Subgroups — #15833496

Order 3328.

614 conjugacy classes of subgroups.

The subgroup lattice has 378 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	#13113751	C₂	1	1	yes	maximal, characteristic, complemented
4	#13113746	C₂	1	1	yes	maximal, characteristic, complemented
5	#13113740	C₂	1	2	yes	maximal, complemented
6	#13113661	C₂	1	1	yes	maximal, characteristic, complemented
7	#13113656	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113650	C₂	1	2	yes	maximal, complemented
9	#13113599	C₂	1	1	yes	maximal, characteristic, complemented
10	#13113297	C₂	1	2	yes	maximal, complemented
11	C₂×C₁₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719184	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	#11718777	C₄	1	1	yes	characteristic
17	#11718776	C₄	1	1	yes	characteristic
18	#11718775	C₄	1	2	yes	—
19	#11718772	—	2	2	no	—
20	C₈×C₂²⋊D₁₃	—	2	2	no	—
21	#11718587	C₂²	1	2	yes	—
22	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
23	#11718358	C₄	1	2	yes	—
24	#11718357	C₄	1	1	yes	characteristic
25	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
26	C₄×#6095	C₂²	1	2	yes	—
27	#11718351	—	2	2	no	—
28	#11718350	—	2	2	no	—
29	#11718349	C₂²	1	2	yes	—
30	C₈×D₅₂	C₂²	1	2	yes	—
31	#11718344	C₄	1	1	yes	characteristic
32	#11718340	C₂²	1	2	yes	—
33	C₈×Dic₂₆	C₂²	1	2	yes	—
34	C₁₃×#322	C₂²	1	1	yes	characteristic
35	C₁₃×#322	C₂²	1	1	yes	characteristic
36	C₁₃×#322	C₂²	1	1	yes	characteristic
37	C₁₃×#322	C₂²	1	1	yes	characteristic
38	#11718051	C₂²	1	2	yes	—
39	#11718051	C₂²	1	2	yes	—
40	#11718050	C₂²	1	2	yes	—
41	#11718049	C₂²	1	1	yes	characteristic
42	#11718049	C₂²	1	1	yes	characteristic
43	#11718048	C₂²	1	2	yes	—
44	#11718048	C₂²	1	2	yes	—
45	C₂²×C₈×C₁₃	—	2	1	no	—
46	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₃	D₄	1	2	yes	—
49	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	#88184	C₂×C₄	1	2	yes	—
51	#88183	—	2	2	no	—
52	C₄×C₂²⋊D₁₃	—	2	2	no	—
53	#88178	C₂×C₄	1	2	yes	—
54	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
55	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
56	C₂×#6095	C₂×C₄	1	2	yes	—
57	C₂×#6094	—	2	2	no	—
58	C₂×C₈×D₁₃	—	2	2	no	—
59	#88130	—	2	2	no	—
60	#88129	C₂×C₄	1	2	yes	—
61	C₄×D₅₂	C₂³	1	2	yes	—
62	#88126	—	2	2	no	—
63	#88124	C₂×C₄	1	2	yes	—
64	C₄×Dic₂₆	C₂³	1	2	yes	—
65	C₄×C₈×C₁₃	—	2	1	no	—
66	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₁₃	D₄	1	2	yes	—
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	C₂²⋊#1055	—	2	2	no	—
76	#88061	C₂×C₄	1	2	yes	—
77	#88061	C₂×C₄	1	2	yes	—
78	#88060	—	2	2	no	—
79	#88060	—	2	2	no	—
80	#88058	C₂×C₄	1	1	yes	characteristic
81	#88058	C₂×C₄	1	1	yes	characteristic
82	#88057	C₂×C₄	1	2	yes	—
83	#88056	C₂×C₄	1	2	yes	—
84	#88056	C₂×C₄	1	2	yes	—
85	#88055	—	2	2	no	—
86	#88054	—	2	2	no	—
