Subgroups — #15863370

Order 3328.

1070 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718782	C₂	1	2	yes	maximal
2	C₂×#11718767	C₂	1	1	yes	maximal, characteristic
3	C₂²×#88058	C₂	1	1	yes	maximal, characteristic
4	#13117459	C₂	1	2	yes	maximal
5	#13115004	C₂	1	4	yes	maximal, complemented
6	#13115000	C₂	1	2	yes	maximal, complemented
7	#13114996	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
12	C₂×#88180	C₂²	1	2	yes	—
13	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
14	C₁₃×#405	C₄	1	8	yes	complemented
15	C₂×C₁₃×#97	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
17	#11718782	—	2	8	no	—
18	#11718767	—	2	4	no	—
19	C₂×#88058	—	2	4	no	—
20	C₂×#88058	C₂²	1	2	yes	—
21	C₂×#88058	C₂²	1	2	yes	—
22	C₂×#88058	C₂²	1	1	yes	characteristic
23	C₂×#88058	C₂²	1	1	yes	characteristic
24	C₂×#88058	C₂²	1	4	yes	—
25	C₂×#88057	C₂²	1	2	yes	—
26	C₂×#88057	C₂²	1	4	yes	—
27	#11718646	—	2	4	no	—
28	#11718611	—	2	4	no	—
29	#11718597	C₂²	1	4	yes	—
30	#11718596	C₂²	1	4	yes	—
31	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
32	C₂³×Dic₁₃	—	2	2	no	—
33	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
34	C₂×C₁₃×M₁₆	D₄	1	4	yes	—
35	C₂×C₁₃×M₁₆	Q₈	1	4	yes	—
36	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₁₃	D₄	1	4	yes	—
38	C₂²×C₈×C₁₃	Q₈	1	4	yes	—
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×#117	C₂×C₄	1	4	yes	—
43	C₁₃×#117	Q₈	1	4	yes	—
44	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
45	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
47	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
48	#88180	—	2	4	no	—
49	#88180	—	2	4	no	—
50	C₂×C₄⋊Dic₁₃	—	2	4	no	—
51	C₂×C₄⋊Dic₁₃	—	2	4	no	—
52	C₂×C₄⋊Dic₁₃	—	2	2	no	—
53	C₂×C₄⋊Dic₁₃	—	2	8	no	—
54	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
55	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
56	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
57	C₂×C₄×Dic₁₃	—	2	2	no	—
58	C₁₃×#97	C₂×C₄	1	8	yes	complemented
59	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	complemented
60	#88071	—	2	8	no	—
61	#88058	—	2	4	no	—
62	#88058	—	2	4	no	—
63	#88058	—	2	2	no	—
64	#88058	—	2	2	no	—
65	#88058	—	2	8	no	—
66	#88057	—	2	4	no	—
67	#88057	—	2	8	no	—
68	#20803	—	13	1	no	maximal, Sylow
69	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
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	2	yes	—
75	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₂²×Dic₁₃	—	2	2	no	—
80	C₂²×Dic₁₃	—	2	4	no	—
81	C₂²×Dic₁₃	—	2	2	no	—
82	C₂²×Dic₁₃	—	4	4	no	—
83	C₂²×Dic₁₃	—	8	4	no	—
84	C₁₃×M₁₆	C₄⋊C₄	1	16	yes	complemented
85	C₂×C₈×C₁₃	C₄⋊C₄	1	16	yes	complemented
86	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
92	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
94	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
95	C₁₃×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
96	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
97	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
98	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
99	C₂²⋊Dic₁₃	—	4	4	no	—
100	C₄⋊Dic₁₃	—	2	4	no	—
101	C₄⋊Dic₁₃	—	2	4	no	—
102	C₄⋊Dic₁₃	—	2	2	no	—
103	C₄⋊Dic₁₃	—	2	2	no	—
104	C₄⋊Dic₁₃	—	2	2	no	—
105	C₄⋊Dic₁₃	—	2	2	no	—
106	C₄⋊Dic₁₃	—	4	4	no	—
107	C₄⋊Dic₁₃	—	8	8	no	—
108	C₄×Dic₁₃	—	4	4	no	—
109	C₂×#428	—	13	2	no	—
110	C₂×#427	—	13	1	no	—
111	C₂²×#107	—	13	1	no	—
112	C₂×#405	D₁₃	1	1	yes	characteristic
113	#2317	—	13	2	no	—
114	#1847	—	13	4	no	—
115	#1846	—	13	2	no	—
116	#1845	—	13	2	no	—
117	C₂³×C₁₃	—	2	2	no	—
118	C₂³×C₁₃	—	2	2	no	—
119	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
120	C₂³×C₁₃	#127	1	2	yes	—
121	C₂³×C₁₃	C₄⋊Q₈	1	2	yes	—
122	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	—	2	4	no	—
124	C₂×C₄×C₁₃	—	2	4	no	—
125	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
