Subgroups — #15881805

Order 3328.

730 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	#13117793	C₂	1	2	yes	maximal
2	#13115606	C₂	1	2	yes	maximal
3	#13115606	C₂	1	2	yes	maximal
4	#13115595	C₂	1	2	yes	maximal
5	#13115004	C₂	1	2	yes	maximal
6	#13115000	C₂	1	1	yes	maximal, characteristic
7	#13114733	C₂	1	2	yes	maximal
8	C₄×#88058	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×#446	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#446	C₄	1	8	yes	complemented
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	#11718891	—	2	2	no	—
17	C₂×#88058	C₂²	1	1	yes	characteristic
18	C₂×#88058	C₂²	1	1	yes	characteristic
19	C₂×#88058	C₂²	1	1	yes	characteristic
20	C₂×#88058	C₂²	1	2	yes	—
21	C₂×#88058	C₂²	1	1	yes	characteristic
22	C₂×#88057	C₂²	1	2	yes	—
23	C₂×#88057	C₂²	1	2	yes	—
24	#11718642	—	2	2	no	—
25	#11718642	—	2	2	no	—
26	C₄⋊C₄×Dic₁₃	—	2	2	no	—
27	#11718597	C₂²	1	2	yes	—
28	#11718597	C₂²	1	2	yes	—
29	#11718596	C₂²	1	2	yes	—
30	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
31	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
32	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
33	#11718143	C₂²	1	2	yes	—
34	#11718143	C₂²	1	2	yes	—
35	#11718143	C₂²	1	2	yes	—
36	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
41	C₄×Q₈×C₁₃	Q₈	1	4	yes	—
42	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
43	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
46	C₂×C₄⋊Dic₁₃	—	2	2	no	—
47	C₂×C₄⋊Dic₁₃	—	2	2	no	—
48	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
49	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
50	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
51	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
52	C₂×C₄×Dic₁₃	—	2	2	no	—
53	C₂×C₄×Dic₁₃	—	2	2	no	—
54	C₂×C₄×Dic₁₃	—	2	2	no	—
55	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
56	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
57	C₁₃×#97	C₂×C₄	1	8	yes	—
58	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
59	#88071	—	2	2	no	—
60	#88071	—	2	2	no	—
61	#88071	—	2	2	no	—
62	#88071	—	2	2	no	—
63	#88071	—	2	2	no	—
64	#88071	—	2	2	no	—
65	#88071	—	2	2	no	—
66	#88058	—	2	2	no	—
67	#88058	—	4	1	no	—
68	#88058	—	4	2	no	—
69	#88058	—	4	1	no	—
70	#88058	Q₈	1	4	yes	complemented
71	#88057	—	2	4	no	—
72	#88057	—	4	2	no	—
73	#22731	—	13	1	no	maximal, Sylow
74	C₂×Q₈×C₁₃	—	4	1	no	—
75	C₂×Q₈×C₁₃	—	4	1	no	—
76	C₂×Q₈×C₁₃	C₄⋊C₄	1	4	yes	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₂²×Dic₁₃	—	2	2	no	—
83	C₂²×Dic₁₃	—	2	2	no	—
84	C₂²×Dic₁₃	—	2	2	no	—
85	C₂²×Dic₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	—	2	1	no	—
88	C₂×C₈×C₁₃	—	2	1	no	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
94	C₂×C₈×C₁₃	#41	1	2	yes	—
95	C₁₃×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
96	C₁₃×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
97	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
98	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
99	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
100	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
101	C₄⋊Dic₁₃	—	2	2	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	—	2	2	no	—
107	C₄⋊Dic₁₃	—	2	2	no	—
108	C₄⋊Dic₁₃	—	4	4	no	—
109	C₄⋊Dic₁₃	—	8	2	no	—
110	C₄⋊Dic₁₃	C₂×Q₈	1	4	yes	—
111	C₄×Dic₁₃	—	4	4	no	—
112	C₄×Dic₁₃	—	8	2	no	—
113	C₄×Dic₁₃	—	8	2	no	—
114	C₂×#446	D₁₃	1	1	yes	characteristic
115	#2355	—	13	2	no	—
116	#1949	—	13	2	no	—
117	#1949	—	13	2	no	—
118	#1947	—	13	2	no	—
119	#1847	—	13	2	no	—
120	#1846	—	13	1	no	—
