Subgroups — #15755634

Order 3328.

472 conjugacy classes of subgroups.

The subgroup lattice has 340 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	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718076	C₂	1	2	yes	maximal
4	#13112675	C₂	1	2	yes	maximal
5	C₁₃×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₂₆	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
8	C₄³×C₁₃	C₂²	1	1	yes	characteristic
9	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
10	#11718594	—	2	1	no	—
11	#11718594	—	2	1	no	—
12	#11718593	—	2	1	no	—
13	C₄×#6101	—	2	1	no	—
14	#11718590	C₄	1	1	yes	characteristic
15	#11718590	C₄	1	1	yes	characteristic
16	#11718589	C₄	1	1	yes	characteristic
17	#11718589	C₄	1	1	yes	characteristic
18	C₂×C₄⋊#1055	—	2	1	no	—
19	C₂×C₄×#1055	C₂²	1	2	yes	—
20	C₁₃×#336	—	2	1	no	—
21	C₁₃×#336	—	2	1	no	—
22	#11718116	—	2	1	no	—
23	#11718076	—	2	4	no	—
24	C₂²×Dic₂₆	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₂×C₄	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	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂³	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×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×#6101	—	2	1	no	—
37	C₂×#6101	—	2	1	no	—
38	C₂×#6101	—	2	1	no	—
39	C₂×C₄×Dic₁₃	—	2	1	no	—
40	C₂×C₄×Dic₁₃	—	2	1	no	—
41	C₂²×#1055	—	2	2	no	—
42	C₄×Dic₂₆	—	2	1	no	—
43	C₄×Dic₂₆	—	2	1	no	—
44	C₄×Dic₂₆	D₄	1	2	yes	—
45	C₄×Dic₂₆	Q₈	1	2	yes	—
46	C₁₃×C₄⋊C₈	—	2	2	no	—
47	C₁₃×C₄⋊C₈	—	2	2	no	—
48	#88071	—	2	1	no	—
49	#88071	—	2	1	no	—
50	#88071	—	2	1	no	—
51	#88071	—	2	1	no	—
52	#88071	—	2	1	no	—
53	#88071	—	2	1	no	—
54	#88071	—	2	1	no	—
55	C₄⋊#1055	—	2	4	no	—
56	C₄×#1055	—	2	2	no	—
57	C₄×#1055	—	2	2	no	—
58	#8570	—	13	1	no	maximal, Sylow
59	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×Dic₁₃	—	2	1	no	—
67	C₂²×Dic₁₃	—	2	1	no	—
68	C₂²×Dic₁₃	—	2	1	no	—
69	C₂²×Dic₁₃	—	2	1	no	—
70	C₂×Dic₂₆	—	2	2	no	—
71	C₂×Dic₂₆	—	2	2	no	—
72	C₂×Dic₂₆	—	2	1	no	—
73	C₂×Dic₂₆	C₄⋊C₄	1	2	yes	—
74	C₂×C₈×C₁₃	—	4	2	no	—
75	C₂×C₈×C₁₃	—	4	2	no	—
76	C₂×C₈×C₁₃	—	4	2	no	—
77	C₄²×C₁₃	—	2	2	no	—
78	C₄²×C₁₃	—	2	2	no	—
79	C₄²×C₁₃	—	2	2	no	—
80	C₄²×C₁₃	—	2	2	no	—
81	C₄²×C₁₃	—	2	2	no	—
82	C₄²×C₁₃	—	4	1	no	—
83	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
85	C₄²×C₁₃	#41	1	1	yes	characteristic
86	C₄²×C₁₃	#41	1	1	yes	characteristic
87	C₄⋊Dic₁₃	—	2	1	no	—
88	C₄⋊Dic₁₃	—	2	2	no	—
89	C₄⋊Dic₁₃	—	2	2	no	—
90	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
91	#6101	—	2	2	no	—
92	#6101	—	2	2	no	—
93	#6101	—	4	2	no	—
94	#6101	—	4	2	no	—
95	C₄×Dic₁₃	—	2	2	no	—
96	C₄×Dic₁₃	—	2	2	no	—
97	C₄×Dic₁₃	—	4	2	no	—
98	C₂×#1055	—	2	4	no	—
99	C₂×#1055	—	4	2	no	—
100	C₂×#1055	—	4	2	no	—
101	#2295	—	13	1	no	—
102	#1831	D₁₃	1	1	yes	characteristic
103	C₄×C₄⋊C₈	—	13	1	no	—
104	C₂×Q₈⋊C₈	—	13	2	no	—
105	#1279	—	13	2	no	—
106	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	1	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	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	#104	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
124	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₁₃	#131	1	2	yes	—
128	C₂×C₄×C₁₃	#131	1	2	yes	—
129	C₂×C₄×C₁₃	#135	1	2	yes	—
130	C₂×C₄×C₁₃	#137	1	2	yes	—
131	C₂×Dic₁₃	—	2	2	no	—
132	C₂×Dic₁₃	—	2	2	no	—
133	C₂×Dic₁₃	—	4	1	no	—
134	C₂×Dic₁₃	—	4	1	no	—
135	C₂×Dic₁₃	—	4	1	no	—
136	C₂×Dic₁₃	—	4	1	no	—
137	C₂×Dic₁₃	—	4	1	no	—
