Subgroups — #15861958

Order 3328.

990 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×C₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718491	C₂	1	1	yes	maximal, characteristic
3	C₄×#88141	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×#1055	C₂	1	1	yes	maximal, characteristic
5	C₈×#6101	C₂	1	2	yes	maximal
6	#13114697	C₂	1	2	yes	maximal
7	C₄⋊C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
8	C₄×#88054	C₂	1	2	yes	maximal
9	#13114581	C₂	1	1	yes	maximal, characteristic
10	#13114580	C₂	1	2	yes	maximal
11	C₈×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
12	C₂×#88141	C₄	1	2	yes	—
13	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×#88054	C₂²	1	2	yes	—
19	C₂×#88054	C₂²	1	2	yes	—
20	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
21	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
22	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
23	C₄⋊C₄×Dic₁₃	C₄	1	2	yes	—
24	C₄×#6101	C₂²	1	2	yes	—
25	C₄²×Dic₁₃	C₂²	1	2	yes	—
26	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
28	C₂×C₄×#1055	C₂²	1	2	yes	—
29	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
30	#11718491	C₄	1	8	yes	complemented
31	C₈×Dic₂₆	C₄	1	8	yes	complemented
32	#11718310	C₄	1	4	yes	—
33	C₁₃×#336	C₂²	1	2	yes	—
34	#11718142	C₂²	1	1	yes	characteristic
35	#11718142	C₂²	1	2	yes	—
36	#11718142	C₂²	1	1	yes	characteristic
37	#11718142	C₂²	1	1	yes	characteristic
38	#11718142	C₂²	1	1	yes	characteristic
39	#11718116	C₂²	1	2	yes	—
40	C₂²×Dic₂₆	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
44	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂×#6101	C₂×C₄	1	2	yes	—
49	C₂×#6101	C₂×C₄	1	4	yes	—
50	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
51	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
52	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
53	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
54	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
55	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
56	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
57	C₂²×#1055	C₂³	1	2	yes	—
58	C₂²×#1055	C₂³	1	1	yes	characteristic
59	C₂²×#1055	C₂³	1	1	yes	characteristic
60	#88141	C₈	1	16	yes	complemented
61	C₄×Dic₂₆	C₂×C₄	1	4	yes	—
62	C₄×Dic₂₆	C₂×C₄	1	4	yes	—
63	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
64	C₄×C₈×C₁₃	—	4	2	no	—
65	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
66	C₁₃×#95	C₂×C₄	1	2	yes	—
67	#88071	C₂×C₄	1	4	yes	—
68	#88054	C₂×C₄	1	8	yes	—
69	#88054	C₂×C₄	1	8	yes	—
70	C₈×Dic₁₃	—	4	2	no	—
71	C₈×Dic₁₃	C₂×C₄	1	8	yes	—
72	C₈×Dic₁₃	C₂×C₄	1	8	yes	—
73	C₄⋊#1055	C₂×C₄	1	4	yes	—
74	C₄×#1055	—	4	2	no	—
75	C₄×#1055	C₂×C₄	1	4	yes	—
76	#20569	—	13	1	no	maximal, Sylow
77	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
