Subgroups — #15767992

Order 3328.

478 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115642	C₂	1	1	yes	maximal, characteristic
2	#13115641	C₂	1	1	yes	maximal, characteristic
3	#13115641	C₂	1	1	yes	maximal, characteristic
4	#13114963	C₂	1	1	yes	maximal, characteristic, complemented
5	#13114963	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112663	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1834	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88128	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₁₃×#384	C₂²	1	1	yes	characteristic
11	C₁₃×#384	C₂²	1	1	yes	characteristic
12	C₂×#88061	—	2	2	no	—
13	C₂×#88049	—	2	2	no	—
14	C₂×#88049	—	2	2	no	—
15	#11718596	C₂²	1	1	yes	characteristic
16	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
17	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
18	C₁₃×#340	—	2	2	no	—
19	#11718121	—	2	2	no	—
20	#11718121	—	2	2	no	—
21	C₂²×D₅₂	—	2	1	no	—
22	C₂²×D₅₂	—	2	1	no	—
23	C₂²×D₅₂	—	2	1	no	—
24	C₂²×C₈×C₁₃	D₄	1	2	yes	—
25	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
28	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
29	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
31	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₃	—	2	1	no	—
33	C₂×C₄⋊Dic₁₃	—	2	1	no	—
34	C₂×C₄⋊Dic₁₃	—	2	1	no	—
35	C₂²×#1055	D₄	1	2	yes	—
36	C₂²×#1055	D₄	1	2	yes	—
37	#88128	—	2	1	no	—
38	#88128	—	2	1	no	—
39	#88128	—	2	1	no	—
40	#88128	Q₈	1	2	yes	—
41	C₄×C₈×C₁₃	—	2	1	no	—
42	C₄×C₈×C₁₃	—	2	1	no	—
43	#88061	—	4	4	no	—
44	#88049	—	4	4	no	—
45	#88049	—	4	4	no	—
46	C₄×#1055	—	2	1	no	—
47	C₄×#1055	—	2	1	no	—
48	C₄×#1055	—	2	1	no	—
49	C₄×#1055	—	2	1	no	—
50	#10678	—	13	1	no	maximal, Sylow
51	C₂³×D₁₃	—	4	1	no	—
52	C₂²×C₄×C₁₃	—	4	1	no	—
53	C₂²×C₄×C₁₃	—	4	1	no	—
54	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₁₃	—	4	1	no	—
58	C₂×D₅₂	—	2	2	no	—
59	C₂×D₅₂	—	2	2	no	—
60	C₂×D₅₂	—	2	2	no	—
61	C₂×D₅₂	—	4	2	no	—
62	C₂×D₅₂	—	4	2	no	—
63	C₂×D₅₂	—	4	2	no	—
64	C₂×D₅₂	—	4	2	no	—
65	C₂×D₅₂	—	4	2	no	—
66	C₂×D₅₂	—	4	2	no	—
67	C₂×D₅₂	—	4	1	no	—
68	C₂×D₅₂	—	4	1	no	—
69	C₂×D₅₂	—	4	1	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₁₃×C₄⋊C₄	—	4	2	no	—
74	C₁₃×C₄⋊C₄	—	4	2	no	—
75	C₁₃×C₄⋊C₄	—	4	2	no	—
76	C₁₃×C₄⋊C₄	—	4	2	no	—
77	C₁₃×C₄⋊C₄	—	4	2	no	—
78	C₁₃×C₄⋊C₄	—	4	2	no	—
79	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
80	C₄²×C₁₃	#41	1	1	yes	characteristic
81	C₄²×C₁₃	#41	1	1	yes	characteristic
82	C₄²×C₁₃	#41	1	1	yes	characteristic
83	C₄⋊Dic₁₃	—	4	2	no	—
84	C₄⋊Dic₁₃	—	4	2	no	—
85	C₄⋊Dic₁₃	—	4	2	no	—
86	C₂×#1055	—	2	2	no	—
87	C₂×#1055	—	2	2	no	—
88	C₂×#1055	—	2	2	no	—
89	C₂×#1055	—	2	2	no	—
90	C₂×#1055	—	2	2	no	—
91	C₂×#1055	—	2	2	no	—
92	#1957	—	13	1	no	—
93	#1957	—	13	1	no	—
94	#1957	—	13	1	no	—
95	#1834	—	13	1	no	—
96	#1834	—	13	1	no	—
97	#1834	D₁₃	1	1	yes	characteristic, complemented
98	#1273	—	13	1	no	—
99	C₂³×C₁₃	#127	1	1	yes	characteristic
100	C₂²×D₁₃	—	4	2	no	—
101	C₂²×D₁₃	—	8	1	no	—
102	C₂²×D₁₃	—	8	1	no	—
103	C₂²×D₁₃	—	8	1	no	—
104	C₂²×D₁₃	—	8	1	no	—
105	C₂²×D₁₃	—	8	1	no	—
