Subgroups — #15832211

Order 3328.

616 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123961	C₂	1	2	yes	maximal
2	#13113625	C₂	1	2	yes	maximal, complemented
3	#13113621	C₂	1	4	yes	maximal, complemented
4	#13113599	C₂	1	2	yes	maximal, complemented
5	#13113592	C₂	1	1	yes	maximal, characteristic, complemented
6	#13113584	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#11718049	C₂	1	2	yes	maximal, complemented
8	C₈×C₁₃×#41	C₂	1	1	yes	maximal, characteristic
9	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
10	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic
11	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic
12	C₈×D₄×C₁₃	C₄	1	2	yes	—
13	C₈×D₄×C₁₃	C₄	1	1	yes	characteristic
14	C₈×D₄×C₁₃	C₄	1	2	yes	—
15	C₈×D₄×C₁₃	C₄	1	1	yes	characteristic
16	C₁₃×#431	C₂²	1	2	yes	—
17	C₁₃×#431	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	#11718685	—	2	2	no	—
21	#11718672	—	2	4	no	—
22	D₄×#1055	—	2	2	no	—
23	#11718587	C₂²	1	4	yes	—
24	#11718586	C₂²	1	2	yes	—
25	#11718585	C₂²	1	2	yes	—
26	C₂×C₄⋊#1055	C₂²	1	4	yes	—
27	#11718050	C₂²	1	2	yes	—
28	#11718050	C₂²	1	4	yes	—
29	#11718049	C₂²	1	2	yes	—
30	#11718049	C₂²	1	4	yes	—
31	#11718049	C₂²	1	1	yes	characteristic
32	#11718049	C₂²	1	2	yes	—
33	#11718049	C₂²	1	1	yes	characteristic
34	C₂×C₁₃×#41	Q₈	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
41	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
42	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
44	C₄×D₄×C₁₃	D₄	1	2	yes	—
45	C₁₃×#117	Q₈	1	1	yes	characteristic
46	C₁₃×#117	Q₈	1	2	yes	—
47	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×#6099	—	2	2	no	—
50	C₂²×#1055	—	2	4	no	—
51	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
52	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
53	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
56	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
57	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
61	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂³	1	2	yes	—
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	C₂²⋊#1055	—	2	4	no	—
66	C₂²⋊#1055	—	2	2	no	—
67	C₄⋊#1055	—	2	2	no	—
68	C₄⋊#1055	—	2	4	no	—
69	C₄⋊#1055	C₂³	1	2	yes	—
70	C₄⋊#1055	C₂³	1	4	yes	—
71	C₄⋊#1055	C₂³	1	2	yes	—
72	#88043	—	2	4	no	—
73	C₄×#1055	—	2	2	no	—
74	#17834	—	13	1	no	maximal, Sylow
75	C₁₃×#41	—	2	4	no	—
76	C₂×Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂×D₄×C₁₃	C₄⋊C₄	1	2	yes	—
78	C₂×D₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₈×C₁₃	—	2	4	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	—	4	2	no	—
88	C₂×C₈×C₁₃	—	4	1	no	—
89	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
96	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
97	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
98	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
99	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
100	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
101	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	2	yes	—
102	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
103	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₃	C₂×D₄	1	2	yes	—
106	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
107	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
108	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
109	#6099	—	8	2	no	—
110	C₂×#1055	—	2	2	no	—
111	C₂×#1055	—	2	2	no	—
112	C₂×#1055	—	2	4	no	—
113	C₂×#1055	—	8	4	no	—
114	#2990	—	13	2	no	—
115	C₈×#41	D₁₃	1	1	yes	characteristic
116	#1574	—	13	2	no	—
117	#1573	—	13	4	no	—
118	#1568	—	13	2	no	—
119	#1566	—	13	1	no	—
120	#1564	—	13	1	no	—
121	C₂×#334	—	13	2	no	—
122	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
124	Q₈×C₁₃	—	2	2	no	—
125	D₄×C₁₃	—	2	4	no	—
126	D₄×C₁₃	#108	1	4	yes	—
127	C₂×C₄×C₁₃	—	2	4	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	1	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	2	4	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
144	C₈×C₁₃	—	2	2	no	—
145	C₈×C₁₃	—	2	1	no	—
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	—	2	2	no	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	#131	1	2	yes	—
150	#1055	—	4	2	no	—
151	#1055	—	4	2	no	—
152	#1055	—	4	4	no	—
153	C₄×#41	C₂×D₁₃	1	1	yes	characteristic
154	#446	—	26	2	no	—
155	C₈×Q₈	Dic₁₃	1	1	yes	characteristic
156	C₈×Q₈	Dic₁₃	1	1	yes	characteristic
157	#435	—	26	4	no	—
158	C₈×D₄	—	26	2	no	—
159	C₈×D₄	Dic₁₃	1	2	yes	—
160	C₈×D₄	Dic₁₃	1	1	yes	characteristic
161	C₈×D₄	Dic₁₃	1	2	yes	—
162	C₈×D₄	Dic₁₃	1	1	yes	characteristic
163	#432	—	13	4	no	—
164	#431	—	13	2	no	—
165	#431	C₂×D₁₃	1	2	yes	—
166	#431	C₂×D₁₃	1	1	yes	characteristic
167	#423	—	13	2	no	—
168	C₂×C₄⋊C₈	—	13	4	no	—
169	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
170	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
171	#335	—	13	2	no	—
172	#335	—	13	4	no	—
173	#334	—	13	1	no	—
174	#334	—	13	2	no	—
175	#334	—	13	4	no	—
176	#334	—	13	1	no	—
177	#334	—	13	2	no	—
178	C₂²×C₁₃	—	2	2	no	—
179	C₂²×C₁₃	—	2	2	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	—	2	1	no	—
182	C₂²×C₁₃	—	2	1	no	—
183	C₂²×C₁₃	C₂×#108	1	4	yes	—
184	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
185	C₄×C₁₃	—	2	2	no	—
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	2	2	no	—
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	2	2	no	—
191	C₄×C₁₃	C₂×#108	1	2	yes	—
192	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
