Subgroups — #15867331

Order 3328.

722 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135×Dic₁₃	C₂	1	1	yes	maximal, characteristic
2	#13123483	C₂	1	2	yes	maximal, complemented
3	#13115730	C₂	1	1	yes	maximal, characteristic
4	#13115589	C₂	1	1	yes	maximal, characteristic, complemented
5	#13115587	C₂	1	1	yes	maximal, characteristic, complemented
6	#13115543	C₂	1	2	yes	maximal, complemented
7	#13115537	C₂	1	2	yes	maximal, complemented
8	#108×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#13114648	C₂	1	2	yes	maximal, complemented
10	#13114524	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#1975	C₂	1	1	yes	maximal, characteristic
12	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
13	#41×Dic₁₃	C₂²	1	2	yes	—
14	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#104	C₂²	1	2	yes	—
16	C₁₃×#408	C₂²	1	2	yes	—
17	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
18	C₂×#88077	C₂²	1	2	yes	—
19	#11718895	C₂²	1	2	yes	—
20	#11718894	C₂²	1	2	yes	—
21	Q₁₆×Dic₁₃	C₄	1	2	yes	complemented
22	SD₁₆×Dic₁₃	C₄	1	4	yes	complemented
23	D₈×Dic₁₃	C₄	1	2	yes	complemented
24	#11718780	C₂²	1	2	yes	—
25	M₁₆×Dic₁₃	C₂²	1	2	yes	—
26	C₂×#88059	C₂²	1	1	yes	characteristic
27	#11718767	C₂²	1	1	yes	characteristic
28	#11718763	C₂²	1	2	yes	—
29	C₂×#88055	C₂²	1	1	yes	characteristic
30	C₂×#88055	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
32	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
33	#11718593	C₂²	1	1	yes	characteristic
34	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
35	#11718381	—	2	4	no	—
36	#11718380	—	2	4	no	—
37	#11718150	C₂²	1	2	yes	—
38	#11718147	C₂²	1	2	yes	—
39	#11718115	C₂²	1	2	yes	—
40	C₂×C₁₃×#41	C₂³	1	2	yes	—
41	C₁₃×#135	—	2	2	no	—
42	C₁₃×#135	—	2	2	no	—
43	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
44	C₂×C₁₃×SD₁₆	C₂×C₄	1	2	yes	—
45	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
47	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	Q₈×Dic₁₃	C₂×C₄	1	4	yes	—
51	D₄×Dic₁₃	—	2	4	no	—
52	D₄×Dic₁₃	C₂×C₄	1	4	yes	—
53	#88180	—	2	2	no	—
54	#88180	C₂³	1	2	yes	—
55	C₂×C₄×Dic₁₃	—	2	2	no	—
56	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
57	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
58	C₂×#6099	C₂³	1	2	yes	—
59	C₂²×#1055	C₂³	1	1	yes	characteristic
60	C₂²×#1055	C₂³	1	1	yes	characteristic
61	C₁₃×#108	—	2	2	no	—
62	C₁₃×#104	—	2	4	no	—
63	C₁₃×C₂²⋊C₈	—	2	4	no	—
64	C₁₃×#97	—	2	2	no	—
65	#88077	—	2	4	no	—
66	#88075	C₂×C₄	1	4	yes	—
67	#88072	C₂×C₄	1	4	yes	—
68	#88071	—	2	2	no	—
69	C₂²⋊#1055	—	2	4	no	—
70	#88059	—	2	2	no	—
71	#88058	C₂×C₄	1	2	yes	—
72	#88057	C₂×C₄	1	2	yes	—
73	#88055	—	2	2	no	—
74	#88055	—	2	2	no	—
75	#88055	D₄	1	2	yes	—
76	#88055	D₄	1	2	yes	—
77	#88054	—	2	4	no	—
78	#88054	—	2	4	no	—
79	C₈×Dic₁₃	—	2	2	no	—
80	C₈×Dic₁₃	—	2	2	no	—
81	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
82	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