106	C₂²×D₁₃	—	8	1	no	—
107	C₂×C₄×C₁₃	—	8	1	no	—
108	C₂×C₄×C₁₃	—	8	1	no	—
109	C₂×C₄×C₁₃	—	8	1	no	—
110	C₂×C₄×C₁₃	—	8	1	no	—
111	C₂×C₄×C₁₃	—	8	1	no	—
112	C₂×C₄×C₁₃	—	8	1	no	—
113	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#135	1	2	yes	—
120	C₂×C₄×C₁₃	#135	1	2	yes	—
121	C₂×C₄×C₁₃	#135	1	2	yes	—
122	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
124	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
125	C₂×Dic₁₃	—	8	1	no	—
126	C₂×Dic₁₃	—	8	1	no	—
127	C₂×Dic₁₃	—	8	1	no	—
128	D₅₂	—	4	4	no	—
129	D₅₂	—	4	4	no	—
130	D₅₂	—	4	4	no	—
131	D₅₂	—	8	2	no	—
132	D₅₂	—	8	2	no	—
133	D₅₂	—	8	2	no	—
134	C₈×C₁₃	—	4	2	no	—
135	#1055	—	4	2	no	—
136	#1055	—	4	2	no	—
137	C₂×#127	—	13	1	no	—
138	C₂×#102	—	26	2	no	—
139	C₂×#102	—	26	2	no	—
140	C₂×#102	—	26	2	no	—
141	C₂×C₄×C₈	—	13	1	no	—
142	C₂×C₄×C₈	—	13	1	no	—
143	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
144	#384	—	13	1	no	—
145	#384	C₂×D₁₃	1	1	yes	characteristic
146	#384	C₂×D₁₃	1	1	yes	characteristic
147	#340	—	2	2	no	—
148	#340	—	26	2	no	—
149	#340	—	26	2	no	—
150	C₂²×C₁₃	#399	1	1	yes	characteristic
151	C₂²×C₁₃	#495	1	1	yes	characteristic
152	C₂²×C₁₃	#495	1	1	yes	characteristic
153	C₂²×C₁₃	#495	1	1	yes	characteristic
154	C₂²×C₁₃	#496	1	1	yes	characteristic
155	C₂²×C₁₃	#496	1	1	yes	characteristic
156	C₂²×C₁₃	#496	1	1	yes	characteristic
157	C₂×D₁₃	—	8	2	no	—
158	C₂×D₁₃	—	8	2	no	—
159	C₂×D₁₃	—	8	2	no	—
160	C₂×D₁₃	—	16	1	no	—
161	C₂×D₁₃	—	16	1	no	—
162	C₂×D₁₃	—	16	1	no	—
163	C₂×D₁₃	—	16	1	no	—
164	C₄×C₁₃	—	16	1	no	—
165	C₄×C₁₃	—	16	1	no	—
166	C₄×C₁₃	#478	1	2	yes	—
167	C₄×C₁₃	#478	1	2	yes	—
168	C₄×C₁₃	#478	1	2	yes	—
169	C₄×C₁₃	#482	1	2	yes	—
170	C₄×C₁₃	#482	1	2	yes	—
171	C₄×C₁₃	#482	1	2	yes	—
172	Dic₁₃	—	16	1	no	—
173	C₂²×D₄	—	26	1	no	—
174	C₂²×D₄	—	26	1	no	—
175	C₂²×D₄	—	26	1	no	—
176	C₂²×C₈	—	13	2	no	—
177	C₂²×C₈	—	13	2	no	—
178	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
179	#127	—	13	2	no	—
180	#127	—	26	1	no	—
181	#127	—	26	1	no	—
182	#127	—	26	1	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	26	1	no	—
190	C₂×C₄⋊C₄	—	26	1	no	—
191	C₂×C₄⋊C₄	—	26	1	no	—
192	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
193	#102	—	52	4	no	—
194	#102	—	52	4	no	—
195	#102	—	52	4	no	—
196	C₄×C₈	—	2	1	no	—
197	C₄×C₈	—	2	1	no	—
198	C₄×C₈	—	26	1	no	—
199	C₄×C₈	—	26	1	no	—
200	C₄×C₈	—	26	1	no	—
201	C₄×C₈	—	26	1	no	—
202	C₂×C₁₃	#1711	1	1	yes	characteristic
203	C₂×C₁₃	#1717	1	1	yes	characteristic
204	C₂×C₁₃	#1717	1	1	yes	characteristic
205	C₂×C₁₃	#1717	1	1	yes	characteristic
206	C₂×C₁₃	#2092	1	1	yes	characteristic
207	C₂×C₁₃	#2092	1	1	yes	characteristic
208	C₂×C₁₃	#2092	1	1	yes	characteristic
209	D₁₃	—	16	2	no	—
210	C₂⁴	—	52	1	no	—
211	C₂×D₄	—	26	2	no	—
212	C₂×D₄	—	26	2	no	—
213	C₂×D₄	—	26	2	no	—
214	C₂×D₄	—	52	1	no	—
215	C₂×D₄	—	52	1	no	—
216	C₂×D₄	—	52	2	no	—
217	C₂×D₄	—	52	2	no	—
218	C₂×D₄	—	52	2	no	—
219	C₂×D₄	—	52	2	no	—
220	C₂×D₄	—	52	1	no	—
221	C₂×D₄	—	52	2	no	—
222	C₂×D₄	—	52	2	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	52	1	no	—
