Subgroups — #15765753

Order 3328.

734 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718740	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718746	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718354	C₂	1	1	yes	maximal, characteristic
4	#13115796	C₂	1	1	yes	maximal, characteristic, complemented
5	#13115641	C₂	1	1	yes	maximal, characteristic
6	#13112663	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂²×#6106	—	2	2	no	—
10	C₂²×#6095	—	2	2	no	—
11	C₂×#88128	C₂²	1	1	yes	characteristic
12	C₂×#88123	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×#384	C₂²	1	1	yes	characteristic
15	#11718746	—	2	2	no	—
16	#11718746	—	2	2	no	—
17	#11718740	—	2	2	no	—
18	#11718740	—	2	2	no	—
19	C₂×#88049	—	2	2	no	—
20	C₂×#88048	—	2	2	no	—
21	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
22	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
23	#11718354	—	2	2	no	—
24	#11718354	—	2	2	no	—
25	C₁₃×#340	—	2	2	no	—
26	#11718121	—	2	2	no	—
27	C₂²×Q₈×C₁₃	—	2	1	no	—
28	C₂²×D₅₂	—	2	1	no	—
29	C₂²×D₅₂	—	2	1	no	—
30	C₂²×D₅₂	—	2	1	no	—
31	C₂²×Dic₂₆	—	2	1	no	—
32	C₂²×C₈×C₁₃	D₄	1	2	yes	—
33	C₁₃×C₄⋊Q₈	—	2	1	no	—
34	C₁₃×C₄⋊Q₈	—	2	1	no	—
35	C₁₃×C₄⋊Q₈	—	2	1	no	—
36	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
37	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×#6106	—	2	8	no	—
44	C₂×#6106	—	4	4	no	—
45	C₂×#6106	—	4	4	no	—
46	C₂×C₄⋊Dic₁₃	—	2	1	no	—
47	C₂×C₄⋊Dic₁₃	—	2	1	no	—
48	C₂²×#1055	D₄	1	2	yes	—
49	C₂²×#1055	D₄	1	2	yes	—
50	C₂×#6095	—	2	8	no	—
51	C₂×#6095	—	4	4	no	—
52	C₂×#6095	—	4	4	no	—
53	#88128	—	2	1	no	—
54	#88128	—	2	1	no	—
55	#88128	—	2	1	no	—
56	#88128	D₄	1	2	yes	—
57	#88123	—	2	1	no	—
58	#88123	—	2	1	no	—
59	#88123	—	2	1	no	—
60	#88123	D₄	1	2	yes	—
61	C₄×C₈×C₁₃	—	2	1	no	—
62	C₄×C₈×C₁₃	—	2	1	no	—
63	#88049	—	4	4	no	—
64	#88048	—	2	8	no	—
65	C₄×#1055	—	2	1	no	—
66	C₄×#1055	—	2	1	no	—
67	C₄×#1055	—	2	1	no	—
68	C₄×#1055	—	2	1	no	—
69	#10315	—	13	1	no	maximal, Sylow
70	C₂³×D₁₃	—	4	1	no	—
71	C₂×Q₈×C₁₃	—	2	2	no	—
72	C₂×Q₈×C₁₃	—	4	1	no	—
73	C₂×Q₈×C₁₃	—	4	2	no	—
74	C₂×Q₈×C₁₃	—	4	2	no	—
75	C₂²×C₄×C₁₃	—	4	1	no	—
76	C₂²×C₄×C₁₃	—	4	1	no	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×Dic₁₃	—	4	1	no	—
81	C₂×D₅₂	—	2	2	no	—
82	C₂×D₅₂	—	2	2	no	—
83	C₂×D₅₂	—	2	2	no	—
84	C₂×D₅₂	—	4	2	no	—
85	C₂×D₅₂	—	4	2	no	—
