Subgroups — #15833682

Order 3328.

606 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125406	C₂	1	1	yes	maximal, characteristic
2	#13123811	C₂	1	2	yes	maximal, complemented
3	#13113716	C₂	1	2	yes	maximal, complemented
4	#13113709	C₂	1	2	yes	maximal, complemented
5	#13113702	C₂	1	2	yes	maximal, complemented
6	#13113695	C₂	1	2	yes	maximal, complemented
7	#13113599	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113297	C₂	1	2	yes	maximal, complemented
9	C₂×C₁₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
10	#11719184	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#11718777	C₄	1	1	yes	characteristic
15	#11718776	C₄	1	1	yes	characteristic
16	#11718775	C₄	1	2	yes	—
17	#11718772	—	2	2	no	—
18	C₈×C₂²⋊D₁₃	—	2	2	no	—
19	#11718587	C₂²	1	2	yes	—
20	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
21	#11718358	C₄	1	2	yes	—
22	#11718357	C₄	1	1	yes	characteristic
23	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
24	C₄×#6095	C₂²	1	2	yes	—
25	#11718351	—	2	2	no	—
26	#11718350	—	2	2	no	—
27	#11718349	C₂²	1	2	yes	—
28	C₈×D₅₂	C₂²	1	2	yes	—
29	#11718344	C₄	1	1	yes	characteristic
30	#11718340	C₂²	1	2	yes	—
31	C₈×Dic₂₆	C₂²	1	2	yes	—
32	C₁₃×#322	C₂²	1	2	yes	—
33	C₁₃×#322	C₂²	1	2	yes	—
34	#11718051	C₂²	1	2	yes	—
35	#11718051	C₂²	1	2	yes	—
36	#11718050	C₂²	1	1	yes	characteristic
37	#11718050	C₂²	1	1	yes	characteristic
38	#11718049	C₂²	1	2	yes	—
39	#11718048	C₂²	1	2	yes	—
40	#11718048	C₂²	1	2	yes	—
41	C₂²×C₈×C₁₃	—	2	1	no	—
42	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	#88184	C₂×C₄	1	2	yes	—
48	#88183	—	2	2	no	—
49	C₄×C₂²⋊D₁₃	—	2	2	no	—
50	#88178	C₂×C₄	1	2	yes	—
51	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
52	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
53	C₂×#6095	C₂×C₄	1	2	yes	—
54	C₂×#6094	—	2	2	no	—
55	C₂×C₈×D₁₃	—	2	2	no	—
56	#88130	—	2	2	no	—
57	#88129	C₂×C₄	1	2	yes	—
58	C₄×D₅₂	C₂³	1	2	yes	—
59	#88126	—	2	2	no	—
60	#88124	C₂×C₄	1	2	yes	—
61	C₄×Dic₂₆	C₂³	1	2	yes	—
62	C₄×C₈×C₁₃	—	2	1	no	—
63	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₁₃	D₄	1	2	yes	—
66	C₄×C₈×C₁₃	C₂³	1	2	yes	—
67	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₁₃	C₂³	1	2	yes	—
70	C₂²⋊#1055	—	2	2	no	—
71	#88061	C₂×C₄	1	2	yes	—
72	#88061	C₂×C₄	1	2	yes	—
73	#88060	—	2	2	no	—
74	#88060	—	2	2	no	—
75	#88058	C₂×C₄	1	1	yes	characteristic
76	#88058	C₂×C₄	1	1	yes	characteristic
77	#88057	C₂×C₄	1	2	yes	—
78	#88056	C₂×C₄	1	2	yes	—
79	#88056	C₂×C₄	1	2	yes	—
80	#88055	—	2	2	no	—
81	#88054	—	2	2	no	—
82	#88054	—	2	2	no	—
83	C₈×Dic₁₃	—	2	2	no	—
84	C₄⋊#1055	C₂³	1	2	yes	—
85	C₄⋊#1055	C₂³	1	2	yes	—
86	#88043	—	2	2	no	—
87	#17982	—	13	1	no	maximal, Sylow
88	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
