Subgroups — #15768715

Order 3328.

1014 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88043	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718142	C₂	1	4	yes	maximal
4	C₂²×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
6	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊Dic₁₃	C₄	1	2	yes	—
8	C₂²×C₄×Dic₁₃	C₂²	1	2	yes	—
9	C₂³×#1055	C₂²	1	2	yes	—
10	C₂×C₁₃×C₄⋊C₈	—	2	2	no	—
11	C₂×C₁₃×C₄⋊C₈	—	2	2	no	—
12	C₂×C₁₃×C₄⋊C₈	C₄	1	4	yes	complemented
13	C₂×#88071	C₄	1	2	yes	—
14	C₄×C₄⋊Dic₁₃	—	2	2	no	—
15	C₄×C₄⋊Dic₁₃	—	2	2	no	—
16	C₂×#88043	—	2	2	no	—
17	C₂×#88043	—	2	2	no	—
18	C₂×#88043	C₄	1	4	yes	complemented
19	#11718142	—	2	8	no	—
20	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂³×Dic₁₃	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₃	—	2	4	no	—
27	C₂²×C₈×C₁₃	—	2	4	no	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₃	D₄	1	2	yes	—
32	C₂×C₄²×C₁₃	D₄	1	2	yes	—
33	C₂×C₄²×C₁₃	D₄	1	2	yes	—
34	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
35	C₂×C₄⋊Dic₁₃	—	2	4	no	—
36	C₂×C₄⋊Dic₁₃	—	2	8	no	—
37	C₂×C₄×Dic₁₃	—	2	2	no	—
38	C₂×C₄×Dic₁₃	—	2	2	no	—
39	C₂×C₄×Dic₁₃	—	2	4	no	—
40	C₂×C₄×Dic₁₃	—	2	4	no	—
41	C₂²×#1055	—	2	2	no	—
42	C₂²×#1055	—	2	2	no	—
43	C₂²×#1055	—	2	2	no	—
44	C₂²×#1055	—	2	2	no	—
45	C₂²×#1055	C₂×C₄	1	4	yes	—
46	C₂²×#1055	D₄	1	4	yes	—
47	C₂²×#1055	Q₈	1	4	yes	—
48	C₁₃×C₄⋊C₈	—	2	8	no	—
49	#88071	—	2	4	no	—
50	#88043	—	2	8	no	—
51	#10835	—	13	1	no	maximal, Sylow
52	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	—	2	4	no	—
54	C₂²×C₄×C₁₃	—	2	4	no	—
55	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
56	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₁₃	#41	1	2	yes	—
70	C₂²×C₄×C₁₃	#41	1	2	yes	—
71	C₂²×Dic₁₃	—	2	4	no	—
72	C₂²×Dic₁₃	—	2	8	no	—
73	C₂²×Dic₁₃	—	2	4	no	—
74	C₂²×Dic₁₃	—	2	8	no	—
75	C₂²×Dic₁₃	M₁₆	1	8	yes	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	—	2	8	no	—
78	C₂×C₈×C₁₃	—	4	4	no	—
79	C₂×C₈×C₁₃	—	4	2	no	—
80	C₂×C₈×C₁₃	—	4	2	no	—
81	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
82	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
83	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
84	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
85	C₄⋊Dic₁₃	—	2	16	no	—
86	C₄×Dic₁₃	—	4	4	no	—
87	C₄×Dic₁₃	—	4	2	no	—
88	C₄×Dic₁₃	—	4	2	no	—
89	C₂×#1055	—	2	4	no	—
90	C₂×#1055	—	2	4	no	—
91	C₂×#1055	—	2	4	no	—
92	C₂×#1055	—	2	4	no	—
93	C₂×#1055	—	2	4	no	—
94	C₂×#1055	—	2	4	no	—
95	C₂×#1055	C₄⋊C₄	1	8	yes	complemented
96	C₂²×C₄⋊C₈	D₁₃	1	1	yes	characteristic
97	C₂²×#97	—	13	1	no	—
98	C₂×C₄×C₄⋊C₄	—	13	1	no	—
99	C₂×#336	—	13	4	no	—
100	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
101	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂³×C₁₃	C₄×D₄	1	2	yes	—
104	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
105	C₂³×C₁₃	C₂×M₁₆	1	4	yes	—
106	C₂³×C₁₃	#131	1	4	yes	—
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	—	2	4	no	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	#95	1	4	yes	—
114	C₂×C₄×C₁₃	#95	1	4	yes	—
115	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	#117	1	2	yes	—
121	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₁₃	#121	1	2	yes	—
124	C₂×C₄×C₁₃	#121	1	2	yes	—
125	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
