Subgroups — #15881922

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×Dic₂₆	C₂	1	1	yes	maximal, characteristic
2	#13123787	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718491	C₂	1	1	yes	maximal, characteristic
4	#13115612	C₂	1	1	yes	maximal, characteristic
5	#13115610	C₂	1	2	yes	maximal
6	#13115608	C₂	1	2	yes	maximal
7	#13114973	C₂	1	1	yes	maximal, characteristic
8	#13114972	C₂	1	2	yes	maximal
9	#108×Dic₁₃	C₂	1	2	yes	maximal
10	C₄×#88062	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#1950	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#108	C₂²	1	2	yes	—
14	C₁₃×#423	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
17	C₂×#88062	C₂²	1	1	yes	characteristic
18	C₂×#88062	C₂²	1	1	yes	characteristic
19	#11718779	C₂²	1	2	yes	—
20	#11718779	C₂²	1	2	yes	—
21	M₁₆×Dic₁₃	C₂²	1	2	yes	—
22	M₁₆×Dic₁₃	C₂²	1	2	yes	—
23	C₂×#88059	C₂²	1	2	yes	—
24	C₂×#88054	C₂²	1	2	yes	—
25	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
26	#11718585	C₂²	1	1	yes	characteristic
27	C₄×#6099	C₂²	1	1	yes	characteristic
28	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
29	#11718492	—	2	1	no	—
30	#11718491	—	2	1	no	—
31	#11718491	C₄	1	4	yes	—
32	#11718491	C₄	1	2	yes	—
33	#11718491	C₄	1	2	yes	—
34	#11718489	—	2	2	no	—
35	#11718362	—	2	2	no	—
36	#11718340	—	2	1	no	—
37	C₈×Dic₂₆	—	2	1	no	—
38	C₁₃×#341	C₂²	1	2	yes	—
39	#11718150	C₂²	1	1	yes	characteristic
40	#11718150	C₂²	1	2	yes	—
41	#11718150	C₂²	1	2	yes	—
42	#11718150	C₂²	1	1	yes	characteristic
43	#11718144	C₂²	1	2	yes	—
44	C₂²×Dic₂₆	Q₈	1	1	yes	characteristic
45	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
46	C₂×C₁₃×M₁₆	C₂³	1	1	yes	characteristic
47	C₂×C₁₃×M₁₆	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
49	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊Dic₁₃	Q₈	1	1	yes	characteristic
51	C₂×#6101	Q₈	1	2	yes	—
52	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
53	C₂×#6099	C₂³	1	1	yes	characteristic
54	C₂×#6099	C₂³	1	2	yes	—
55	C₂×#6099	C₂³	1	1	yes	characteristic
56	C₂²×#1055	C₂³	1	1	yes	characteristic
57	C₂²×#1055	C₂³	1	1	yes	characteristic
58	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
60	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₂₆	D₄	1	2	yes	—
62	C₄×Dic₂₆	D₄	1	1	yes	characteristic
63	C₄×Dic₂₆	D₄	1	1	yes	characteristic
64	C₁₃×#108	—	2	4	no	—
65	C₁₃×C₄⋊C₈	—	2	1	no	—
66	C₁₃×C₄⋊C₈	—	2	1	no	—
67	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
68	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
69	C₁₃×#97	—	2	1	no	—
70	C₄×C₈×C₁₃	—	2	1	no	—
71	#88062	—	2	2	no	—
72	#88062	—	2	2	no	—
73	#88059	—	2	4	no	—
74	#88055	—	2	2	no	—
75	#88055	—	2	2	no	—
76	#88054	—	2	2	no	—
77	#88054	—	2	2	no	—
78	#88054	—	2	2	no	—
79	#88054	—	2	2	no	—
80	#88054	C₂×C₄	1	4	yes	—
81	#88054	C₂×C₄	1	4	yes	—
82	C₈×Dic₁₃	—	2	2	no	—
83	C₈×Dic₁₃	—	2	2	no	—
84	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
85	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
86	C₄⋊#1055	—	2	1	no	—
87	C₄⋊#1055	—	2	1	no	—
88	#88043	—	2	1	no	—
89	C₄×#1055	—	2	1	no	—
