Subgroups — #15768767

Order 3328.

998 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	C₂²×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718121	C₂	1	4	yes	maximal
4	C₂×C₁₃×#382	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
6	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
8	C₂²×C₄×Dic₁₃	—	2	1	no	—
9	C₂³×#1055	C₂²	1	2	yes	—
10	C₁₃×#382	—	2	2	no	—
11	C₁₃×#382	C₄	1	4	yes	complemented
12	C₂×C₁₃×#95	—	2	1	no	—
13	C₂×C₁₃×#95	—	2	1	no	—
14	C₂×#88071	—	2	1	no	—
15	C₄×C₄⋊Dic₁₃	—	2	2	no	—
16	C₄×C₄⋊Dic₁₃	C₄	1	4	yes	complemented
17	C₂×C₄⋊#1055	—	2	2	no	—
18	C₂×C₄⋊#1055	—	2	2	no	—
19	C₂×C₄⋊#1055	C₄	1	4	yes	complemented
20	#11718121	—	2	8	no	—
21	C₂³×C₄×C₁₃	—	2	2	no	—
22	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂³×Dic₁₃	—	2	2	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
28	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
29	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₃	D₄	1	2	yes	—
34	C₂×C₄²×C₁₃	D₄	1	2	yes	—
35	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
36	C₂×C₄⋊Dic₁₃	—	2	2	no	—
37	C₂×C₄⋊Dic₁₃	—	2	2	no	—
38	C₂×C₄⋊Dic₁₃	—	2	2	no	—
39	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊Dic₁₃	D₄	1	4	yes	—
41	C₂×C₄⋊Dic₁₃	Q₈	1	4	yes	—
42	C₂×C₄×Dic₁₃	—	2	4	no	—
43	C₂×C₄×Dic₁₃	—	4	4	no	—
44	C₂²×#1055	—	2	2	no	—
45	C₂²×#1055	—	2	2	no	—
46	C₂²×#1055	—	2	4	no	—
47	C₂²×#1055	—	2	4	no	—
48	C₂²×#1055	C₂×C₄	1	4	yes	—
49	C₁₃×#95	—	2	4	no	—
50	C₁₃×#95	—	2	4	no	—
51	#88071	—	2	4	no	—
52	C₄⋊#1055	—	2	8	no	—
53	#10850	—	13	1	no	maximal, Sylow
54	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	—	2	4	no	—
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	—	2	4	no	—
59	C₂²×C₄×C₁₃	—	4	4	no	—
60	C₂²×C₄×C₁₃	—	4	4	no	—
61	C₂²×C₄×C₁₃	—	4	2	no	—
62	C₂²×C₄×C₁₃	—	4	2	no	—
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₁₃	#41	1	2	yes	—
78	C₂²×C₄×C₁₃	#41	1	2	yes	—
79	C₂²×Dic₁₃	—	2	4	no	—
80	C₂²×Dic₁₃	—	2	4	no	—
81	C₂²×Dic₁₃	—	2	4	no	—
82	C₂²×Dic₁₃	—	4	4	no	—
83	C₂²×Dic₁₃	—	4	4	no	—
84	C₁₃×C₄⋊C₄	—	2	8	no	—
85	C₁₃×C₄⋊C₄	—	4	2	no	—
86	C₁₃×C₄⋊C₄	—	4	2	no	—
87	C₄²×C₁₃	C₄²	1	4	yes	—
88	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
89	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
90	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
91	C₄⋊Dic₁₃	—	2	4	no	—
92	C₄⋊Dic₁₃	—	2	4	no	—
93	C₄⋊Dic₁₃	—	2	4	no	—
94	C₄⋊Dic₁₃	C₄⋊C₄	1	8	yes	—
95	C₄×Dic₁₃	—	4	8	no	—
96	C₂×#1055	—	2	4	no	—
97	C₂×#1055	—	2	8	no	—
98	C₂×#1055	—	4	4	no	—
99	C₂×#1055	—	4	2	no	—
100	C₂×#1055	—	4	2	no	—
101	C₂²×C₄⋊C₈	—	13	1	no	—
102	C₂×#382	D₁₃	1	1	yes	characteristic
103	C₂×C₄×C₄⋊C₄	—	13	1	no	—