82	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
83	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
84	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
85	C₂×Dic₂₆	C₄²	1	4	yes	—
86	C₂×Dic₂₆	C₂×C₈	1	8	yes	—
87	C₂×C₈×C₁₃	—	2	1	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	—	2	1	no	—
90	C₂×C₈×C₁₃	—	2	1	no	—
91	C₂×C₈×C₁₃	—	2	1	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
96	C₁₃×C₄⋊C₄	C₂×C₈	1	8	yes	—
97	C₄²×C₁₃	—	2	2	no	—
98	C₄²×C₁₃	—	2	2	no	—
99	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
100	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
101	C₄⋊Dic₁₃	C₄²	1	4	yes	—
102	#6101	C₄²	1	8	yes	—
103	#6101	C₂×C₈	1	16	yes	—
104	C₄×Dic₁₃	—	2	1	no	—
105	C₄×Dic₁₃	—	2	1	no	—
106	C₄×Dic₁₃	—	2	1	no	—
107	C₄×Dic₁₃	—	2	1	no	—
108	C₄×Dic₁₃	—	4	2	no	—
109	C₄×Dic₁₃	—	4	4	no	—
110	C₄×Dic₁₃	C₂×C₈	1	8	yes	—
111	C₄×Dic₁₃	C₂×C₈	1	16	yes	—
112	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
113	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
114	C₂×#1055	—	2	1	no	—
115	C₂×#1055	—	2	2	no	—
116	C₂×#1055	—	2	1	no	—
117	C₂×#1055	—	2	1	no	—
118	C₂×#1055	—	2	2	no	—
119	C₂×#1055	—	2	1	no	—
120	C₂×#1055	C₂²×C₄	1	2	yes	—
121	C₂×#1055	C₂²×C₄	1	2	yes	—
122	C₂×#1055	C₂²×C₄	1	4	yes	—
123	C₂×C₈×Q₈	—	13	1	no	—
124	C₂×C₈×Q₈	—	13	1	no	—
125	C₄²×Q₈	—	13	1	no	—
126	C₈×C₄⋊C₄	—	13	2	no	—
127	C₈×C₄⋊C₄	—	13	1	no	—
128	C₈×C₄⋊C₄	—	13	2	no	—
129	C₈×C₄⋊C₄	D₁₃	1	1	yes	characteristic
130	C₄×C₄⋊C₈	—	13	1	no	—
131	C₄×C₄⋊C₈	—	13	2	no	—
132	#1744	—	13	1	no	—
133	#1744	—	13	2	no	—
134	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	—	2	4	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	2	4	no	—
138	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
139	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
140	C₂×C₄×C₁₃	C₂²×C₈	1	8	yes	—
141	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
142	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
145	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
146	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
147	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
148	C₂×Dic₁₃	—	2	2	no	—
149	C₂×Dic₁₃	—	2	2	no	—
150	C₂×Dic₁₃	—	2	2	no	—
151	C₂×Dic₁₃	—	2	4	no	—
152	C₂×Dic₁₃	—	2	2	no	—
153	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
154	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
155	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
156	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
157	C₂×Dic₁₃	C₂²×C₈	1	8	yes	—
158	Dic₂₆	C₄×C₈	1	16	yes	complemented
159	C₈×C₁₃	—	2	2	no	—
160	C₈×C₁₃	—	2	2	no	—
161	C₈×C₁₃	—	2	4	no	—
162	#1055	—	2	2	no	—
163	#1055	—	2	4	no	—
164	#1055	—	2	2	no	—
165	C₂×C₄×Q₈	—	13	2	no	—
166	C₂×C₄×Q₈	—	13	1	no	—
167	C₈×Q₈	—	13	8	no	—
168	C₈×Q₈	—	13	8	no	—
169	C₂×C₄⋊C₈	—	13	2	no	—
170	C₂×C₄⋊C₈	—	13	2	no	—