91	C₂×C₂²⋊D₁₃	—	2	2	no	—
92	C₂×D₅₂	C₂²×C₄	1	2	yes	—
93	C₂×C₄×D₁₃	—	2	2	no	—
94	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₁₃	—	2	4	no	—
96	C₂×C₈×C₁₃	—	2	2	no	—
97	C₂×C₈×C₁₃	—	4	1	no	—
98	C₂×C₈×C₁₃	—	4	1	no	—
99	C₂×C₈×C₁₃	—	4	1	no	—
100	C₂×C₈×C₁₃	—	4	1	no	—
101	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
107	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₃	#41	1	1	yes	characteristic
110	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
111	C₂²⋊Dic₁₃	—	2	2	no	—
112	#6103	—	2	2	no	—
113	#6103	—	2	2	no	—
114	#6103	—	2	2	no	—
115	#6103	—	2	2	no	—
116	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
117	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
118	#6101	—	2	2	no	—
119	#6101	—	2	2	no	—
120	#6101	—	2	2	no	—
121	#6101	—	2	2	no	—
122	C₄×Dic₁₃	—	2	2	no	—
123	C₂×#1055	—	2	2	no	—
124	C₂×#1055	—	2	2	no	—
125	Dic₅₂	—	4	1	no	—
126	D₁₀₄	—	4	1	no	—
127	#6095	—	4	2	no	—
128	#6094	—	4	4	no	—
129	C₈×D₁₃	—	4	4	no	—
130	#3168	—	13	1	no	—
131	#2966	—	13	2	no	—
132	#1588	—	13	2	no	—
133	#1587	—	13	2	no	—
134	#1586	—	13	2	no	—
135	#1585	—	13	2	no	—
136	#1568	—	13	1	no	—
137	#1494	—	13	2	no	—
138	C₂×#322	D₁₃	1	1	yes	characteristic, complemented
139	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
140	C₂²×D₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	—	2	2	no	—
142	C₂×C₄×C₁₃	—	2	1	no	—
143	C₂×C₄×C₁₃	—	2	1	no	—
144	C₂×C₄×C₁₃	—	2	1	no	—
145	C₂×C₄×C₁₃	—	2	1	no	—
146	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
152	C₂²⋊D₁₃	—	4	4	no	—
153	C₂×Dic₁₃	—	2	2	no	—
154	C₂×Dic₁₃	—	2	2	no	—
155	C₂×Dic₁₃	—	2	2	no	—
156	D₅₂	—	2	2	no	—
157	D₅₂	—	2	2	no	—
158	C₄×D₁₃	—	4	2	no	—
159	C₄×D₁₃	—	4	2	no	—
160	Dic₂₆	—	2	2	no	—
161	Dic₂₆	—	2	2	no	—
162	C₈×C₁₃	—	2	4	no	—
163	C₈×C₁₃	—	2	1	no	—
164	C₈×C₁₃	—	2	1	no	—
165	C₈×C₁₃	—	2	1	no	—
166	C₈×C₁₃	—	2	1	no	—
167	C₈×C₁₃	—	2	2	no	—
168	C₈×C₁₃	—	2	2	no	—
169	#1055	—	4	2	no	—
170	#1055	—	4	2	no	—
171	#529	—	13	2	no	—
172	#499	—	13	1	no	—
173	#495	—	13	1	no	—
174	#488	—	13	2	no	—
175	#467	—	13	1	no	—
176	#466	—	13	1	no	—
177	#465	—	13	2	no	—
178	#446	—	13	2	no	—
179	C₈×Q₈	—	13	2	no	—
180	C₄×Q₁₆	—	13	1	no	—
181	C₄×SD₁₆	—	13	2	no	—
182	C₄×D₈	—	13	1	no	—
183	#436	—	13	2	no	—
184	#435	—	26	2	no	—
185	C₈×D₄	—	13	2	no	—
186	C₈×D₄	—	26	2	no	—
187	#433	—	26	2	no	—
188	#432	—	13	2	no	—
189	#405	—	26	2	no	—
190	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
191	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
192	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
193	#335	—	13	1	no	—
194	#335	—	13	1	no	—
195	#334	—	13	2	no	—
196	Q₈⋊C₈	—	13	2	no	—
197	Q₈⋊C₈	—	13	2	no	—
198	D₄⋊C₈	—	13	2	no	—
199	D₄⋊C₈	—	13	2	no	—