126	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
127	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
128	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₁₃	#127	1	2	yes	—
130	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
131	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
132	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
133	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
139	C₂×Dic₁₃	—	4	2	no	—
140	C₂×Dic₁₃	—	4	2	no	—
141	C₂×Dic₁₃	—	4	4	no	—
142	C₂×Dic₁₃	—	4	4	no	—
143	C₂×Dic₁₃	—	4	8	no	—
144	C₂×Dic₁₃	—	4	4	no	—
145	C₂×Dic₁₃	—	8	8	no	—
146	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
147	C₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
148	C₈×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
149	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
150	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
151	C₂×#117	—	13	2	no	—
152	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
153	C₂²×C₄⋊C₄	—	13	2	no	—
154	#428	—	26	8	no	—
155	#427	—	26	4	no	—
156	C₂×#107	—	13	1	no	—
157	C₂×#107	—	13	2	no	—
158	C₂×#107	—	13	4	no	—
159	C₂×#107	—	13	1	no	—
160	C₂×#107	—	13	2	no	—
161	C₂×#107	—	26	4	no	—
162	C₂×#106	—	13	4	no	—
163	C₂×#106	—	13	2	no	—
164	#405	Dic₁₃	1	8	yes	complemented
165	C₂×#97	C₂×D₁₃	1	2	yes	—
166	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
167	#389	—	26	4	no	—
168	#385	—	26	4	no	—
169	#384	—	13	4	no	—
170	#382	—	13	4	no	—
171	C₂²×C₁₃	—	2	2	no	—
172	C₂²×C₁₃	—	2	2	no	—
173	C₂²×C₁₃	—	2	4	no	—
174	C₂²×C₁₃	—	2	4	no	—
175	C₂²×C₁₃	#384	1	4	yes	—
176	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
177	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
178	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
179	C₂²×C₁₃	#576	1	2	yes	—
180	C₂²×C₁₃	#578	1	2	yes	—
181	C₄×C₁₃	—	2	8	no	—
182	C₄×C₁₃	#384	1	4	yes	—
183	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
184	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
185	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
186	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
187	Dic₁₃	—	8	2	no	—
188	Dic₁₃	—	8	4	no	—
189	Dic₁₃	—	8	2	no	—
190	C₂³×C₄	—	26	2	no	—
191	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
192	C₂×M₁₆	Dic₂₆	1	4	yes	—
193	C₂×M₁₆	D₅₂	1	4	yes	—
194	C₂×M₁₆	C₂×Dic₁₃	1	4	yes	—
195	C₂²×C₈	Dic₂₆	1	4	yes	—
196	C₂²×C₈	D₅₂	1	4	yes	—
197	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
198	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
199	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
201	#117	—	26	4	no	—
202	#117	—	26	4	no	—
203	#117	Dic₂₆	1	4	yes	—
204	#117	C₂×Dic₁₃	1	4	yes	—
205	C₂×C₄⋊C₄	—	13	4	no	—
206	C₂×C₄⋊C₄	—	13	2	no	—
207	C₂×C₄⋊C₄	—	13	2	no	—
208	C₂×C₄⋊C₄	—	26	4	no	—
209	C₂×C₄⋊C₄	—	26	4	no	—
210	C₂×C₄⋊C₄	—	26	2	no	—
211	C₂×C₄⋊C₄	—	26	8	no	—
212	C₂×C₄⋊C₄	D₅₂	1	2	yes	—
213	C₂×C₂²⋊C₄	—	26	2	no	—
214	C₂×C₂²⋊C₄	D₅₂	1	2	yes	—
215	C₂×C₄²	—	26	2	no	—
216	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
217	#107	—	26	4	no	—
218	#107	—	26	4	no	—
219	#107	—	26	2	no	—
220	#107	—	26	2	no	—
221	#107	—	26	8	no	—
222	#106	—	26	4	no	—
223	#106	—	26	8	no	—
224	#97	C₂×Dic₁₃	1	8	yes	complemented
225	C₄×C₈	C₂×Dic₁₃	1	8	yes	complemented
226	#95	—	26	8	no	—
227	C₂×C₁₃	—	2	4	no	—
228	C₂×C₁₃	C₂×#384	1	1	yes	characteristic
229	C₂×C₁₃	#2913	1	2	yes	—
230	C₂×C₁₃	#3155	1	1	yes	characteristic
231	C₂×C₁₃	#3156	1	1	yes	characteristic
232	C₂×C₁₃	#3164	1	2	yes	—
233	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
234	C₂²×C₄	—	26	2	no	—
235	C₂²×C₄	—	26	4	no	—
236	C₂²×C₄	—	26	2	no	—
237	C₂²×C₄	—	52	4	no	—
238	C₂²×C₄	—	104	4	no	—
239	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
240	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
241	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
