Subgroups — #15756754

Order 3328.

556 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
2	#13114816	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718049	C₂	1	2	yes	maximal, complemented
4	#13112671	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
7	C₂³×#1055	—	2	1	no	—
8	C₂×C₁₃×C₂²⋊C₈	C₄	1	1	yes	characteristic
9	C₂×C₁₃×C₂²⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
11	C₂×C₂²⋊#1055	—	2	1	no	—
12	C₂×C₄⋊#1055	—	2	4	no	—
13	C₂×C₄⋊#1055	—	2	1	no	—
14	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊#1055	C₂²	1	2	yes	—
17	C₁₃×#336	C₄	1	2	yes	—
18	#11718116	—	2	1	no	—
19	#11718049	—	2	4	no	—
20	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₃	Q₈	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₁₃	—	2	2	no	—
26	C₂×C₄²×C₁₃	—	2	2	no	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×#1055	—	2	4	no	—
32	C₂²×#1055	—	2	2	no	—
33	C₂²×#1055	—	2	1	no	—
34	C₂²×#1055	—	2	1	no	—
35	C₂²×#1055	—	4	2	no	—
36	C₂²×#1055	—	4	2	no	—
37	C₁₃×C₂²⋊C₈	—	2	2	no	—
38	C₁₃×C₂²⋊C₈	C₈	1	4	yes	complemented
39	C₄×C₈×C₁₃	—	2	2	no	—
40	C₄×C₈×C₁₃	—	2	2	no	—
41	C₂²⋊#1055	—	4	2	no	—
42	C₄⋊#1055	—	2	4	no	—
43	C₄⋊#1055	—	2	2	no	—
44	C₄⋊#1055	—	2	2	no	—
45	C₄⋊#1055	—	2	2	no	—
46	C₄⋊#1055	D₄	1	4	yes	—
47	#8830	—	13	1	no	maximal, Sylow
48	C₂⁴×C₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	—	2	2	no	—
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	—	2	1	no	—
52	C₂²×C₄×C₁₃	—	2	2	no	—
53	C₂²×C₄×C₁₃	—	2	1	no	—
54	C₂²×C₄×C₁₃	—	2	1	no	—
55	C₂²×C₄×C₁₃	—	2	2	no	—
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	—	4	1	no	—
58	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
63	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₈×C₁₃	—	2	2	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	complemented
74	C₄²×C₁₃	—	2	2	no	—
75	C₄²×C₁₃	—	2	1	no	—
76	C₄²×C₁₃	—	2	2	no	—
77	C₄²×C₁₃	—	2	1	no	—
78	C₄²×C₁₃	C₂×D₄	1	2	yes	—
79	C₄²×C₁₃	#41	1	2	yes	—
80	C₂×#1055	—	2	4	no	—
81	C₂×#1055	—	4	8	no	—
82	C₂×#1055	—	4	2	no	—
83	C₂×#1055	—	4	2	no	—
84	C₂×#1055	—	4	1	no	—
85	C₂×#1055	—	4	1	no	—
86	C₂×#1055	—	4	1	no	—
87	C₂×#1055	—	4	1	no	—
88	C₂×#1055	—	4	1	no	—
89	C₂×#1055	—	4	1	no	—
90	C₂²×C₄⋊C₈	—	13	1	no	—
91	#1797	—	13	1	no	—
92	C₄×C₂²⋊C₈	D₁₃	1	1	yes	characteristic
93	C₂×#334	—	13	2	no	—
94	#1277	—	13	2	no	—
95	C₂³×C₁₃	—	2	1	no	—
96	C₂³×C₁₃	—	2	1	no	—
97	C₂³×C₁₃	—	2	1	no	—
98	C₂³×C₁₃	—	2	1	no	—
99	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
100	C₂³×C₁₃	#106	1	2	yes	—
101	C₂³×C₁₃	#108	1	2	yes	—
102	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	1	no	—
108	C₂×C₄×C₁₃	—	2	1	no	—
109	C₂×C₄×C₁₃	—	2	4	no	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	—	4	4	no	—
119	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
120	C₂×C₄×C₁₃	#106	1	2	yes	—
121	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₁₃	#131	1	2	yes	—
128	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
