Subgroups — #13131574

Order 1664.

544 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#6122	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₁₃⋊M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88135	C₂	1	1	yes	maximal, characteristic
4	#11719169	C₂	1	1	yes	maximal, characteristic, complemented
5	#11719129	C₂	1	1	yes	maximal, characteristic, complemented
6	#11719126	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#6122	C₂	1	1	yes	maximal, characteristic, complemented
8	#11719066	C₂	1	2	yes	maximal, complemented
9	#11719065	C₂	1	2	yes	maximal, complemented
10	#11719064	C₂	1	2	yes	maximal, complemented
11	#11719063	C₂	1	2	yes	maximal, complemented
12	C₂²×C₂²⋊D₁₃	C₄	1	1	yes	characteristic
13	C₂³×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
15	C₂×#6122	—	2	4	no	—
16	C₂×#6122	C₂²	1	1	yes	characteristic
17	C₂×#6122	C₂²	1	1	yes	characteristic
18	C₂×#6122	C₂²	1	1	yes	characteristic
19	C₂×#6122	C₂²	1	1	yes	characteristic
20	C₂×#6122	C₂²	1	2	yes	complemented
21	C₂²×#1057	C₂²	1	2	yes	complemented
22	D₁₃⋊M₁₆	—	2	4	no	—
23	C₂×#6118	C₂²	1	2	yes	complemented
24	C₂×D₁₃⋊C₈	C₂²	1	2	yes	complemented
25	C₂×C₁₃×C₂²⋊C₄	C₄	1	1	yes	characteristic
26	C₂×#6103	C₄	1	1	yes	characteristic
27	C₂×#6101	C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
29	#88135	C₄	1	1	yes	characteristic
30	#88135	C₄	1	1	yes	characteristic
31	#88135	C₄	1	1	yes	characteristic
32	#88135	C₄	1	1	yes	characteristic
33	#88135	C₂²	1	1	yes	characteristic
34	#88135	C₂²	1	1	yes	characteristic
35	#88135	C₂²	1	1	yes	characteristic
36	#88135	C₂²	1	1	yes	characteristic
37	C₂²⋊#1057	C₂²	1	2	yes	complemented
38	C₂²⋊#1057	C₂²	1	2	yes	complemented
39	#88112	C₂²	1	2	yes	complemented
40	#88111	C₂²	1	2	yes	complemented
41	#88111	C₂²	1	2	yes	complemented
42	#88110	C₂²	1	2	yes	complemented
43	C₄⋊#1057	C₂²	1	2	yes	complemented
44	C₄×#1057	C₂²	1	2	yes	complemented
45	C₂⁴×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₂³×D₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₂²⋊D₁₃	—	2	4	no	—
50	C₂×C₂²⋊D₁₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊D₁₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₂²⋊D₁₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₂²⋊D₁₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×Dic₁₃	—	2	2	no	—
55	C₂²×Dic₁₃	—	2	2	no	—
56	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
58	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
59	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
60	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
61	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
62	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₁₃	—	2	2	no	—
64	C₂×C₄×D₁₃	—	2	2	no	—
65	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
69	#6122	—	2	2	no	—
70	#6122	—	2	2	no	—
71	#6122	—	2	2	no	—
72	#6122	—	2	4	no	—
