Subgroups — #15861761

Order 3328.

994 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718586	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718585	C₂	1	1	yes	maximal, characteristic
3	#13114705	C₂	1	2	yes	maximal, complemented
4	C₄⋊C₄×#1055	C₂	1	2	yes	maximal, complemented
5	#13114689	C₂	1	2	yes	maximal, complemented
6	C₄×C₄⋊#1055	C₂	1	2	yes	maximal, complemented
7	#13114620	C₂	1	4	yes	maximal
8	C₄×C₁₃×#117	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×#117	C₄	1	2	yes	—
10	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×#6099	C₂²	1	2	yes	—
12	C₄×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
13	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×C₂²⋊C₄	C₄	1	2	yes	—
16	C₄×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
17	C₁₃×#376	C₄	1	2	yes	—
18	C₄³×C₁₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊#1055	C₂²	1	2	yes	—
20	#11718586	C₄	1	8	yes	complemented
21	#11718585	C₄	1	8	yes	complemented
22	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊#1055	C₂²	1	2	yes	—
24	C₂×C₄⋊#1055	C₂²	1	2	yes	—
25	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
26	C₂×#88043	C₂²	1	4	yes	—
27	C₂×C₄×#1055	C₂²	1	4	yes	—
28	C₂×C₄×#1055	C₂²	1	2	yes	—
29	#11718116	C₂²	1	4	yes	—
30	#11718116	C₂²	1	4	yes	—
31	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₁₃×#117	—	2	8	no	—
35	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×#6099	—	4	2	no	—
49	C₂×#6099	—	4	2	no	—
50	C₂×#6099	C₂×C₄	1	8	yes	—
51	C₂²×#1055	C₂³	1	4	yes	—
52	C₂²×#1055	C₂³	1	2	yes	—
53	C₂²×#1055	C₂³	1	2	yes	—
54	C₁₃×#95	C₂×C₄	1	2	yes	—
55	C₁₃×#95	C₂×C₄	1	2	yes	—
56	C₁₃×#95	C₂×C₄	1	4	yes	—
57	C₂²⋊#1055	C₂×C₄	1	8	yes	—
58	C₄⋊#1055	C₂×C₄	1	8	yes	—
59	C₄⋊#1055	C₂×C₄	1	8	yes	—
60	C₄⋊#1055	C₂×C₄	1	4	yes	—
61	C₄⋊#1055	C₂×C₄	1	4	yes	—
62	#88043	—	4	4	no	—
63	C₄×#1055	—	4	4	no	—
64	C₄×#1055	C₂×C₄	1	8	yes	—
65	#20550	—	13	1	no	maximal, Sylow
66	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	—	2	4	no	—
68	C₂²×C₄×C₁₃	—	2	4	no	—
69	C₂²×C₄×C₁₃	—	2	1	no	—
70	C₂²×C₄×C₁₃	—	2	1	no	—
71	C₂²×C₄×C₁₃	—	2	1	no	—
72	C₂²×C₄×C₁₃	—	2	1	no	—
73	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
74	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
75	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
85	C₁₃×C₄⋊C₄	—	2	4	no	—
86	C₁₃×C₄⋊C₄	M₁₆	1	8	yes	complemented
87	C₁₃×C₂²⋊C₄	—	2	8	no	—
88	C₄²×C₁₃	—	2	8	no	—
89	C₄²×C₁₃	—	2	8	no	—
90	C₄²×C₁₃	—	2	2	no	—
91	C₄²×C₁₃	—	2	2	no	—
92	C₄²×C₁₃	—	2	2	no	—
93	C₄²×C₁₃	—	2	2	no	—
94	C₄²×C₁₃	C₄²	1	4	yes	—
95	C₄²×C₁₃	C₄²	1	4	yes	—
96	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
97	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
98	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
99	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
100	#6099	—	4	8	no	—
101	C₂×#1055	—	2	4	no	—
102	C₂×#1055	—	2	4	no	—
103	C₂×#1055	—	2	2	no	—
104	C₂×#1055	—	2	2	no	—
105	C₂×#1055	—	2	2	no	—
106	C₂×#1055	—	2	2	no	—
107	C₂×#1055	C₂²×C₄	1	4	yes	—
108	C₂×#1055	C₂²×C₄	1	4	yes	—
109	C₂×#1055	C₂²×C₄	1	8	yes	—
110	C₂×#431	—	13	1	no	—
111	C₂×#423	—	13	1	no	—
112	C₄×#117	D₁₃	1	1	yes	characteristic
113	#1770	—	13	2	no	—
114	C₈×C₄⋊C₄	—	13	2	no	—
115	#1767	—	13	2	no	—
116	C₄×C₄⋊C₈	—	13	2	no	—
117	#1753	—	13	4	no	—
118	C₂³×C₁₃	—	2	1	no	—
119	C₂³×C₁₃	—	2	1	no	—
120	C₂³×C₁₃	—	2	2	no	—
121	C₂³×C₁₃	—	2	2	no	—
122	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
123	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	—	2	4	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	4	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	8	no	—
135	C₂×C₄×C₁₃	—	2	8	no	—
