Subgroups — #15881821

Order 3328.

876 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718491	C₂	1	2	yes	maximal
2	C₄²×Dic₂₆	C₂	1	1	yes	maximal, characteristic
3	#13115553	C₂	1	4	yes	maximal
4	#13115552	C₂	1	2	yes	maximal
5	#13114951	C₂	1	2	yes	maximal
6	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
7	#13114684	C₂	1	2	yes	maximal
8	C₁₃×#1831	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₂₆	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₂₆	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₄³×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×#88054	C₂²	1	4	yes	—
15	C₂×#88054	C₂²	1	4	yes	—
16	C₂×C₈×Dic₁₃	C₂²	1	4	yes	—
17	C₄×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
18	C₄×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
19	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
20	C₄×#6101	C₄	1	2	yes	—
21	C₄×#6101	C₄	1	2	yes	—
22	C₄×#6101	C₂²	1	2	yes	—
23	C₄²×Dic₁₃	C₂²	1	2	yes	—
24	C₂×C₄⋊#1055	C₂²	1	2	yes	—
25	C₂×C₄×#1055	C₂²	1	2	yes	—
26	#11718491	—	2	8	no	—
27	C₁₃×#336	C₂²	1	2	yes	—
28	C₁₃×#336	C₂²	1	2	yes	—
29	#11718142	C₂²	1	4	yes	—
30	#11718116	C₂²	1	1	yes	characteristic
31	#11718116	C₂²	1	1	yes	characteristic
32	C₂²×Dic₂₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×#6101	C₂×C₄	1	2	yes	—
45	C₂×#6101	C₂×C₄	1	2	yes	—
46	C₂×#6101	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
48	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
49	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
50	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
51	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
52	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
53	C₂²×#1055	C₂³	1	2	yes	—
54	C₂²×#1055	C₂³	1	2	yes	—
55	C₄×Dic₂₆	—	2	4	no	—
56	C₄×Dic₂₆	C₈	1	8	yes	complemented
57	C₄×Dic₂₆	D₄	1	4	yes	—
58	C₄×Dic₂₆	Q₈	1	4	yes	—
59	C₁₃×C₄⋊C₈	—	2	4	no	—
60	#88071	C₂×C₄	1	2	yes	—
61	#88071	C₂×C₄	1	2	yes	—
62	#88071	C₂×C₄	1	2	yes	—
63	#88054	—	2	8	no	—
64	#88054	—	2	8	no	—
65	C₈×Dic₁₃	—	2	8	no	—
66	C₄⋊#1055	—	2	4	no	—
67	C₄×#1055	—	2	4	no	—
68	#22734	—	13	1	no	maximal, Sylow
69	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
77	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
78	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
79	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
80	C₂×Dic₂₆	C₄⋊C₄	1	4	yes	—
81	C₂×Dic₂₆	C₂×C₈	1	4	yes	—
82	C₂×Dic₂₆	M₁₆	1	4	yes	—
83	C₂×C₈×C₁₃	—	2	4	no	—
84	C₂×C₈×C₁₃	—	2	4	no	—
85	C₂×C₈×C₁₃	—	2	4	no	—
86	C₄²×C₁₃	—	2	2	no	—
87	C₄²×C₁₃	—	2	2	no	—
88	C₄²×C₁₃	—	2	2	no	—
89	C₄²×C₁₃	—	2	2	no	—
90	C₄²×C₁₃	—	4	1	no	—
91	C₄²×C₁₃	C₂×C₈	1	4	yes	—
92	C₄²×C₁₃	C₂×D₄	1	2	yes	—
93	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
94	C₄⋊Dic₁₃	—	2	2	no	—
95	C₄⋊Dic₁₃	C₄⋊C₄	1	4	yes	—
96	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
97	#6101	—	2	4	no	—
98	#6101	C₄⋊C₄	1	8	yes	—
99	#6101	C₂×C₈	1	8	yes	—
100	C₄×Dic₁₃	—	2	4	no	—
101	C₄×Dic₁₃	—	2	4	no	—
102	C₄×Dic₁₃	—	2	4	no	—
103	C₄×Dic₁₃	—	4	2	no	—
104	C₄×Dic₁₃	C₂×C₈	1	8	yes	—
105	C₄×Dic₁₃	C₂×D₄	1	4	yes	—
106	C₄×Dic₁₃	C₂×Q₈	1	4	yes	—
107	C₂×#1055	—	2	2	no	—
108	C₂×#1055	—	2	2	no	—
109	C₂×#1055	—	2	2	no	—
110	C₂×#1055	—	2	2	no	—
111	C₂×#1055	—	2	2	no	—
112	C₂×#1055	#41	1	4	yes	—
113	C₂×C₈×Q₈	—	13	2	no	—
114	C₄²×Q₈	—	13	1	no	—
115	#1939	—	13	2	no	—
116	#1939	—	13	4	no	—
117	#1831	—	13	2	no	—
118	#1831	D₁₃	1	1	yes	characteristic
119	C₄×C₄⋊C₈	—	13	1	no	—
120	C₄×C₄⋊C₈	—	13	2	no	—
