Subgroups — #15834751

Order 3328.

592 conjugacy classes of subgroups.

The subgroup lattice has 368 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	#13123834	C₂	1	2	yes	maximal, complemented
3	#13113737	C₂	1	2	yes	maximal, complemented
4	#13113706	C₂	1	2	yes	maximal, complemented
5	#13113699	C₂	1	2	yes	maximal, complemented
6	#13113647	C₂	1	2	yes	maximal, complemented
7	#13113242	C₂	1	2	yes	maximal, complemented
8	#13113125	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	#11719184	C₂²	1	2	yes	—
11	C₁₃×#431	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	#11718777	C₄	1	1	yes	characteristic
14	#11718776	C₄	1	1	yes	characteristic
15	#11718775	C₄	1	2	yes	—
16	#11718586	C₂²	1	1	yes	characteristic
17	C₄×#6099	C₂²	1	1	yes	characteristic
18	#11718499	C₂²	1	2	yes	—
19	#11718495	C₂²	1	2	yes	—
20	#11718492	C₂²	1	2	yes	—
21	#11718491	C₂²	1	2	yes	—
22	#11718390	—	2	2	no	—
23	#11718389	—	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	#11718351	—	2	2	no	—
30	#11718350	—	2	2	no	—
31	#11718344	C₄	1	1	yes	characteristic
32	C₁₃×#335	C₂²	1	1	yes	characteristic
33	C₁₃×#335	C₂²	1	1	yes	characteristic
34	C₁₃×#334	C₂²	1	2	yes	—
35	#11718076	C₂²	1	2	yes	—
36	#11718076	C₂²	1	2	yes	—
37	#11718075	C₂²	1	2	yes	—
38	#11718075	C₂²	1	2	yes	—
39	#11718047	C₂²	1	2	yes	—
40	#11718046	C₂²	1	2	yes	—
41	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	#88184	C₂×C₄	1	2	yes	—
45	#88183	—	2	2	no	—
46	C₄×C₂²⋊D₁₃	—	2	2	no	—
47	#88178	C₂×C₄	1	2	yes	—
48	C₂×#6099	D₄	1	1	yes	characteristic
49	C₂×#6099	D₄	1	1	yes	characteristic
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	C₂³	1	2	yes	—
62	C₁₃×C₄⋊C₈	C₂³	1	2	yes	—
63	C₁₃×C₂²⋊C₈	—	2	2	no	—
64	C₄×C₈×C₁₃	—	2	2	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	1	yes	characteristic
68	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	C₂²⋊#1055	—	2	1	no	—
70	#88061	C₂×C₄	1	2	yes	—
71	#88061	C₂×C₄	1	2	yes	—
72	#88060	—	2	2	no	—
73	#88060	—	2	2	no	—
74	#88058	C₂×C₄	1	1	yes	characteristic
75	#88058	C₂×C₄	1	1	yes	characteristic
76	#88057	C₂×C₄	1	2	yes	—
77	#88056	C₂×C₄	1	2	yes	—
78	#88056	C₂×C₄	1	2	yes	—
79	#88055	—	2	2	no	—
80	#88054	—	2	2	no	—
81	#88054	—	2	2	no	—
82	C₈×Dic₁₃	—	2	2	no	—
83	C₄⋊#1055	—	2	1	no	—
84	#88043	D₄	1	2	yes	—
85	C₄×#1055	C₂³	1	2	yes	—
86	C₄×#1055	C₂³	1	2	yes	—
87	#18074	—	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	2	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	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₃	#41	1	1	yes	characteristic
106	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
107	C₂²⋊Dic₁₃	—	2	2	no	—
108	#6103	—	2	2	no	—
109	#6103	—	2	2	no	—
110	#6103	—	2	2	no	—
111	#6103	—	2	2	no	—
112	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
113	C₄⋊Dic₁₃	C₂²×C₄	1	2	yes	—
114	#6101	—	2	2	no	—
115	#6101	—	2	2	no	—
116	#6101	—	2	2	no	—
117	#6101	—	2	2	no	—
118	C₄×Dic₁₃	—	2	2	no	—
119	#6099	—	4	1	no	—
120	#6099	—	4	1	no	—
121	C₂×#1055	—	2	2	no	—
122	C₂×#1055	C₂×D₄	1	2	yes	—
123	C₂×#1055	C₂×D₄	1	2	yes	—
124	Dic₅₂	—	4	1	no	—
125	D₁₀₄	—	4	1	no	—
126	#6095	—	2	4	no	—
127	#6094	—	4	4	no	—
128	C₈×D₁₃	—	4	4	no	—
129	#3168	—	13	1	no	—
130	#2969	—	13	2	no	—
131	#1592	—	13	2	no	—