87	#88054	—	2	2	no	—
88	C₈×Dic₁₃	—	2	2	no	—
89	C₄⋊#1055	C₂³	1	2	yes	—
90	C₄⋊#1055	C₂³	1	2	yes	—
91	#88043	—	2	2	no	—
92	#17963	—	13	1	no	maximal, Sylow
93	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
96	C₂×C₂²⋊D₁₃	—	2	2	no	—
97	C₂×D₅₂	C₂²×C₄	1	2	yes	—
98	C₂×C₄×D₁₃	—	2	2	no	—
99	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₁₃	—	2	2	no	—
101	C₂×C₈×C₁₃	—	2	2	no	—
102	C₂×C₈×C₁₃	—	2	1	no	—
103	C₂×C₈×C₁₃	—	2	1	no	—
104	C₂×C₈×C₁₃	—	4	2	no	—
105	C₂×C₈×C₁₃	—	4	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	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
112	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₁₃	#41	1	1	yes	characteristic
115	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
116	C₂²⋊Dic₁₃	—	2	2	no	—
117	#6103	—	2	2	no	—
118	#6103	—	2	2	no	—
119	#6103	—	2	2	no	—
120	#6103	—	2	2	no	—
121	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
122	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
123	#6101	—	2	2	no	—
124	#6101	—	2	2	no	—
125	#6101	—	2	2	no	—
126	#6101	—	2	2	no	—
127	C₄×Dic₁₃	—	2	2	no	—
128	C₂×#1055	—	2	2	no	—
129	C₂×#1055	—	2	2	no	—
130	Dic₅₂	—	2	2	no	—
131	D₁₀₄	—	2	2	no	—
132	#6095	—	2	4	no	—
133	#6094	—	4	4	no	—
134	C₈×D₁₃	—	4	4	no	—
135	#3168	—	13	1	no	—
136	#2966	—	13	2	no	—
137	#1594	—	13	1	no	—
138	#1593	—	13	1	no	—
139	#1592	—	13	2	no	—
140	#1580	—	13	1	no	—
141	#1579	—	13	1	no	—
142	SD₁₆⋊C₈	—	13	2	no	—
143	#1568	—	13	1	no	—
144	#1494	—	13	2	no	—
145	C₂×#322	D₁₃	1	1	yes	characteristic, complemented
146	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
147	C₂²×D₁₃	—	2	2	no	—
148	C₂×C₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	2	1	no	—
150	C₂×C₄×C₁₃	—	2	1	no	—
151	C₂×C₄×C₁₃	—	2	2	no	—
152	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
158	C₂²⋊D₁₃	—	4	4	no	—
159	C₂×Dic₁₃	—	2	2	no	—
160	C₂×Dic₁₃	—	2	2	no	—
161	C₂×Dic₁₃	—	2	2	no	—
162	D₅₂	—	2	2	no	—
163	D₅₂	—	2	2	no	—
164	C₄×D₁₃	—	4	2	no	—
165	C₄×D₁₃	—	4	2	no	—
166	Dic₂₆	—	2	2	no	—
167	Dic₂₆	—	2	2	no	—
168	C₈×C₁₃	—	2	1	no	—
169	C₈×C₁₃	—	2	1	no	—
170	C₈×C₁₃	—	2	2	no	—
171	C₈×C₁₃	—	2	2	no	—
172	C₈×C₁₃	—	4	1	no	—
173	C₈×C₁₃	—	4	1	no	—
174	C₈×C₁₃	#131	1	2	yes	—
175	C₈×C₁₃	#131	1	2	yes	—
176	#1055	—	4	2	no	—
177	#1055	—	4	2	no	—
178	#529	—	13	2	no	—
179	#499	—	13	1	no	—
180	#495	—	13	1	no	—
181	#488	—	13	2	no	—
182	#467	—	13	1	no	—
183	#466	—	13	1	no	—
184	#465	—	13	2	no	—
185	#446	—	13	2	no	—
186	C₈×Q₈	—	13	2	no	—
187	C₄×Q₁₆	—	13	1	no	—
188	C₄×SD₁₆	—	13	2	no	—
189	C₄×D₈	—	13	1	no	—
190	#436	—	13	2	no	—
191	#435	—	26	2	no	—
192	C₈×D₄	—	13	2	no	—
193	C₈×D₄	—	26	2	no	—
194	#433	—	26	2	no	—
195	#432	—	13	2	no	—
196	#405	—	26	2	no	—
197	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
198	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
