Subgroups — #16015535

Order 3328.

806 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718537	C₂	1	2	yes	maximal
2	C₂×#11718535	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718539	C₂	1	1	yes	maximal, characteristic
4	#13120958	C₂	1	2	yes	maximal
5	#13115003	C₂	1	2	yes	maximal
6	#13115002	C₂	1	2	yes	maximal
7	#13115000	C₂	1	1	yes	maximal, characteristic
8	#13114999	C₂	1	1	yes	maximal, characteristic
9	#13114994	C₂	1	1	yes	maximal, characteristic
10	#13114522	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#1845	C₂	1	1	yes	maximal, characteristic
12	C₂×#88144	C₂²	1	2	yes	—
13	C₂×#88142	C₂²	1	2	yes	—
14	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#107	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#384	C₂²	1	2	yes	—
19	C₂×#88058	C₂²	1	1	yes	characteristic
20	C₂×#88058	C₂²	1	1	yes	characteristic
21	C₂×#88057	C₂²	1	2	yes	—
22	C₂×#88055	C₂²	1	1	yes	characteristic
23	C₂×#88055	C₂²	1	1	yes	characteristic
24	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
25	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
26	#11718645	—	2	4	no	—
27	#11718640	—	2	4	no	—
28	#11718638	C₂²	1	2	yes	—
29	#11718637	C₂²	1	2	yes	—
30	C₂×#88048	C₂²	1	2	yes	—
31	C₂×#88048	C₂²	1	2	yes	—
32	C₂×#88047	C₂²	1	2	yes	—
33	C₂×#88047	C₂²	1	2	yes	—
34	#11718597	C₂²	1	2	yes	—
35	#11718593	C₂²	1	1	yes	characteristic
36	C₂×#88043	C₂²	1	1	yes	characteristic
37	#11718539	—	2	4	no	—
38	#11718537	—	2	8	no	—
39	#11718535	—	2	4	no	—
40	C₂²×Dic₂₆	—	2	2	no	—
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
44	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
45	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₄⋊Dic₁₃	—	2	2	no	—
48	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
49	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
50	C₂×#6101	—	2	2	no	—
51	C₂×#6101	—	2	2	no	—
52	C₂×#6101	—	2	4	no	—
53	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
55	C₂²×#1055	C₂³	1	1	yes	characteristic
56	C₂²×#1055	C₂³	1	1	yes	characteristic
57	#88144	—	2	4	no	—
58	#88144	—	2	4	no	—
59	#88142	—	2	4	no	—
60	#88142	—	2	4	no	—
61	C₁₃×#107	—	2	2	no	—
62	C₁₃×#107	—	2	4	no	—
63	C₁₃×#107	—	2	2	no	—
64	C₄×C₈×C₁₃	Q₈	1	4	yes	complemented
65	#88071	—	2	4	no	—
66	#88071	—	2	4	no	—
67	#88071	D₄	1	2	yes	—
68	#88071	D₄	1	2	yes	—
69	#88058	—	2	2	no	—
70	#88058	—	2	2	no	—
71	#88057	—	2	4	no	—
72	#88055	D₄	1	4	yes	—
73	#88055	Q₈	1	4	yes	complemented
74	C₈×Dic₁₃	D₄	1	4	yes	—
75	C₈×Dic₁₃	Q₈	1	4	yes	complemented
76	#88048	—	2	4	no	—
77	#88048	—	2	4	no	—
78	#88047	—	2	4	no	—
79	#88047	—	2	4	no	—
80	#88043	Q₈	1	4	yes	complemented
81	#33319	—	13	1	no	maximal, Sylow
82	C₂²×C₄×C₁₃	—	2	2	no	—
83	C₂²×C₄×C₁₃	—	2	2	no	—
84	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
86	C₂²×Dic₁₃	—	2	2	no	—
87	C₂²×Dic₁₃	—	2	2	no	—
88	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
91	C₂×Dic₂₆	—	4	4	no	—
92	C₂×Dic₂₆	—	8	4	no	—
93	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
96	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
97	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
98	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	—	2	2	no	—
102	C₁₃×C₄⋊C₄	—	2	2	no	—
103	C₁₃×C₄⋊C₄	—	8	4	no	—
104	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
105	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
106	C₄⋊Dic₁₃	—	2	2	no	—
107	C₄⋊Dic₁₃	—	2	2	no	—
108	C₄⋊Dic₁₃	—	2	2	no	—
109	C₄⋊Dic₁₃	—	2	2	no	—
110	C₄⋊Dic₁₃	—	4	4	no	—
111	#6101	—	4	4	no	—
112	#6101	—	4	4	no	—
113	#6101	—	8	4	no	—
114	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