83	C₄×#1055	—	2	2	no	—
84	#21295	—	13	1	no	maximal, Sylow
85	C₁₃×#41	—	2	2	no	—
86	C₁₃×#41	—	2	2	no	—
87	C₁₃×#41	—	2	2	no	—
88	C₁₃×#41	—	2	2	no	—
89	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
90	C₂×D₄×C₁₃	—	2	2	no	—
91	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₁₃	—	2	1	no	—
93	C₂²×C₄×C₁₃	—	2	2	no	—
94	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₂²×Dic₁₃	—	2	1	no	—
96	C₂²×Dic₁₃	—	2	4	no	—
97	C₂²×Dic₁₃	#41	1	2	yes	—
98	C₁₃×Q₁₆	—	2	2	no	—
99	C₁₃×SD₁₆	—	2	4	no	—
100	C₁₃×D₈	—	2	2	no	—
101	C₁₃×M₁₆	—	2	2	no	—
102	C₁₃×M₁₆	—	2	2	no	—
103	C₂×C₈×C₁₃	—	2	1	no	—
104	C₂×C₈×C₁₃	—	2	1	no	—
105	C₂×C₈×C₁₃	—	2	2	no	—
106	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₁₃	#41	1	2	yes	—
110	C₂×C₈×C₁₃	#41	1	2	yes	—
111	C₄²×C₁₃	—	2	1	no	—
112	C₄²×C₁₃	#41	1	2	yes	—
113	C₂²⋊Dic₁₃	—	2	2	no	—
114	C₂²⋊Dic₁₃	—	2	4	no	—
115	C₄⋊Dic₁₃	—	2	4	no	—
116	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
117	C₄×Dic₁₃	—	2	2	no	—
118	C₄×Dic₁₃	—	2	1	no	—
119	C₄×Dic₁₃	—	2	4	no	—
120	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
121	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
122	C₄×Dic₁₃	C₂×D₄	1	1	yes	characteristic
123	C₄×Dic₁₃	#41	1	2	yes	—
124	#6099	—	2	2	no	—
125	#6099	—	2	2	no	—
126	C₂×#1055	—	2	1	no	—
127	C₂×#1055	—	2	1	no	—
128	C₂×#1055	—	2	1	no	—
129	C₂×#1055	—	2	1	no	—
130	C₂×#1055	—	2	2	no	—
131	C₂×#1055	C₂²×C₄	1	2	yes	—
132	C₂×#1055	C₂×D₄	1	2	yes	—
133	C₄×#135	—	13	1	no	—
134	#2923	—	13	2	no	—
135	#1975	—	13	1	no	—
136	#1975	D₁₃	1	1	yes	characteristic
137	#1946	—	13	1	no	—
138	#1946	—	13	1	no	—
139	#1937	—	13	2	no	—
140	#1937	—	13	2	no	—
141	C₄×#108	—	13	1	no	—
142	#1759	—	13	2	no	—
143	#1726	—	13	1	no	—
144	C₂³×C₁₃	—	2	2	no	—
145	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
146	Q₈×C₁₃	—	2	2	no	—
147	Q₈×C₁₃	—	2	2	no	—
148	D₄×C₁₃	—	2	2	no	—
149	D₄×C₁₃	—	2	2	no	—
150	D₄×C₁₃	—	4	2	no	—
151	D₄×C₁₃	—	4	2	no	—
152	C₂×C₄×C₁₃	—	2	2	no	—
153	C₂×C₄×C₁₃	—	2	2	no	—
154	C₂×C₄×C₁₃	—	4	2	no	—
155	C₂×C₄×C₁₃	—	4	2	no	—
156	C₂×C₄×C₁₃	—	4	2	no	—
157	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
161	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
162	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
163	C₂×Dic₁₃	—	2	4	no	—
164	C₂×Dic₁₃	—	2	2	no	—
165	C₂×Dic₁₃	—	4	2	no	—
166	C₂×Dic₁₃	—	4	2	no	—
167	C₂×Dic₁₃	—	8	4	no	—
168	C₂×Dic₁₃	C₄×D₄	1	2	yes	—
169	C₂×Dic₁₃	C₄×D₄	1	2	yes	—
170	C₈×C₁₃	—	2	2	no	—
171	C₈×C₁₃	—	4	2	no	—
172	C₈×C₁₃	#117	1	2	yes	—
173	C₈×C₁₃	#117	1	2	yes	—
174	#1055	—	2	2	no	—
175	#1055	—	2	2	no	—
176	#1055	—	4	2	no	—
177	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
178	C₄×#41	—	13	2	no	—
179	C₄×Q₁₆	—	13	2	no	—
180	C₄×SD₁₆	—	13	4	no	—
181	C₄×D₈	—	13	2	no	—
182	#433	—	26	4	no	—
183	#432	—	26	4	no	—
184	C₂×#108	—	13	1	no	—
185	C₂×#108	C₂×D₁₃	1	1	yes	characteristic
186	#427	—	13	1	no	—
187	#424	—	13	2	no	—