121	#1776	—	13	2	no	—
122	C₄×#107	—	13	1	no	—
123	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
124	Q₈×C₁₃	—	4	4	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	1	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
135	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
137	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂×Dic₁₃	—	2	4	no	—
144	C₂×Dic₁₃	—	4	2	no	—
145	C₂×Dic₁₃	—	4	2	no	—
146	C₂×Dic₁₃	—	4	2	no	—
147	C₂×Dic₁₃	—	4	2	no	—
148	C₂×Dic₁₃	—	4	2	no	—
149	C₂×Dic₁₃	—	4	2	no	—
150	C₂×Dic₁₃	—	4	2	no	—
151	C₂×Dic₁₃	—	4	2	no	—
152	C₂×Dic₁₃	—	4	2	no	—
153	C₂×Dic₁₃	—	4	2	no	—
154	C₂×Dic₁₃	—	4	2	no	—
155	C₈×C₁₃	—	2	4	no	—
156	C₈×C₁₃	—	2	2	no	—
157	C₈×C₁₃	C₄×Q₈	1	4	yes	—
158	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
159	#446	Dic₁₃	1	8	yes	complemented
160	C₂×#107	—	13	1	no	—
161	C₂×#107	—	13	1	no	—
162	C₂×#107	—	13	1	no	—
163	C₂×#107	—	13	1	no	—
164	C₂×#107	—	13	2	no	—
165	C₂×#106	—	13	2	no	—
166	C₂×#106	—	13	2	no	—
167	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
168	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
169	C₂×#97	C₂×D₁₃	1	2	yes	—
170	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
171	#391	—	26	2	no	—
172	#387	—	26	2	no	—
173	#387	—	26	2	no	—
174	#384	—	13	2	no	—
175	#382	—	13	2	no	—
176	#382	—	13	2	no	—
177	C₄×C₄⋊C₄	—	13	2	no	—
178	C₄×C₄⋊C₄	—	13	2	no	—
179	C₄×C₄⋊C₄	—	13	2	no	—
180	C₄×C₄⋊C₄	—	26	2	no	—
181	#340	—	13	2	no	—
182	#340	—	13	2	no	—
183	#340	—	13	2	no	—
184	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
185	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
186	C₂²×C₁₃	#550	1	1	yes	characteristic
187	C₂²×C₁₃	#575	1	1	yes	characteristic
188	C₂²×C₁₃	#575	1	1	yes	characteristic
189	C₂²×C₁₃	#576	1	1	yes	characteristic
190	C₂²×C₁₃	#578	1	1	yes	characteristic
191	C₄×C₁₃	—	2	4	no	—
192	C₄×C₁₃	—	2	2	no	—
193	C₄×C₁₃	—	2	2	no	—
194	C₄×C₁₃	—	2	4	no	—
195	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
196	C₄×C₁₃	#520	1	1	yes	characteristic
197	C₄×C₁₃	#557	1	1	yes	characteristic
198	C₄×C₁₃	Q₈²	1	1	yes	characteristic
199	Dic₁₃	—	4	4	no	—
200	Dic₁₃	—	8	2	no	—
201	Dic₁₃	—	8	2	no	—
202	Dic₁₃	—	8	2	no	—
203	C₂²×Q₈	D₅₂	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
205	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
206	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
207	C₄×Q₈	Dic₂₆	1	4	yes	—
208	C₄×Q₈	C₂×Dic₁₃	1	4	yes	—
209	C₂×C₄⋊C₄	—	13	2	no	—
210	C₂×C₄⋊C₄	—	13	2	no	—
211	C₂×C₄⋊C₄	—	13	2	no	—
212	C₂×C₄⋊C₄	—	13	2	no	—
213	C₂×C₄⋊C₄	—	26	2	no	—
214	C₂×C₄⋊C₄	—	26	2	no	—
215	C₂×C₄⋊C₄	D₅₂	1	2	yes	—
216	C₂×C₄⋊C₄	D₅₂	1	1	yes	characteristic
217	C₂×C₄²	—	26	2	no	—
218	C₂×C₄²	—	26	2	no	—
219	C₂×C₄²	—	26	2	no	—
220	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
221	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
222	#107	—	13	4	no	—
223	#107	—	26	2	no	—
224	#107	—	52	1	no	—
225	#107	—	52	2	no	—
226	#107	—	52	1	no	—
227	#106	—	26	4	no	—
228	#106	—	52	2	no	—
229	C₄⋊C₈	C₂×Dic₁₃	1	4	yes	—
230	C₄⋊C₈	C₂×Dic₁₃	1	8	yes	—
231	#97	C₂×Dic₁₃	1	8	yes	—
232	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
233	#95	—	26	2	no	—
234	#95	—	26	2	no	—
235	#95	—	26	2	no	—
236	#95	—	26	2	no	—
237	#95	—	26	2	no	—
238	#95	—	26	2	no	—
239	#95	—	26	2	no	—
240	C₂×C₁₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
241	C₂×C₁₃	#2911	1	1	yes	characteristic
242	C₂×C₁₃	#2913	1	1	yes	characteristic
243	C₂×C₁₃	#3389	1	1	yes	characteristic
244	C₂×C₁₃	#3390	1	1	yes	characteristic
245	C₂×C₁₃	#3402	1	1	yes	characteristic