136	C₂×C₄×C₁₃	—	2	4	no	—
137	C₂×C₄×C₁₃	—	2	4	no	—
138	C₂×C₄×C₁₃	—	2	4	no	—
139	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
140	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
141	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
142	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
143	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
144	C₂×C₄×C₁₃	C₂×M₁₆	1	8	yes	—
145	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
146	C₂×C₄×C₁₃	#131	1	4	yes	—
147	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
151	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
152	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
153	#1055	—	2	8	no	—
154	#1055	—	2	4	no	—
155	#1055	—	2	4	no	—
156	C₂²×M₁₆	—	13	2	no	—
157	C₂×#117	Dic₁₃	1	2	yes	—
158	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
159	#431	—	13	8	no	—
160	#423	—	13	8	no	—
161	C₂×C₄⋊C₈	—	13	2	no	—
162	C₂×C₄⋊C₈	—	13	2	no	—
163	C₂×C₄⋊C₈	—	13	1	no	—
164	C₂×C₄⋊C₈	—	13	1	no	—
165	C₂×C₂²⋊C₈	—	13	2	no	—
166	C₂×#97	—	13	4	no	—
167	C₂×C₄×C₈	—	13	2	no	—
168	C₂×C₄×C₈	—	13	4	no	—
169	C₄×C₄⋊C₄	Dic₁₃	1	2	yes	—
170	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
171	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
172	C₄×C₂²⋊C₄	Dic₁₃	1	2	yes	—
173	C₄×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
174	#376	Dic₁₃	1	2	yes	—
175	C₄³	C₂×D₁₃	1	2	yes	—
176	#336	—	13	4	no	—
177	#336	—	13	4	no	—
178	C₂²×C₁₃	—	2	2	no	—
179	C₂²×C₁₃	—	2	2	no	—
180	C₂²×C₁₃	—	2	2	no	—
181	C₂²×C₁₃	—	4	2	no	—
182	C₂²×C₁₃	—	4	2	no	—
183	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
184	C₂²×C₁₃	#520	1	1	yes	characteristic
185	C₂²×C₁₃	#520	1	1	yes	characteristic
186	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
187	C₂²×C₁₃	C₂×#131	1	2	yes	—
188	C₄×C₁₃	—	2	2	no	—
189	C₄×C₁₃	—	2	2	no	—
190	C₄×C₁₃	—	2	8	no	—
191	C₄×C₁₃	—	2	2	no	—
192	C₄×C₁₃	—	4	4	no	—
193	C₄×C₁₃	C₄×M₁₆	1	4	yes	—
194	C₄×C₁₃	#405	1	4	yes	—
195	C₄×C₁₃	#518	1	1	yes	characteristic
196	C₄×C₁₃	#519	1	1	yes	characteristic
197	C₄×C₁₃	#520	1	1	yes	characteristic
198	C₄×C₁₃	#520	1	1	yes	characteristic
199	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
200	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
201	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
202	C₂×M₁₆	—	13	8	no	—
203	C₂×M₁₆	—	52	2	no	—
204	C₂×M₁₆	—	52	2	no	—
205	C₂²×C₈	—	13	4	no	—
206	C₂²×C₈	—	13	2	no	—
207	C₂²×C₈	—	13	2	no	—
208	#117	—	2	8	no	—
209	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
210	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
211	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
212	C₂×C₄²	C₄×D₁₃	1	4	yes	—
213	C₂×C₄²	C₄×D₁₃	1	4	yes	—
214	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
215	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
216	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
217	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
218	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
219	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
220	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
221	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
222	C₄⋊C₈	—	13	8	no	—
223	C₄⋊C₈	—	13	8	no	—
224	C₄⋊C₈	—	13	4	no	—
225	C₄⋊C₈	—	13	4	no	—
226	C₂²⋊C₈	—	13	8	no	—
227	#97	—	52	4	no	—
228	C₄×C₈	—	13	8	no	—
229	C₄×C₈	—	52	4	no	—
230	#95	C₂×Dic₁₃	1	2	yes	—
231	#95	C₂×Dic₁₃	1	2	yes	—
232	#95	C₂×Dic₁₃	1	4	yes	—
233	C₂×C₁₃	—	4	2	no	—
234	C₂×C₁₃	#2274	1	1	yes	characteristic
235	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
236	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
237	C₂×C₁₃	#2990	1	2	yes	—