104	C₂×#340	—	13	4	no	—
105	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂³×C₁₃	#121	1	2	yes	—
109	C₂³×C₁₃	C₂²⋊Q₈	1	2	yes	—
110	C₂³×C₁₃	#135	1	4	yes	—
111	C₂³×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
112	C₂³×C₁₃	#137	1	2	yes	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	4	no	—
119	C₂×C₄×C₁₃	—	2	4	no	—
120	C₂×C₄×C₁₃	—	4	4	no	—
121	C₂×C₄×C₁₃	—	4	4	no	—
122	C₂×C₄×C₁₃	—	4	4	no	—
123	C₂×C₄×C₁₃	—	8	4	no	—
124	C₂×C₄×C₁₃	—	8	4	no	—
125	C₂×C₄×C₁₃	#95	1	4	yes	—
126	C₂×C₄×C₁₃	#95	1	4	yes	—
127	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	#117	1	2	yes	—
133	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
135	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
136	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
137	C₂×C₄×C₁₃	#121	1	2	yes	—
138	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
139	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
141	C₂×C₄×C₁₃	#125	1	2	yes	—
142	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
144	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
145	C₂×Dic₁₃	—	2	8	no	—
146	C₂×Dic₁₃	—	4	4	no	—
147	C₂×Dic₁₃	—	4	4	no	—
148	C₂×Dic₁₃	—	4	8	no	—
149	C₂×Dic₁₃	—	4	8	no	—
150	#1055	—	4	4	no	—
151	C₂³×C₈	—	13	2	no	—
152	C₂²×C₄⋊C₄	—	13	2	no	—
153	C₂²×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
154	C₂²×C₄²	—	26	1	no	—
155	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
156	C₂×C₄⋊C₈	—	13	4	no	—
157	C₂×C₄⋊C₈	—	26	2	no	—
158	C₂×C₄⋊C₈	—	26	2	no	—
159	#382	—	2	2	no	—
160	#382	Dic₁₃	1	4	yes	complemented
161	C₄×C₄⋊C₄	—	13	4	no	—
162	C₄×C₄⋊C₄	—	26	2	no	—
163	C₂×#95	—	2	1	no	—
164	C₂×#95	—	2	1	no	—
165	C₂×#95	—	26	1	no	—
166	#340	—	26	8	no	—
167	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
168	C₂²×C₁₃	#380	1	2	yes	—
169	C₂²×C₁₃	#389	1	2	yes	—
170	C₂²×C₁₃	#390	1	1	yes	characteristic
171	C₂²×C₁₃	#391	1	1	yes	characteristic
172	C₂²×C₁₃	#416	1	2	yes	—
173	C₂²×C₁₃	#416	1	2	yes	—
174	C₂²×C₁₃	#417	1	2	yes	—
175	C₂²×C₁₃	#418	1	1	yes	characteristic
176	C₂²×C₁₃	#419	1	1	yes	characteristic
177	C₂²×C₁₃	#427	1	2	yes	—
178	C₂²×C₁₃	#428	1	2	yes	—
179	C₂²×C₁₃	#463	1	4	yes	—
180	C₂²×C₁₃	#464	1	4	yes	—
181	C₂²×C₁₃	#478	1	4	yes	—
182	C₂²×C₁₃	#479	1	4	yes	—
183	C₄×C₁₃	—	2	4	no	—
184	C₄×C₁₃	—	2	4	no	—
185	C₄×C₁₃	—	8	4	no	—
186	C₄×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
187	C₄×C₁₃	#382	1	1	yes	characteristic
188	C₄×C₁₃	#384	1	1	yes	characteristic
189	C₄×C₁₃	#389	1	2	yes	—
190	C₄×C₁₃	#390	1	1	yes	characteristic
191	C₄×C₁₃	#391	1	1	yes	characteristic
192	Dic₁₃	—	4	8	no	—
193	C₂³×C₄	—	2	2	no	—
194	C₂³×C₄	—	26	2	no	—
195	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