171	C₂×C₄⋊C₈	—	13	1	no	—
172	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
173	C₂×C₄×C₈	—	13	2	no	—
174	C₂×C₄×C₈	—	13	2	no	—
175	C₂×C₄×C₈	—	13	1	no	—
176	C₂×C₄×C₈	—	13	2	no	—
177	C₂×C₄×C₈	—	13	2	no	—
178	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
179	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
180	C₄×C₄⋊C₄	—	13	2	no	—
181	C₄×C₄⋊C₄	—	13	2	no	—
182	C₄×C₄⋊C₄	—	13	4	no	—
183	C₄×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
184	C₄³	—	13	2	no	—
185	C₄³	—	13	1	no	—
186	#336	—	13	1	no	—
187	#336	—	13	1	no	—
188	#336	—	13	2	no	—
189	#336	—	13	1	no	—
190	#336	—	13	1	no	—
191	#336	—	13	2	no	—
192	#336	C₂×D₁₃	1	2	yes	—
193	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
194	C₂²×C₁₃	#519	1	1	yes	characteristic
195	C₂²×C₁₃	#520	1	1	yes	characteristic
196	C₂²×C₁₃	C₂³×C₈	1	2	yes	—
197	C₂²×C₁₃	#568	1	2	yes	—
198	C₄×C₁₃	—	2	2	no	—
199	C₄×C₁₃	—	2	8	no	—
200	C₄×C₁₃	C₂×C₄×C₈	1	4	yes	—
201	C₄×C₁₃	#519	1	1	yes	characteristic
202	C₄×C₁₃	#519	1	1	yes	characteristic
203	C₄×C₁₃	#520	1	1	yes	characteristic
204	C₄×C₁₃	#520	1	1	yes	characteristic
205	Dic₁₃	—	2	4	no	—
206	Dic₁₃	—	2	4	no	—
207	Dic₁₃	—	2	4	no	—
208	Dic₁₃	C₂×C₄×C₈	1	8	yes	—
209	C₂²×Q₈	—	13	1	no	—
210	C₂²×C₈	—	13	1	no	—
211	C₂²×C₈	—	13	2	no	—
212	C₂²×C₈	—	13	1	no	—
213	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
215	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
216	C₄×Q₈	—	13	16	no	—
217	C₄×Q₈	—	13	4	no	—
218	C₄×Q₈	—	13	4	no	—
219	C₂×C₄⋊C₄	—	13	2	no	—
220	C₂×C₄⋊C₄	—	13	4	no	—
221	C₂×C₄⋊C₄	—	13	1	no	—
222	C₂×C₄⋊C₄	C₄×D₁₃	1	2	yes	—
223	C₂×C₄²	—	13	4	no	—
224	C₂×C₄²	—	13	2	no	—
225	C₂×C₄²	—	13	2	no	—
226	C₂×C₄²	—	13	1	no	—
227	C₂×C₄²	—	13	1	no	—
228	C₂×C₄²	—	13	2	no	—
229	C₂×C₄²	—	13	4	no	—
230	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
231	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
232	C₄⋊C₈	—	13	8	no	—
233	C₄⋊C₈	—	13	8	no	—
234	C₄⋊C₈	—	13	4	no	—
235	C₄⋊C₈	C₄×D₁₃	1	4	yes	—
236	C₄×C₈	—	4	2	no	—
237	C₄×C₈	—	13	8	no	—
238	C₄×C₈	—	13	8	no	—
239	C₄×C₈	—	13	4	no	—
240	C₄×C₈	—	52	2	no	—
241	C₄×C₈	—	52	2	no	—
242	C₄×C₈	C₄×D₁₃	1	4	yes	—
243	#95	—	13	4	no	—
244	#95	C₄×D₁₃	1	2	yes	—
245	C₂×C₁₃	#2276	1	1	yes	characteristic
246	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
247	C₂×C₁₃	#2875	1	1	yes	characteristic
248	C₂×C₁₃	#2982	1	2	yes	—
249	C₂×C₁₃	#2988	1	2	yes	—
250	C₂×Q₈	—	13	4	no	—
251	C₂×Q₈	—	13	8	no	—
252	C₂²×C₄	—	13	2	no	—
253	C₂²×C₄	—	13	2	no	—
254	C₂²×C₄	—	13	2	no	—
255	C₂²×C₄	—	13	2	no	—
256	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₄×D₁₃	1	4	yes	—
258	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
259	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	13	2	no	—
267	C₂×C₈	—	13	4	no	—
268	C₂×C₈	—	13	2	no	—
269	C₂×C₈	—	26	2	no	—
270	C₂×C₈	—	26	1	no	—