200	#322	C₂×D₁₃	1	2	yes	—
201	#322	C₂×D₁₃	1	2	yes	—
202	C₂²×C₁₃	—	2	1	no	—
203	C₂²×C₁₃	—	2	1	no	—
204	C₂²×C₁₃	#443	1	2	yes	—
205	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
206	C₂×D₁₃	—	4	2	no	—
207	C₂×D₁₃	—	4	2	no	—
208	C₂×D₁₃	—	4	2	no	—
209	C₄×C₁₃	—	2	1	no	—
210	C₄×C₁₃	—	2	1	no	—
211	C₄×C₁₃	—	2	1	no	—
212	C₄×C₁₃	—	2	1	no	—
213	C₄×C₁₃	#568	1	1	yes	characteristic
214	C₄×C₁₃	#568	1	1	yes	characteristic
215	C₄×C₁₃	#575	1	1	yes	characteristic
216	C₄×C₁₃	#575	1	1	yes	characteristic
217	Dic₁₃	—	4	2	no	—
218	Dic₁₃	—	4	2	no	—
219	Dic₁₃	—	4	2	no	—
220	C₂×Q₁₆	—	13	1	no	—
221	C₂×SD₁₆	—	13	2	no	—
222	C₂×D₈	—	13	1	no	—
223	C₂×M₁₆	—	26	2	no	—
224	C₂²×C₈	—	2	1	no	—
225	C₂²×C₈	—	26	2	no	—
226	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
227	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
228	C₂²×C₈	D₅₂	1	1	yes	characteristic
229	C₂²×C₈	D₅₂	1	1	yes	characteristic
230	#126	—	26	2	no	—
231	#125	—	13	2	no	—
232	#124	—	13	2	no	—
233	#123	—	26	2	no	—
234	C₂²⋊Q₈	—	13	2	no	—
235	#121	—	13	2	no	—
236	C₄×Q₈	—	13	2	no	—
237	C₄×D₄	—	13	2	no	—
238	C₄×D₄	—	26	2	no	—
239	#117	—	26	2	no	—
240	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
241	#107	—	13	1	no	—
242	#107	—	13	1	no	—
243	#106	—	13	2	no	—
244	C₄⋊C₈	—	13	2	no	—
245	C₄⋊C₈	—	13	2	no	—
246	C₄⋊C₈	—	26	2	no	—
247	C₄⋊C₈	—	26	2	no	—
248	#103	—	13	2	no	—
249	#103	—	13	2	no	—
250	#102	—	13	2	no	—
251	#102	—	13	2	no	—
252	C₂²⋊C₈	—	26	2	no	—
253	C₂²⋊C₈	—	26	2	no	—
254	C₂²⋊C₈	—	26	2	no	—
255	#97	—	26	2	no	—
256	#97	—	26	2	no	—
257	C₄×C₈	—	2	1	no	—
258	C₄×C₈	—	26	2	no	—
259	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
260	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
261	C₄×C₈	D₅₂	1	2	yes	—
262	C₄×C₈	C₂²×D₁₃	1	2	yes	—
263	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
265	C₄×C₈	C₂²×D₁₃	1	2	yes	—
266	C₂×C₁₃	—	2	2	no	—
267	C₂×C₁₃	#2932	1	1	yes	characteristic
268	C₂×C₁₃	#2943	1	1	yes	characteristic
269	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
270	D₁₃	—	8	2	no	—
271	C₂×Q₈	—	13	2	no	—
272	C₂×D₄	—	13	2	no	—
273	C₂×D₄	—	26	2	no	—
274	C₂²×C₄	—	26	2	no	—
275	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
277	C₂²×C₄	#6127	1	1	yes	characteristic
278	Q₁₆	—	52	1	no	—
279	SD₁₆	—	52	2	no	—
280	D₈	—	52	1	no	—
281	M₁₆	—	52	4	no	—
282	C₂×C₈	—	2	4	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	4	1	no	—
287	C₂×C₈	—	4	1	no	—
288	C₂×C₈	—	26	2	no	—
289	C₂×C₈	—	26	2	no	—
290	C₂×C₈	—	52	4	no	—
291	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
292	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
293	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
294	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