188	#424	—	13	2	no	—
189	C₂×#104	—	13	2	no	—
190	C₂×#104	C₂×D₁₃	1	2	yes	—
191	#416	—	13	2	no	—
192	#408	—	13	2	no	—
193	#408	C₂×D₁₃	1	2	yes	—
194	C₄×M₁₆	—	13	2	no	—
195	C₂×#97	—	13	1	no	—
196	C₂×#97	—	13	1	no	—
197	C₂×#97	C₂×D₁₃	1	1	yes	characteristic
198	C₂×C₄×C₈	—	13	1	no	—
199	C₂×C₄×C₈	—	13	1	no	—
200	C₂×C₄×C₈	—	13	1	no	—
201	#376	—	13	1	no	—
202	#341	—	13	2	no	—
203	#339	—	13	2	no	—
204	#339	—	13	2	no	—
205	C₂²×C₁₃	—	2	2	no	—
206	C₂²×C₁₃	—	4	4	no	—
207	C₂²×C₁₃	—	4	2	no	—
208	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
209	C₂²×C₁₃	#529	1	1	yes	characteristic
210	C₂²×C₁₃	#529	1	1	yes	characteristic
211	C₄×C₁₃	—	4	2	no	—
212	C₄×C₁₃	—	4	2	no	—
213	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
214	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
215	C₄×C₁₃	#532	1	1	yes	characteristic
216	C₄×C₁₃	#532	1	1	yes	characteristic
217	Dic₁₃	—	2	2	no	—
218	Dic₁₃	—	2	2	no	—
219	Dic₁₃	—	4	2	no	—
220	Dic₁₃	—	4	4	no	—
221	C₂×#41	C₂²×D₁₃	1	2	yes	—
222	#135	—	2	2	no	—
223	#135	—	2	2	no	—
224	C₂×Q₁₆	C₄×D₁₃	1	1	yes	characteristic
225	C₂×SD₁₆	C₄×D₁₃	1	2	yes	—
226	C₂×D₈	C₄×D₁₃	1	1	yes	characteristic
227	C₂×M₁₆	—	13	2	no	—
228	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
229	C₂²×C₈	—	13	1	no	—
230	C₂²×C₈	—	13	1	no	—
231	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
232	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
233	C₄×Q₈	—	13	4	no	—
234	C₄×D₄	—	13	4	no	—
235	C₄×D₄	—	26	4	no	—
236	#117	—	13	2	no	—
237	#117	—	26	2	no	—
238	C₂×C₄²	—	13	1	no	—
239	C₂×C₄²	—	13	1	no	—
240	C₂×C₄²	—	26	2	no	—
241	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
242	#108	—	2	2	no	—
243	#108	—	26	2	no	—
244	#107	—	13	2	no	—
245	#106	—	13	2	no	—
246	C₄⋊C₈	—	26	4	no	—
247	C₄⋊C₈	—	26	4	no	—
248	#104	—	2	4	no	—
249	#104	—	26	4	no	—
250	#103	—	13	4	no	—
251	#102	—	13	4	no	—
252	C₂²⋊C₈	—	2	4	no	—
253	C₂²⋊C₈	—	26	4	no	—
254	#97	—	2	2	no	—
255	#97	—	13	2	no	—
256	#97	—	13	2	no	—
257	#97	—	26	2	no	—
258	#97	—	26	2	no	—
259	C₄×C₈	—	13	2	no	—
260	C₄×C₈	—	13	2	no	—
261	C₄×C₈	—	26	2	no	—
262	C₄×C₈	—	26	2	no	—
263	C₄×C₈	—	26	2	no	—
264	#95	—	26	2	no	—
265	C₂×C₁₃	—	2	1	no	—
266	C₂×C₁₃	—	2	1	no	—
267	C₂×C₁₃	—	8	2	no	—
268	C₂×C₁₃	#2293	1	1	yes	characteristic
269	C₂×C₁₃	#2957	1	2	yes	—
270	#41	—	2	2	no	—
271	#41	—	2	2	no	—
272	#41	—	2	2	no	—
273	#41	—	2	2	no	—
274	C₂×Q₈	C₂×C₄×D₁₃	1	2	yes	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	C₂×C₄×D₁₃	1	2	yes	—
277	C₂²×C₄	—	2	1	no	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	13	2	no	—
280	C₂²×C₄	—	26	4	no	—
281	C₂²×C₄	—	26	1	no	—
282	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
283	Q₁₆	—	2	2	no	—
284	SD₁₆	—	2	4	no	—
285	D₈	—	2	2	no	—
286	M₁₆	—	2	2	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	26	2	no	—
289	M₁₆	—	26	2	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	13	2	no	—
294	C₂×C₈	—	13	2	no	—
295	C₂×C₈	—	26	1	no	—
