Subgroups — #15762150

Order 3328.

586 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718746	C₂	1	1	yes	maximal, characteristic
2	#13115789	C₂	1	1	yes	maximal, characteristic, complemented
3	#13115762	C₂	1	1	yes	maximal, characteristic
4	#13114701	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718072	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718077	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂²×#6106	—	2	1	no	—
10	C₂²×#6106	—	2	1	no	—
11	C₂×#88128	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×#88061	—	2	2	no	—
15	C₂×C₈×Dic₁₃	—	2	1	no	—
16	#11718746	—	2	2	no	—
17	#11718746	C₄	1	4	yes	complemented
18	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
20	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
21	C₁₃×#332	—	2	2	no	—
22	#11718142	—	2	1	no	—
23	#11718121	—	2	2	no	—
24	#11718077	—	2	2	no	—
25	#11718072	—	2	2	no	—
26	C₂²×Q₈×C₁₃	—	2	1	no	—
27	C₂²×D₅₂	—	2	1	no	—
28	C₂²×D₅₂	—	2	1	no	—
29	C₂²×D₅₂	—	2	1	no	—
30	C₂²×C₈×C₁₃	—	2	1	no	—
31	C₂²×C₈×C₁₃	—	2	1	no	—
32	C₁₃×C₄⋊Q₈	—	2	1	no	—
33	C₁₃×C₄⋊Q₈	—	2	1	no	—
34	C₁₃×C₄⋊Q₈	—	2	1	no	—
35	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
36	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×#6106	—	2	4	no	—
40	C₂×#6106	—	2	4	no	—
41	C₂×#6106	—	4	2	no	—
42	C₂×#6106	—	4	2	no	—
43	C₂×#6106	—	4	2	no	—
44	C₂×#6106	—	4	2	no	—
45	C₂×C₄⋊Dic₁₃	D₄	1	2	yes	—
46	C₂×C₄×Dic₁₃	—	2	1	no	—
47	C₂²×#1055	—	2	1	no	—
48	C₂²×#1055	D₄	1	1	yes	characteristic
49	C₂²×#1055	D₄	1	1	yes	characteristic
50	#88128	—	2	1	no	—
51	#88128	—	2	1	no	—
52	#88128	—	2	1	no	—
53	#88128	C₂×C₄	1	2	yes	—
54	C₁₃×C₄⋊C₈	—	2	1	no	—
55	C₁₃×C₄⋊C₈	—	2	1	no	—
56	C₁₃×C₄⋊C₈	—	2	1	no	—
57	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
58	#88071	—	2	1	no	—
59	#88061	—	4	4	no	—
60	C₈×Dic₁₃	—	4	2	no	—
61	C₄⋊#1055	—	2	1	no	—
62	C₄⋊#1055	—	2	1	no	—
63	C₄×#1055	C₂×C₄	1	2	yes	—
64	C₄×#1055	D₄	1	2	yes	—
65	#9732	—	13	1	no	maximal, Sylow
66	C₂³×D₁₃	—	4	1	no	—
67	C₂×Q₈×C₁₃	—	2	2	no	—
68	C₂×Q₈×C₁₃	—	4	1	no	—
69	C₂×Q₈×C₁₃	—	4	1	no	—
70	C₂×Q₈×C₁₃	—	4	1	no	—
71	C₂×Q₈×C₁₃	—	4	1	no	—
72	C₂×Q₈×C₁₃	—	4	1	no	—
73	C₂²×C₄×C₁₃	—	4	1	no	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
77	C₂²×Dic₁₃	—	2	2	no	—
78	C₂×D₅₂	—	2	2	no	—
79	C₂×D₅₂	—	2	2	no	—
80	C₂×D₅₂	—	2	2	no	—
81	C₂×D₅₂	—	4	1	no	—
82	C₂×D₅₂	—	4	1	no	—
83	C₂×D₅₂	—	4	2	no	—
84	C₂×D₅₂	—	4	2	no	—
85	C₂×D₅₂	—	4	1	no	—
86	C₂×D₅₂	—	4	1	no	—
87	C₂×D₅₂	—	4	1	no	—
88	C₂×D₅₂	—	4	1	no	—
89	C₂×D₅₂	—	4	1	no	—
90	C₂×D₅₂	—	8	2	no	—
91	C₂×C₈×C₁₃	—	2	2	no	—
92	C₂×C₈×C₁₃	—	4	1	no	—
93	C₂×C₈×C₁₃	—	4	1	no	—
94	C₂×C₈×C₁₃	—	4	1	no	—
95	C₂×C₈×C₁₃	—	4	2	no	—
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	4	1	no	—
99	C₁₃×C₄⋊C₄	—	4	1	no	—