238	C₂×C₁₃	#2991	1	2	yes	—
239	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	C₄×Dic₁₃	1	4	yes	—
247	C₂²×C₄	C₄×Dic₁₃	1	4	yes	—
248	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
249	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
250	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
251	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
252	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
253	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
255	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
256	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
257	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
258	M₁₆	—	52	8	no	—
259	C₂×C₈	—	13	4	no	—
260	C₂×C₈	—	13	4	no	—
261	C₂×C₈	—	13	8	no	—
262	C₂×C₈	—	26	2	no	—
263	C₂×C₈	—	26	4	no	—
264	C₂×C₈	—	26	2	no	—
265	C₂×C₈	—	26	4	no	—
266	C₂×C₈	—	26	2	no	—
267	C₂×C₈	—	26	2	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	#6099	1	8	yes	complemented
270	C₂²⋊C₄	—	2	8	no	—
271	C₄²	—	2	8	no	—
272	C₄²	—	2	8	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	C₄×Dic₁₃	1	4	yes	—
278	C₄²	C₄×Dic₁₃	1	4	yes	—
279	C₄²	C₂×C₄×D₁₃	1	2	yes	—
280	C₄²	C₂×C₄×D₁₃	1	2	yes	—
281	C₄²	C₂×C₄×D₁₃	1	2	yes	—
282	C₄²	C₂×C₄×D₁₃	1	2	yes	—
283	C₁₃	#20550	1	1	yes	characteristic, complemented, Sylow
284	C₂³	—	2	2	no	—
285	C₂³	—	2	2	no	—
286	C₂³	—	2	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	C₂×C₄×Dic₁₃	1	2	yes	—
289	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	8	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	C₂×#6099	1	8	yes	—
306	C₂×C₄	C₂×#6099	1	4	yes	—
307	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
308	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
309	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
310	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
311	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
312	C₂×C₄	#88202	1	4	yes	—
313	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
315	C₂×C₄	#88256	1	1	yes	characteristic
316	C₂×C₄	#88256	1	1	yes	characteristic
317	C₂×C₄	#88257	1	1	yes	characteristic
318	C₂×C₄	#88257	1	1	yes	characteristic
319	C₈	—	26	8	no	—
320	C₈	—	26	4	no	—
321	C₈	—	26	4	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	4	2	no	—
327	C₂²	C₂²×#6099	1	2	yes	—
328	C₂²	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
329	C₂²	C₂×#88202	1	2	yes	—
330	C₂²	#11719532	1	1	yes	characteristic
331	C₂²	#11719532	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	8	no	—
334	C₄	—	2	2	no	—
335	C₄	—	2	2	no	—
336	C₄	—	4	4	no	—
337	C₄	C₄×#6099	1	4	yes	—
338	C₄	#11718658	1	4	yes	—
339	C₄	#11719221	1	1	yes	characteristic
340	C₄	#11719226	1	1	yes	characteristic
341	C₄	#11719228	1	1	yes	characteristic
342	C₄	#11719228	1	1	yes	characteristic
343	C₂	—	4	2	no	—
344	C₂	#13117192	1	1	yes	characteristic
345	C₂	C₂×C₄×#6099	1	1	yes	characteristic
346	C₂	C₂×#11718658	1	1	yes	characteristic
347	C₂	#13123961	1	2	yes	—
348	C₂	#13123968	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂²×C₄²
Fitting	C₄×C₁₃×#117	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#117	D₁₃
hypocenter	C₁₃	#20550
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. #13114705	C₂	semidirect
4. C₄⋊C₄×#1055	C₂	semidirect
5. #13114689	C₂	semidirect
6. C₄×C₄⋊#1055	C₂	semidirect
20. #11718586	C₄	semidirect
21. #11718585	C₄	semidirect
86. C₁₃×C₄⋊C₄	M₁₆	semidirect
269. C₄⋊C₄	#6099	semidirect
283. C₁₃	#20550	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718586	2
C₂×#11718585	2
#13114705	2
C₄⋊C₄×#1055	2
#13114689	2
C₄×C₄⋊#1055	2
#13114620	2
C₄×C₁₃×#117	2
#20550	13