296	C₂×C₈	—	26	1	no	—
297	C₂×C₈	—	26	1	no	—
298	C₂×C₈	—	26	1	no	—
299	C₂×C₈	—	26	2	no	—
300	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
301	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
302	C₂×C₈	D₄×D₁₃	1	2	yes	—
303	C₂×C₈	#6129	1	2	yes	—
304	C₂×C₈	#6129	1	2	yes	—
305	C₄⋊C₄	—	13	4	no	—
306	C₄⋊C₄	—	26	4	no	—
307	C₂²⋊C₄	—	26	4	no	—
308	C₂²⋊C₄	—	26	2	no	—
309	C₄²	—	2	1	no	—
310	C₄²	—	13	2	no	—
311	C₄²	—	13	1	no	—
312	C₄²	—	13	1	no	—
313	C₄²	—	13	2	no	—
314	C₄²	—	26	4	no	—
315	C₄²	—	26	2	no	—
316	C₄²	—	26	1	no	—
317	C₄²	#6129	1	2	yes	—
318	C₁₃	#21295	1	1	yes	characteristic, complemented, Sylow
319	C₂³	—	2	2	no	—
320	C₂³	#41×D₁₃	1	1	yes	characteristic
321	Q₈	—	2	2	no	—
322	Q₈	—	2	2	no	—
323	D₄	—	2	2	no	—
324	D₄	—	2	2	no	—
325	D₄	—	4	2	no	—
326	D₄	—	4	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	13	2	no	—
333	C₂×C₄	—	13	2	no	—
334	C₂×C₄	—	26	4	no	—
335	C₂×C₄	—	26	2	no	—
336	C₂×C₄	—	52	2	no	—
337	C₂×C₄	—	52	2	no	—
338	C₂×C₄	—	104	4	no	—
339	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
341	C₂×C₄	C₂×#6129	1	1	yes	characteristic
342	C₂×C₄	C₂×#6129	1	1	yes	characteristic
343	C₂×C₄	C₂×#6129	1	1	yes	characteristic
344	C₂×C₄	#41×D₁₃	1	1	yes	characteristic
345	C₈	—	2	2	no	—
346	C₈	—	4	2	no	—
347	C₈	—	26	2	no	—
348	C₈	—	26	2	no	—
349	C₈	—	52	2	no	—
350	C₈	#88131	1	2	yes	—
351	C₈	#88131	1	2	yes	—
352	C₂²	—	2	2	no	—
353	C₂²	—	4	2	no	—
354	C₂²	—	4	4	no	—
355	C₂²	C₄×D₄×D₁₃	1	1	yes	characteristic
356	C₂²	#11719305	1	1	yes	characteristic
357	C₂²	#11719385	1	1	yes	characteristic
358	C₄	—	4	2	no	—
359	C₄	—	4	2	no	—
360	C₄	—	26	2	no	—
361	C₄	—	26	2	no	—
362	C₄	—	52	2	no	—
363	C₄	—	52	4	no	—
364	C₄	C₂×#88131	1	1	yes	characteristic
365	C₄	C₄×#6129	1	1	yes	characteristic
366	C₄	#11719301	1	1	yes	characteristic
367	C₄	#11719421	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	2	1	no	—
370	C₂	—	8	2	no	—
371	C₂	#13117298	1	1	yes	characteristic
372	C₂	#13123750	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#1975	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂×C₁₃×#135	C₂²
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#1975	D₁₃
hypocenter	C₁₃	#21295
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1975

Complements

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

N	H ≅ G/N	type
2. #13123483	C₂	semidirect
4. #13115589	C₂	semidirect
5. #13115587	C₂	semidirect
6. #13115543	C₂	semidirect
7. #13115537	C₂	semidirect
8. #108×Dic₁₃	C₂	semidirect
9. #13114648	C₂	semidirect
10. #13114524	C₂	semidirect
21. Q₁₆×Dic₁₃	C₄	semidirect
22. SD₁₆×Dic₁₃	C₄	semidirect
23. D₈×Dic₁₃	C₄	semidirect
318. C₁₃	#21295	semidirect

Maximal subgroups

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

subgroup	index
#135×Dic₁₃	2
#13123483	2
#13115730	2
#13115589	2
#13115587	2
#13115543	2
#13115537	2
#108×Dic₁₃	2
#13114648	2
#13114524	2
C₁₃×#1975	2
#21295	13