193	C₄×C₁₃	#568	1	1	yes	characteristic
194	C₄×C₁₃	#577	1	2	yes	—
195	C₂×#41	Dic₂₆	1	1	yes	characteristic
196	C₂×M₁₆	—	26	2	no	—
197	C₂²×C₈	—	26	4	no	—
198	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
199	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
200	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
201	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
202	C₄×Q₈	D₅₂	1	1	yes	characteristic
203	C₄×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
204	C₄×D₄	D₅₂	1	1	yes	characteristic
205	C₄×D₄	D₅₂	1	2	yes	—
206	C₄×D₄	C₂×Dic₁₃	1	2	yes	—
207	C₄×D₄	C₂×Dic₁₃	1	1	yes	characteristic
208	#117	Dic₂₆	1	1	yes	characteristic
209	#117	Dic₂₆	1	2	yes	—
210	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
211	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
212	C₄⋊C₈	—	13	2	no	—
213	C₄⋊C₈	—	13	4	no	—
214	C₄⋊C₈	—	13	2	no	—
215	C₄⋊C₈	—	26	4	no	—
216	C₄⋊C₈	—	26	2	no	—
217	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
218	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
219	C₄⋊C₈	C₂×Dic₁₃	1	1	yes	characteristic
220	C₄⋊C₈	C₂×Dic₁₃	1	1	yes	characteristic
221	C₂²⋊C₈	—	26	2	no	—
222	C₂²⋊C₈	—	26	4	no	—
223	C₂²⋊C₈	C₂×Dic₁₃	1	2	yes	—
224	C₂²⋊C₈	C₂×Dic₁₃	1	2	yes	—
225	C₂²⋊C₈	C₂×Dic₁₃	1	2	yes	—
226	#97	—	26	4	no	—
227	C₄×C₈	—	26	2	no	—
228	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
229	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
230	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
231	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
232	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
233	C₄×C₈	C₂²×D₁₃	1	2	yes	—
234	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
235	C₂×C₁₃	—	2	4	no	—
236	C₂×C₁₃	—	4	1	no	—
237	C₂×C₁₃	#2906	1	1	yes	characteristic
238	C₂×C₁₃	#2912	1	1	yes	characteristic
239	C₂×C₁₃	C₂²×#108	1	1	yes	characteristic
240	#41	—	2	4	no	—
241	C₂×Q₈	C₄⋊Dic₁₃	1	1	yes	characteristic
242	C₂×D₄	C₄⋊Dic₁₃	1	2	yes	—
243	C₂×D₄	C₄⋊Dic₁₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
245	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₅₂	1	2	yes	—
247	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
248	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
249	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
250	M₁₆	—	104	2	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	26	2	no	—
256	C₂×C₈	—	26	2	no	—
257	C₂×C₈	—	26	4	no	—
258	C₂×C₈	—	104	4	no	—
259	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
261	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
262	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
263	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
264	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
265	C₄⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
266	C₄⋊C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
267	C₄⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
268	C₄⋊C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
269	C₂²⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
270	C₂²⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
271	C₂²⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
272	C₄²	C₂×Dic₂₆	1	2	yes	—
273	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
274	C₄²	C₂×D₅₂	1	1	yes	characteristic
275	C₄²	C₂×D₅₂	1	2	yes	—
276	C₄²	C₂²×Dic₁₃	1	2	yes	—
277	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
278	C₄²	C₂³×D₁₃	1	1	yes	characteristic
279	C₁₃	#17834	1	1	yes	characteristic, complemented, Sylow
280	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
281	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
282	Q₈	—	2	2	no	—
283	D₄	—	2	4	no	—
284	D₄	#88059	1	4	yes	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
294	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
295	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
296	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
297	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
298	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
299	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
300	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
301	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	2	2	no	—
306	C₈	—	2	1	no	—
307	C₈	—	52	2	no	—
308	C₈	—	52	2	no	—
309	C₈	—	52	4	no	—
310	C₈	#88202	1	2	yes	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	C₂×#88059	1	4	yes	—
317	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	2	no	—
324	C₄	C₂×#88059	1	2	yes	—
325	C₄	#11719456	1	2	yes	—
326	C₄	C₂×#88202	1	1	yes	characteristic
327	C₄	#11719524	1	1	yes	characteristic
328	C₂	—	2	4	no	—
329	C₂	—	4	1	no	—
330	C₂	#13123404	1	1	yes	characteristic
331	C₂	#13123431	1	1	yes	characteristic
332	C₂	C₂²×#88059	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₈×C₁₃×#41	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂×C₁₃×#41	Q₈
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₈×#41	D₁₃
hypocenter	C₁₃	#17834
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. #13113625	C₂	semidirect
3. #13113621	C₂	semidirect
4. #13113599	C₂	semidirect
5. #13113592	C₂	semidirect
6. #13113584	C₂	semidirect
7. C₂×#11718049	C₂	semidirect
279. C₁₃	#17834	semidirect

Maximal subgroups

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

subgroup	index
#13123961	2
#13113625	2
#13113621	2
#13113599	2
#13113592	2
#13113584	2
C₂×#11718049	2
C₈×C₁₃×#41	2
#17834	13