73	#6122	D₄	1	4	yes	complemented
74	C₂×#1057	—	2	4	no	—
75	C₂×#1057	C₂³	1	2	yes	—
76	C₂×#1057	C₂³	1	2	yes	—
77	C₂×#1057	C₂³	1	2	yes	—
78	C₂×#1057	C₂³	1	2	yes	—
79	#6118	—	2	4	no	—
80	D₁₃⋊C₈	—	2	4	no	—
81	C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
82	C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
83	C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
84	C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
85	#6103	C₂×C₄	1	1	yes	characteristic
86	#6103	C₂×C₄	1	1	yes	characteristic
87	#6103	C₂×C₄	1	1	yes	characteristic
88	#6103	C₂×C₄	1	1	yes	characteristic
89	#6101	C₂×C₄	1	1	yes	characteristic
90	#6101	C₂×C₄	1	1	yes	characteristic
91	#6101	C₂×C₄	1	1	yes	characteristic
92	#6101	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
94	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
96	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
97	D₄×M₁₆	—	13	1	no	maximal, Sylow
98	C₂³×C₁₃	—	2	2	no	—
99	C₂³×C₁₃	—	2	2	no	—
100	C₂³×C₁₃	—	4	1	no	—
101	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₂²×D₁₃	—	2	2	no	—
105	C₂²×D₁₃	—	2	2	no	—
106	C₂²×D₁₃	—	4	1	no	—
107	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	4	1	no	—
111	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
115	C₂²⋊D₁₃	—	2	2	no	—
116	C₂²⋊D₁₃	—	2	2	no	—
117	C₂²⋊D₁₃	—	2	2	no	—
118	C₂²⋊D₁₃	M₁₆	1	4	yes	complemented
119	C₂×Dic₁₃	—	2	2	no	—
120	C₂×Dic₁₃	—	2	2	no	—
121	C₂×Dic₁₃	—	2	2	no	—
122	C₂×Dic₁₃	—	2	2	no	—
123	C₂×Dic₁₃	—	2	1	no	—
124	C₂×Dic₁₃	—	2	1	no	—
125	C₂×Dic₁₃	—	2	1	no	—
126	C₂×Dic₁₃	—	2	1	no	—
127	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
130	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
131	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
132	C₂×Dic₁₃	C₂×D₄	1	2	yes	—
133	C₂×Dic₁₃	#41	1	2	yes	—
134	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
135	C₄×D₁₃	—	2	2	no	—
136	C₄×D₁₃	—	2	2	no	—
137	C₄×D₁₃	—	2	1	no	—
138	C₄×D₁₃	—	2	1	no	—
139	C₄×D₁₃	—	2	1	no	—
140	C₄×D₁₃	—	2	1	no	—
141	#1057	—	2	2	no	—
142	#1057	—	2	2	no	—
143	#1057	—	2	2	no	—
144	#1057	C₂×D₄	1	4	yes	complemented
145	C₂²×M₁₆	—	13	1	no	—
146	C₂²×M₁₆	—	13	1	no	—
147	C₂×C₄×D₄	—	13	1	no	—
148	#436	—	13	2	no	—
149	#435	—	13	2	no	—
150	#435	—	13	2	no	—
151	C₈×D₄	—	13	2	no	—
152	#423	—	13	1	no	—
153	#407	—	13	1	no	—
154	#407	—	13	1	no	—
155	C₄×M₁₆	—	13	1	no	—
156	C₂²×C₁₃	—	2	1	no	—
157	C₂²×C₁₃	—	2	1	no	—
158	C₂²×C₁₃	—	2	1	no	—
159	C₂²×C₁₃	—	2	1	no	—
160	C₂²×C₁₃	—	2	1	no	—
161	C₂²×C₁₃	—	2	1	no	—
162	C₂²×C₁₃	—	2	1	no	—
163	C₂²×C₁₃	—	4	4	no	—
164	C₂²×C₁₃	C₄×D₄	1	2	yes	—
165	C₂²×C₁₃	C₂×M₁₆	1	2	yes	complemented
166	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
167	C₂×D₁₃	—	2	1	no	—
168	C₂×D₁₃	—	2	1	no	—
169	C₂×D₁₃	—	2	1	no	—
170	C₂×D₁₃	—	2	1	no	—
171	C₂×D₁₃	—	2	1	no	—
172	C₂×D₁₃	—	4	4	no	—
173	C₂×D₁₃	C₂×M₁₆	1	2	yes	complemented
174	C₄×C₁₃	—	2	1	no	—
175	C₄×C₁₃	—	2	1	no	—