128	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#125	1	2	yes	—
130	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
132	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
133	C₂×Dic₁₃	—	2	8	no	—
134	C₂×Dic₁₃	—	2	16	no	—
135	C₂×Dic₁₃	—	4	4	no	—
136	C₂×Dic₁₃	—	4	4	no	—
137	C₂×Dic₁₃	—	4	8	no	—
138	C₈×C₁₃	—	4	4	no	—
139	#1055	—	2	8	no	—
140	C₂³×C₈	—	13	2	no	—
141	C₂³×C₈	C₂×D₁₃	1	2	yes	—
142	C₂²×C₄⋊C₄	—	13	2	no	—
143	C₂²×C₄²	—	13	2	no	—
144	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
145	C₂×C₄⋊C₈	—	2	2	no	—
146	C₂×C₄⋊C₈	—	2	2	no	—
147	C₂×C₄⋊C₈	Dic₁₃	1	4	yes	complemented
148	C₂×#97	—	13	4	no	—
149	C₂×#97	—	26	2	no	—
150	C₂×#97	—	26	2	no	—
151	C₄×C₄⋊C₄	—	26	2	no	—
152	C₄×C₄⋊C₄	—	26	2	no	—
153	C₂×#95	—	13	2	no	—
154	#336	—	26	8	no	—
155	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
156	C₂²×C₁₃	C₄×C₄⋊C₄	1	2	yes	—
157	C₂²×C₁₃	#389	1	2	yes	—
158	C₂²×C₁₃	#390	1	1	yes	characteristic
159	C₂²×C₁₃	#391	1	1	yes	characteristic
160	C₂²×C₁₃	C₄×M₁₆	1	2	yes	—
161	C₂²×C₁₃	#405	1	2	yes	—
162	C₂²×C₁₃	#407	1	2	yes	—
163	C₂²×C₁₃	#408	1	2	yes	—
164	C₂²×C₁₃	#423	1	2	yes	—
165	C₂²×C₁₃	#424	1	2	yes	—
166	C₂²×C₁₃	#435	1	8	yes	—
167	C₂²×C₁₃	#446	1	8	yes	—
168	C₄×C₁₃	—	2	4	no	—
169	C₄×C₁₃	—	2	4	no	—
170	C₄×C₁₃	#380	1	2	yes	—
171	C₄×C₁₃	#382	1	1	yes	characteristic
172	C₄×C₁₃	#384	1	1	yes	characteristic
173	C₄×C₁₃	#389	1	2	yes	—
174	C₄×C₁₃	#390	1	1	yes	characteristic
175	C₄×C₁₃	#391	1	1	yes	characteristic
176	Dic₁₃	—	4	8	no	—
177	C₂³×C₄	—	13	4	no	—
178	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
179	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
180	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
181	C₂²×C₈	—	2	4	no	—
182	C₂²×C₈	—	2	4	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	13	4	no	—
186	C₂²×C₈	—	13	4	no	—
187	C₂²×C₈	—	13	4	no	—
188	C₂²×C₈	—	26	2	no	—
189	C₂²×C₈	—	26	2	no	—
190	C₂²×C₈	—	26	2	no	—
191	C₂²×C₈	—	26	2	no	—
192	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
193	C₂×C₄⋊C₄	—	26	8	no	—
194	C₂×C₄⋊C₄	—	26	4	no	—
195	C₂×C₄²	—	26	2	no	—
196	C₂×C₄²	—	26	4	no	—
197	C₂×C₄²	—	26	4	no	—
198	C₂×C₄²	—	26	2	no	—
199	C₂×C₄²	Dic₂₆	1	2	yes	—
200	C₂×C₄²	C₄×D₁₃	1	2	yes	—
201	C₂×C₄²	D₅₂	1	2	yes	—
202	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
203	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
204	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
205	C₄⋊C₈	—	2	8	no	—
206	#97	—	26	8	no	—
207	#95	—	26	4	no	—
208	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
209	C₂×C₁₃	#1728	1	1	yes	characteristic
210	C₂×C₁₃	#1729	1	1	yes	characteristic
211	C₂×C₁₃	#1771	1	4	yes	—
212	C₂×C₁₃	#1940	1	4	yes	—
213	C₂×C₁₃	#1955	1	2	yes	—
214	C₂×C₁₃	#1985	1	2	yes	—
215	C₂⁴	C₂×C₄×D₁₃	1	1	yes	characteristic
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	13	8	no	—
219	C₂²×C₄	—	26	8	no	—
220	C₂²×C₄	—	26	8	no	—
221	C₂²×C₄	—	26	4	no	—
222	C₂²×C₄	—	26	4	no	—
223	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
224	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
225	C₂²×C₄	#6101	1	2	yes	—
226	C₂²×C₄	#6101	1	2	yes	—
227	C₂²×C₄	#6103	1	2	yes	—
228	C₂²×C₄	#6103	1	2	yes	—
229	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₁₃	1	2	yes	—
233	C₂²×C₄	#6129	1	2	yes	—
