	4	no	—
329	C₈	—	26	4	no	—
330	C₂²	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
331	C₂²	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
332	C₂²	C₂×#88202	1	2	yes	—
333	C₂²	C₂×#88202	1	2	yes	—
334	C₂²	#41×Dic₁₃	1	1	yes	characteristic
335	C₄	—	2	4	no	—
336	C₄	—	2	2	no	—
337	C₄	—	2	4	no	—
338	C₄	—	2	4	no	—
339	C₄	—	2	4	no	—
340	C₄	—	4	4	no	—
341	C₄	#11718658	1	4	yes	—
342	C₄	C₄×Q₈×D₁₃	1	1	yes	characteristic
343	C₄	C₄×Q₈×D₁₃	1	1	yes	characteristic
344	C₄	C₄×#6131	1	1	yes	characteristic
345	C₄	C₄×#6131	1	1	yes	characteristic
346	C₂	C₄×Q₈×Dic₁₃	1	1	yes	characteristic
347	C₂	C₂×#11718658	1	1	yes	characteristic
348	C₂	C₄×#88202	1	1	yes	characteristic
349	C₂	#13123771	1	2	yes	—
350	C₂	#13123812	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂²×C₄²
Fitting	C₁₃×#2295	C₂
socle, prime radical	C₂³×C₁₃	C₂³×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#2295	D₁₃
hypocenter	C₁₃	#20516
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2295

Complements

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

N	H ≅ G/N	type
20. #11718685	C₄	semidirect
21. Q₈×#1055	C₄	semidirect
78. C₄×#1055	Q₈	semidirect
298. C₁₃	#20516	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718685	2
C₂×Q₈×#1055	2
#13114711	2
#13114705	2
C₄⋊C₄×#1055	2
#13114693	2
#13114689	2
#13114523	2
C₄²×#1055	2
C₁₃×#2295	2
#20516	13