132	#1587	—	13	2	no	—
133	#1586	—	13	2	no	—
134	SD₁₆⋊C₈	—	13	2	no	—
135	#1564	D₁₃	1	1	yes	characteristic, complemented
136	#1482	—	13	2	no	—
137	#1451	—	13	1	no	—
138	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
139	C₂²×D₁₃	—	2	2	no	—
140	C₂×C₄×C₁₃	—	2	1	no	—
141	C₂×C₄×C₁₃	—	2	1	no	—
142	C₂×C₄×C₁₃	—	2	1	no	—
143	C₂×C₄×C₁₃	—	2	1	no	—
144	C₂×C₄×C₁₃	—	4	1	no	—
145	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
151	C₂²⋊D₁₃	—	8	2	no	—
152	C₂×Dic₁₃	—	2	2	no	—
153	C₂×Dic₁₃	—	2	2	no	—
154	C₂×Dic₁₃	—	2	2	no	—
155	D₅₂	—	2	2	no	—
156	D₅₂	—	2	2	no	—
157	C₄×D₁₃	—	4	2	no	—
158	C₄×D₁₃	—	4	2	no	—
159	Dic₂₆	—	2	2	no	—
160	Dic₂₆	—	2	2	no	—
161	C₈×C₁₃	—	2	1	no	—
162	C₈×C₁₃	—	2	1	no	—
163	C₈×C₁₃	—	2	1	no	—
164	C₈×C₁₃	—	4	2	no	—
165	C₈×C₁₃	—	4	2	no	—
166	C₈×C₁₃	#131	1	2	yes	—
167	#1055	—	2	2	no	—
168	#1055	—	2	2	no	—
169	#1055	—	4	2	no	—
170	#529	—	13	2	no	—
171	#499	—	13	1	no	—
172	#495	—	13	1	no	—
173	#488	—	13	2	no	—
174	#467	—	13	1	no	—
175	#466	—	13	1	no	—
176	#465	—	13	2	no	—
177	#446	—	13	2	no	—
178	C₈×Q₈	—	13	2	no	—
179	C₄×Q₁₆	—	13	1	no	—
180	C₄×SD₁₆	—	13	2	no	—
181	C₄×D₈	—	13	1	no	—
182	#436	—	13	2	no	—
183	#436	—	26	2	no	—
184	#435	—	26	2	no	—
185	C₈×D₄	—	13	2	no	—
186	#433	—	26	2	no	—
187	#431	—	13	1	no	—
188	#431	C₂×D₁₃	1	2	yes	—
189	#405	—	26	2	no	—
190	C₄×M₁₆	—	13	1	no	—
191	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
192	#335	C₂×D₁₃	1	1	yes	characteristic
193	#335	C₂×D₁₃	1	1	yes	characteristic
194	#334	C₂×D₁₃	1	2	yes	—
195	Q₈⋊C₈	—	13	2	no	—
196	Q₈⋊C₈	—	13	2	no	—
197	D₄⋊C₈	—	13	2	no	—
198	D₄⋊C₈	—	13	2	no	—
199	#322	—	13	2	no	—
200	#322	—	13	2	no	—
201	C₂²×C₁₃	—	2	1	no	—
202	C₂²×C₁₃	—	2	1	no	—
203	C₂²×C₁₃	—	2	1	no	—
204	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
205	C₂×D₁₃	—	4	2	no	—
206	C₂×D₁₃	—	4	2	no	—
207	C₂×D₁₃	—	4	2	no	—
208	C₄×C₁₃	—	2	1	no	—
209	C₄×C₁₃	—	2	1	no	—
210	C₄×C₁₃	—	2	1	no	—
211	C₄×C₁₃	—	2	1	no	—
212	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
213	C₄×C₁₃	#568	1	1	yes	characteristic
214	C₄×C₁₃	#575	1	1	yes	characteristic
215	C₄×C₁₃	#575	1	1	yes	characteristic
216	Dic₁₃	—	4	2	no	—
217	Dic₁₃	—	4	2	no	—
218	Dic₁₃	—	4	2	no	—
219	C₂×Q₁₆	—	13	1	no	—
220	C₂×SD₁₆	—	13	2	no	—
221	C₂×D₈	—	13	1	no	—
222	C₂×M₁₆	—	13	1	no	—
223	C₂×M₁₆	—	13	1	no	—
224	C₂×M₁₆	—	26	2	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	#126	—	26	2	no	—
229	#125	—	13	2	no	—
230	#124	—	13	2	no	—
231	#123	—	26	2	no	—
232	C₂²⋊Q₈	—	13	2	no	—
233	#121	—	13	2	no	—
234	C₄×Q₈	—	13	2	no	—
235	C₄×D₄	—	13	2	no	—
236	C₄×D₄	—	26	2	no	—
237	#117	—	26	2	no	—
238	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
239	#107	—	13	1	no	—
240	#107	—	13	1	no	—
241	#106	—	13	2	no	—
242	C₄⋊C₈	—	26	2	no	—
243	C₄⋊C₈	—	26	2	no	—
244	C₄⋊C₈	—	26	1	no	—
245	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
246	C₄⋊C₈	C₂²×D₁₃	1	2	yes	—
247	#103	—	13	2	no	—
248	#103	—	13	2	no	—
249	#102	—	13	2	no	—
250	#102	—	13	2	no	—
251	C₂²⋊C₈	—	2	2	no	—
252	C₂²⋊C₈	—	26	2	no	—
253	C₂²⋊C₈	—	26	2	no	—
254	C₂²⋊C₈	—	26	1	no	—
255	#97	—	13	2	no	—
256	#97	—	26	2	no	—
257	C₄×C₈	—	2	2	no	—
258	C₄×C₈	—	13	2	no	—
259	C₄×C₈	—	13	2	no	—
260	C₄×C₈	—	26	2	no	—
261	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