226	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
227	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	26	2	no	—
233	C₂×C₈	—	26	2	no	—
234	C₂×C₈	—	26	2	no	—
235	C₂×C₈	—	26	2	no	—
236	C₂×C₈	—	26	2	no	—
237	C₂×C₈	—	26	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	52	2	no	—
245	C₄⋊C₄	—	52	2	no	—
246	C₄⋊C₄	—	52	2	no	—
247	C₄²	#6127	1	1	yes	characteristic
248	C₄²	#6127	1	1	yes	characteristic
249	C₄²	Q₈×D₁₃	1	1	yes	characteristic
250	C₄²	#6131	1	1	yes	characteristic
251	C₁₃	#10678	1	1	yes	characteristic, complemented, Sylow
252	C₂³	—	52	2	no	—
253	C₂³	—	104	1	no	—
254	C₂³	—	104	1	no	—
255	C₂³	—	104	1	no	—
256	C₂³	—	104	1	no	—
257	C₂³	—	104	1	no	—
258	C₂³	—	104	1	no	—
259	C₂³	#88194	1	1	yes	characteristic
260	D₄	—	52	4	no	—
261	D₄	—	52	4	no	—
262	D₄	—	52	4	no	—
263	D₄	—	104	2	no	—
264	D₄	—	104	2	no	—
265	D₄	—	104	2	no	—
266	C₂×C₄	—	8	1	no	—
267	C₂×C₄	—	8	1	no	—
268	C₂×C₄	—	8	1	no	—
269	C₂×C₄	—	8	1	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	104	1	no	—
273	C₂×C₄	—	104	1	no	—
274	C₂×C₄	—	104	1	no	—
275	C₂×C₄	#88148	1	1	yes	characteristic
276	C₂×C₄	#88148	1	1	yes	characteristic
277	C₂×C₄	#88150	1	1	yes	characteristic
278	C₂×C₄	#88150	1	1	yes	characteristic
279	C₂×C₄	#88165	1	2	yes	—
280	C₂×C₄	#88168	1	2	yes	—
281	C₂×C₄	#88170	1	2	yes	—
282	C₂×C₄	#88173	1	2	yes	—
283	C₂×C₄	#88183	1	1	yes	characteristic
284	C₂×C₄	#88200	1	1	yes	characteristic
285	C₂×C₄	#88203	1	2	yes	—
286	C₂×C₄	#88204	1	2	yes	—
287	C₈	—	4	2	no	—
288	C₈	—	52	2	no	—
289	C₈	—	52	2	no	—
290	C₂²	—	104	2	no	—
291	C₂²	—	104	2	no	—
292	C₂²	—	104	2	no	—
293	C₂²	—	208	1	no	—
294	C₂²	—	208	1	no	—
295	C₂²	—	208	1	no	—
296	C₂²	—	208	1	no	—
297	C₂²	#11718653	1	1	yes	characteristic
298	C₂²	#11718719	1	1	yes	characteristic
299	C₂²	#11718720	1	1	yes	characteristic
300	C₂²	#11718815	1	1	yes	characteristic
301	C₂²	#11718829	1	1	yes	characteristic
302	C₂²	#11718837	1	1	yes	characteristic
303	C₂²	#11718852	1	1	yes	characteristic
304	C₄	—	16	1	no	—
305	C₄	—	16	1	no	—
306	C₄	—	208	1	no	—
307	C₄	#11718527	1	2	yes	—
308	C₄	#11718532	1	2	yes	—
309	C₄	#11718546	1	2	yes	—
310	C₄	#11718552	1	2	yes	—
311	C₄	#11718664	1	2	yes	—
312	C₄	#11718727	1	2	yes	—
313	C₂	—	208	2	no	—
314	C₂	#13114449	1	1	yes	characteristic
315	C₂	#13114476	1	1	yes	characteristic
316	C₂	#13114477	1	1	yes	characteristic
317	C₂	#13114478	1	1	yes	characteristic
318	C₂	#13116354	1	1	yes	characteristic
319	C₂	#13116358	1	1	yes	characteristic
320	C₂	#13116360	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88194
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1834	C₂
socle	C₂³×C₁₃	#127
prime radical	C₂×#88128	C₂²
hypercenter	#1834	D₁₃
hypocenter	C₁₃	#10678
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
4. #13114963	C₂	semidirect
5. #13114963	C₂	semidirect
6. #13112663	C₂	semidirect
7. C₁₃×#1834	C₂	semidirect
97. #1834	D₁₃	semidirect
251. C₁₃	#10678	semidirect

Maximal subgroups

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

subgroup	index
#13115642	2
#13115641	2
#13114963	2
#13112663	2
C₁₃×#1834	2
#10678	13