196	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
197	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
198	C₂²×C₈	—	13	4	no	—
199	C₂²×C₈	—	26	2	no	—
200	C₂²×C₈	—	26	4	no	—
201	C₂²×C₈	—	26	4	no	—
202	C₂²×C₈	—	26	2	no	—
203	C₂×C₄⋊C₄	—	2	4	no	—
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	13	4	no	—
207	C₂×C₄⋊C₄	—	13	4	no	—
208	C₂×C₄⋊C₄	—	13	4	no	—
209	C₂×C₄⋊C₄	—	26	2	no	—
210	C₂×C₄⋊C₄	—	26	2	no	—
211	C₂×C₄⋊C₄	—	26	2	no	—
212	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
213	C₂×C₄²	—	26	4	no	—
214	C₂×C₄²	—	52	4	no	—
215	C₂×C₄²	Dic₂₆	1	2	yes	—
216	C₂×C₄²	C₄×D₁₃	1	2	yes	—
217	C₂×C₄²	C₄×D₁₃	1	2	yes	—
218	C₂×C₄²	D₅₂	1	2	yes	—
219	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
220	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
221	C₄⋊C₈	—	26	8	no	—
222	#95	—	2	4	no	—
223	#95	—	2	4	no	—
224	#95	—	26	4	no	—
225	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
226	C₂×C₁₃	#1735	1	1	yes	characteristic
227	C₂×C₁₃	#1736	1	1	yes	characteristic
228	C₂×C₁₃	#1804	1	2	yes	—
229	C₂×C₁₃	#1805	1	2	yes	—
230	C₂×C₁₃	#1949	1	4	yes	—
231	C₂×C₁₃	#1983	1	2	yes	—
232	C₂×C₁₃	#1989	1	2	yes	—
233	C₂⁴	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	4	4	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	26	4	no	—
243	C₂²×C₄	—	26	4	no	—
244	C₂²×C₄	—	26	4	no	—
245	C₂²×C₄	—	52	4	no	—
246	C₂²×C₄	—	52	4	no	—
247	C₂²×C₄	#6101	1	2	yes	—
248	C₂²×C₄	#6101	1	2	yes	—
249	C₂²×C₄	#6103	1	2	yes	—
250	C₂²×C₄	#6103	1	2	yes	—
251	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
252	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
253	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
257	C₂²×C₄	D₄×D₁₃	1	2	yes	—
258	C₂²×C₄	#6129	1	2	yes	—
259	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
260	C₂²×C₄	#6131	1	2	yes	—
261	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
262	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
263	C₂×C₈	—	26	8	no	—
264	C₂×C₈	—	26	4	no	—
265	C₂×C₈	—	52	4	no	—
266	C₂×C₈	—	52	2	no	—
267	C₂×C₈	—	52	2	no	—
268	C₄⋊C₄	—	2	8	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	13	8	no	—
272	C₄⋊C₄	—	26	4	no	—
273	C₄⋊C₄	—	26	4	no	—
274	C₄⋊C₄	—	26	4	no	—
275	C₄²	—	52	8	no	—
276	C₄²	C₄×Dic₁₃	1	4	yes	—
277	C₄²	#6101	1	4	yes	—
278	C₄²	C₄⋊Dic₁₃	1	4	yes	—
279	C₄²	#6103	1	4	yes	—
280	C₁₃	#10850	1	1	yes	characteristic, complemented, Sylow
281	C₂³	C₂×#6101	1	1	yes	characteristic
282	C₂³	C₂×#6103	1	1	yes	characteristic
283	C₂³	#88192	1	2	yes	—
284	C₂³	#88200	1	2	yes	—
285	C₂³	#88203	1	2	yes	—
286	C₂³	#88204	1	4	yes	—
287	C₂³	#88205	1	2	yes	—