86	C₂×D₅₂	—	4	2	no	—
87	C₂×D₅₂	—	4	2	no	—
88	C₂×D₅₂	—	4	2	no	—
89	C₂×D₅₂	—	4	2	no	—
90	C₂×D₅₂	—	4	1	no	—
91	C₂×D₅₂	—	4	1	no	—
92	C₂×D₅₂	—	4	1	no	—
93	C₂×Dic₂₆	—	2	2	no	—
94	C₂×Dic₂₆	—	4	2	no	—
95	C₂×Dic₂₆	—	4	2	no	—
96	C₂×Dic₂₆	—	4	1	no	—
97	C₂×C₈×C₁₃	—	2	2	no	—
98	C₂×C₈×C₁₃	—	2	2	no	—
99	C₂×C₈×C₁₃	SD₁₆	1	4	yes	complemented
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	—	2	2	no	—
102	C₁₃×C₄⋊C₄	—	4	1	no	—
103	C₁₃×C₄⋊C₄	—	4	1	no	—
104	C₁₃×C₄⋊C₄	—	4	2	no	—
105	C₁₃×C₄⋊C₄	—	4	2	no	—
106	C₁₃×C₄⋊C₄	—	4	2	no	—
107	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₁₃	#41	1	1	yes	characteristic
111	#6106	—	4	16	no	—
112	C₄⋊Dic₁₃	—	2	2	no	—
113	C₄⋊Dic₁₃	—	2	2	no	—
114	C₄⋊Dic₁₃	—	4	1	no	—
115	C₄⋊Dic₁₃	—	4	1	no	—
116	C₂×#1055	—	2	2	no	—
117	C₂×#1055	—	2	2	no	—
118	C₂×#1055	—	2	2	no	—
119	C₂×#1055	—	2	2	no	—
120	C₂×#1055	SD₁₆	1	4	yes	complemented
121	C₂×#1055	SD₁₆	1	4	yes	complemented
122	#6095	—	4	16	no	—
123	C₂×#498	—	13	1	no	—
124	C₂×#492	—	13	1	no	—
125	C₂×#492	—	13	1	no	—
126	#1990	—	13	1	no	—
127	#1990	D₁₃	1	1	yes	characteristic, complemented
128	#1957	—	13	1	no	—
129	#1273	—	13	1	no	—
130	C₂³×C₁₃	#127	1	1	yes	characteristic
131	C₂²×D₁₃	—	4	2	no	—
132	C₂²×D₁₃	—	8	1	no	—
133	C₂²×D₁₃	—	8	1	no	—
134	C₂²×D₁₃	—	8	1	no	—
135	C₂²×D₁₃	—	8	1	no	—
136	C₂²×D₁₃	—	8	1	no	—
137	C₂²×D₁₃	—	8	1	no	—
138	Q₈×C₁₃	—	4	4	no	—
139	Q₈×C₁₃	—	8	2	no	—
140	C₂×C₄×C₁₃	—	4	2	no	—
141	C₂×C₄×C₁₃	—	8	1	no	—
142	C₂×C₄×C₁₃	—	8	1	no	—
143	C₂×C₄×C₁₃	—	8	1	no	—
144	C₂×C₄×C₁₃	—	8	1	no	—
145	C₂×C₄×C₁₃	—	8	1	no	—
146	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
147	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
148	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
149	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
150	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
151	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
153	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
154	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
155	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
156	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
157	C₂×C₄×C₁₃	#137	1	2	yes	—
158	C₂×Dic₁₃	—	4	2	no	—
159	C₂×Dic₁₃	—	8	1	no	—
160	C₂×Dic₁₃	—	8	1	no	—
161	D₅₂	—	4	4	no	—
162	D₅₂	—	4	4	no	—
163	D₅₂	—	4	4	no	—