132	C₈×C₁₃	—	2	4	no	—
133	C₈×C₁₃	—	4	2	no	—
134	#1055	—	4	4	no	—
135	#1055	—	8	1	no	—
136	#1055	—	8	1	no	—
137	C₂³×C₈	—	26	1	no	—
138	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
139	C₂×C₄⋊C₈	—	13	2	no	—
140	C₂×C₄⋊C₈	—	13	1	no	—
141	C₂×C₄⋊C₈	—	13	1	no	—
142	C₂×C₄⋊C₈	—	26	4	no	—
143	C₂×C₄⋊C₈	—	26	1	no	—
144	C₂×C₂²⋊C₈	—	26	1	no	—
145	C₂×C₂²⋊C₈	Dic₁₃	1	1	yes	characteristic
146	C₂×C₂²⋊C₈	Dic₁₃	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
148	#336	—	26	1	no	—
149	#336	Dic₁₃	1	2	yes	—
150	#334	—	26	4	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	—	2	2	no	—
154	C₂²×C₁₃	—	2	2	no	—
155	C₂²×C₁₃	—	2	2	no	—
156	C₂²×C₁₃	—	2	2	no	—
157	C₂²×C₁₃	#334	1	4	yes	complemented
158	C₂²×C₁₃	#385	1	1	yes	characteristic
159	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
160	C₂²×C₁₃	#424	1	1	yes	characteristic
161	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
162	C₂²×C₁₃	#428	1	1	yes	characteristic
163	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
164	C₂²×C₁₃	#430	1	1	yes	characteristic
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	2	2	no	—
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	—	4	1	no	—
169	C₄×C₁₃	C₈×D₄	1	2	yes	—
170	C₄×C₁₃	#435	1	2	yes	—
171	C₄×C₁₃	#448	1	1	yes	characteristic
172	C₄×C₁₃	#450	1	1	yes	characteristic
173	C₄×C₁₃	#481	1	1	yes	characteristic
174	C₄×C₁₃	#483	1	1	yes	characteristic
175	C₂³×C₄	Dic₂₆	1	1	yes	characteristic
176	C₂³×C₄	D₅₂	1	1	yes	characteristic
177	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
178	C₂²×C₈	—	26	4	no	—
179	C₂²×C₈	—	26	2	no	—
180	C₂²×C₈	—	26	1	no	—
181	C₂²×C₈	—	26	1	no	—
182	C₂²×C₈	—	52	2	no	—
183	C₂²×C₈	—	52	2	no	—
184	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
185	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
189	C₂×C₄²	D₅₂	1	1	yes	characteristic
190	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
191	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
192	C₄⋊C₈	—	13	4	no	—
193	C₄⋊C₈	—	26	4	no	—
194	C₄⋊C₈	—	26	2	no	—
195	C₄⋊C₈	—	26	2	no	—
196	C₄⋊C₈	—	26	2	no	—
197	C₂²⋊C₈	—	2	2	no	—
198	C₂²⋊C₈	—	52	2	no	—
199	C₂²⋊C₈	#1055	1	4	yes	complemented
200	C₄×C₈	—	2	2	no	—
201	C₄×C₈	—	2	2	no	—
202	C₂×C₁₃	—	2	4	no	—
203	C₂×C₁₃	C₂×#334	1	1	yes	characteristic
204	C₂×C₁₃	#1565	1	1	yes	characteristic
205	C₂×C₁₃	#1570	1	1	yes	characteristic
206	C₂×C₁₃	#1573	1	1	yes	characteristic
207	C₂×C₁₃	#1850	1	1	yes	characteristic
208	C₂×C₁₃	#1853	1	1	yes	characteristic
209	C₂×C₁₃	#1855	1	1	yes	characteristic
210	C₂⁴	C₄⋊Dic₁₃	1	1	yes	characteristic
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	4	1	no	—
220	C₂²×C₄	#6095	1	2	yes	—
221	C₂²×C₄	#6095	1	2	yes	—
222	C₂²×C₄	C₂×#1055	1	2	yes	—
223	C₂²×C₄	#6099	1	2	yes	—
224	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
225	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