295	C₂×C₈	C₂×D₅₂	1	2	yes	—
296	C₂×C₈	C₂×D₅₂	1	2	yes	—
297	C₄⋊C₄	—	13	2	no	—
298	C₄⋊C₄	—	13	2	no	—
299	C₄⋊C₄	—	26	2	no	—
300	C₄⋊C₄	—	26	2	no	—
301	C₄⋊C₄	—	26	2	no	—
302	C₄⋊C₄	—	26	2	no	—
303	C₂²⋊C₄	—	26	2	no	—
304	C₂²⋊C₄	—	26	2	no	—
305	C₂²⋊C₄	—	26	2	no	—
306	C₂²⋊C₄	—	26	2	no	—
307	C₂²⋊C₄	—	26	2	no	—
308	C₄²	—	26	2	no	—
309	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
310	C₄²	C₂×D₅₂	1	1	yes	characteristic
311	C₄²	#6127	1	1	yes	characteristic
312	C₄²	C₂³×D₁₃	1	1	yes	characteristic
313	C₁₃	#17982	1	1	yes	characteristic, complemented, Sylow
314	C₂³	—	26	2	no	—
315	C₂³	C₄×D₅₂	1	1	yes	characteristic
316	Q₈	—	26	2	no	—
317	Q₈	—	26	2	no	—
318	D₄	—	26	2	no	—
319	D₄	—	26	2	no	—
320	D₄	—	52	4	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	26	2	no	—
327	C₂×C₄	—	26	2	no	—
328	C₂×C₄	—	26	2	no	—
329	C₂×C₄	—	52	2	no	—
330	C₂×C₄	—	52	2	no	—
331	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
332	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
333	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
334	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
335	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
336	C₂×C₄	C₂×#6127	1	1	yes	characteristic
337	C₈	—	2	4	no	—
338	C₈	—	2	1	no	—
339	C₈	—	2	1	no	—
340	C₈	—	2	1	no	—
341	C₈	—	2	1	no	—
342	C₈	—	2	2	no	—
343	C₈	—	2	2	no	—
344	C₈	—	52	2	no	—
345	C₈	—	52	2	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	52	2	no	—
349	C₂²	—	52	2	no	—
350	C₂²	—	52	2	no	—
351	C₂²	#11718361	1	2	yes	—
352	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	52	2	no	—
358	C₄	—	52	2	no	—
359	C₄	—	52	2	no	—
360	C₄	#11719450	1	1	yes	characteristic
361	C₄	#11719450	1	1	yes	characteristic
362	C₄	#11719453	1	1	yes	characteristic
363	C₄	#11719453	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	104	2	no	—
366	C₂	#13123555	1	1	yes	characteristic
367	C₂	#13123638	1	1	yes	characteristic
368	C₂	C₂×#11718361	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₁₃×#322	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#11718776	C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×#322	D₁₃
hypocenter	C₁₃	#17982
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #13123811	C₂	semidirect
3. #13113716	C₂	semidirect
4. #13113709	C₂	semidirect
5. #13113702	C₂	semidirect
6. #13113695	C₂	semidirect
7. #13113599	C₂	semidirect
8. #13113297	C₂	semidirect
9. C₂×C₁₃×#322	C₂	semidirect
138. C₂×#322	D₁₃	semidirect
313. C₁₃	#17982	semidirect

Maximal subgroups

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

subgroup	index
#13125406	2
#13123811	2
#13113716	2
#13113709	2
#13113702	2
#13113695	2
#13113599	2
#13113297	2
C₂×C₁₃×#322	2
#17982	13