Subgroups — #15834932

Order 3328.

606 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	#13125406	C₂	1	1	yes	maximal, characteristic
2	#13123998	C₂	1	2	yes	maximal, complemented
3	#13113728	C₂	1	2	yes	maximal, complemented
4	#13113709	C₂	1	2	yes	maximal, complemented
5	#13113702	C₂	1	2	yes	maximal, complemented
6	C₈×#6095	C₂	1	2	yes	maximal, complemented
7	C₂×#11718049	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113250	C₂	1	2	yes	maximal, complemented
9	C₈×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#11719184	C₂²	1	2	yes	—
11	C₁₃×#431	C₂²	1	1	yes	characteristic
12	C₄×C₁₃×M₁₆	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	C₂×C₄⋊#1055	C₂²	1	2	yes	—
18	#11718501	—	2	2	no	—
19	#11718500	—	2	2	no	—
20	#11718391	—	2	2	no	—
21	#11718386	—	2	2	no	—
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	#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	C₈²×C₁₃	C₂²	1	2	yes	—
34	#11718051	C₂²	1	2	yes	—
35	#11718051	C₂²	1	2	yes	—
36	#11718049	C₂²	1	1	yes	characteristic
37	#11718049	C₂²	1	2	yes	—
38	#11718049	C₂²	1	1	yes	characteristic
39	#11718048	C₂²	1	2	yes	—
40	#11718048	C₂²	1	2	yes	—
41	C₂×C₁₃×M₁₆	—	2	1	no	—
42	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	#88184	C₂×C₄	1	2	yes	—
46	#88183	—	2	2	no	—
47	C₄×C₂²⋊D₁₃	—	2	2	no	—
48	#88178	C₂×C₄	1	2	yes	—
49	C₂²×#1055	—	2	2	no	—
50	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
51	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
52	C₂×#6095	C₂×C₄	1	2	yes	—
53	C₂×#6094	—	2	2	no	—
54	C₂×C₈×D₁₃	—	2	2	no	—
55	#88130	—	2	2	no	—
56	#88129	C₂×C₄	1	2	yes	—
57	C₄×D₅₂	C₂³	1	2	yes	—
58	#88126	—	2	2	no	—
59	#88124	C₂×C₄	1	2	yes	—
60	C₄×Dic₂₆	C₂³	1	2	yes	—
61	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
62	C₁₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
63	C₁₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
64	C₁₃×#97	—	2	1	no	—
65	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₁₃	C₂³	1	2	yes	—
68	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	2	yes	—
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	—	2	2	no	—
85	C₄⋊#1055	C₂³	1	2	yes	—
86	C₄⋊#1055	C₂³	1	2	yes	—
87	#18088	—	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	C₁₃×M₁₆	—	2	4	no	—
96	C₂×C₈×C₁₃	—	2	2	no	—
97	C₂×C₈×C₁₃	—	2	2	no	—
98	C₂×C₈×C₁₃	—	4	1	no	—
99	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
105	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₃	#41	1	1	yes	characteristic
108	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
109	C₂²⋊Dic₁₃	—	2	2	no	—
110	#6103	—	2	2	no	—
111	#6103	—	2	2	no	—
112	#6103	—	2	2	no	—
113	#6103	—	2	2	no	—
114	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
115	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
116	#6101	—	2	2	no	—
117	#6101	—	2	2	no	—
118	#6101	—	2	2	no	—
119	#6101	—	2	2	no	—
120	C₄×Dic₁₃	—	2	2	no	—
121	C₂×#1055	—	2	2	no	—
122	C₂×#1055	—	2	2	no	—
123	C₂×#1055	—	8	2	no	—
124	Dic₅₂	—	4	1	no	—
125	D₁₀₄	—	4	1	no	—
126	#6095	—	2	4	no	—
127	#6094	—	8	2	no	—
128	C₈×D₁₃	—	8	2	no	—
129	#3168	—	13	1	no	—
130	#2994	—	13	2	no	—
131	#1590	—	13	2	no	—
132	#1587	—	13	2	no	—
133	#1586	—	13	2	no	—
134	C₈×SD₁₆	—	13	2	no	—
135	C₂×#334	—	13	1	no	—
136	#1483	—	13	2	no	—
137	C₈×M₁₆	D₁₃	1	1	yes	characteristic, complemented
138	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