262	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
265	C₂×C₁₃	—	4	1	no	—
266	C₂×C₁₃	#2935	1	1	yes	characteristic
267	C₂×C₁₃	#2943	1	1	yes	characteristic
268	C₂×C₁₃	#2957	1	1	yes	characteristic
269	D₁₃	—	8	2	no	—
270	C₂×Q₈	—	13	2	no	—
271	C₂×D₄	—	13	2	no	—
272	C₂×D₄	—	26	2	no	—
273	C₂²×C₄	—	26	2	no	—
274	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
275	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
276	C₂²×C₄	#6131	1	1	yes	characteristic
277	Q₁₆	—	52	1	no	—
278	SD₁₆	—	26	4	no	—
279	D₈	—	52	1	no	—
280	M₁₆	—	52	4	no	—
281	M₁₆	—	52	1	no	—
282	M₁₆	—	52	1	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	4	1	no	—
287	C₂×C₈	—	13	2	no	—
288	C₂×C₈	—	13	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	C₄⋊C₄	—	13	2	no	—
296	C₄⋊C₄	—	13	2	no	—
297	C₄⋊C₄	—	26	2	no	—
298	C₄⋊C₄	—	26	2	no	—
299	C₄⋊C₄	—	26	2	no	—
300	C₄⋊C₄	—	26	2	no	—
301	C₂²⋊C₄	—	26	2	no	—
302	C₂²⋊C₄	—	26	2	no	—
303	C₂²⋊C₄	—	26	2	no	—
304	C₂²⋊C₄	—	26	2	no	—
305	C₂²⋊C₄	—	26	2	no	—
306	C₄²	—	26	2	no	—
307	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
308	C₄²	D₄×D₁₃	1	1	yes	characteristic
309	C₄²	#6131	1	1	yes	characteristic
310	C₄²	C₂³×D₁₃	1	1	yes	characteristic
311	C₁₃	#18074	1	1	yes	characteristic, complemented, Sylow
312	C₂³	—	26	2	no	—
313	C₂³	#88147	1	1	yes	characteristic
314	Q₈	—	26	2	no	—
315	Q₈	—	26	2	no	—
316	D₄	—	26	2	no	—
317	D₄	—	26	2	no	—
318	D₄	—	104	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	26	2	no	—
325	C₂×C₄	—	26	2	no	—
326	C₂×C₄	—	26	2	no	—
327	C₂×C₄	—	52	2	no	—
328	C₂×C₄	—	52	2	no	—
329	C₂×C₄	#88147	1	1	yes	characteristic
330	C₂×C₄	#88147	1	1	yes	characteristic
331	C₂×C₄	#88147	1	1	yes	characteristic
332	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
334	C₂×C₄	C₂×#6131	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	4	2	no	—
339	C₈	—	4	2	no	—
340	C₈	—	26	2	no	—
341	C₈	—	26	2	no	—
342	C₈	—	52	2	no	—
343	C₈	#88155	1	2	yes	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	52	2	no	—
348	C₂²	—	52	2	no	—
349	C₂²	—	52	2	no	—
350	C₂²	C₂×#88147	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	52	2	no	—
356	C₄	—	52	2	no	—
357	C₄	—	52	2	no	—
358	C₄	C₂×#88155	1	1	yes	characteristic
359	C₄	#11719450	1	1	yes	characteristic
360	C₄	#11719473	1	1	yes	characteristic
361	C₄	#11719473	1	1	yes	characteristic
362	C₂	—	4	1	no	—
363	C₂	—	104	2	no	—
364	C₂	#13123579	1	1	yes	characteristic
365	C₂	#13123636	1	1	yes	characteristic
366	C₂	#13123752	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1564	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#11718776	C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1564	D₁₃
hypocenter	C₁₃	#18074
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
2. #13123834	C₂	semidirect
3. #13113737	C₂	semidirect
4. #13113706	C₂	semidirect
5. #13113699	C₂	semidirect
6. #13113647	C₂	semidirect
7. #13113242	C₂	semidirect
8. #13113125	C₂	semidirect
9. C₁₃×#1564	C₂	semidirect
135. #1564	D₁₃	semidirect
311. C₁₃	#18074	semidirect

Maximal subgroups

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

subgroup	index
#13125406	2
#13123834	2
#13113737	2
#13113706	2
#13113699	2
#13113647	2
#13113242	2
#13113125	2
C₁₃×#1564	2
#18074	13