Subgroups — #15834708

Order 3328.

600 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	#13125406	C₂	1	1	yes	maximal, characteristic
2	#13123840	C₂	1	2	yes	maximal, complemented
3	#13113734	C₂	1	1	yes	maximal, characteristic, complemented
4	#13113722	C₂	1	1	yes	maximal, characteristic, complemented
5	#13113681	C₂	1	2	yes	maximal, complemented
6	#13113674	C₂	1	2	yes	maximal, complemented
7	C₈×Dic₅₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
9	#13113584	C₂	1	1	yes	maximal, characteristic, complemented
10	#13113245	C₂	1	2	yes	maximal, complemented
11	C₈×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#11719184	C₂²	1	2	yes	—
13	C₁₃×#431	C₂²	1	1	yes	characteristic
14	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	#11718789	—	2	2	no	—
17	#11718788	—	2	2	no	—
18	#11718777	C₄	1	1	yes	characteristic
19	#11718776	C₄	1	1	yes	characteristic
20	#11718775	C₄	1	2	yes	—
21	#11718586	C₂²	1	2	yes	—
22	#11718370	—	2	2	no	—
23	#11718369	—	2	2	no	—
24	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
25	#11718358	C₄	1	2	yes	—
26	#11718357	C₄	1	1	yes	characteristic
27	C₄×D₁₀₄	C₂²	1	1	yes	characteristic
28	C₄×#6095	C₂²	1	2	yes	—
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	C₈²×C₁₃	C₂²	1	1	yes	characteristic
37	C₈²×C₁₃	C₂²	1	1	yes	characteristic
38	#11718051	C₂²	1	2	yes	—
39	#11718051	C₂²	1	2	yes	—
40	#11718050	C₂²	1	1	yes	characteristic
41	#11718050	C₂²	1	1	yes	characteristic
42	#11718049	C₂²	1	2	yes	—
43	#11718048	C₂²	1	2	yes	—
44	#11718048	C₂²	1	2	yes	—
45	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
46	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
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	C₁₃×C₄⋊C₈	—	2	1	no	—
66	C₁₃×C₂²⋊C₈	—	2	1	no	—
67	C₁₃×#97	D₄	1	1	yes	characteristic
68	C₁₃×#97	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂×C₄	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	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
76	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
77	C₂²⋊#1055	—	2	2	no	—
78	#88061	C₂×C₄	1	2	yes	—
79	#88061	C₂×C₄	1	2	yes	—
80	#88060	—	2	2	no	—
81	#88060	—	2	2	no	—
82	#88058	C₂×C₄	1	1	yes	characteristic
83	#88058	C₂×C₄	1	1	yes	characteristic
84	#88057	C₂×C₄	1	2	yes	—
85	#88056	C₂×C₄	1	2	yes	—
86	#88056	C₂×C₄	1	2	yes	—
87	#88055	—	2	2	no	—
88	#88054	—	2	2	no	—
89	#88054	—	2	2	no	—
90	C₈×Dic₁₃	—	2	2	no	—
91	C₄⋊#1055	C₂³	1	2	yes	—
92	C₄⋊#1055	C₂³	1	2	yes	—
93	C₄×#1055	—	2	2	no	—
94	#18069	—	13	1	no	maximal, Sylow
95	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
98	C₂×C₂²⋊D₁₃	—	2	2	no	—
99	C₂×D₅₂	C₂²×C₄	1	2	yes	—
100	C₂×C₄×D₁₃	—	2	2	no	—
101	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
102	C₁₃×M₁₆	—	2	2	no	—
103	C₁₃×M₁₆	—	2	2	no	—
104	C₂×C₈×C₁₃	—	2	2	no	—
105	C₂×C₈×C₁₃	—	2	1	no	—
106	C₂×C₈×C₁₃	—	2	1	no	—
