Subgroups — #15861975

Order 3328.

826 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718340	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic
3	#13117313	C₂	1	1	yes	maximal, characteristic
4	#13114707	C₂	1	2	yes	maximal
5	#13114703	C₂	1	1	yes	maximal, characteristic
6	C₈×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
7	#13114697	C₂	1	2	yes	maximal
8	#13114692	C₂	1	2	yes	maximal
9	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
10	#13114586	C₂	1	2	yes	maximal
11	C₁₃×#1744	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₄³×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×#88055	C₂²	1	2	yes	—
17	C₂×#88055	C₂²	1	2	yes	—
18	C₂×#88054	C₂²	1	2	yes	—
19	C₂×#88054	C₂²	1	2	yes	—
20	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
21	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
22	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
23	#11718594	C₄	1	4	yes	—
24	#11718593	C₂²	1	2	yes	—
25	C₄×#6101	C₂²	1	2	yes	—
26	#11718590	C₄	1	2	yes	—
27	#11718589	C₄	1	2	yes	—
28	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
30	C₂×C₄×#1055	C₂²	1	2	yes	—
31	#11718340	C₄	1	8	yes	complemented
32	C₈×Dic₂₆	C₄	1	8	yes	complemented
33	C₁₃×#336	C₂²	1	1	yes	characteristic
34	C₁₃×#336	C₂²	1	1	yes	characteristic
35	C₁₃×#336	C₂²	1	1	yes	characteristic
36	C₁₃×#336	C₂²	1	1	yes	characteristic
37	#11718142	C₂²	1	2	yes	—
38	#11718142	C₂²	1	2	yes	—
39	#11718116	C₂²	1	2	yes	—
40	C₂²×Dic₂₆	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
52	C₂×#6101	C₂×C₄	1	4	yes	—
53	C₂×#6101	C₂×C₄	1	2	yes	—
54	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
55	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
56	C₂²×#1055	C₂³	1	2	yes	—
57	C₂²×#1055	C₂³	1	2	yes	—
58	C₄×Dic₂₆	C₂×C₄	1	4	yes	—
59	C₄×Dic₂₆	C₂×C₄	1	4	yes	—
60	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
61	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
62	#88071	C₂×C₄	1	4	yes	—
63	#88071	C₂×C₄	1	4	yes	—
64	#88071	C₂×C₄	1	4	yes	—
65	#88071	C₂×C₄	1	2	yes	—
66	#88055	—	4	2	no	—
67	#88055	C₂×C₄	1	8	yes	—
68	#88054	C₂×C₄	1	8	yes	—
69	#88054	C₂×C₄	1	8	yes	—
70	C₈×Dic₁₃	—	4	2	no	—
71	C₈×Dic₁₃	C₂×C₄	1	8	yes	—
72	C₄⋊#1055	C₂×C₄	1	4	yes	—
73	C₄⋊#1055	C₂×C₄	1	4	yes	—
74	C₄×#1055	—	4	2	no	—
75	#20571	—	13	1	no	maximal, Sylow
76	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₁₃	C₂²×C₄	1	4	yes	—
82	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
83	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
84	C₂×Dic₂₆	—	2	4	no	—
85	C₂×Dic₂₆	C₄²	1	4	yes	—
86	C₂×C₈×C₁₃	—	2	1	no	—
87	C₂×C₈×C₁₃	—	2	1	no	—
88	C₂×C₈×C₁₃	—	2	1	no	—
89	C₂×C₈×C₁₃	—	2	1	no	—
90	C₂×C₈×C₁₃	—	2	1	no	—
91	C₂×C₈×C₁₃	—	2	1	no	—
92	C₂×C₈×C₁₃	—	2	1	no	—
93	C₂×C₈×C₁₃	—	2	1	no	—
94	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
95	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
98	C₄²×C₁₃	—	2	4	no	—
99	C₄²×C₁₃	—	2	4	no	—
100	C₄²×C₁₃	—	2	1	no	—
101	C₄²×C₁₃	—	2	1	no	—
102	C₄²×C₁₃	—	2	1	no	—
103	C₄²×C₁₃	—	2	1	no	—
104	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
105	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
106	C₄⋊Dic₁₃	—	2	4	no	—
107	C₄⋊Dic₁₃	C₄²	1	4	yes	—
108	#6101	—	2	8	no	—
109	#6101	C₄²	1	8	yes	—
110	C₄×Dic₁₃	—	2	2	no	—
111	C₄×Dic₁₃	—	2	2	no	—
112	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
113	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
114	C₂×#1055	—	2	2	no	—
115	C₂×#1055	—	2	2	no	—
116	C₂×#1055	—	2	2	no	—