139	C₂²×D₁₃	—	2	2	no	—
140	C₂×C₄×C₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	—	2	1	no	—
142	C₂×C₄×C₁₃	—	2	1	no	—
143	C₂×C₄×C₁₃	—	2	2	no	—
144	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
150	C₂²⋊D₁₃	—	4	4	no	—
151	C₂×Dic₁₃	—	2	2	no	—
152	C₂×Dic₁₃	—	2	2	no	—
153	C₂×Dic₁₃	—	2	2	no	—
154	D₅₂	—	2	2	no	—
155	D₅₂	—	2	2	no	—
156	C₄×D₁₃	—	4	2	no	—
157	C₄×D₁₃	—	4	2	no	—
158	Dic₂₆	—	2	2	no	—
159	Dic₂₆	—	2	2	no	—
160	C₈×C₁₃	—	2	4	no	—
161	C₈×C₁₃	—	2	1	no	—
162	C₈×C₁₃	—	2	1	no	—
163	C₈×C₁₃	—	2	1	no	—
164	C₈×C₁₃	—	2	2	no	—
165	C₈×C₁₃	—	2	2	no	—
166	C₈×C₁₃	#131	1	2	yes	—
167	#1055	—	4	2	no	—
168	#1055	—	4	2	no	—
169	#529	—	13	2	no	—
170	#499	—	13	1	no	—
171	#495	—	13	1	no	—
172	#488	—	13	2	no	—
173	#467	—	13	1	no	—
174	#466	—	13	1	no	—
175	#465	—	13	2	no	—
176	#446	—	13	2	no	—
177	C₈×Q₈	—	13	2	no	—
178	C₄×Q₁₆	—	13	1	no	—
179	C₄×SD₁₆	—	13	2	no	—
180	C₄×D₈	—	13	1	no	—
181	#436	—	13	2	no	—
182	#435	—	26	2	no	—
183	C₈×D₄	—	13	2	no	—
184	C₈×D₄	—	26	2	no	—
185	#433	—	26	2	no	—
186	#431	C₂×D₁₃	1	1	yes	characteristic
187	#424	—	26	2	no	—
188	C₂×C₄⋊C₈	—	13	2	no	—
189	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
190	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
191	#334	—	13	1	no	—
192	#334	—	13	2	no	—
193	#334	—	13	1	no	—
194	Q₈⋊C₈	—	13	2	no	—
195	Q₈⋊C₈	—	13	2	no	—
196	D₄⋊C₈	—	13	2	no	—
197	D₄⋊C₈	—	13	2	no	—
198	#322	C₂×D₁₃	1	2	yes	—
199	C₈²	C₂×D₁₃	1	2	yes	—
200	C₂²×C₁₃	—	2	1	no	—
201	C₂²×C₁₃	—	2	1	no	—
202	C₂²×C₁₃	#443	1	2	yes	—
203	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
204	C₂×D₁₃	—	4	2	no	—
205	C₂×D₁₃	—	4	2	no	—
206	C₂×D₁₃	—	4	2	no	—
207	C₄×C₁₃	—	2	1	no	—
208	C₄×C₁₃	—	2	1	no	—
209	C₄×C₁₃	—	2	1	no	—
210	C₄×C₁₃	—	2	1	no	—
211	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
212	C₄×C₁₃	#568	1	1	yes	characteristic
213	C₄×C₁₃	#577	1	2	yes	—
214	Dic₁₃	—	4	2	no	—
215	Dic₁₃	—	4	2	no	—
216	Dic₁₃	—	4	2	no	—
217	C₂×Q₁₆	—	13	1	no	—
218	C₂×SD₁₆	—	13	2	no	—
219	C₂×D₈	—	13	1	no	—
220	C₂×M₁₆	—	2	1	no	—
221	C₂×M₁₆	—	26	2	no	—
222	C₂²×C₈	—	26	2	no	—
223	C₂²×C₈	—	26	2	no	—
224	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
225	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
226	#126	—	26	2	no	—
227	#125	—	13	2	no	—
228	#124	—	13	2	no	—
229	#123	—	26	2	no	—
230	C₂²⋊Q₈	—	13	2	no	—
231	#121	—	13	2	no	—
232	C₄×Q₈	—	13	2	no	—
233	C₄×D₄	—	13	2	no	—
234	C₄×D₄	—	26	2	no	—
235	#117	—	26	2	no	—
236	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
237	#107	—	13	1	no	—
238	#107	—	13	1	no	—
239	#106	—	13	2	no	—
240	C₄⋊C₈	—	13	2	no	—
241	C₄⋊C₈	—	13	2	no	—
242	C₄⋊C₈	—	26	2	no	—
243	C₄⋊C₈	—	26	2	no	—
244	C₄⋊C₈	—	26	2	no	—
245	C₄⋊C₈	D₅₂	1	2	yes	—
246	#103	—	13	2	no	—
247	#103	—	13	2	no	—
248	#102	—	13	2	no	—
249	#102	—	13	2	no	—
250	C₂²⋊C₈	—	26	2	no	—
251	C₂²⋊C₈	—	26	2	no	—
252	C₂²⋊C₈	D₅₂	1	1	yes	characteristic
253	C₂²⋊C₈	D₅₂	1	1	yes	characteristic
254	#97	—	2	1	no	—
255	#97	—	26	2	no	—
256	C₄×C₈	—	26	2	no	—
257	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
258	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
259	C₄×C₈	C₂²×D₁₃	1	2	yes	—
260	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
262	C₄×C₈	C₂²×D₁₃	1	2	yes	—
263	C₂×C₁₃	—	2	2	no	—
264	C₂×C₁₃	#2935	1	1	yes	characteristic
265	C₂×C₁₃	#2952	1	1	yes	characteristic