107	C₂×C₈×C₁₃	—	4	1	no	—
108	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₁₃	#41	1	1	yes	characteristic
119	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
120	C₂²⋊Dic₁₃	—	2	2	no	—
121	#6103	—	2	2	no	—
122	#6103	—	2	2	no	—
123	#6103	—	2	2	no	—
124	#6103	—	2	2	no	—
125	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
126	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
127	#6101	—	2	2	no	—
128	#6101	—	2	2	no	—
129	#6101	—	2	2	no	—
130	#6101	—	2	2	no	—
131	C₄×Dic₁₃	—	2	2	no	—
132	C₂×#1055	—	2	2	no	—
133	C₂×#1055	—	2	2	no	—
134	Dic₅₂	—	2	2	no	—
135	D₁₀₄	—	2	2	no	—
136	#6095	—	4	2	no	—
137	#6094	—	4	4	no	—
138	C₈×D₁₃	—	4	4	no	—
139	#3168	—	13	1	no	—
140	#2969	—	13	2	no	—
141	#1591	—	13	1	no	—
142	#1589	—	13	1	no	—
143	#1583	—	13	2	no	—
144	#1582	—	13	2	no	—
145	C₈×Q₁₆	—	13	1	no	—
146	C₈×D₈	—	13	1	no	—
147	#1564	—	13	1	no	—
148	#1482	—	13	2	no	—
149	C₈×M₁₆	D₁₃	1	1	yes	characteristic, complemented
150	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
151	C₂²×D₁₃	—	2	2	no	—
152	C₂×C₄×C₁₃	—	2	1	no	—
153	C₂×C₄×C₁₃	—	2	1	no	—
154	C₂×C₄×C₁₃	—	2	1	no	—
155	C₂×C₄×C₁₃	—	2	1	no	—
156	C₂×C₄×C₁₃	—	4	1	no	—
157	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
161	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
163	C₂²⋊D₁₃	—	8	2	no	—
164	C₂×Dic₁₃	—	2	2	no	—
165	C₂×Dic₁₃	—	2	2	no	—
166	C₂×Dic₁₃	—	2	2	no	—
167	D₅₂	—	2	2	no	—
168	D₅₂	—	2	2	no	—
169	C₄×D₁₃	—	4	2	no	—
170	C₄×D₁₃	—	4	2	no	—
171	Dic₂₆	—	2	2	no	—
172	Dic₂₆	—	2	2	no	—
173	C₈×C₁₃	—	2	1	no	—
174	C₈×C₁₃	—	2	1	no	—
175	C₈×C₁₃	—	2	1	no	—
176	C₈×C₁₃	—	2	1	no	—
177	C₈×C₁₃	—	2	1	no	—
178	C₈×C₁₃	—	4	1	no	—
179	C₈×C₁₃	—	4	1	no	—
180	C₈×C₁₃	#131	1	2	yes	—
181	C₈×C₁₃	#135	1	2	yes	—
182	C₈×C₁₃	#135	1	2	yes	—
183	#1055	—	4	2	no	—
184	#1055	—	4	2	no	—
185	#529	—	13	2	no	—
186	#499	—	13	1	no	—
187	#495	—	13	1	no	—
188	#488	—	13	2	no	—
189	#467	—	13	1	no	—
190	#466	—	13	1	no	—
191	#465	—	13	2	no	—
192	#446	—	13	2	no	—
193	C₈×Q₈	—	13	2	no	—
194	C₄×Q₁₆	—	13	1	no	—
195	C₄×SD₁₆	—	13	2	no	—
196	C₄×D₈	—	13	1	no	—
197	#436	—	13	2	no	—
198	#436	—	26	2	no	—
199	#435	—	26	2	no	—
200	C₈×D₄	—	13	2	no	—
201	#433	—	26	2	no	—
202	#431	—	13	2	no	—
203	#431	C₂×D₁₃	1	1	yes	characteristic
204	#405	—	26	2	no	—
205	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
206	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
207	#335	—	13	1	no	—
208	#335	—	13	1	no	—
209	#334	—	13	2	no	—
210	Q₈⋊C₈	—	13	2	no	—
211	Q₈⋊C₈	—	13	2	no	—
212	D₄⋊C₈	—	13	2	no	—
213	D₄⋊C₈	—	13	2	no	—
214	#322	C₂×D₁₃	1	1	yes	characteristic
215	#322	C₂×D₁₃	1	1	yes	characteristic
216	C₈²	C₂×D₁₃	1	1	yes	characteristic
217	C₈²	C₂×D₁₃	1	1	yes	characteristic
218	C₂²×C₁₃	—	2	1	no	—
219	C₂²×C₁₃	—	2	1	no	—
220	C₂²×C₁₃	—	2	1	no	—
221	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
222	C₂×D₁₃	—	4	2	no	—