271	C₂×C₈	—	26	2	no	—
272	C₂×C₈	—	26	1	no	—
273	C₂×C₈	—	26	1	no	—
274	C₂×C₈	—	26	1	no	—
275	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
276	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
277	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
278	C₄⋊C₄	—	13	8	no	—
279	C₄⋊C₄	—	13	4	no	—
280	C₄⋊C₄	—	13	16	no	—
281	C₄⋊C₄	C₈×D₁₃	1	8	yes	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	13	16	no	—
285	C₄²	—	13	8	no	—
286	C₄²	—	13	4	no	—
287	C₄²	—	13	4	no	—
288	C₄²	—	26	1	no	—
289	C₄²	—	26	1	no	—
290	C₄²	—	26	1	no	—
291	C₄²	—	26	1	no	—
292	C₄²	—	52	4	no	—
293	C₄²	—	52	2	no	—
294	C₄²	C₂×C₄×D₁₃	1	2	yes	—
295	C₄²	C₂×C₄×D₁₃	1	2	yes	—
296	C₁₃	#20569	1	1	yes	characteristic, complemented, Sylow
297	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
298	Q₈	—	13	16	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	13	4	no	—
303	C₂×C₄	—	13	8	no	—
304	C₂×C₄	—	13	4	no	—
305	C₂×C₄	—	13	4	no	—
306	C₂×C₄	—	13	4	no	—
307	C₂×C₄	—	26	2	no	—
308	C₂×C₄	—	26	2	no	—
309	C₂×C₄	—	26	4	no	—
310	C₂×C₄	—	26	2	no	—
311	C₂×C₄	—	26	2	no	—
312	C₂×C₄	C₄²×D₁₃	1	2	yes	—
313	C₂×C₄	C₄²×D₁₃	1	2	yes	—
314	C₂×C₄	C₂×C₈×D₁₃	1	8	yes	—
315	C₂×C₄	C₂×C₈×D₁₃	1	4	yes	—
316	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
318	C₂×C₄	#88251	1	1	yes	characteristic
319	C₂×C₄	#88254	1	1	yes	characteristic
320	C₂×C₄	#88257	1	1	yes	characteristic
321	C₂×C₄	#88257	1	1	yes	characteristic
322	C₈	—	2	2	no	—
323	C₈	—	2	2	no	—
324	C₈	—	2	4	no	—
325	C₈	—	26	4	no	—
326	C₈	—	26	2	no	—
327	C₈	—	26	2	no	—
328	C₂²	C₂×C₄²×D₁₃	1	1	yes	characteristic
329	C₂²	#11719251	1	1	yes	characteristic
330	C₂²	#11719277	1	1	yes	characteristic
331	C₂²	C₂²×C₈×D₁₃	1	2	yes	—
332	C₂²	#11719455	1	2	yes	—
333	C₄	—	2	8	no	—
334	C₄	—	2	2	no	—
335	C₄	—	13	8	no	—
336	C₄	—	26	4	no	—
337	C₄	—	26	4	no	—
338	C₄	—	26	4	no	—
339	C₄	C₄×C₈×D₁₃	1	4	yes	—
340	C₄	#11719209	1	1	yes	characteristic
341	C₄	#11719221	1	1	yes	characteristic
342	C₄	#11719251	1	1	yes	characteristic
343	C₄	#11719277	1	1	yes	characteristic
344	C₂	#13117200	1	1	yes	characteristic
345	C₂	C₂×C₄×C₈×D₁₃	1	1	yes	characteristic
346	C₂	#13123263	1	1	yes	characteristic
347	C₂	#13123915	1	2	yes	—
348	C₂	#13123943	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂×C₁₃	C₂²×C₄×C₈
Fitting	C₈×C₁₃×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₁₃	C₂³×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₈×C₄⋊C₄	D₁₃
hypocenter	C₁₃	#20569
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
30. #11718491	C₄	semidirect
31. C₈×Dic₂₆	C₄	semidirect
60. #88141	C₈	semidirect
158. Dic₂₆	C₄×C₈	semidirect
296. C₁₃	#20569	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₂₆	2
C₂×#11718491	2
C₄×#88141	2
C₄⋊C₄×#1055	2
C₈×#6101	2
#13114697	2
C₄⋊C₈×Dic₁₃	2
C₄×#88054	2
#13114581	2
#13114580	2
C₈×C₁₃×C₄⋊C₄	2
#20569	13