226	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
229	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	26	4	no	—
236	C₂×C₈	—	52	8	no	—
237	C₂×C₈	—	52	2	no	—
238	C₂×C₈	—	52	2	no	—
239	C₂×C₈	—	52	1	no	—
240	C₂×C₈	—	52	1	no	—
241	C₂×C₈	—	52	1	no	—
242	C₂×C₈	—	52	1	no	—
243	C₂×C₈	—	52	1	no	—
244	C₂×C₈	—	52	1	no	—
245	C₂×C₈	C₂×#1055	1	4	yes	complemented
246	C₄²	—	2	2	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	2	1	no	—
249	C₄²	—	2	2	no	—
250	C₄²	D₄×D₁₃	1	2	yes	—
251	C₄²	#6129	1	2	yes	—
252	C₁₃	#8830	1	1	yes	characteristic, complemented, Sylow
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	2	1	no	—
257	C₂³	C₄⋊#1055	1	2	yes	—
258	C₂³	#88057	1	2	yes	—
259	C₂³	#88059	1	2	yes	—
260	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	C₄⋊#1055	1	2	yes	—
278	C₂×C₄	#88057	1	2	yes	—
279	C₂×C₄	#88132	1	2	yes	—
280	C₂×C₄	#88136	1	1	yes	characteristic
281	C₂×C₄	#88140	1	1	yes	characteristic
282	C₂×C₄	C₂×#6095	1	1	yes	characteristic
283	C₂×C₄	C₂×#6095	1	1	yes	characteristic
284	C₂×C₄	#88162	1	1	yes	characteristic
285	C₂×C₄	#88163	1	1	yes	characteristic
286	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
287	C₂×C₄	C₂×#6099	1	1	yes	characteristic
288	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
289	C₂×C₄	#88202	1	2	yes	—
290	C₈	—	2	4	no	—
291	C₈	—	4	2	no	—
292	C₈	—	52	4	no	—
293	C₈	—	104	1	no	—
294	C₈	—	104	1	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	#11718049	1	4	yes	complemented
302	C₂²	C₂×C₄⋊#1055	1	1	yes	characteristic
303	C₂²	#11718611	1	1	yes	characteristic
304	C₂²	#11718660	1	1	yes	characteristic
305	C₂²	C₂×#88057	1	1	yes	characteristic
306	C₂²	C₂×#88059	1	1	yes	characteristic
307	C₂²	#11718782	1	1	yes	characteristic
308	C₂²	#11718786	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	4	1	no	—
313	C₄	#11718382	1	1	yes	characteristic
314	C₄	#11718392	1	1	yes	characteristic
315	C₄	#11718396	1	1	yes	characteristic
316	C₄	#11718399	1	1	yes	characteristic
317	C₄	D₄×#1055	1	2	yes	—
318	C₄	#11718672	1	2	yes	—
319	C₂	—	2	4	no	—
320	C₂	C₂×#11718049	1	1	yes	characteristic
321	C₂	#13113588	1	1	yes	characteristic
322	C₂	#13113607	1	1	yes	characteristic
323	C₂	#13113621	1	1	yes	characteristic
324	C₂	#13115016	1	1	yes	characteristic
325	C₂	#13115031	1	1	yes	characteristic
326	C₂	#13115043	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×C₂²⋊C₈	C₂
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×C₂²⋊C₈	D₁₃
hypocenter	C₁₃	#8830
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#11718049	C₂	semidirect
4. #13112671	C₂	semidirect
38. C₁₃×C₂²⋊C₈	C₈	semidirect
73. C₂×C₈×C₁₃	C₂×C₈	semidirect
157. C₂²×C₁₃	#334	semidirect
199. C₂²⋊C₈	#1055	semidirect
245. C₂×C₈	C₂×#1055	semidirect
252. C₁₃	#8830	semidirect
301. C₂²	#11718049	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊#1055	2
#13114816	2
C₂×#11718049	2
#13112671	2
C₄×C₁₃×C₂²⋊C₈	2
#8830	13