223	C₂×D₁₃	—	4	2	no	—
224	C₂×D₁₃	—	4	2	no	—
225	C₄×C₁₃	—	2	1	no	—
226	C₄×C₁₃	—	2	1	no	—
227	C₄×C₁₃	—	2	1	no	—
228	C₄×C₁₃	—	2	1	no	—
229	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
230	C₄×C₁₃	#568	1	1	yes	characteristic
231	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
232	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
233	Dic₁₃	—	4	2	no	—
234	Dic₁₃	—	4	2	no	—
235	Dic₁₃	—	4	2	no	—
236	C₂×Q₁₆	—	13	1	no	—
237	C₂×SD₁₆	—	13	2	no	—
238	C₂×D₈	—	13	1	no	—
239	C₂×M₁₆	—	26	2	no	—
240	C₂×M₁₆	D₅₂	1	1	yes	characteristic
241	C₂×M₁₆	D₅₂	1	1	yes	characteristic
242	C₂²×C₈	—	26	2	no	—
243	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
244	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
245	#126	—	26	2	no	—
246	#125	—	13	2	no	—
247	#124	—	13	2	no	—
248	#123	—	26	2	no	—
249	C₂²⋊Q₈	—	13	2	no	—
250	#121	—	13	2	no	—
251	C₄×Q₈	—	13	2	no	—
252	C₄×D₄	—	13	2	no	—
253	C₄×D₄	—	26	2	no	—
254	#117	—	26	2	no	—
255	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
256	#107	—	13	1	no	—
257	#107	—	13	1	no	—
258	#106	—	13	2	no	—
259	C₄⋊C₈	—	2	1	no	—
260	C₄⋊C₈	—	13	2	no	—
261	C₄⋊C₈	—	13	2	no	—
262	C₄⋊C₈	—	26	2	no	—
263	C₄⋊C₈	—	26	2	no	—
264	#103	—	13	2	no	—
265	#103	—	13	2	no	—
266	#102	—	13	2	no	—
267	#102	—	13	2	no	—
268	C₂²⋊C₈	—	2	1	no	—
269	C₂²⋊C₈	—	26	2	no	—
270	C₂²⋊C₈	—	26	2	no	—
271	C₂²⋊C₈	—	26	2	no	—
272	#97	—	26	2	no	—
273	#97	D₅₂	1	1	yes	characteristic
274	#97	D₅₂	1	1	yes	characteristic
275	C₄×C₈	—	26	2	no	—
276	C₄×C₈	—	26	2	no	—
277	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
278	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
281	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
282	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
283	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
285	C₂×C₁₃	—	4	1	no	—
286	C₂×C₁₃	#2935	1	1	yes	characteristic
287	C₂×C₁₃	C₄×#135	1	1	yes	characteristic
288	C₂×C₁₃	#2957	1	1	yes	characteristic
289	D₁₃	—	8	2	no	—
290	C₂×Q₈	—	13	2	no	—
291	C₂×D₄	—	13	2	no	—
292	C₂×D₄	—	26	2	no	—
293	C₂²×C₄	—	26	2	no	—
294	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
295	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
296	C₂²×C₄	#6127	1	1	yes	characteristic
297	Q₁₆	—	26	2	no	—
298	SD₁₆	—	52	2	no	—
299	D₈	—	26	2	no	—
300	M₁₆	—	2	2	no	—
301	M₁₆	—	2	2	no	—
302	M₁₆	—	52	4	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	4	1	no	—
307	C₂×C₈	—	26	2	no	—
308	C₂×C₈	—	26	2	no	—
309	C₂×C₈	—	52	4	no	—
310	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
311	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
312	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
313	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
314	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
315	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