234	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
235	C₂²×C₄	#6131	1	2	yes	—
236	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
239	C₂×C₈	—	2	8	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	—	4	4	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	13	8	no	—
245	C₂×C₈	—	26	4	no	—
246	C₂×C₈	—	26	4	no	—
247	C₂×C₈	—	26	4	no	—
248	C₂×C₈	—	26	4	no	—
249	C₂×C₈	—	26	4	no	—
250	C₂×C₈	—	26	4	no	—
251	C₄⋊C₄	—	26	16	no	—
252	C₄²	—	52	4	no	—
253	C₄²	—	52	2	no	—
254	C₄²	—	52	2	no	—
255	C₄²	#6101	1	4	yes	—
256	C₄²	C₄⋊Dic₁₃	1	4	yes	—
257	C₄²	#6103	1	4	yes	—
258	C₄²	C₂²⋊Dic₁₃	1	4	yes	—
259	C₁₃	#10835	1	1	yes	characteristic, complemented, Sylow
260	C₂³	#88141	1	2	yes	—
261	C₂³	#88147	1	2	yes	—
262	C₂³	#88155	1	4	yes	—
263	C₂³	M₁₆×D₁₃	1	4	yes	—
264	C₂³	C₂×C₄×Dic₁₃	1	1	yes	characteristic
265	C₂³	C₂×#6101	1	1	yes	characteristic
266	C₂³	C₂×#6103	1	1	yes	characteristic
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	26	16	no	—
274	C₂×C₄	—	26	8	no	—
275	C₂×C₄	—	52	8	no	—
276	C₂×C₄	—	52	4	no	—
277	C₂×C₄	—	52	4	no	—
278	C₂×C₄	#88071	1	4	yes	—
279	C₂×C₄	#88071	1	4	yes	—
280	C₂×C₄	#88142	1	2	yes	—
281	C₂×C₄	#88144	1	2	yes	—
282	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
283	C₂×C₄	#88146	1	2	yes	—
284	C₂×C₄	#88149	1	2	yes	—
285	C₂×C₄	#88151	1	2	yes	—
286	C₂×C₄	C₂×#6101	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
288	C₂×C₄	C₂×#6103	1	1	yes	characteristic
289	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
290	C₂×C₄	#88190	1	1	yes	characteristic
291	C₂×C₄	#88192	1	2	yes	—
292	C₂×C₄	#88194	1	1	yes	characteristic
293	C₂×C₄	#88198	1	1	yes	characteristic
294	C₂×C₄	Q₈×Dic₁₃	1	2	yes	—
295	C₂×C₄	#88200	1	2	yes	—
296	C₂×C₄	#88201	1	1	yes	characteristic
297	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
298	C₈	—	4	4	no	—
299	C₈	—	26	8	no	—
300	C₂²	#11718489	1	8	yes	—
301	C₂²	#11718497	1	8	yes	—
302	C₂²	#11718634	1	2	yes	—
303	C₂²	#11718635	1	1	yes	characteristic
304	C₂²	C₄⋊C₄×Dic₁₃	1	2	yes	—
305	C₂²	#11718649	1	1	yes	characteristic
306	C₂²	#11718762	1	2	yes	—
307	C₂²	#11718763	1	2	yes	—
308	C₂²	#11718771	1	2	yes	—
309	C₂²	M₁₆×Dic₁₃	1	2	yes	—
310	C₂²	#11718779	1	2	yes	—
311	C₂²	#11718787	1	2	yes	—
312	C₂²	C₂×#88071	1	1	yes	characteristic
313	C₄	—	2	4	no	—
314	C₄	—	2	4	no	—
315	C₄	—	52	8	no	—
316	C₄	#11718637	1	1	yes	characteristic
317	C₄	#11718638	1	1	yes	characteristic
318	C₄	#11718646	1	2	yes	—
319	C₄	#11718648	1	2	yes	—
320	C₄	#11718882	1	1	yes	characteristic
321	C₄	#11718891	1	1	yes	characteristic
322	C₂	#13113065	1	1	yes	characteristic
323	C₂	#13114529	1	1	yes	characteristic
324	C₂	#13114532	1	1	yes	characteristic
325	C₂	#13114713	1	4	yes	—
326	C₂	#13115557	1	4	yes	—
327	C₂	#13115633	1	2	yes	—
328	C₂	#13115775	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₁₃
commutator	C₂³×C₁₃	C₂×C₄²
Fitting	C₂²×C₁₃×C₄⋊C₈	C₂
socle, prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂²×C₄⋊C₈	D₁₃
hypocenter	C₁₃	#10835
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
12. C₂×C₁₃×C₄⋊C₈	C₄	semidirect
18. C₂×#88043	C₄	semidirect
95. C₂×#1055	C₄⋊C₄	semidirect
147. C₂×C₄⋊C₈	Dic₁₃	semidirect
259. C₁₃	#10835	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88043	2
C₂×C₄×C₄⋊Dic₁₃	2
C₂×#11718142	2
C₂²×C₁₃×C₄⋊C₈	2
#10835	13