Subgroups — #15751013

Order 3328.

406 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13117313	C₂	1	1	yes	maximal, characteristic
2	#13114953	C₂	1	1	yes	maximal, characteristic
3	#13112661	C₂	1	2	yes	maximal
4	#13112658	C₂	1	2	yes	maximal
5	C₁₃×#1839	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₂₆	C₄	1	1	yes	characteristic
7	C₄³×C₁₃	C₂²	1	1	yes	characteristic
8	C₄×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
9	#11718594	—	2	1	no	—
10	#11718594	—	2	1	no	—
11	#11718593	—	2	1	no	—
12	C₄×#6101	—	2	1	no	—
13	#11718590	C₄	1	1	yes	characteristic
14	#11718590	C₂²	1	1	yes	characteristic
15	#11718589	C₄	1	1	yes	characteristic
16	#11718589	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊#1055	—	2	1	no	—
18	C₁₃×#336	—	2	1	no	—
19	C₁₃×#336	—	2	1	no	—
20	C₁₃×#336	C₂²	1	2	yes	—
21	#11718116	—	2	1	no	—
22	#11718116	C₂²	1	2	yes	—
23	C₂²×Dic₂₆	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×#6101	—	2	1	no	—
36	C₂×#6101	—	2	1	no	—
37	C₂×#6101	—	2	1	no	—
38	C₂×C₄×Dic₁₃	—	2	1	no	—
39	C₂×C₄×Dic₁₃	—	2	1	no	—
40	C₂²×#1055	—	2	2	no	—
41	C₄×Dic₂₆	—	2	2	no	—
42	C₄×Dic₂₆	C₈	1	4	yes	complemented
43	#88071	—	2	1	no	—
44	#88071	—	2	1	no	—
45	#88071	—	2	1	no	—
46	#88071	—	2	1	no	—
47	#88071	—	2	1	no	—
48	#88071	—	2	1	no	—
49	#88071	—	2	1	no	—
50	C₄⋊#1055	—	4	2	no	—
51	#7596	—	13	1	no	maximal, Sylow
52	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₁₃	—	2	1	no	—
60	C₂²×Dic₁₃	—	2	1	no	—
61	C₂²×Dic₁₃	—	2	1	no	—
62	C₂²×Dic₁₃	—	2	1	no	—
63	C₂×Dic₂₆	—	2	1	no	—
64	C₂×Dic₂₆	—	4	1	no	—
65	C₂×Dic₂₆	—	4	1	no	—
66	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
67	C₂×C₈×C₁₃	—	4	2	no	—
68	C₂×C₈×C₁₃	—	4	2	no	—
69	C₂×C₈×C₁₃	—	4	2	no	—
70	C₄²×C₁₃	—	2	1	no	—
71	C₄²×C₁₃	—	2	1	no	—
72	C₄²×C₁₃	—	2	2	no	—
73	C₄²×C₁₃	—	4	1	no	—
74	C₄²×C₁₃	—	4	1	no	—
75	C₄²×C₁₃	—	4	1	no	—
76	C₄²×C₁₃	—	4	1	no	—
77	C₄²×C₁₃	C₂×C₈	1	2	yes	—
78	C₄²×C₁₃	M₁₆	1	2	yes	—
79	C₄⋊Dic₁₃	—	2	1	no	—
80	C₄⋊Dic₁₃	—	4	1	no	—
81	C₄⋊Dic₁₃	—	4	1	no	—
82	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
83	#6101	—	2	4	no	—
84	#6101	—	4	2	no	—
85	#6101	—	4	1	no	—
86	#6101	—	4	1	no	—
87	C₄×Dic₁₃	—	2	4	no	—
88	C₄×Dic₁₃	—	4	2	no	—
89	C₂×#1055	—	4	2	no	—
90	C₂×#1055	—	4	2	no	—
91	C₂×#1055	—	4	2	no	—
92	#2295	—	13	1	no	—
93	#1839	D₁₃	1	1	yes	characteristic
94	#1831	—	13	1	no	—
95	#1272	—	13	2	no	—
96	#1271	—	13	2	no	—
97	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
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	1	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
115	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
116	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
120	C₂×C₄×C₁₃	#137	1	2	yes	—
121	C₂×Dic₁₃	—	2	2	no	—