115	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
116	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
117	C₄×Dic₁₃	C₂×Q₈	1	2	yes	—
118	C₂×#1055	C₂×D₄	1	2	yes	—
119	C₂×#1055	C₂×D₄	1	2	yes	—
120	C₂×#1055	C₂×Q₈	1	2	yes	—
121	C₂×#1055	C₂×Q₈	1	2	yes	—
122	C₂×#1055	C₂×Q₈	1	2	yes	—
123	C₂×#1055	C₂×Q₈	1	2	yes	—
124	C₂×#501	—	13	2	no	—
125	C₂×#500	—	13	1	no	—
126	C₂×#499	—	13	1	no	—
127	#2660	—	13	2	no	—
128	#1847	—	13	2	no	—
129	#1847	—	13	2	no	—
130	#1846	—	13	1	no	—
131	#1846	—	13	1	no	—
132	#1845	—	13	1	no	—
133	#1845	D₁₃	1	1	yes	characteristic
134	#1726	—	13	1	no	—
135	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	4	4	no	—
141	C₂×C₄×C₁₃	—	4	4	no	—
142	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
143	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
149	C₂×Dic₁₃	—	2	2	no	—
150	C₂×Dic₁₃	—	2	2	no	—
151	C₂×Dic₁₃	—	2	2	no	—
152	C₂×Dic₁₃	—	2	2	no	—
153	C₂×Dic₁₃	—	4	2	no	—
154	C₂×Dic₁₃	—	4	2	no	—
155	C₂×Dic₁₃	—	4	4	no	—
156	C₂×Dic₁₃	—	4	4	no	—
157	C₂×Dic₁₃	#127	1	2	yes	—
158	C₂×Dic₁₃	#127	1	2	yes	—
159	C₂×Dic₁₃	C₄⋊Q₈	1	4	yes	—
160	Dic₂₆	—	8	8	no	—
161	C₈×C₁₃	C₄⋊Q₈	1	4	yes	—
162	C₈×C₁₃	C₄⋊Q₈	1	4	yes	—
163	#1055	C₄⋊Q₈	1	4	yes	—
164	#1055	C₄⋊Q₈	1	4	yes	—
165	C₂×C₄⋊Q₈	—	13	2	no	—
166	C₂×#125	—	13	2	no	—
167	#501	—	26	8	no	—
168	#500	—	26	4	no	—
169	#499	—	26	4	no	—
170	C₂×#107	—	13	1	no	—
171	C₂×#107	—	13	1	no	—
172	C₂×#107	—	13	2	no	—
173	C₂×#107	—	13	2	no	—
174	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
175	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
176	C₂×#107	C₂×D₁₃	1	2	yes	—
177	C₂×#106	—	13	2	no	—
178	C₂×#106	—	13	2	no	—
179	C₂×#106	—	13	2	no	—
180	C₂×#97	—	13	1	no	—
181	C₂×#97	—	13	1	no	—
182	C₂×#97	—	13	1	no	—
183	C₂×C₄×C₈	—	13	1	no	—
184	C₂×C₄×C₈	—	13	1	no	—
185	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
186	#398	—	26	4	no	—
187	#395	—	26	4	no	—
188	#384	—	13	2	no	—
189	#384	C₂×D₁₃	1	2	yes	—
190	#382	—	13	2	no	—
191	#382	—	13	2	no	—
192	#376	—	13	1	no	—
193	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
194	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
195	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
196	C₂²×C₁₃	#576	1	2	yes	—
197	C₂²×C₁₃	#578	1	2	yes	—
198	C₄×C₁₃	—	2	4	no	—
199	C₄×C₁₃	—	8	2	no	—
200	C₄×C₁₃	—	8	2	no	—
201	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
202	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
203	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
204	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
205	Dic₁₃	—	2	2	no	—
206	Dic₁₃	—	2	2	no	—
207	Dic₁₃	—	2	4	no	—
208	Dic₁₃	—	8	2	no	—
209	Dic₁₃	—	8	2	no	—
210	C₂²×Q₈	—	26	2	no	—
211	C₂²×C₈	—	13	1	no	—
212	C₂²×C₈	—	13	1	no	—
213	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
215	C₄⋊Q₈	—	26	4	no	—
216	C₄⋊Q₈	—	26	4	no	—
217	#125	—	26	4	no	—
218	#125	—	26	4	no	—
219	C₂×C₄⋊C₄	—	2	4	no	—
220	C₂×C₄⋊C₄	—	13	2	no	—
221	C₂×C₄⋊C₄	—	13	2	no	—
222	C₂×C₄⋊C₄	—	26	2	no	—
223	C₂×C₄⋊C₄	—	26	2	no	—
224	C₂×C₄⋊C₄	—	26	2	no	—
225	C₂×C₄⋊C₄	—	26	4	no	—
226	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
227	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
228	C₂×C₄²	—	13	1	no	—
229	C₂×C₄²	—	13	1	no	—
230	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
231	#107	—	2	2	no	—
232	#107	—	2	2	no	—
233	#107	—	2	4	no	—
234	#107	—	26	4	no	—
235	#107	—	26	4	no	—
236	#107	—	26	2	no	—
237	#107	—	26	2	no	—
238	#106	—	26	4	no	—
239	#106	—	26	4	no	—
240	#106	—	26	4	no	—
241	#97	—	13	4	no	—
242	#97	—	13	4	no	—
243	#97	—	13	4	no	—
244	C₄×C₈	—	13	4	no	—