100	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₃	#41	1	1	yes	characteristic
104	#6106	—	4	8	no	—
105	#6106	—	4	8	no	—
106	C₄⋊Dic₁₃	—	2	2	no	—
107	C₄⋊Dic₁₃	SD₁₆	1	4	yes	complemented
108	C₄×Dic₁₃	—	4	1	no	—
109	C₄×Dic₁₃	—	4	1	no	—
110	C₂×#1055	—	2	1	no	—
111	C₂×#1055	—	2	1	no	—
112	C₂×#1055	—	4	1	no	—
113	C₂×#1055	—	4	2	no	—
114	C₂×#1055	C₂²⋊C₄	1	2	yes	—
115	C₂×#1055	C₂²⋊C₄	1	2	yes	—
116	C₂×#1055	D₈	1	2	yes	—
117	C₂×#1055	SD₁₆	1	2	yes	—
118	C₂×#492	—	13	1	no	—
119	#1988	—	13	1	no	—
120	#1982	—	13	1	no	—
121	C₈×C₄⋊C₄	—	13	1	no	—
122	C₂×#334	—	13	1	no	—
123	C₂×#332	D₁₃	1	1	yes	characteristic, complemented
124	C₂×#331	—	13	1	no	—
125	C₂³×C₁₃	#121	1	1	yes	characteristic
126	C₂²×D₁₃	—	4	2	no	—
127	C₂²×D₁₃	—	8	1	no	—
128	C₂²×D₁₃	—	8	1	no	—
129	C₂²×D₁₃	—	8	2	no	—
130	C₂²×D₁₃	—	8	2	no	—
131	Q₈×C₁₃	—	4	2	no	—
132	Q₈×C₁₃	—	4	2	no	—
133	Q₈×C₁₃	—	8	1	no	—
134	Q₈×C₁₃	—	8	1	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	8	1	no	—
138	C₂×C₄×C₁₃	—	8	1	no	—
139	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
143	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
144	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
148	C₂×C₄×C₁₃	#135	1	2	yes	—
149	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×Dic₁₃	—	2	4	no	—
152	C₂×Dic₁₃	—	4	2	no	—
153	C₂×Dic₁₃	—	4	2	no	—
154	D₅₂	—	4	2	no	—
155	D₅₂	—	4	2	no	—
156	D₅₂	—	4	4	no	—
157	D₅₂	—	8	2	no	—
158	D₅₂	—	8	2	no	—
159	D₅₂	—	8	1	no	—
160	D₅₂	—	8	1	no	—
161	D₅₂	—	8	1	no	—
162	D₅₂	—	8	1	no	—
163	C₈×C₁₃	—	4	2	no	—
164	C₈×C₁₃	—	8	1	no	—
165	#1055	—	2	2	no	—
166	#1055	—	8	1	no	—
167	#1055	#102	1	4	yes	complemented
168	C₂²×SD₁₆	—	26	1	no	—
169	C₂²×SD₁₆	—	26	1	no	—
170	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
171	C₂×#127	—	13	1	no	—
172	#492	—	13	4	no	—
173	#492	—	26	2	no	—
174	C₂×C₄⋊C₈	—	13	1	no	—
175	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
176	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
177	C₂×#102	—	26	2	no	—
178	C₂×C₄×C₈	—	13	1	no	—
179	C₂×C₄×C₈	—	26	1	no	—
180	C₄×C₄⋊C₄	—	13	1	no	—
181	#340	—	26	2	no	—
182	#336	—	26	1	no	—
183	#334	—	26	2	no	—
184	#332	—	2	2	no	—
185	#331	—	26	2	no	—
186	C₂²×C₁₃	#390	1	1	yes	characteristic
187	C₂²×C₁₃	#459	1	1	yes	characteristic
188	C₂²×C₁₃	#460	1	1	yes	characteristic
189	C₂²×C₁₃	#465	1	2	yes	—
190	C₂²×C₁₃	#471	1	2	yes	—
191	C₂×D₁₃	—	8	2	no	—
192	C₂×D₁₃	—	8	2	no	—
193	C₂×D₁₃	—	8	2	no	—
194	C₂×D₁₃	—	16	2	no	—
195	C₂×D₁₃	—	16	2	no	—
196	C₄×C₁₃	—	2	1	no	—
197	C₄×C₁₃	—	2	1	no	—
198	C₄×C₁₃	—	8	2	no	—
199	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
200	C₄×C₁₃	#418	1	1	yes	characteristic
201	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
202	C₄×C₁₃	#459	1	1	yes	characteristic
203	C₄×C₁₃	#460	1	1	yes	characteristic