242	C₂²×C₄	C₂×D₅₂	1	4	yes	—
243	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₅₂	1	2	yes	—
245	C₂²×C₄	C₂×D₅₂	1	2	yes	—
246	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
247	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
248	M₁₆	C₄⋊Dic₁₃	1	16	yes	complemented
249	C₂×C₈	C₄⋊Dic₁₃	1	16	yes	complemented
250	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
251	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
252	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
253	C₂×C₈	C₂×D₅₂	1	2	yes	—
254	C₂×C₈	C₂×D₅₂	1	4	yes	—
255	C₂×C₈	C₂×D₅₂	1	2	yes	—
256	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
257	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
258	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
259	C₄⋊C₄	—	26	2	no	—
260	C₄⋊C₄	—	26	4	no	—
261	C₄⋊C₄	—	26	2	no	—
262	C₄⋊C₄	—	26	2	no	—
263	C₄⋊C₄	—	26	4	no	—
264	C₄⋊C₄	—	26	2	no	—
265	C₄⋊C₄	—	52	4	no	—
266	C₄⋊C₄	—	104	8	no	—
267	C₄⋊C₄	C₄⋊Dic₁₃	1	8	yes	—
268	C₂²⋊C₄	—	52	4	no	—
269	C₂²⋊C₄	C₄⋊Dic₁₃	1	8	yes	—
270	C₄²	—	52	4	no	—
271	C₄²	C₂×Dic₂₆	1	4	yes	—
272	C₄²	C₂²×Dic₁₃	1	4	yes	—
273	C₁₃	#20803	1	1	yes	characteristic, complemented, Sylow
274	C₂³	—	2	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	#88123	1	2	yes	—
277	C₂³	#88128	1	2	yes	—
278	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
279	C₂³	C₂²×D₅₂	1	1	yes	characteristic
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	52	2	no	—
283	C₂×C₄	—	52	2	no	—
284	C₂×C₄	—	52	4	no	—
285	C₂×C₄	—	52	4	no	—
286	C₂×C₄	—	52	8	no	—
287	C₂×C₄	—	52	4	no	—
288	C₂×C₄	—	104	8	no	—
289	C₂×C₄	#88123	1	4	yes	—
290	C₂×C₄	#88123	1	4	yes	—
291	C₂×C₄	#88123	1	2	yes	—
292	C₂×C₄	#88128	1	2	yes	—
293	C₂×C₄	C₂×C₄⋊Dic₁₃	1	8	yes	—
294	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
295	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
296	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
297	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
298	C₂×C₄	C₂²×Dic₂₆	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	C₂²×D₅₂	1	1	yes	characteristic
302	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
303	C₈	C₂×C₄⋊Dic₁₃	1	4	yes	—
304	C₈	C₂×C₄⋊Dic₁₃	1	4	yes	—
305	C₈	C₂×C₄⋊Dic₁₃	1	8	yes	—
306	C₂²	—	2	4	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	4	no	—
310	C₂²	#11718596	1	4	yes	—
311	C₂²	C₂×#88123	1	1	yes	characteristic
312	C₂²	C₂×#88128	1	1	yes	characteristic
313	C₂²	#11719457	1	2	yes	—
314	C₂²	#11719458	1	2	yes	—
315	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
316	C₄	—	2	8	no	—
317	C₄	—	104	2	no	—
318	C₄	—	104	4	no	—
319	C₄	—	104	2	no	—
320	C₄	#11718596	1	4	yes	—
321	C₄	C₂×#88123	1	1	yes	characteristic
322	C₄	C₂×#88123	1	1	yes	characteristic
323	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
324	C₄	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
325	C₂	—	2	4	no	—
326	C₂	C₂×#11718596	1	1	yes	characteristic
327	C₂	#13123436	1	2	yes	—
328	C₂	#13125322	1	1	yes	characteristic
329	C₂	#13125330	1	1	yes	characteristic
330	C₂	#13125375	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#405	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#405	D₁₃
hypocenter	C₁₃	#20803
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
5. #13115004	C₂	semidirect
6. #13115000	C₂	semidirect
7. #13114996	C₂	semidirect
14. C₁₃×#405	C₄	semidirect
58. C₁₃×#97	C₂×C₄	semidirect
59. C₄×C₈×C₁₃	C₂×C₄	semidirect
84. C₁₃×M₁₆	C₄⋊C₄	semidirect
85. C₂×C₈×C₁₃	C₄⋊C₄	semidirect
164. #405	Dic₁₃	semidirect
224. #97	C₂×Dic₁₃	semidirect
225. C₄×C₈	C₂×Dic₁₃	semidirect
248. M₁₆	C₄⋊Dic₁₃	semidirect
249. C₂×C₈	C₄⋊Dic₁₃	semidirect
273. C₁₃	#20803	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718782	2
C₂×#11718767	2
C₂²×#88058	2
#13117459	2
#13115004	2
#13115000	2
#13114996	2
C₂×C₁₃×#405	2
#20803	13