246	C₂×C₁₃	#3403	1	1	yes	characteristic
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	C₄⋊Dic₁₃	1	4	yes	—
250	C₂²×C₄	—	26	2	no	—
251	C₂²×C₄	—	26	2	no	—
252	C₂²×C₄	—	26	2	no	—
253	C₂²×C₄	—	26	2	no	—
254	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₅₂	1	2	yes	—
257	C₂²×C₄	C₂×D₅₂	1	2	yes	—
258	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	Q₈×D₁₃	1	2	yes	—
264	C₂×C₈	#6131	1	2	yes	—
265	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
266	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
267	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
268	C₄⋊C₄	—	13	4	no	—
269	C₄⋊C₄	—	26	2	no	—
270	C₄⋊C₄	—	26	2	no	—
271	C₄⋊C₄	—	26	2	no	—
272	C₄⋊C₄	—	26	2	no	—
273	C₄⋊C₄	—	26	2	no	—
274	C₄⋊C₄	—	26	2	no	—
275	C₄⋊C₄	—	26	2	no	—
276	C₄⋊C₄	—	52	4	no	—
277	C₄⋊C₄	—	104	2	no	—
278	C₄⋊C₄	C₄⋊Dic₁₃	1	8	yes	—
279	C₄⋊C₄	C₄⋊Dic₁₃	1	4	yes	—
280	C₄²	—	52	4	no	—
281	C₄²	—	104	2	no	—
282	C₄²	—	104	2	no	—
283	C₄²	C₂×Dic₂₆	1	2	yes	—
284	C₄²	C₂×Dic₂₆	1	4	yes	—
285	C₄²	C₂²×Dic₁₃	1	2	yes	—
286	C₄²	C₂²×Dic₁₃	1	4	yes	—
287	C₁₃	#22731	1	1	yes	characteristic, complemented, Sylow
288	C₂³	C₂²×D₅₂	1	1	yes	characteristic
289	Q₈	—	4	4	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	26	4	no	—
299	C₂×C₄	—	52	2	no	—
300	C₂×C₄	—	52	2	no	—
301	C₂×C₄	—	52	2	no	—
302	C₂×C₄	—	52	2	no	—
303	C₂×C₄	—	52	2	no	—
304	C₂×C₄	—	52	2	no	—
305	C₂×C₄	—	52	2	no	—
306	C₂×C₄	—	52	2	no	—
307	C₂×C₄	—	52	2	no	—
308	C₂×C₄	—	52	2	no	—
309	C₂×C₄	—	52	2	no	—
310	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
311	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
312	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
313	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
314	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
315	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
316	C₂×C₄	C₂×#6131	1	1	yes	characteristic
317	C₂×C₄	#88256	1	1	yes	characteristic
318	C₂×C₄	#88257	1	1	yes	characteristic
319	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
320	C₈	—	2	4	no	—
321	C₈	—	2	2	no	—
322	C₈	Q₈×Dic₁₃	1	4	yes	—
323	C₂²	Q₈×D₅₂	1	1	yes	characteristic
324	C₂²	#11719282	1	1	yes	characteristic
325	C₂²	#11719453	1	1	yes	characteristic
326	C₂²	#11719453	1	1	yes	characteristic
327	C₂²	#11719457	1	1	yes	characteristic
328	C₂²	#11719458	1	1	yes	characteristic
329	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
330	C₄	—	2	4	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	2	4	no	—
334	C₄	—	52	4	no	—
335	C₄	—	104	2	no	—
336	C₄	—	104	2	no	—
337	C₄	—	104	2	no	—
338	C₄	Q₈×Dic₂₆	1	1	yes	characteristic
339	C₄	#11719275	1	1	yes	characteristic
340	C₄	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
341	C₄	#11719532	1	1	yes	characteristic
342	C₂	Q₈×C₄⋊Dic₁₃	1	1	yes	characteristic
343	C₂	#13123426	1	1	yes	characteristic
344	C₂	#13123436	1	1	yes	characteristic
345	C₂	#13127796	1	1	yes	characteristic
346	C₂	#13127805	1	1	yes	characteristic
347	C₂	#13127917	1	1	yes	characteristic
348	C₂	#13127926	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#446	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#446	D₁₃
hypocenter	C₁₃	#22731
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
11. C₁₃×#446	C₄	semidirect
70. #88058	Q₈	semidirect
159. #446	Dic₁₃	semidirect
287. C₁₃	#22731	semidirect

Maximal subgroups

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

subgroup	index
#13117793	2
#13115606	2
#13115595	2
#13115004	2
#13115000	2
#13114733	2
C₄×#88058	2
C₂×C₁₃×#446	2
#22731	13