199	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
200	#335	—	13	2	no	—
201	#334	—	13	1	no	—
202	#334	—	13	1	no	—
203	Q₈⋊C₈	—	13	2	no	—
204	Q₈⋊C₈	—	13	2	no	—
205	D₄⋊C₈	—	13	2	no	—
206	D₄⋊C₈	—	13	2	no	—
207	#322	C₂×D₁₃	1	1	yes	characteristic
208	#322	C₂×D₁₃	1	1	yes	characteristic
209	#322	C₂×D₁₃	1	1	yes	characteristic
210	#322	C₂×D₁₃	1	1	yes	characteristic
211	C₂²×C₁₃	—	2	1	no	—
212	C₂²×C₁₃	—	2	1	no	—
213	C₂²×C₁₃	#444	1	2	yes	—
214	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
215	C₂×D₁₃	—	4	2	no	—
216	C₂×D₁₃	—	4	2	no	—
217	C₂×D₁₃	—	4	2	no	—
218	C₄×C₁₃	—	2	1	no	—
219	C₄×C₁₃	—	2	1	no	—
220	C₄×C₁₃	—	2	1	no	—
221	C₄×C₁₃	—	2	1	no	—
222	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
223	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
224	C₄×C₁₃	#575	1	2	yes	—
225	Dic₁₃	—	4	2	no	—
226	Dic₁₃	—	4	2	no	—
227	Dic₁₃	—	4	2	no	—
228	C₂×Q₁₆	—	13	1	no	—
229	C₂×SD₁₆	—	13	2	no	—
230	C₂×D₈	—	13	1	no	—
231	C₂×M₁₆	—	26	2	no	—
232	C₂²×C₈	—	2	1	no	—
233	C₂²×C₈	—	26	2	no	—
234	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
235	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
236	C₂²×C₈	D₅₂	1	2	yes	—
237	#126	—	26	2	no	—
238	#125	—	13	2	no	—
239	#124	—	13	2	no	—
240	#123	—	26	2	no	—
241	C₂²⋊Q₈	—	13	2	no	—
242	#121	—	13	2	no	—
243	C₄×Q₈	—	13	2	no	—
244	C₄×D₄	—	13	2	no	—
245	C₄×D₄	—	26	2	no	—
246	#117	—	26	2	no	—
247	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
248	#107	—	13	1	no	—
249	#107	—	13	1	no	—
250	#106	—	13	2	no	—
251	C₄⋊C₈	—	13	2	no	—
252	C₄⋊C₈	—	13	2	no	—
253	C₄⋊C₈	—	26	2	no	—
254	C₄⋊C₈	—	26	2	no	—
255	#103	—	13	2	no	—
256	#103	—	13	2	no	—
257	#102	—	13	2	no	—
258	#102	—	13	2	no	—
259	C₂²⋊C₈	—	26	2	no	—
260	C₂²⋊C₈	—	26	2	no	—
261	C₂²⋊C₈	—	26	2	no	—
262	#97	—	26	2	no	—
263	#97	—	26	2	no	—
264	C₄×C₈	—	2	1	no	—
265	C₄×C₈	—	26	2	no	—
266	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
267	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
268	C₄×C₈	D₅₂	1	2	yes	—
269	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
271	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
272	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
275	C₂×C₁₃	—	2	2	no	—
276	C₂×C₁₃	#2931	1	1	yes	characteristic
277	C₂×C₁₃	#2943	1	1	yes	characteristic
278	C₂×C₁₃	C₂×#444	1	1	yes	characteristic
279	D₁₃	—	8	2	no	—
280	C₂×Q₈	—	13	2	no	—
281	C₂×D₄	—	13	2	no	—
282	C₂×D₄	—	26	2	no	—
283	C₂²×C₄	—	26	2	no	—
284	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
285	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
286	C₂²×C₄	#6127	1	1	yes	characteristic
287	Q₁₆	—	26	2	no	—
288	SD₁₆	—	26	4	no	—
289	D₈	—	26	2	no	—
290	M₁₆	—	52	4	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	4	2	no	—
296	C₂×C₈	—	4	2	no	—
297	C₂×C₈	—	26	2	no	—