164	D₅₂	—	8	2	no	—
165	D₅₂	—	8	2	no	—
166	D₅₂	—	8	2	no	—
167	Dic₂₆	—	4	4	no	—
168	Dic₂₆	—	8	2	no	—
169	C₈×C₁₃	—	2	4	no	—
170	#1055	—	2	4	no	—
171	#1055	—	2	4	no	—
172	C₂²×SD₁₆	—	26	2	no	—
173	C₂²×SD₁₆	—	26	2	no	—
174	C₂×C₄⋊Q₈	—	13	1	no	—
175	C₂×C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
176	C₂×#127	—	13	1	no	—
177	#498	—	26	2	no	—
178	#498	—	26	2	no	—
179	#492	—	26	2	no	—
180	#492	—	26	2	no	—
181	#492	—	26	2	no	—
182	#492	—	26	2	no	—
183	C₂×#106	—	26	2	no	—
184	C₂×#102	—	26	2	no	—
185	C₂×C₄×C₈	—	13	1	no	—
186	C₂×C₄×C₈	—	13	1	no	—
187	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
188	#384	C₂×D₁₃	1	1	yes	characteristic
189	#340	—	2	2	no	—
190	#340	—	26	2	no	—
191	C₂²×C₁₃	#392	1	1	yes	characteristic
192	C₂²×C₁₃	#492	1	1	yes	characteristic
193	C₂²×C₁₃	#492	1	1	yes	characteristic
194	C₂²×C₁₃	#492	1	1	yes	characteristic
195	C₂²×C₁₃	#496	1	1	yes	characteristic
196	C₂²×C₁₃	#496	1	1	yes	characteristic
197	C₂²×C₁₃	#497	1	1	yes	characteristic
198	C₂×D₁₃	—	8	2	no	—
199	C₂×D₁₃	—	8	2	no	—
200	C₂×D₁₃	—	8	2	no	—
201	C₂×D₁₃	—	16	1	no	—
202	C₂×D₁₃	—	16	1	no	—
203	C₂×D₁₃	—	16	1	no	—
204	C₂×D₁₃	—	16	1	no	—
205	C₄×C₁₃	—	8	2	no	—
206	C₄×C₁₃	—	16	1	no	—
207	C₄×C₁₃	#448	1	2	yes	—
208	C₄×C₁₃	#450	1	2	yes	—
209	C₄×C₁₃	#450	1	2	yes	—
210	C₄×C₁₃	#460	1	2	yes	—
211	C₄×C₁₃	#460	1	2	yes	—
212	C₄×C₁₃	#461	1	2	yes	—
213	Dic₁₃	—	8	2	no	—
214	C₂²×Q₈	—	2	1	no	—
215	C₂²×Q₈	—	26	1	no	—
216	C₂²×D₄	—	26	1	no	—
217	C₂²×D₄	—	26	1	no	—
218	C₂²×D₄	—	26	1	no	—
219	C₂×SD₁₆	—	26	8	no	—
220	C₂×SD₁₆	—	26	8	no	—
221	C₂×SD₁₆	—	52	4	no	—
222	C₂×SD₁₆	—	52	4	no	—
223	C₂×SD₁₆	—	52	4	no	—
224	C₂×SD₁₆	—	52	4	no	—
225	C₂²×C₈	—	13	2	no	—
226	C₂²×C₈	—	13	2	no	—
227	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	—	13	2	no	—
232	C₄⋊Q₈	—	26	1	no	—
233	C₄⋊Q₈	—	26	1	no	—
234	C₄⋊Q₈	—	26	1	no	—
235	C₄⋊Q₈	D₅₂	1	2	yes	—
236	#127	—	13	2	no	—
237	#127	—	26	1	no	—
238	#127	—	26	1	no	—
239	#127	—	26	1	no	—
240	C₂×C₄⋊C₄	—	2	1	no	—
241	C₂×C₄⋊C₄	—	2	1	no	—
242	C₂×C₄⋊C₄	—	2	1	no	—
243	C₂×C₄⋊C₄	—	2	1	no	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	—	26	1	no	—
246	C₂×C₄⋊C₄	—	26	1	no	—
247	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
248	#106	—	26	8	no	—
249	#102	—	52	4	no	—
250	C₄×C₈	—	2	1	no	—
251	C₄×C₈	—	2	1	no	—
252	C₄×C₈	—	26	1	no	—