121	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	1	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	2	1	no	—
133	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
137	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
138	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
139	C₂×C₄×C₁₃	#131	1	2	yes	—
140	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
143	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
144	C₂×Dic₁₃	—	2	2	no	—
145	C₂×Dic₁₃	—	2	2	no	—
146	C₂×Dic₁₃	—	2	2	no	—
147	C₂×Dic₁₃	—	2	2	no	—
148	C₂×Dic₁₃	—	2	2	no	—
149	C₂×Dic₁₃	—	2	2	no	—
150	C₂×Dic₁₃	—	2	2	no	—
151	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
152	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
153	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
154	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
155	C₂×Dic₁₃	C₂×M₁₆	1	4	yes	—
156	Dic₂₆	C₄⋊C₈	1	16	yes	complemented
157	C₈×C₁₃	—	4	4	no	—
158	#1055	—	2	4	no	—
159	#1055	—	4	2	no	—
160	C₂×C₄×Q₈	—	13	1	no	—
161	C₂×C₄×Q₈	—	13	1	no	—
162	C₂×C₄×Q₈	—	13	1	no	—
163	C₈×Q₈	—	26	8	no	—
164	C₂×C₄⋊C₈	—	13	4	no	—
165	C₂×C₄⋊C₈	—	13	4	no	—
166	C₂×C₄⋊C₈	—	13	2	no	—
167	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
168	C₂×C₄×C₈	—	13	4	no	—
169	C₂×C₄×C₈	—	13	2	no	—
170	C₄×C₄⋊C₄	—	13	2	no	—
171	C₄×C₄⋊C₄	—	13	2	no	—
172	C₄×C₄⋊C₄	—	13	1	no	—
173	C₄×C₄⋊C₄	—	13	1	no	—
174	C₄×C₄⋊C₄	—	13	2	no	—
175	C₄×C₄⋊C₄	—	13	1	no	—
176	C₄³	—	13	2	no	—
177	C₄³	C₂×D₁₃	1	1	yes	characteristic
178	#336	—	13	4	no	—
179	#336	—	13	1	no	—
180	#336	—	13	1	no	—
181	#336	C₂×D₁₃	1	2	yes	—
182	#336	C₂×D₁₃	1	2	yes	—
183	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
184	C₂²×C₁₃	C₄×#41	1	1	yes	characteristic
185	C₂²×C₁₃	#519	1	1	yes	characteristic
186	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
187	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
188	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
189	C₂²×C₁₃	#568	1	1	yes	characteristic
190	C₄×C₁₃	—	2	2	no	—
191	C₄×C₁₃	—	2	2	no	—
192	C₄×C₁₃	—	2	2	no	—
193	C₄×C₁₃	—	2	2	no	—
194	C₄×C₁₃	—	4	1	no	—
195	C₄×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
196	C₄×C₁₃	#548	1	2	yes	—
197	C₄×C₁₃	#557	1	2	yes	—
198	Dic₁₃	—	2	4	no	—
199	Dic₁₃	—	2	4	no	—
200	Dic₁₃	—	4	2	no	—
201	Dic₁₃	C₂×C₄⋊C₈	1	8	yes	—
202	C₂²×Q₈	—	13	1	no	—
203	C₂²×C₈	—	13	2	no	—
204	C₂²×C₈	—	13	2	no	—
205	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
206	C₄×Q₈	—	13	8	no	—
207	C₄×Q₈	—	13	4	no	—
208	C₄×Q₈	—	13	4	no	—
209	C₄×Q₈	—	26	4	no	—
210	C₂×C₄⋊C₄	—	13	2	no	—
211	C₂×C₄⋊C₄	—	13	2	no	—
212	C₂×C₄⋊C₄	—	13	1	no	—
213	C₂×C₄⋊C₄	—	13	1	no	—
214	C₂×C₄⋊C₄	—	13	2	no	—
215	C₂×C₄⋊C₄	—	13	1	no	—
216	C₂×C₄²	—	13	2	no	—
217	C₂×C₄²	—	13	2	no	—
218	C₂×C₄²	—	13	2	no	—
219	C₂×C₄²	—	13	2	no	—
220	C₂×C₄²	—	13	2	no	—
221	C₂×C₄²	—	13	2	no	—
222	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
223	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
224	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
225	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
226	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
227	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
228	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
229	C₄⋊C₈	—	2	4	no	—
230	C₄⋊C₈	—	26	4	no	—
231	C₄⋊C₈	—	26	8	no	—
232	C₄⋊C₈	—	26	8	no	—
233	C₄×C₈	—	26	8	no	—
234	C₄×C₈	—	26	4	no	—
235	#95	—	13	2	no	—
236	#95	—	13	2	no	—
237	#95	—	13	2	no	—
238	C₂×C₁₃	#2351	1	1	yes	characteristic
239	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