298	C₂×C₈	—	26	2	no	—
299	C₂×C₈	—	52	4	no	—
300	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
301	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
302	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
303	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
304	C₂×C₈	C₂×D₅₂	1	2	yes	—
305	C₂×C₈	C₂×D₅₂	1	2	yes	—
306	C₄⋊C₄	—	13	2	no	—
307	C₄⋊C₄	—	13	2	no	—
308	C₄⋊C₄	—	26	2	no	—
309	C₄⋊C₄	—	26	2	no	—
310	C₄⋊C₄	—	26	2	no	—
311	C₄⋊C₄	—	26	2	no	—
312	C₂²⋊C₄	—	26	2	no	—
313	C₂²⋊C₄	—	26	2	no	—
314	C₂²⋊C₄	—	26	2	no	—
315	C₂²⋊C₄	—	26	2	no	—
316	C₂²⋊C₄	—	26	2	no	—
317	C₄²	—	26	2	no	—
318	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
319	C₄²	C₂×D₅₂	1	1	yes	characteristic
320	C₄²	#6127	1	1	yes	characteristic
321	C₄²	C₂³×D₁₃	1	1	yes	characteristic
322	C₁₃	#17963	1	1	yes	characteristic, complemented, Sylow
323	C₂³	—	26	2	no	—
324	C₂³	C₄×D₅₂	1	1	yes	characteristic
325	Q₈	—	26	2	no	—
326	Q₈	—	26	2	no	—
327	D₄	—	26	2	no	—
328	D₄	—	26	2	no	—
329	D₄	—	52	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	26	2	no	—
335	C₂×C₄	—	26	2	no	—
336	C₂×C₄	—	26	2	no	—
337	C₂×C₄	—	52	2	no	—
338	C₂×C₄	—	52	2	no	—
339	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
340	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
341	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
342	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
343	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
344	C₂×C₄	C₂×#6127	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	2	2	no	—
348	C₈	—	2	2	no	—
349	C₈	—	4	1	no	—
350	C₈	—	4	1	no	—
351	C₈	—	52	2	no	—
352	C₈	—	52	2	no	—
353	C₈	#88155	1	2	yes	—
354	C₈	#88155	1	2	yes	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	52	2	no	—
358	C₂²	—	52	2	no	—
359	C₂²	—	52	2	no	—
360	C₂²	#11718378	1	2	yes	—
361	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	52	2	no	—
367	C₄	—	52	2	no	—
368	C₄	—	52	2	no	—
369	C₄	C₂×#88155	1	1	yes	characteristic
370	C₄	C₂×#88155	1	1	yes	characteristic
371	C₄	#11719453	1	2	yes	—
372	C₂	—	2	2	no	—
373	C₂	—	104	2	no	—
374	C₂	#13123549	1	1	yes	characteristic
375	C₂	#13123638	1	1	yes	characteristic
376	C₂	C₂×#11718378	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	#17963
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. #13113751	C₂	semidirect
4. #13113746	C₂	semidirect
5. #13113740	C₂	semidirect
6. #13113661	C₂	semidirect
7. #13113656	C₂	semidirect
8. #13113650	C₂	semidirect
9. #13113599	C₂	semidirect
10. #13113297	C₂	semidirect
11. C₂×C₁₃×#322	C₂	semidirect
145. C₂×#322	D₁₃	semidirect
322. C₁₃	#17963	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
#13113751	2
#13113746	2
#13113740	2
#13113661	2
#13113656	2
#13113650	2
#13113599	2
#13113297	2
C₂×C₁₃×#322	2
#17963	13