253	C₄×C₈	—	26	1	no	—
254	C₄×C₈	—	26	1	no	—
255	C₄×C₈	—	26	1	no	—
256	C₂×C₁₃	#1705	1	1	yes	characteristic
257	C₂×C₁₃	#1714	1	1	yes	characteristic
258	C₂×C₁₃	#1714	1	1	yes	characteristic
259	C₂×C₁₃	#1715	1	1	yes	characteristic
260	C₂×C₁₃	#2013	1	1	yes	characteristic
261	C₂×C₁₃	#2014	1	1	yes	characteristic
262	C₂×C₁₃	#2014	1	1	yes	characteristic
263	D₁₃	—	16	2	no	—
264	C₂⁴	—	52	1	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	4	1	no	—
267	C₂×Q₈	—	4	2	no	—
268	C₂×Q₈	—	4	2	no	—
269	C₂×Q₈	—	26	2	no	—
270	C₂×Q₈	—	52	1	no	—
271	C₂×Q₈	—	52	2	no	—
272	C₂×Q₈	—	52	2	no	—
273	C₂×D₄	—	26	2	no	—
274	C₂×D₄	—	26	2	no	—
275	C₂×D₄	—	26	2	no	—
276	C₂×D₄	—	52	1	no	—
277	C₂×D₄	—	52	1	no	—
278	C₂×D₄	—	52	2	no	—
279	C₂×D₄	—	52	2	no	—
280	C₂×D₄	—	52	2	no	—
281	C₂×D₄	—	52	2	no	—
282	C₂×D₄	—	52	1	no	—
283	C₂×D₄	—	52	2	no	—
284	C₂×D₄	—	52	2	no	—
285	C₂²×C₄	—	4	1	no	—
286	C₂²×C₄	—	4	1	no	—
287	C₂²×C₄	—	52	1	no	—
288	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
289	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
290	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
291	SD₁₆	—	52	16	no	—
292	SD₁₆	—	52	16	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	13	4	no	—
296	C₂×C₈	—	13	4	no	—
297	C₂×C₈	—	26	2	no	—
298	C₂×C₈	—	26	2	no	—
299	C₂×C₈	—	26	2	no	—
300	C₂×C₈	—	26	2	no	—
301	C₂×C₈	#6106	1	4	yes	complemented
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	4	1	no	—
306	C₄⋊C₄	—	4	2	no	—
307	C₄⋊C₄	—	4	2	no	—
308	C₄⋊C₄	—	4	2	no	—
309	C₄⋊C₄	—	26	2	no	—
310	C₄⋊C₄	—	26	2	no	—
311	C₄⋊C₄	—	52	1	no	—
312	C₄⋊C₄	—	52	1	no	—
313	C₄²	C₂×D₅₂	1	1	yes	characteristic
314	C₄²	#6127	1	1	yes	characteristic
315	C₄²	D₄×D₁₃	1	1	yes	characteristic
316	C₄²	D₄×D₁₃	1	1	yes	characteristic
317	C₁₃	#10315	1	1	yes	characteristic, complemented, Sylow
318	C₂³	—	52	2	no	—
319	C₂³	—	104	1	no	—
320	C₂³	—	104	1	no	—
321	C₂³	—	104	1	no	—
322	C₂³	—	104	1	no	—
323	C₂³	—	104	1	no	—
324	C₂³	—	104	1	no	—
325	C₂³	#88194	1	1	yes	characteristic
326	Q₈	—	4	4	no	—
327	Q₈	—	8	2	no	—
328	Q₈	—	52	4	no	—
329	Q₈	—	104	2	no	—
330	D₄	—	52	4	no	—
331	D₄	—	52	4	no	—
332	D₄	—	52	4	no	—
333	D₄	—	104	2	no	—
334	D₄	—	104	2	no	—
335	D₄	—	104	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	8	1	no	—
338	C₂×C₄	—	8	1	no	—
339	C₂×C₄	—	8	1	no	—
340	C₂×C₄	—	8	1	no	—