122	C₂×Dic₁₃	—	2	2	no	—
123	C₂×Dic₁₃	—	4	1	no	—
124	C₂×Dic₁₃	—	4	1	no	—
125	C₂×Dic₁₃	—	4	1	no	—
126	C₂×Dic₁₃	—	4	1	no	—
127	C₂×Dic₁₃	—	4	1	no	—
128	C₂×Dic₁₃	—	4	1	no	—
129	C₂×Dic₁₃	—	4	1	no	—
130	C₂×Dic₁₃	—	4	1	no	—
131	C₂×Dic₁₃	—	4	1	no	—
132	C₂×Dic₁₃	—	4	1	no	—
133	Dic₂₆	—	4	2	no	—
134	Dic₂₆	—	4	2	no	—
135	C₈×C₁₃	—	8	2	no	—
136	#1055	—	8	2	no	—
137	C₂×C₄×Q₈	—	13	1	no	—
138	C₂×C₄⋊C₈	—	26	1	no	—
139	#391	—	13	1	no	—
140	#391	—	13	1	no	—
141	#389	—	13	1	no	—
142	#389	—	13	1	no	—
143	#387	—	26	1	no	—
144	#387	—	26	1	no	—
145	C₄×C₄⋊C₄	—	13	1	no	—
146	C₄×C₄⋊C₄	—	26	1	no	—
147	#376	—	26	1	no	—
148	C₄³	C₂×D₁₃	1	1	yes	characteristic
149	#336	—	2	1	no	—
150	#336	—	2	1	no	—
151	#336	—	13	2	no	—
152	#336	—	26	1	no	—
153	#336	C₂×D₁₃	1	2	yes	—
154	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
155	C₂²×C₁₃	#410	1	1	yes	characteristic
156	C₂²×C₁₃	#413	1	1	yes	characteristic
157	C₂²×C₁₃	#417	1	1	yes	characteristic
158	C₂²×C₁₃	#419	1	1	yes	characteristic
159	C₂²×C₁₃	#421	1	1	yes	characteristic
160	C₂²×C₁₃	#421	1	1	yes	characteristic
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	—	2	2	no	—
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	—	4	1	no	—
169	C₄×C₁₃	—	4	1	no	—
170	Dic₁₃	—	4	2	no	—
171	Dic₁₃	—	4	2	no	—
172	Dic₁₃	—	8	1	no	—
173	Dic₁₃	—	8	1	no	—
174	C₂²×Q₈	—	13	1	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	26	2	no	—
177	C₄×Q₈	—	13	4	no	—
178	C₄×Q₈	—	26	2	no	—
179	C₂×C₄⋊C₄	—	13	1	no	—
180	C₂×C₄⋊C₄	—	13	1	no	—
181	C₂×C₄⋊C₄	—	13	1	no	—
182	C₂×C₄⋊C₄	—	26	1	no	—
183	C₂×C₄⋊C₄	—	26	1	no	—
184	C₂×C₄⋊C₄	—	26	1	no	—
185	C₂×C₄²	—	26	1	no	—
186	C₂×C₄²	—	26	1	no	—
187	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
188	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
189	C₂×C₄²	D₅₂	1	1	yes	characteristic
190	C₂×C₄²	D₅₂	1	1	yes	characteristic
191	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
192	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
193	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
194	C₄⋊C₈	—	52	2	no	—
195	#95	—	26	1	no	—
196	#95	—	26	1	no	—
197	#95	—	26	1	no	—
198	#95	—	26	1	no	—
199	#95	—	26	1	no	—
200	#95	—	26	1	no	—
201	#95	—	26	1	no	—
202	C₂×C₁₃	#1458	1	1	yes	characteristic
203	C₂×C₁₃	#1477	1	1	yes	characteristic
204	C₂×C₁₃	#1479	1	1	yes	characteristic
205	C₂×C₁₃	#1512	1	1	yes	characteristic
206	C₂×C₁₃	#1515	1	1	yes	characteristic
207	C₂×C₁₃	#1535	1	1	yes	characteristic
208	C₂×C₁₃	#1559	1	1	yes	characteristic
209	C₂×Q₈	—	13	2	no	—
210	C₂×Q₈	—	26	1	no	—
211	C₂×Q₈	—	52	1	no	—
212	C₂×Q₈	—	52	1	no	—
213	C₂²×C₄	—	26	1	no	—
214	C₂²×C₄	—	26	1	no	—
215	C₂²×C₄	—	26	1	no	—
216	C₂²×C₄	—	26	1	no	—
217	C₂²×C₄	#6103	1	1	yes	characteristic