176	C₄×C₁₃	—	2	1	no	—
177	C₄×C₁₃	—	2	1	no	—
178	Dic₁₃	—	2	1	no	—
179	Dic₁₃	—	2	1	no	—
180	Dic₁₃	—	2	1	no	—
181	Dic₁₃	C₄×D₄	1	2	yes	—
182	Dic₁₃	C₂×M₁₆	1	2	yes	complemented
183	Dic₁₃	C₂²×D₄	1	1	yes	characteristic
184	Dic₁₃	C₂×#41	1	1	yes	characteristic
185	C₂²×D₄	—	13	1	no	—
186	C₂³×C₄	—	13	1	no	—
187	C₂³×C₄	—	13	1	no	—
188	C₂×M₁₆	—	13	1	no	—
189	C₂×M₁₆	—	13	1	no	—
190	C₂×M₁₆	—	13	1	no	—
191	C₂×M₁₆	—	13	2	no	—
192	C₂×M₁₆	—	13	2	no	—
193	C₂×M₁₆	—	13	1	no	—
194	C₂×M₁₆	—	26	4	no	—
195	C₂×M₁₆	—	26	4	no	—
196	C₂²×C₈	—	13	2	no	—
197	C₂²×C₈	—	13	2	no	—
198	C₄×D₄	—	13	1	no	—
199	C₄×D₄	—	13	1	no	—
200	C₄×D₄	—	13	1	no	—
201	C₄×D₄	—	13	1	no	—
202	C₄×D₄	—	13	1	no	—
203	C₄×D₄	—	13	1	no	—
204	C₄×D₄	—	13	1	no	—
205	C₄×D₄	—	13	1	no	—
206	C₂×C₄⋊C₄	—	13	1	no	—
207	C₂×C₂²⋊C₄	—	13	1	no	—
208	C₂×C₂²⋊C₄	#261	1	1	yes	characteristic
209	C₂×C₄²	—	13	1	no	—
210	C₄⋊C₈	—	13	2	no	—
211	C₄⋊C₈	—	13	2	no	—
212	C₂²⋊C₈	—	13	2	no	—
213	C₂²⋊C₈	—	13	2	no	—
214	C₂²⋊C₈	—	13	2	no	—
215	C₂²⋊C₈	—	13	2	no	—
216	#97	—	13	2	no	—
217	C₄×C₈	—	13	2	no	—
218	C₂×C₁₃	—	2	2	no	—
219	C₂×C₁₃	—	2	2	no	—
220	C₂×C₁₃	—	4	1	no	—
221	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
222	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
223	C₂×C₁₃	#568	1	1	yes	characteristic
224	D₁₃	—	2	2	no	—
225	D₁₃	—	4	1	no	—
226	C₂⁴	—	13	1	no	—
227	C₂⁴	C₂×#261	1	1	yes	characteristic
228	C₂×D₄	—	13	1	no	—
229	C₂×D₄	—	13	1	no	—
230	C₂×D₄	—	13	1	no	—
231	C₂×D₄	—	13	1	no	—
232	C₂×D₄	—	26	4	no	—
233	C₂²×C₄	—	13	1	no	—
234	C₂²×C₄	—	13	1	no	—
235	C₂²×C₄	—	13	1	no	—
236	C₂²×C₄	—	13	1	no	—
237	C₂²×C₄	—	13	1	no	—
238	C₂²×C₄	—	13	1	no	—
239	C₂²×C₄	—	13	1	no	—
240	C₂²×C₄	—	13	1	no	—
241	C₂²×C₄	—	13	1	no	—
242	C₂²×C₄	—	13	1	no	—
243	C₂²×C₄	—	13	1	no	—
244	C₂²×C₄	—	26	2	no	—
245	C₂²×C₄	—	26	2	no	—
246	C₂²×C₄	—	26	2	no	—
247	C₂²×C₄	—	26	2	no	—
248	C₂²×C₄	C₂×#261	1	1	yes	characteristic
249	C₂²×C₄	C₂×#261	1	1	yes	characteristic
250	M₁₆	—	13	4	no	—
251	M₁₆	—	26	4	no	—
252	M₁₆	—	26	4	no	—
253	M₁₆	—	26	2	no	—
254	M₁₆	—	26	2	no	—
255	M₁₆	—	26	2	no	—
256	C₂×C₈	—	13	2	no	—
257	C₂×C₈	—	13	2	no	—
258	C₂×C₈	—	13	2	no	—
259	C₂×C₈	—	13	2	no	—
260	C₂×C₈	—	26	4	no	—
261	C₂×C₈	—	26	4	no	—
262	C₄⋊C₄	—	13	1	no	—
263	C₄⋊C₄	—	13	1	no	—
264	C₄⋊C₄	—	13	1	no	—
265	C₄⋊C₄	—	13	1	no	—
266	C₂²⋊C₄	—	13	1	no	—
267	C₂²⋊C₄	—	13	1	no	—
268	C₂²⋊C₄	—	13	1	no	—
269	C₂²⋊C₄	—	13	1	no	—
270	C₂²⋊C₄	C₂×#261	1	1	yes	characteristic
271	C₂²⋊C₄	C₂×#261	1	1	yes	characteristic
272	C₂²⋊C₄	C₂×#261	1	1	yes	characteristic
273	C₂²⋊C₄	C₂×#261	1	1	yes	characteristic
274	C₄²	—	13	1	no	—
275	C₄²	—	13	1	no	—
276	C₄²	—	13	1	no	—
277	C₄²	—	13	1	no	—
278	C₁₃	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
279	C₂³	—	2	2	no	—
280	C₂³	—	2	2	no	—
281	C₂³	—	4	1	no	—
282	C₂³	—	13	1	no	—
283	C₂³	—	13	1	no	—
284	C₂³	—	26	2	no	—
285	C₂³	—	26	2	no	—
286	C₂³	—	52	1	no	—
287	C₂³	C₂²×#261	1	1	yes	characteristic