90	C₄×#1055	C₂×C₄	1	2	yes	—
91	C₄×#1055	C₂×C₄	1	2	yes	—
92	#22746	—	13	1	no	maximal, Sylow
93	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
96	C₂²×Dic₁₃	C₂×Q₈	1	2	yes	—
97	C₂²×Dic₁₃	C₂×Q₈	1	2	yes	—
98	C₂×Dic₂₆	—	2	4	no	—
99	C₂×Dic₂₆	C₄⋊C₄	1	1	yes	characteristic
100	C₂×Dic₂₆	C₄⋊C₄	1	2	yes	—
101	C₂×Dic₂₆	C₄⋊C₄	1	1	yes	characteristic
102	C₁₃×M₁₆	—	2	1	no	—
103	C₁₃×M₁₆	—	2	2	no	—
104	C₁₃×M₁₆	—	2	1	no	—
105	C₁₃×M₁₆	—	2	1	no	—
106	C₁₃×M₁₆	—	2	2	no	—
107	C₁₃×M₁₆	#41	1	2	yes	—
108	C₂×C₈×C₁₃	—	2	1	no	—
109	C₂×C₈×C₁₃	—	2	2	no	—
110	C₂×C₈×C₁₃	—	2	2	no	—
111	C₂×C₈×C₁₃	—	2	1	no	—
112	C₂×C₈×C₁₃	—	2	2	no	—
113	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
114	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
118	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
119	C₄⋊Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
120	C₄⋊Dic₁₃	C₄⋊C₄	1	1	yes	characteristic
121	#6101	C₄⋊C₄	1	2	yes	—
122	#6101	C₄⋊C₄	1	2	yes	—
123	#6101	C₄⋊C₄	1	2	yes	—
124	#6101	C₄⋊C₄	1	2	yes	—
125	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
126	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
127	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
128	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
129	#6099	—	2	1	no	—
130	#6099	—	2	2	no	—
131	#6099	—	2	1	no	—
132	#6099	—	2	2	no	—
133	#6099	—	2	1	no	—
134	#6099	#41	1	2	yes	—
135	C₂×#1055	—	2	1	no	—
136	C₂×#1055	—	2	1	no	—
137	C₂×#1055	—	2	1	no	—
138	C₂×#1055	—	2	1	no	—
139	C₂×#1055	—	2	1	no	—
140	C₂×#1055	—	2	1	no	—
141	C₂×#1055	—	2	2	no	—
142	C₂×#1055	C₂²×C₄	1	2	yes	—
143	C₂×#1055	C₂²×C₄	1	2	yes	—
144	Q₈×M₁₆	—	13	1	no	—
145	Q₈×M₁₆	—	13	1	no	—
146	C₂×C₈×Q₈	—	13	1	no	—
147	#1950	—	13	2	no	—
148	#1950	—	13	1	no	—
149	#1950	—	13	2	no	—
150	#1950	D₁₃	1	1	yes	characteristic
151	#1837	—	13	1	no	—
152	#1837	—	13	2	no	—
153	C₄×#108	—	13	2	no	—
154	C₄×#108	—	13	1	no	—
155	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
156	C₂×C₄×C₁₃	—	2	2	no	—
157	C₂×C₄×C₁₃	—	4	1	no	—
158	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
163	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
166	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
167	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
168	C₂×Dic₁₃	—	2	4	no	—
169	C₂×Dic₁₃	—	4	2	no	—
170	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
171	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
172	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
173	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
174	Dic₂₆	—	2	2	no	—
175	Dic₂₆	—	2	4	no	—
176	Dic₂₆	#108	1	4	yes	complemented
177	C₈×C₁₃	—	2	4	no	—
178	C₈×C₁₃	—	2	2	no	—
179	C₈×C₁₃	—	4	2	no	—
180	C₈×C₁₃	—	4	1	no	—
181	#1055	—	2	2	no	—
182	#1055	—	2	2	no	—
183	#1055	—	2	2	no	—
184	#1055	—	4	1	no	—
185	#1055	—	4	2	no	—
186	C₂×C₄×Q₈	—	13	1	no	—
187	#446	—	26	1	no	—
188	#446	—	26	2	no	—
189	#446	—	26	2	no	—
190	#446	—	26	1	no	—
191	C₈×Q₈	—	13	2	no	—
192	C₈×Q₈	—	13	4	no	—
193	C₈×Q₈	—	13	2	no	—
194	C₈×Q₈	—	26	1	no	—
195	C₈×Q₈	—	26	1	no	—
196	C₂×#108	—	13	1	no	—
197	C₂×#108	—	13	1	no	—