341	C₂×C₄	—	8	1	no	—
342	C₂×C₄	—	52	2	no	—
343	C₂×C₄	—	104	1	no	—
344	C₂×C₄	—	104	1	no	—
345	C₂×C₄	#88136	1	1	yes	characteristic
346	C₂×C₄	#88136	1	1	yes	characteristic
347	C₂×C₄	#88138	1	1	yes	characteristic
348	C₂×C₄	#88138	1	1	yes	characteristic
349	C₂×C₄	#88165	1	2	yes	—
350	C₂×C₄	SD₁₆×D₁₃	1	2	yes	—
351	C₂×C₄	SD₁₆×D₁₃	1	2	yes	—
352	C₂×C₄	#88172	1	2	yes	—
353	C₂×C₄	#88184	1	1	yes	characteristic
354	C₂×C₄	#88191	1	1	yes	characteristic
355	C₂×C₄	C₂×#6106	1	2	yes	—
356	C₂×C₄	#88203	1	2	yes	—
357	C₈	—	2	4	no	—
358	C₈	—	26	4	no	—
359	C₈	—	26	4	no	—
360	C₂²	—	104	2	no	—
361	C₂²	—	104	2	no	—
362	C₂²	—	104	2	no	—
363	C₂²	—	208	1	no	—
364	C₂²	—	208	1	no	—
365	C₂²	—	208	1	no	—
366	C₂²	—	208	1	no	—
367	C₂²	#11718621	1	1	yes	characteristic
368	C₂²	#11718719	1	1	yes	characteristic
369	C₂²	#11718746	1	1	yes	characteristic
370	C₂²	#11718815	1	1	yes	characteristic
371	C₂²	#11718836	1	1	yes	characteristic
372	C₂²	#11718836	1	1	yes	characteristic
373	C₂²	#11718851	1	1	yes	characteristic
374	C₄	—	8	2	no	—
375	C₄	—	16	1	no	—
376	C₄	—	104	2	no	—
377	C₄	#11718423	1	2	yes	—
378	C₄	#11718437	1	2	yes	—
379	C₄	#11718467	1	2	yes	—
380	C₄	#11718479	1	2	yes	—
381	C₄	#11718688	1	2	yes	—
382	C₄	#11718707	1	2	yes	—
383	C₂	—	208	2	no	—
384	C₂	#13114424	1	1	yes	characteristic
385	C₂	#13114459	1	1	yes	characteristic
386	C₂	#13114460	1	1	yes	characteristic
387	C₂	#13114466	1	1	yes	characteristic
388	C₂	#13115920	1	1	yes	characteristic
389	C₂	#13115923	1	1	yes	characteristic
390	C₂	#13115925	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88194
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1990	C₂
socle	C₂³×C₁₃	#127
prime radical	C₂×#88128	C₂²
hypercenter	#1990	D₁₃
hypocenter	C₁₃	#10315
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1990

Complements

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

N	H ≅ G/N	type
1. C₂×#11718740	C₂	semidirect
4. #13115796	C₂	semidirect
6. #13112663	C₂	semidirect
7. C₁₃×#1990	C₂	semidirect
99. C₂×C₈×C₁₃	SD₁₆	semidirect
120. C₂×#1055	SD₁₆	semidirect
121. C₂×#1055	SD₁₆	semidirect
127. #1990	D₁₃	semidirect
301. C₂×C₈	#6106	semidirect
317. C₁₃	#10315	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718740	2
C₂×#11718746	2
C₂×#11718354	2
#13115796	2
#13115641	2
#13112663	2
C₁₃×#1990	2
#10315	13