204	C₄×C₁₃	#492	1	1	yes	characteristic
205	C₄×C₁₃	#495	1	1	yes	characteristic
206	Dic₁₃	—	4	4	no	—
207	C₂²×Q₈	—	2	1	no	—
208	C₂²×D₄	—	26	1	no	—
209	C₂²×D₄	—	26	1	no	—
210	C₂²×D₄	—	26	1	no	—
211	C₂×SD₁₆	—	26	4	no	—
212	C₂×SD₁₆	—	26	4	no	—
213	C₂×SD₁₆	—	52	2	no	—
214	C₂×SD₁₆	—	52	2	no	—
215	C₂×SD₁₆	—	52	2	no	—
216	C₂×SD₁₆	—	52	2	no	—
217	C₂²×C₈	—	2	1	no	—
218	C₂²×C₈	—	2	1	no	—
219	C₂²×C₈	—	13	1	no	—
220	C₂²×C₈	—	13	1	no	—
221	C₂²×C₈	—	26	1	no	—
222	C₄⋊Q₈	—	2	1	no	—
223	C₄⋊Q₈	—	2	1	no	—
224	C₄⋊Q₈	—	2	1	no	—
225	C₄⋊Q₈	C₄×D₁₃	1	2	yes	—
226	#127	—	13	2	no	—
227	#127	—	26	1	no	—
228	#127	—	26	1	no	—
229	#127	—	26	1	no	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₄⋊C₄	—	2	1	no	—
232	C₂×C₄⋊C₄	—	13	2	no	—
233	C₂×C₄²	—	26	1	no	—
234	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
235	C₄⋊C₈	—	2	1	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	2	1	no	—
238	C₄⋊C₈	—	26	1	no	—
239	C₄⋊C₈	—	26	1	no	—
240	C₄⋊C₈	C₂²⋊D₁₃	1	2	yes	—
241	#102	—	52	4	no	—
242	C₄×C₈	—	13	2	no	—
243	C₄×C₈	—	13	2	no	—
244	C₄×C₈	—	52	2	no	—
245	#95	—	26	1	no	—
246	C₂×C₁₃	#1665	1	2	yes	—
247	C₂×C₁₃	#1666	1	2	yes	—
248	C₂×C₁₃	#1968	1	1	yes	characteristic
249	C₂×C₁₃	#1977	1	1	yes	characteristic
250	C₂×C₁₃	#1982	1	1	yes	characteristic
251	D₁₃	—	16	2	no	—
252	C₂⁴	—	52	1	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	—	4	1	no	—
256	C₂×Q₈	—	4	1	no	—
257	C₂×Q₈	—	4	1	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×D₄	—	26	2	no	—
260	C₂×D₄	—	26	2	no	—
261	C₂×D₄	—	26	2	no	—
262	C₂×D₄	—	52	1	no	—
263	C₂×D₄	—	52	1	no	—
264	C₂×D₄	—	52	1	no	—
265	C₂×D₄	—	52	2	no	—
266	C₂×D₄	—	52	1	no	—
267	C₂×D₄	—	52	1	no	—
268	C₂×D₄	—	52	2	no	—
269	C₂×D₄	—	52	1	no	—
270	C₂×D₄	—	52	1	no	—
271	C₂×D₄	—	104	2	no	—
272	C₂²×C₄	—	4	1	no	—
273	C₂²×C₄	—	26	2	no	—
274	C₂²×C₄	#6127	1	1	yes	characteristic
275	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
276	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
277	SD₁₆	—	52	8	no	—
278	SD₁₆	—	52	8	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	4	2	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	13	2	no	—
285	C₂×C₈	—	13	2	no	—
286	C₂×C₈	—	13	2	no	—
287	C₂×C₈	—	13	2	no	—
288	C₂×C₈	—	26	1	no	—
289	C₂×C₈	—	26	1	no	—
290	C₂×C₈	—	52	2	no	—
291	C₂×C₈	—	52	1	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	13	4	no	—
297	C₄⋊C₄	—	26	2	no	—
298	C₄²	—	52	1	no	—
299	C₄²	—	52	1	no	—
300	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
301	C₄²	D₄×D₁₃	1	1	yes	characteristic
302	C₄²	#6129	1	1	yes	characteristic
303	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
304	C₁₃	#9732	1	1	yes	characteristic, complemented, Sylow
305	C₂³	—	52	2	no	—
306	C₂³	—	104	2	no	—
307	C₂³	—	104	1	no	—