117	C₂×#1055	—	2	2	no	—
118	C₂×#1055	C₂²×C₄	1	4	yes	—
119	C₂×#1055	C₂²×C₄	1	4	yes	—
120	C₂×#446	—	13	1	no	—
121	C₂×C₈×Q₈	—	13	1	no	—
122	#2295	—	13	1	no	—
123	#1771	—	13	2	no	—
124	#1770	—	13	1	no	—
125	C₈×C₄⋊C₄	—	13	1	no	—
126	C₈×C₄⋊C₄	—	13	2	no	—
127	#1768	—	13	2	no	—
128	C₄×C₄⋊C₈	—	13	1	no	—
129	#1746	—	13	2	no	—
130	#1744	D₁₃	1	1	yes	characteristic
131	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
139	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
140	C₂×C₄×C₁₃	#131	1	4	yes	—
141	C₂×C₄×C₁₃	#131	1	4	yes	—
142	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
145	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
146	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
147	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
148	C₂×Dic₁₃	—	2	4	no	—
149	C₂×Dic₁₃	—	2	4	no	—
150	C₂×Dic₁₃	—	2	4	no	—
151	C₂×Dic₁₃	—	2	4	no	—
152	C₂×Dic₁₃	—	2	4	no	—
153	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
154	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
155	Dic₂₆	—	2	8	no	—
156	C₈×C₁₃	—	2	2	no	—
157	C₈×C₁₃	—	2	2	no	—
158	C₈×C₁₃	—	2	2	no	—
159	C₈×C₁₃	—	2	2	no	—
160	#1055	—	2	4	no	—
161	#1055	—	2	4	no	—
162	C₂×C₄×Q₈	—	13	1	no	—
163	#446	—	13	8	no	—
164	C₈×Q₈	—	13	8	no	—
165	C₂×C₄⋊C₈	—	13	2	no	—
166	C₂×C₄⋊C₈	—	13	2	no	—
167	C₂×C₄⋊C₈	—	13	1	no	—
168	C₂×C₄⋊C₈	—	13	1	no	—
169	C₂×#97	—	13	2	no	—
170	C₂×#97	—	13	2	no	—
171	C₂×C₄×C₈	—	13	2	no	—
172	C₂×C₄×C₈	—	13	2	no	—
173	C₂×C₄×C₈	—	13	2	no	—
174	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
175	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
176	#391	—	13	2	no	—
177	#389	—	13	2	no	—
178	#387	—	13	4	no	—
179	C₄×C₄⋊C₄	—	13	2	no	—
180	C₄×C₄⋊C₄	—	13	1	no	—
181	#376	—	13	2	no	—
182	C₄³	C₂×D₁₃	1	1	yes	characteristic
183	#336	—	13	2	no	—
184	#336	—	13	2	no	—
185	#336	—	13	2	no	—
186	#336	C₂×D₁₃	1	1	yes	characteristic
187	#336	C₂×D₁₃	1	1	yes	characteristic
188	#336	C₂×D₁₃	1	1	yes	characteristic
189	#336	C₂×D₁₃	1	1	yes	characteristic
190	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
191	C₂²×C₁₃	#519	1	1	yes	characteristic
192	C₂²×C₁₃	#520	1	1	yes	characteristic
193	C₂²×C₁₃	C₂×#131	1	2	yes	—
194	C₂²×C₁₃	C₂×#131	1	2	yes	—
195	C₄×C₁₃	—	2	2	no	—
196	C₄×C₁₃	—	2	4	no	—
197	C₄×C₁₃	—	4	2	no	—
198	C₄×C₁₃	#405	1	4	yes	—
199	C₄×C₁₃	#519	1	1	yes	characteristic
200	C₄×C₁₃	#519	1	1	yes	characteristic
201	C₄×C₁₃	#520	1	1	yes	characteristic
202	C₄×C₁₃	#520	1	1	yes	characteristic
203	Dic₁₃	—	2	4	no	—
204	Dic₁₃	—	2	4	no	—
205	Dic₁₃	—	4	4	no	—
206	C₂²×Q₈	—	13	1	no	—
207	C₂²×C₈	—	13	2	no	—
208	C₂²×C₈	—	13	2	no	—
209	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
210	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
211	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
212	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
213	C₄×Q₈	—	13	4	no	—
214	C₄×Q₈	—	13	4	no	—
215	C₂×C₄⋊C₄	—	13	2	no	—
216	C₂×C₄⋊C₄	—	13	1	no	—
217	C₂×C₄⋊C₄	—	13	4	no	—
218	C₂×C₄⋊C₄	—	13	2	no	—
219	C₂×C₄²	—	13	2	no	—
220	C₂×C₄²	—	13	2	no	—
221	C₂×C₄²	C₄×D₁₃	1	2	yes	—
222	C₂×C₄²	C₄×D₁₃	1	2	yes	—
223	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
224	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
225	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
226	C₄⋊C₈	—	13	8	no	—
227	C₄⋊C₈	—	13	8	no	—
228	C₄⋊C₈	—	13	4	no	—
229	C₄⋊C₈	—	13	4	no	—
230	#97	—	13	8	no	—
231	#97	—	52	2	no	—
232	C₄×C₈	—	13	8	no	—
233	C₄×C₈	—	52	2	no	—
234	C₄×C₈	—	52	2	no	—