266	C₂×C₁₃	C₂×#443	1	1	yes	characteristic
267	D₁₃	—	8	2	no	—
268	C₂×Q₈	—	13	2	no	—
269	C₂×D₄	—	13	2	no	—
270	C₂×D₄	—	26	2	no	—
271	C₂²×C₄	—	26	2	no	—
272	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
274	C₂²×C₄	#6127	1	1	yes	characteristic
275	Q₁₆	—	52	1	no	—
276	SD₁₆	—	26	4	no	—
277	D₈	—	52	1	no	—
278	M₁₆	—	2	4	no	—
279	M₁₆	—	104	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	4	1	no	—
283	C₂×C₈	—	26	2	no	—
284	C₂×C₈	—	26	2	no	—
285	C₂×C₈	—	104	2	no	—
286	C₂×C₈	—	104	2	no	—
287	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
288	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
289	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
290	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
291	C₂×C₈	C₂×D₅₂	1	2	yes	—
292	C₂×C₈	C₂×D₅₂	1	2	yes	—
293	C₄⋊C₄	—	13	2	no	—
294	C₄⋊C₄	—	13	2	no	—
295	C₄⋊C₄	—	26	2	no	—
296	C₄⋊C₄	—	26	2	no	—
297	C₄⋊C₄	—	26	2	no	—
298	C₄⋊C₄	—	26	2	no	—
299	C₂²⋊C₄	—	26	2	no	—
300	C₂²⋊C₄	—	26	2	no	—
301	C₂²⋊C₄	—	26	2	no	—
302	C₂²⋊C₄	—	26	2	no	—
303	C₂²⋊C₄	—	26	2	no	—
304	C₄²	—	26	2	no	—
305	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
306	C₄²	C₂×D₅₂	1	1	yes	characteristic
307	C₄²	#6127	1	1	yes	characteristic
308	C₄²	C₂³×D₁₃	1	1	yes	characteristic
309	C₁₃	#18088	1	1	yes	characteristic, complemented, Sylow
310	C₂³	—	26	2	no	—
311	C₂³	C₄×D₅₂	1	1	yes	characteristic
312	Q₈	—	26	2	no	—
313	Q₈	—	26	2	no	—
314	D₄	—	26	2	no	—
315	D₄	—	26	2	no	—
316	D₄	—	52	4	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	26	2	no	—
322	C₂×C₄	—	26	2	no	—
323	C₂×C₄	—	26	2	no	—
324	C₂×C₄	—	52	2	no	—
325	C₂×C₄	—	52	2	no	—
326	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
327	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
328	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
329	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
330	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
331	C₂×C₄	C₂×#6127	1	1	yes	characteristic
332	C₈	—	2	4	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	2	2	no	—
337	C₈	—	2	2	no	—
338	C₈	—	52	2	no	—
339	C₈	—	52	2	no	—
340	C₈	#88161	1	2	yes	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	52	2	no	—
344	C₂²	—	52	2	no	—
345	C₂²	—	52	2	no	—
346	C₂²	#11718361	1	2	yes	—
347	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	52	2	no	—
353	C₄	—	52	2	no	—
354	C₄	—	52	2	no	—
355	C₄	C₂×#88161	1	1	yes	characteristic
356	C₄	#11719450	1	1	yes	characteristic
357	C₄	#11719456	1	2	yes	—
358	C₂	—	2	2	no	—
359	C₂	—	104	2	no	—
360	C₂	#13123582	1	1	yes	characteristic
361	C₂	#13123721	1	1	yes	characteristic
362	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	C₈×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#11718776	C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₈×M₁₆	D₁₃
hypocenter	C₁₃	#18088
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
2. #13123998	C₂	semidirect
3. #13113728	C₂	semidirect
4. #13113709	C₂	semidirect
5. #13113702	C₂	semidirect
6. C₈×#6095	C₂	semidirect
7. C₂×#11718049	C₂	semidirect
8. #13113250	C₂	semidirect
9. C₈×C₁₃×M₁₆	C₂	semidirect
137. C₈×M₁₆	D₁₃	semidirect
309. C₁₃	#18088	semidirect

Maximal subgroups

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

subgroup	index
#13125406	2
#13123998	2
#13113728	2
#13113709	2
#13113702	2
C₈×#6095	2
C₂×#11718049	2
#13113250	2
C₈×C₁₃×M₁₆	2
#18088	13