218	C₂²×C₄	#6103	1	1	yes	characteristic
219	C₂²×C₄	#6103	1	1	yes	characteristic
220	C₂²×C₄	#6103	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	52	2	no	—
228	C₂×C₈	—	52	2	no	—
229	C₂×C₈	—	52	2	no	—
230	C₄⋊C₄	—	13	2	no	—
231	C₄⋊C₄	—	26	1	no	—
232	C₄⋊C₄	—	26	4	no	—
233	C₄⋊C₄	—	52	1	no	—
234	C₄⋊C₄	—	52	1	no	—
235	C₄⋊C₄	—	52	2	no	—
236	C₄⋊C₄	—	52	1	no	—
237	C₄⋊C₄	—	52	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	4	1	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	4	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	26	4	no	—
246	C₄²	—	52	2	no	—
247	C₄²	C₈×D₁₃	1	2	yes	—
248	C₄²	#6094	1	2	yes	—
249	C₁₃	#7596	1	1	yes	characteristic, complemented, Sylow
250	C₂³	C₂×#6103	1	1	yes	characteristic
251	Q₈	—	52	2	no	—
252	Q₈	—	52	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
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	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	26	2	no	—
270	C₂×C₄	—	26	2	no	—
271	C₂×C₄	—	52	1	no	—
272	C₂×C₄	—	52	1	no	—
273	C₂×C₄	—	52	1	no	—
274	C₂×C₄	—	52	1	no	—
275	C₂×C₄	—	52	1	no	—
276	C₂×C₄	—	52	1	no	—
277	C₂×C₄	—	52	1	no	—
278	C₂×C₄	—	52	1	no	—
279	C₂×C₄	—	52	1	no	—
280	C₂×C₄	—	52	1	no	—
281	C₂×C₄	#88060	1	2	yes	—
282	C₂×C₄	#88060	1	2	yes	—
283	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
284	C₂×C₄	C₂×#6094	1	1	yes	characteristic
285	C₂×C₄	#88162	1	1	yes	characteristic
286	C₂×C₄	#88163	1	1	yes	characteristic
287	C₂×C₄	#88205	1	2	yes	—
288	C₈	—	8	2	no	—
289	C₈	—	104	2	no	—
290	C₂²	#11718665	1	1	yes	characteristic
291	C₂²	#11718666	1	1	yes	characteristic
292	C₂²	#11718667	1	1	yes	characteristic
293	C₂²	C₂×#88060	1	1	yes	characteristic
294	C₂²	#11718793	1	1	yes	characteristic
295	C₂²	#11718794	1	1	yes	characteristic
296	C₂²	#11718800	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	52	2	no	—
307	C₄	—	52	2	no	—
308	C₄	—	104	1	no	—
309	C₄	—	104	1	no	—
310	C₂	#13113148	1	1	yes	characteristic
311	C₂	#13113224	1	1	yes	characteristic
312	C₂	#13113230	1	1	yes	characteristic
313	C₂	#13113375	1	1	yes	characteristic
314	C₂	#13113386	1	1	yes	characteristic
315	C₂	#13113468	1	1	yes	characteristic
316	C₂	#13113567	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6103
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1839	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1839	D₁₃
hypocenter	C₁₃	#7596
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1839

Complements

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

N	H ≅ G/N	type
42. C₄×Dic₂₆	C₈	semidirect
249. C₁₃	#7596	semidirect

Maximal subgroups

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

subgroup	index
#13117313	2
#13114953	2
#13112661	2
#13112658	2
C₁₃×#1839	2
#7596	13