198	C₂×#108	—	13	2	no	—
199	C₂×#108	C₂×D₁₃	1	2	yes	—
200	#423	—	13	2	no	—
201	#423	—	13	2	no	—
202	#423	—	13	1	no	—
203	#423	C₂×D₁₃	1	1	yes	characteristic
204	C₂×C₄⋊C₈	—	13	2	no	—
205	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
206	C₄×M₁₆	—	13	2	no	—
207	C₄×M₁₆	—	13	2	no	—
208	C₄×M₁₆	—	13	1	no	—
209	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
210	C₂×C₄×C₈	—	13	2	no	—
211	C₂×C₄×C₈	—	13	1	no	—
212	#341	—	13	2	no	—
213	#341	—	13	1	no	—
214	#341	—	13	2	no	—
215	#341	—	13	1	no	—
216	#341	—	13	2	no	—
217	#341	C₂×D₁₃	1	2	yes	—
218	C₂²×C₁₃	—	2	1	no	—
219	C₂²×C₁₃	—	2	1	no	—
220	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
221	C₂²×C₁₃	#557	1	1	yes	characteristic
222	C₂²×C₁₃	#557	1	1	yes	characteristic
223	C₄×C₁₃	—	2	1	no	—
224	C₄×C₁₃	—	2	1	no	—
225	C₄×C₁₃	—	2	1	no	—
226	C₄×C₁₃	C₂×#108	1	2	yes	—
227	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
228	C₄×C₁₃	#519	1	1	yes	characteristic
229	C₄×C₁₃	#548	1	1	yes	characteristic
230	C₄×C₁₃	#548	1	1	yes	characteristic
231	Dic₁₃	—	2	2	no	—
232	Dic₁₃	—	2	2	no	—
233	Dic₁₃	—	2	2	no	—
234	Dic₁₃	C₂×#108	1	4	yes	—
235	C₂²×Q₈	—	13	1	no	—
236	C₂×M₁₆	—	13	1	no	—
237	C₂×M₁₆	—	13	1	no	—
238	C₂×M₁₆	—	13	2	no	—
239	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
240	C₂×M₁₆	C₂²×D₁₃	1	1	yes	characteristic
241	C₂×M₁₆	C₂²×D₁₃	1	1	yes	characteristic
242	C₂²×C₈	—	13	1	no	—
243	C₂²×C₈	—	13	1	no	—
244	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
245	C₄×Q₈	—	13	1	no	—
246	C₄×Q₈	—	13	2	no	—
247	C₄×Q₈	—	13	1	no	—
248	C₄×Q₈	—	13	1	no	—
249	C₄×Q₈	—	13	2	no	—
250	C₄×Q₈	—	13	1	no	—
251	C₂×C₄⋊C₄	—	13	2	no	—
252	C₂×C₄⋊C₄	—	13	1	no	—
253	C₂×C₄²	—	13	2	no	—
254	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
255	#108	—	2	4	no	—
256	#108	—	26	4	no	—
257	#108	—	26	2	no	—
258	#108	—	26	2	no	—
259	C₄⋊C₈	—	2	1	no	—
260	C₄⋊C₈	—	2	1	no	—
261	C₄⋊C₈	—	13	4	no	—
262	C₄⋊C₈	—	13	4	no	—
263	C₄⋊C₈	—	26	2	no	—
264	C₄⋊C₈	—	26	2	no	—
265	C₄⋊C₈	—	26	2	no	—
266	C₄⋊C₈	—	26	2	no	—
267	C₄⋊C₈	—	26	1	no	—
268	C₄⋊C₈	—	26	1	no	—
269	C₄⋊C₈	C₄×D₁₃	1	2	yes	—
270	C₄⋊C₈	C₄×D₁₃	1	2	yes	—
271	#97	—	2	1	no	—
272	#97	—	26	2	no	—
273	#97	—	26	2	no	—
274	#97	—	26	1	no	—
275	C₄×C₈	—	2	1	no	—
276	C₄×C₈	—	13	4	no	—
277	C₄×C₈	—	13	4	no	—
278	C₄×C₈	—	13	2	no	—
279	C₄×C₈	—	13	2	no	—
280	C₄×C₈	—	26	2	no	—
281	C₄×C₈	—	26	2	no	—
282	C₄×C₈	—	26	1	no	—
283	C₂×C₁₃	—	2	1	no	—
284	C₂×C₁₃	—	2	1	no	—
285	C₂×C₁₃	#2351	1	1	yes	characteristic
286	C₂×C₁₃	C₂²×#108	1	1	yes	characteristic
287	C₂×C₁₃	#2916	1	1	yes	characteristic
288	C₂×Q₈	—	13	1	no	—
289	C₂×Q₈	—	13	2	no	—
290	C₂×Q₈	—	13	1	no	—
291	C₂×Q₈	—	26	4	no	—
292	C₂²×C₄	—	13	2	no	—
293	C₂²×C₄	—	13	2	no	—
294	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
295	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
296	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
297	M₁₆	—	2	1	no	—
298	M₁₆	—	2	2	no	—
299	M₁₆	—	2	1	no	—
300	M₁₆	—	2	1	no	—
301	M₁₆	—	2	2	no	—
302	M₁₆	—	13	2	no	—
303	M₁₆	—	26	1	no	—
304	M₁₆	—	26	1	no	—
305	M₁₆	—	26	2	no	—
306	M₁₆	—	26	2	no	—
307	M₁₆	—	26	1	no	—