308	C₂³	—	104	1	no	—
309	C₂³	—	104	2	no	—
310	C₂³	#88138	1	1	yes	characteristic
311	Q₈	—	4	2	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	8	1	no	—
314	Q₈	—	8	1	no	—
315	D₄	—	52	2	no	—
316	D₄	—	52	2	no	—
317	D₄	—	52	4	no	—
318	D₄	—	104	2	no	—
319	D₄	—	104	2	no	—
320	D₄	—	104	1	no	—
321	D₄	—	104	1	no	—
322	D₄	—	104	1	no	—
323	D₄	—	104	1	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	8	1	no	—
327	C₂×C₄	—	8	1	no	—
328	C₂×C₄	—	26	4	no	—
329	C₂×C₄	—	52	2	no	—
330	C₂×C₄	—	52	2	no	—
331	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
332	C₂×C₄	#88135	1	1	yes	characteristic
333	C₂×C₄	#88139	1	1	yes	characteristic
334	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
335	C₂×C₄	#88165	1	1	yes	characteristic
336	C₂×C₄	SD₁₆×D₁₃	1	1	yes	characteristic
337	C₂×C₄	SD₁₆×D₁₃	1	2	yes	—
338	C₂×C₄	#88168	1	1	yes	characteristic
339	C₂×C₄	#88173	1	2	yes	—
340	C₂×C₄	C₄×C₂²⋊D₁₃	1	1	yes	characteristic
341	C₂×C₄	#88193	1	1	yes	characteristic
342	C₂×C₄	#88194	1	1	yes	characteristic
343	C₈	—	4	2	no	—
344	C₈	—	8	1	no	—
345	C₈	—	13	4	no	—
346	C₈	—	26	2	no	—
347	C₈	—	104	1	no	—
348	C₂²	—	104	2	no	—
349	C₂²	—	104	2	no	—
350	C₂²	—	104	2	no	—
351	C₂²	—	208	2	no	—
352	C₂²	—	208	2	no	—
353	C₂²	#11718415	1	2	yes	—
354	C₂²	#11718423	1	1	yes	characteristic
355	C₂²	#11718447	1	1	yes	characteristic
356	C₂²	#11718468	1	2	yes	—
357	C₂²	#11718618	1	1	yes	characteristic
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	8	2	no	—
361	C₄	—	52	4	no	—
362	C₄	D₁₃×#102	1	1	yes	characteristic
363	C₄	#11718431	1	1	yes	characteristic
364	C₄	#11718449	1	2	yes	—
365	C₄	#11718811	1	1	yes	characteristic
366	C₄	#11718824	1	1	yes	characteristic
367	C₄	#11718836	1	1	yes	characteristic
368	C₄	#11718852	1	1	yes	characteristic
369	C₂	—	208	2	no	—
370	C₂	#13114180	1	2	yes	—
371	C₂	#13114187	1	2	yes	—
372	C₂	#13115687	1	1	yes	characteristic
373	C₂	#13115740	1	1	yes	characteristic
374	C₂	#13115761	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88138
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#332	C₂
socle	C₂³×C₁₃	#121
prime radical	C₂×#88128	C₂²
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#332	D₁₃
hypocenter	C₁₃	#9732
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#332

Complements

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

N	H ≅ G/N	type
2. #13115789	C₂	semidirect
4. #13114701	C₂	semidirect
5. C₂×#11718072	C₂	semidirect
7. C₂×C₁₃×#332	C₂	semidirect
17. #11718746	C₄	semidirect
107. C₄⋊Dic₁₃	SD₁₆	semidirect
123. C₂×#332	D₁₃	semidirect
167. #1055	#102	semidirect
304. C₁₃	#9732	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718746	2
#13115789	2
#13115762	2
#13114701	2
C₂×#11718072	2
C₂×#11718077	2
C₂×C₁₃×#332	2
#9732	13