288	C₂³	C₂²×#261	1	1	yes	characteristic
289	C₂³	C₂²×#261	1	1	yes	characteristic
290	D₄	—	13	4	no	—
291	D₄	—	26	2	no	—
292	D₄	—	26	2	no	—
293	D₄	—	26	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	13	2	no	—
297	C₂×C₄	—	13	2	no	—
298	C₂×C₄	—	13	1	no	—
299	C₂×C₄	—	13	1	no	—
300	C₂×C₄	—	13	1	no	—
301	C₂×C₄	—	13	1	no	—
302	C₂×C₄	—	13	1	no	—
303	C₂×C₄	—	13	1	no	—
304	C₂×C₄	—	26	1	no	—
305	C₂×C₄	—	26	1	no	—
306	C₂×C₄	—	26	2	no	—
307	C₂×C₄	—	26	2	no	—
308	C₂×C₄	—	26	1	no	—
309	C₂×C₄	—	26	1	no	—
310	C₂×C₄	—	26	1	no	—
311	C₂×C₄	—	26	1	no	—
312	C₂×C₄	—	26	2	no	—
313	C₂×C₄	—	26	2	no	—
314	C₂×C₄	—	26	1	no	—
315	C₂×C₄	—	26	1	no	—
316	C₂×C₄	—	26	2	no	—
317	C₂×C₄	—	26	2	no	—
318	C₂×C₄	C₂²×#261	1	1	yes	characteristic
319	C₂×C₄	C₂²×#261	1	1	yes	characteristic
320	C₂×C₄	C₂²×#261	1	1	yes	characteristic
321	C₂×C₄	C₂²×#261	1	1	yes	characteristic
322	C₈	—	13	4	no	—
323	C₈	—	26	2	no	—
324	C₈	—	26	2	no	—
325	C₈	—	26	2	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	4	4	no	—
334	C₂²	—	13	2	no	—
335	C₂²	—	26	1	no	—
336	C₂²	—	26	1	no	—
337	C₂²	—	26	1	no	—
338	C₂²	—	26	1	no	—
339	C₂²	—	26	1	no	—
340	C₂²	—	52	4	no	—
341	C₂²	D₁₃⋊M₁₆	1	2	yes	complemented
342	C₂²	D₄×#261	1	2	yes	—
343	C₂²	C₂³×#261	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	13	2	no	—
349	C₄	—	13	2	no	—
350	C₄	—	13	1	no	—
351	C₄	—	13	1	no	—
352	C₄	—	26	1	no	—
353	C₄	—	26	1	no	—
354	C₄	—	26	1	no	—
355	C₂	—	2	2	no	—
356	C₂	—	2	2	no	—
357	C₂	—	4	1	no	—
358	C₂	—	26	2	no	—
359	C₂	—	52	1	no	—
360	C₂	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
361	C₂	#11719621	1	1	yes	characteristic
362	C₂	C₂×D₄×#261	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₂×C₁₃×C₂²⋊C₄	C₄
prime radical	C₂²×C₂²⋊D₁₃	C₄
cosocle	C₂×Dic₁₃	C₂⁴
hypercenter	C₂×C₂²⋊C₄	#261
hypocenter	C₁₃	D₄×M₁₆
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#6122	C₂	semidirect
2. C₂×D₁₃⋊M₁₆	C₂	semidirect
4. #11719169	C₂	semidirect
5. #11719129	C₂	semidirect
6. #11719126	C₂	semidirect
7. C₄×#6122	C₂	semidirect
8. #11719066	C₂	semidirect
9. #11719065	C₂	semidirect
10. #11719064	C₂	semidirect
11. #11719063	C₂	semidirect
20. C₂×#6122	C₂²	semidirect
21. C₂²×#1057	C₂²	semidirect
23. C₂×#6118	C₂²	semidirect
24. C₂×D₁₃⋊C₈	C₂²	semidirect
37. C₂²⋊#1057	C₂²	semidirect
38. C₂²⋊#1057	C₂²	semidirect
39. #88112	C₂²	semidirect
40. #88111	C₂²	semidirect
41. #88111	C₂²	semidirect
42. #88110	C₂²	semidirect
43. C₄⋊#1057	C₂²	semidirect
44. C₄×#1057	C₂²	semidirect
73. #6122	D₄	semidirect
118. C₂²⋊D₁₃	M₁₆	semidirect
144. #1057	C₂×D₄	semidirect
165. C₂²×C₁₃	C₂×M₁₆	semidirect
173. C₂×D₁₃	C₂×M₁₆	semidirect
182. Dic₁₃	C₂×M₁₆	semidirect
278. C₁₃	D₄×M₁₆	semidirect
341. C₂²	D₁₃⋊M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#6122	2
C₂×D₁₃⋊M₁₆	2
C₂×#88135	2
#11719169	2
#11719129	2
#11719126	2
C₄×#6122	2
#11719066	2
#11719065	2
#11719064	2
#11719063	2
D₄×M₁₆	13