308	M₁₆	#6129	1	2	yes	—
309	C₂×C₈	—	2	1	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	1	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	13	2	no	—
315	C₂×C₈	—	13	2	no	—
316	C₂×C₈	—	26	1	no	—
317	C₂×C₈	—	26	1	no	—
318	C₂×C₈	—	26	1	no	—
319	C₂×C₈	—	26	1	no	—
320	C₂×C₈	—	26	1	no	—
321	C₂×C₈	—	26	1	no	—
322	C₂×C₈	—	26	2	no	—
323	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
324	C₄⋊C₄	—	13	2	no	—
325	C₄⋊C₄	—	13	2	no	—
326	C₄⋊C₄	—	13	2	no	—
327	C₄⋊C₄	—	13	2	no	—
328	C₄⋊C₄	—	13	2	no	—
329	C₄⋊C₄	—	13	1	no	—
330	C₄⋊C₄	—	13	1	no	—
331	C₄²	—	13	2	no	—
332	C₄²	—	13	2	no	—
333	C₄²	—	13	2	no	—
334	C₄²	—	13	2	no	—
335	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
336	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
337	C₄²	D₄×D₁₃	1	1	yes	characteristic
338	C₄²	D₄×D₁₃	1	1	yes	characteristic
339	C₁₃	#22746	1	1	yes	characteristic, complemented, Sylow
340	C₂³	C₂×Q₈×D₁₃	1	1	yes	characteristic
341	Q₈	—	13	4	no	—
342	Q₈	—	26	4	no	—
343	Q₈	—	26	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	13	2	no	—
347	C₂×C₄	—	13	2	no	—
348	C₂×C₄	—	13	2	no	—
349	C₂×C₄	—	13	2	no	—
350	C₂×C₄	—	26	4	no	—
351	C₂×C₄	—	52	2	no	—
352	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
353	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
354	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
355	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
356	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
357	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
358	C₂×C₄	C₂×#6129	1	1	yes	characteristic
359	C₂×C₄	#88254	1	1	yes	characteristic
360	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
361	C₂×C₄	#88257	1	1	yes	characteristic
362	C₈	—	2	4	no	—
363	C₈	—	2	2	no	—
364	C₈	—	4	2	no	—
365	C₈	—	4	1	no	—
366	C₈	—	26	2	no	—
367	C₈	—	26	2	no	—
368	C₈	—	26	2	no	—
369	C₈	—	52	1	no	—
370	C₈	—	52	2	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
374	C₂²	#11719390	1	1	yes	characteristic
375	C₂²	#11719426	1	1	yes	characteristic
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	13	4	no	—
380	C₄	—	26	2	no	—
381	C₄	—	26	2	no	—
382	C₄	—	26	2	no	—
383	C₄	D₁₃×#108	1	2	yes	—
384	C₄	C₄×#6129	1	1	yes	characteristic
385	C₄	#11719277	1	1	yes	characteristic
386	C₄	#11719303	1	1	yes	characteristic
387	C₄	#11719338	1	1	yes	characteristic
388	C₂	—	2	1	no	—
389	C₂	—	2	1	no	—
390	C₂	#13117755	1	1	yes	characteristic
391	C₂	C₂×D₁₃×#108	1	1	yes	characteristic
392	C₂	#13123446	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#1950	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₁₃	C₂²×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#1950	D₁₃
hypocenter	C₁₃	#22746
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1950

Complements

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

N	H ≅ G/N	type
176. Dic₂₆	#108	semidirect
339. C₁₃	#22746	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₂₆	2
#13123787	2
C₂×#11718491	2
#13115612	2
#13115610	2
#13115608	2
#13114973	2
#13114972	2
#108×Dic₁₃	2
C₄×#88062	2
C₁₃×#1950	2
#22746	13