288	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	8	4	no	—
300	C₂×C₄	—	8	4	no	—
301	C₂×C₄	—	26	8	no	—
302	C₂×C₄	—	52	4	no	—
303	C₂×C₄	—	52	4	no	—
304	C₂×C₄	—	52	8	no	—
305	C₂×C₄	—	52	8	no	—
306	C₂×C₄	#88071	1	4	yes	—
307	C₂×C₄	#88071	1	4	yes	—
308	C₂×C₄	#88141	1	2	yes	—
309	C₂×C₄	#88142	1	2	yes	—
310	C₂×C₄	#88144	1	2	yes	—
311	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
312	C₂×C₄	#88146	1	2	yes	—
313	C₂×C₄	#88147	1	2	yes	—
314	C₂×C₄	#88149	1	2	yes	—
315	C₂×C₄	#88151	1	2	yes	—
316	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
317	C₂×C₄	C₂×#6101	1	1	yes	characteristic
318	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
319	C₂×C₄	C₂×#6103	1	1	yes	characteristic
320	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
321	C₂×C₄	#88190	1	1	yes	characteristic
322	C₂×C₄	#88194	1	1	yes	characteristic
323	C₂×C₄	#88198	1	1	yes	characteristic
324	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
325	C₂×C₄	#88201	1	1	yes	characteristic
326	C₈	—	52	4	no	—
327	C₂²	#11718634	1	2	yes	—
328	C₂²	#11718648	1	2	yes	—
329	C₂²	#11718656	1	2	yes	—
330	C₂²	#11718659	1	2	yes	—
331	C₂²	#11718662	1	1	yes	characteristic
332	C₂²	#11718663	1	2	yes	—
333	C₂²	#11718665	1	1	yes	characteristic
334	C₂²	#11718715	1	4	yes	—
335	C₂²	#11718718	1	4	yes	—
336	C₂²	#11718724	1	4	yes	—
337	C₂²	#11718727	1	4	yes	—
338	C₂²	C₂×#88071	1	1	yes	characteristic
339	C₂²	#11718882	1	1	yes	characteristic
340	C₂²	#11718891	1	1	yes	characteristic
341	C₂²	#11718893	1	2	yes	—
342	C₂²	#11718894	1	2	yes	—
343	C₄	—	2	4	no	—
344	C₄	—	2	4	no	—
345	C₄	—	8	4	no	—
346	C₄	—	52	8	no	—
347	C₄	#11718635	1	1	yes	characteristic
348	C₄	C₄⋊C₄×Dic₁₃	1	2	yes	—
349	C₄	#11718637	1	1	yes	characteristic
350	C₄	#11718638	1	1	yes	characteristic
351	C₄	#11718646	1	2	yes	—
352	C₄	#11718649	1	1	yes	characteristic
353	C₂	#13113065	1	1	yes	characteristic
354	C₂	#13114552	1	1	yes	characteristic
355	C₂	#13114557	1	1	yes	characteristic
356	C₂	#13114843	1	2	yes	—
357	C₂	#13114848	1	2	yes	—
358	C₂	#13115602	1	4	yes	—
359	C₂	#13115769	1	2	yes	—
360	C₂	#13115794	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×#382	C₂
socle, prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×#382	D₁₃
hypocenter	C₁₃	#10850
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂×#382

Complements

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

N	H ≅ G/N	type
11. C₁₃×#382	C₄	semidirect
16. C₄×C₄⋊Dic₁₃	C₄	semidirect
19. C₂×C₄⋊#1055	C₄	semidirect
160. #382	Dic₁₃	semidirect
280. C₁₃	#10850	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊#1055	2
C₂×C₄×C₄⋊Dic₁₃	2
C₂×#11718121	2
C₂×C₁₃×#382	2
#10850	13