235	C₄×C₈	C₄×D₁₃	1	4	yes	—
236	C₄×C₈	C₄×D₁₃	1	4	yes	—
237	#95	—	13	4	no	—
238	#95	—	13	4	no	—
239	#95	—	13	2	no	—
240	#95	—	13	4	no	—
241	C₂×C₁₃	#2276	1	1	yes	characteristic
242	C₂×C₁₃	C₂×#405	1	1	yes	characteristic
243	C₂×C₁₃	C₄×#131	1	1	yes	characteristic
244	C₂×C₁₃	#2986	1	2	yes	—
245	C₂×C₁₃	#2994	1	2	yes	—
246	C₂×Q₈	—	13	4	no	—
247	C₂×Q₈	—	26	4	no	—
248	C₂²×C₄	—	13	2	no	—
249	C₂²×C₄	—	13	2	no	—
250	C₂²×C₄	—	13	4	no	—
251	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
252	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
254	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
255	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	13	4	no	—
265	C₂×C₈	—	13	4	no	—
266	C₂×C₈	—	26	2	no	—
267	C₂×C₈	—	26	2	no	—
268	C₂×C₈	—	26	2	no	—
269	C₂×C₈	—	26	2	no	—
270	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
271	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
272	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
273	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
274	C₄⋊C₄	—	13	8	no	—
275	C₄⋊C₄	—	13	4	no	—
276	C₄⋊C₄	—	26	8	no	—
277	C₄⋊C₄	—	26	4	no	—
278	C₄²	—	2	4	no	—
279	C₄²	—	2	4	no	—
280	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	2	1	no	—
284	C₄²	—	13	4	no	—
285	C₄²	—	13	4	no	—
286	C₄²	—	26	2	no	—
287	C₄²	—	26	2	no	—
288	C₄²	C₂×C₄×D₁₃	1	2	yes	—
289	C₄²	C₂×C₄×D₁₃	1	2	yes	—
290	C₁₃	#20571	1	1	yes	characteristic, complemented, Sylow
291	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
292	Q₈	—	26	8	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	13	4	no	—
300	C₂×C₄	—	13	4	no	—
301	C₂×C₄	—	26	4	no	—
302	C₂×C₄	—	26	4	no	—
303	C₂×C₄	—	26	4	no	—
304	C₂×C₄	—	26	4	no	—
305	C₂×C₄	—	26	4	no	—
306	C₂×C₄	C₄²×D₁₃	1	2	yes	—
307	C₂×C₄	C₄²×D₁₃	1	2	yes	—
308	C₂×C₄	#88155	1	4	yes	—
309	C₂×C₄	#88161	1	4	yes	—
310	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
312	C₂×C₄	#88256	1	1	yes	characteristic
313	C₂×C₄	#88257	1	1	yes	characteristic
314	C₂×C₄	#88257	1	1	yes	characteristic
315	C₂×C₄	#88257	1	1	yes	characteristic
316	C₈	—	2	2	no	—
317	C₈	—	2	2	no	—
318	C₈	—	2	2	no	—
319	C₈	—	2	2	no	—
320	C₈	—	26	4	no	—
321	C₈	—	26	4	no	—
322	C₂²	C₂×C₄²×D₁₃	1	1	yes	characteristic
323	C₂²	#11719221	1	1	yes	characteristic
324	C₂²	#11719228	1	1	yes	characteristic
325	C₂²	C₂×#88155	1	2	yes	—
326	C₂²	C₂×#88161	1	2	yes	—
327	C₄	—	2	4	no	—
328	C₄	—	2	2	no	—
329	C₄	—	4	2	no	—
330	C₄	—	26	4	no	—
331	C₄	—	26	4	no	—
332	C₄	—	52	4	no	—
333	C₄	#11718369	1	4	yes	—
334	C₄	#11719221	1	1	yes	characteristic
335	C₄	#11719221	1	1	yes	characteristic
336	C₄	#11719228	1	1	yes	characteristic
337	C₄	#11719228	1	1	yes	characteristic
338	C₂	#13117199	1	1	yes	characteristic
339	C₂	C₂×#11718369	1	1	yes	characteristic
340	C₂	C₄×#88155	1	1	yes	characteristic
341	C₂	#13123937	1	2	yes	—
342	C₂	#13123998	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂²×C₄²
Fitting	C₁₃×#1744	C₂
socle, prime radical	C₂³×C₁₃	C₂³×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#1744	D₁₃
hypocenter	C₁₃	#20571
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1744

Complements

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

N	H ≅ G/N	type
31. #11718340	C₄	semidirect
32. C₈×Dic₂₆	C₄	semidirect
290. C₁₃	#20571	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718340	2
C₂×C₈×Dic₂₆	2
#13117313	2
#13114707	2
#13114703	2
C₈×C₄⋊Dic₁₃	2
#13114697	2
#13114692	2
C₄×C₄⋊#1055	2
#13114586	2
C₁₃×#1744	2
#20571	13