316	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
317	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
318	C₄⋊C₄	—	13	2	no	—
319	C₄⋊C₄	—	13	2	no	—
320	C₄⋊C₄	—	26	2	no	—
321	C₄⋊C₄	—	26	2	no	—
322	C₄⋊C₄	—	26	2	no	—
323	C₄⋊C₄	—	26	2	no	—
324	C₂²⋊C₄	—	26	2	no	—
325	C₂²⋊C₄	—	26	2	no	—
326	C₂²⋊C₄	—	26	2	no	—
327	C₂²⋊C₄	—	26	2	no	—
328	C₂²⋊C₄	—	26	2	no	—
329	C₄²	—	26	2	no	—
330	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
331	C₄²	C₂×D₅₂	1	1	yes	characteristic
332	C₄²	#6127	1	1	yes	characteristic
333	C₄²	C₂³×D₁₃	1	1	yes	characteristic
334	C₁₃	#18069	1	1	yes	characteristic, complemented, Sylow
335	C₂³	—	26	2	no	—
336	C₂³	C₄×D₅₂	1	1	yes	characteristic
337	Q₈	—	26	2	no	—
338	Q₈	—	26	2	no	—
339	D₄	—	26	2	no	—
340	D₄	—	26	2	no	—
341	D₄	—	104	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	26	2	no	—
348	C₂×C₄	—	26	2	no	—
349	C₂×C₄	—	26	2	no	—
350	C₂×C₄	—	52	2	no	—
351	C₂×C₄	—	52	2	no	—
352	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
353	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
354	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
355	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
356	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
357	C₂×C₄	C₂×#6127	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	—	2	1	no	—
363	C₈	—	4	1	no	—
364	C₈	—	4	1	no	—
365	C₈	—	52	2	no	—
366	C₈	—	52	2	no	—
367	C₈	#88158	1	2	yes	—
368	C₈	#88158	1	2	yes	—
369	C₈	#88161	1	2	yes	—
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	—	52	2	no	—
374	C₂²	—	52	2	no	—
375	C₂²	—	52	2	no	—
376	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	52	2	no	—
382	C₄	—	52	2	no	—
383	C₄	—	52	2	no	—
384	C₄	C₂×#88158	1	1	yes	characteristic
385	C₄	C₂×#88158	1	1	yes	characteristic
386	C₄	C₂×#88161	1	1	yes	characteristic
387	C₄	#11719450	1	1	yes	characteristic
388	C₂	—	4	1	no	—
389	C₂	—	104	2	no	—
390	C₂	#13123582	1	1	yes	characteristic
391	C₂	C₄×#88158	1	1	yes	characteristic
392	C₂	#13123754	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₈×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#11718776	C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₈×M₁₆	D₁₃
hypocenter	C₁₃	#18069
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
2. #13123840	C₂	semidirect
3. #13113734	C₂	semidirect
4. #13113722	C₂	semidirect
5. #13113681	C₂	semidirect
6. #13113674	C₂	semidirect
7. C₈×Dic₅₂	C₂	semidirect
8. C₈×D₁₀₄	C₂	semidirect
9. #13113584	C₂	semidirect
10. #13113245	C₂	semidirect
11. C₈×C₁₃×M₁₆	C₂	semidirect
149. C₈×M₁₆	D₁₃	semidirect
334. C₁₃	#18069	semidirect

Maximal subgroups

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

subgroup	index
#13125406	2
#13123840	2
#13113734	2
#13113722	2
#13113681	2
#13113674	2
C₈×Dic₅₂	2
C₈×D₁₀₄	2
#13113584	2
#13113245	2
C₈×C₁₃×M₁₆	2
#18069	13