245	C₄×C₈	—	13	4	no	—
246	C₄×C₈	Dic₂₆	1	4	yes	complemented
247	#95	—	13	2	no	—
248	#95	—	13	2	no	—
249	#95	—	26	4	no	—
250	#95	—	26	4	no	—
251	C₂×C₁₃	#2868	1	1	yes	characteristic
252	C₂×C₁₃	#3155	1	1	yes	characteristic
253	C₂×C₁₃	#3156	1	1	yes	characteristic
254	C₂×C₁₃	#3164	1	2	yes	—
255	C₂×C₁₃	#3164	1	2	yes	—
256	C₂×Q₈	—	52	4	no	—
257	C₂×Q₈	—	104	4	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	13	2	no	—
261	C₂²×C₄	—	13	1	no	—
262	C₂²×C₄	—	13	1	no	—
263	C₂²×C₄	—	26	2	no	—
264	C₂²×C₄	—	26	2	no	—
265	C₂²×C₄	D₄×D₁₃	1	2	yes	—
266	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
267	C₂×C₈	—	13	2	no	—
268	C₂×C₈	—	13	2	no	—
269	C₂×C₈	—	13	2	no	—
270	C₂×C₈	—	13	2	no	—
271	C₂×C₈	—	13	2	no	—
272	C₂×C₈	—	13	2	no	—
273	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
274	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
275	C₂×C₈	D₄×D₁₃	1	2	yes	—
276	C₂×C₈	D₄×D₁₃	1	2	yes	—
277	C₂×C₈	Q₈×D₁₃	1	2	yes	—
278	C₂×C₈	Q₈×D₁₃	1	2	yes	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	8	4	no	—
284	C₄⋊C₄	—	26	2	no	—
285	C₄⋊C₄	—	26	2	no	—
286	C₄⋊C₄	—	26	2	no	—
287	C₄⋊C₄	—	26	2	no	—
288	C₄⋊C₄	—	52	4	no	—
289	C₄⋊C₄	—	52	4	no	—
290	C₄⋊C₄	—	52	4	no	—
291	C₄⋊C₄	—	104	4	no	—
292	C₄²	—	13	2	no	—
293	C₄²	—	13	2	no	—
294	C₄²	—	13	2	no	—
295	C₄²	—	13	2	no	—
296	C₄²	C₂×Dic₂₆	1	2	yes	—
297	C₄²	Q₈×D₁₃	1	2	yes	—
298	C₁₃	#33319	1	1	yes	characteristic, complemented, Sylow
299	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
300	Q₈	—	104	8	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	13	4	no	—
308	C₂×C₄	—	13	2	no	—
309	C₂×C₄	—	13	2	no	—
310	C₂×C₄	—	26	2	no	—
311	C₂×C₄	—	26	2	no	—
312	C₂×C₄	—	26	2	no	—
313	C₂×C₄	—	26	2	no	—
314	C₂×C₄	—	52	4	no	—
315	C₂×C₄	—	52	4	no	—
316	C₂×C₄	—	52	2	no	—
317	C₂×C₄	—	52	2	no	—
318	C₂×C₄	#88142	1	4	yes	—
319	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
320	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
321	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
322	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
323	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
324	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
325	C₈	—	13	4	no	—
326	C₈	—	13	4	no	—
327	C₈	#88142	1	4	yes	—
328	C₈	#88142	1	4	yes	—
329	C₂²	C₂×#88142	1	1	yes	characteristic
330	C₂²	D₁₃×#127	1	1	yes	characteristic
331	C₂²	D₁₃×C₄⋊Q₈	1	1	yes	characteristic
332	C₂²	#11719462	1	2	yes	—
333	C₂²	#11719471	1	2	yes	—
334	C₄	—	2	4	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	—	26	2	no	—
338	C₄	—	26	2	no	—
339	C₄	—	26	4	no	—
340	C₄	—	104	2	no	—
341	C₄	—	104	2	no	—
342	C₄	C₂×#88142	1	1	yes	characteristic
343	C₄	C₂×#88142	1	1	yes	characteristic
344	C₄	D₁₃×C₄⋊Q₈	1	1	yes	characteristic
345	C₄	D₁₃×C₄⋊Q₈	1	1	yes	characteristic
346	C₂	#13123233	1	1	yes	characteristic
347	C₂	#13125316	1	1	yes	characteristic
348	C₂	#13125324	1	1	yes	characteristic
349	C₂	#13125369	1	2	yes	—
350	C₂	#13125372	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator, cosocle	C₂²×C₄×C₁₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#1845	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
hypercenter	#1845	D₁₃
hypocenter	C₁₃	#33319
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
64. C₄×C₈×C₁₃	Q₈	semidirect
73. #88055	Q₈	semidirect
75. C₈×Dic₁₃	Q₈	semidirect
80. #88043	Q₈	semidirect
246. C₄×C₈	Dic₂₆	semidirect
298. C₁₃	#33319	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718537	2
C₂×#11718535	2
C₂×#11718539	2
#13120958	2
#13115003	2
#13115002	2
#13115000	2
#13114999	2
#13114994	2
#13114522	2
C₁₃×#1845	2
#33319	13