240	C₂×C₁₃	#2906	1	1	yes	characteristic
241	C₂×C₁₃	C₈×#41	1	1	yes	characteristic
242	C₂×C₁₃	#2971	1	1	yes	characteristic
243	C₂×C₁₃	#2982	1	1	yes	characteristic
244	C₂×C₁₃	#2984	1	1	yes	characteristic
245	C₂×Q₈	—	13	4	no	—
246	C₂×Q₈	—	13	4	no	—
247	C₂×Q₈	—	13	4	no	—
248	C₂²×C₄	—	13	2	no	—
249	C₂²×C₄	—	13	2	no	—
250	C₂²×C₄	—	13	2	no	—
251	C₂²×C₄	—	13	2	no	—
252	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
253	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
258	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	13	4	no	—
263	C₂×C₈	—	26	2	no	—
264	C₂×C₈	—	26	2	no	—
265	C₂×C₈	—	26	2	no	—
266	C₂×C₈	—	26	2	no	—
267	C₂×C₈	—	26	2	no	—
268	C₄⋊C₄	—	13	8	no	—
269	C₄⋊C₄	—	13	4	no	—
270	C₄⋊C₄	—	13	8	no	—
271	C₄⋊C₄	—	13	4	no	—
272	C₄⋊C₄	—	26	4	no	—
273	C₄⋊C₄	—	26	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	13	8	no	—
280	C₄²	—	13	4	no	—
281	C₄²	—	13	4	no	—
282	C₄²	—	26	4	no	—
283	C₄²	—	26	4	no	—
284	C₄²	—	26	4	no	—
285	C₄²	—	52	2	no	—
286	C₄²	C₈×D₁₃	1	4	yes	—
287	C₄²	D₄×D₁₃	1	2	yes	—
288	C₄²	Q₈×D₁₃	1	2	yes	—
289	C₁₃	#22734	1	1	yes	characteristic, complemented, Sylow
290	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
291	Q₈	—	13	16	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	13	4	no	—
304	C₂×C₄	—	13	4	no	—
305	C₂×C₄	—	13	4	no	—
306	C₂×C₄	—	13	4	no	—
307	C₂×C₄	—	13	4	no	—
308	C₂×C₄	—	26	2	no	—
309	C₂×C₄	—	26	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	—	26	2	no	—
315	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
316	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
317	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
318	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
319	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
320	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
321	C₂×C₄	#88161	1	2	yes	—
322	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
323	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
324	C₂×C₄	#41×D₁₃	1	2	yes	—
325	C₂×C₄	#88257	1	2	yes	—
326	C₈	—	4	4	no	—
327	C₈	—	26	4	no	—
328	C₈	—	52	2	no	—
329	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
330	C₂²	#11719221	1	1	yes	characteristic
331	C₂²	#11719227	1	1	yes	characteristic
332	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
333	C₂²	C₂×M₁₆×D₁₃	1	1	yes	characteristic
334	C₂²	C₂×#88161	1	1	yes	characteristic
335	C₂²	#11719450	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	2	no	—
339	C₄	—	2	2	no	—
340	C₄	—	4	1	no	—
341	C₄	—	13	8	no	—
342	C₄	—	26	4	no	—
343	C₄	—	26	4	no	—
344	C₄	—	52	2	no	—
345	C₄	D₁₃×C₄⋊C₈	1	4	yes	—
346	C₄	#11719382	1	2	yes	—
347	C₄	#11719390	1	2	yes	—
348	C₂	#13117753	1	1	yes	characteristic
349	C₂	C₂×D₁₃×C₄⋊C₈	1	1	yes	characteristic
350	C₂	#13123405	1	1	yes	characteristic
351	C₂	#13123792	1	1	yes	characteristic
352	C₂	#13123853	1	1	yes	characteristic
353	C₂	#13123916	1	1	yes	characteristic
354	C₂	#13123926	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#1831	C₂
socle, prime radical	C₂³×C₁₃	C₂³×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#1831	D₁₃
hypocenter	C₁₃	#22734
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1831

Complements

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

N	H ≅ G/N	type
56. C₄×Dic₂₆	C₈	semidirect
156. Dic₂₆	C₄⋊C₈	semidirect
289. C₁₃	#22734	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718491	2
C₄²×Dic₂₆	2
#13115553	2
#13115552	2
#13114951	2
C₄×C₄⋊#1055	2
#13114684	2
C₁₃×#1831	2
#22734	13