138	C₂×Dic₁₃	—	4	1	no	—
139	C₂×Dic₁₃	—	4	1	no	—
140	C₂×Dic₁₃	—	4	1	no	—
141	C₂×Dic₁₃	—	4	1	no	—
142	C₂×Dic₁₃	—	4	1	no	—
143	Dic₂₆	—	2	4	no	—
144	Dic₂₆	—	4	2	no	—
145	C₈×C₁₃	—	8	2	no	—
146	#1055	—	4	4	no	—
147	C₂×C₄×Q₈	—	13	1	no	—
148	C₂×C₄⋊C₈	—	26	1	no	—
149	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
150	C₂×C₄×C₈	—	13	2	no	—
151	#391	—	13	1	no	—
152	#391	—	13	1	no	—
153	#389	—	13	1	no	—
154	#389	—	13	1	no	—
155	#387	—	26	1	no	—
156	#387	—	26	1	no	—
157	C₄×C₄⋊C₄	—	13	1	no	—
158	C₄×C₄⋊C₄	—	26	1	no	—
159	#376	—	26	1	no	—
160	C₄³	C₂×D₁₃	1	1	yes	characteristic
161	#336	—	2	1	no	—
162	#336	—	2	1	no	—
163	#336	—	26	1	no	—
164	Q₈⋊C₈	—	26	4	no	—
165	C₂²×C₁₃	#380	1	1	yes	characteristic
166	C₂²×C₁₃	#408	1	1	yes	characteristic
167	C₂²×C₁₃	#408	1	1	yes	characteristic
168	C₂²×C₁₃	#416	1	1	yes	characteristic
169	C₂²×C₁₃	#419	1	1	yes	characteristic
170	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
171	C₂²×C₁₃	#421	1	1	yes	characteristic
172	C₄×C₁₃	—	2	1	no	—
173	C₄×C₁₃	—	2	1	no	—
174	C₄×C₁₃	—	2	2	no	—
175	C₄×C₁₃	—	4	1	no	—
176	C₄×C₁₃	—	4	1	no	—
177	C₄×C₁₃	—	4	1	no	—
178	C₄×C₁₃	#424	1	2	yes	—
179	C₄×C₁₃	#433	1	2	yes	—
180	C₄×C₁₃	#464	1	2	yes	—
181	C₄×C₁₃	#479	1	2	yes	—
182	Dic₁₃	—	4	2	no	—
183	Dic₁₃	—	4	2	no	—
184	Dic₁₃	—	8	1	no	—
185	Dic₁₃	—	8	1	no	—
186	C₂²×Q₈	—	13	1	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	26	2	no	—
189	C₄×Q₈	—	13	2	no	—
190	C₄×Q₈	—	13	2	no	—
191	C₄×Q₈	—	26	1	no	—
192	C₄×Q₈	—	26	1	no	—
193	C₂×C₄⋊C₄	—	13	1	no	—
194	C₂×C₄⋊C₄	—	13	1	no	—
195	C₂×C₄⋊C₄	—	13	1	no	—
196	C₂×C₄⋊C₄	—	26	1	no	—
197	C₂×C₄⋊C₄	—	26	1	no	—
198	C₂×C₄⋊C₄	—	26	1	no	—
199	C₂×C₄²	—	26	1	no	—
200	C₂×C₄²	—	26	1	no	—
201	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
202	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
203	C₂×C₄²	D₅₂	1	1	yes	characteristic
204	C₂×C₄²	D₅₂	1	1	yes	characteristic
205	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
206	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
207	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
208	C₄⋊C₈	—	2	2	no	—
209	C₄⋊C₈	—	2	2	no	—
210	C₄⋊C₈	—	26	4	no	—
211	C₄×C₈	—	26	2	no	—
212	C₄×C₈	—	26	2	no	—
213	#95	—	26	1	no	—
214	#95	—	26	1	no	—
215	#95	—	26	1	no	—
216	#95	—	26	1	no	—
217	#95	—	26	1	no	—
218	#95	—	26	1	no	—
219	#95	—	26	1	no	—
220	C₂×C₁₃	#1482	1	1	yes	characteristic
221	C₂×C₁₃	#1485	1	1	yes	characteristic
222	C₂×C₁₃	#1494	1	1	yes	characteristic
223	C₂×C₁₃	#1497	1	1	yes	characteristic
224	C₂×C₁₃	#1798	1	1	yes	characteristic
225	C₂×C₁₃	#1805	1	1	yes	characteristic
226	C₂×C₁₃	#1806	1	1	yes	characteristic
227	C₂×Q₈	—	13	2	no	—
228	C₂×Q₈	—	26	1	no	—
229	C₂×Q₈	—	26	2	no	—
230	C₂×Q₈	—	26	2	no	—
231	C₂²×C₄	—	26	1	no	—
232	C₂²×C₄	—	26	1	no	—
233	C₂²×C₄	—	26	1	no	—
234	C₂²×C₄	—	26	1	no	—
235	C₂²×C₄	#6103	1	1	yes	characteristic
236	C₂²×C₄	#6103	1	1	yes	characteristic
237	C₂²×C₄	#6103	1	1	yes	characteristic
238	C₂²×C₄	#6103	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	26	4	no	—
246	C₂×C₈	—	52	2	no	—
247	C₂×C₈	—	52	2	no	—
248	C₄⋊C₄	—	13	2	no	—
249	C₄⋊C₄	—	26	1	no	—
250	C₄⋊C₄	—	26	2	no	—
251	C₄⋊C₄	—	26	2	no	—
252	C₄⋊C₄	—	26	2	no	—
253	C₄⋊C₄	—	26	2	no	—
254	C₄⋊C₄	—	52	2	no	—
255	C₄⋊C₄	—	52	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	4	1	no	—
262	C₄²	—	26	2	no	—
263	C₄²	—	26	2	no	—
264	C₄²	—	52	2	no	—
265	C₄²	D₄×D₁₃	1	1	yes	characteristic
266	C₄²	#6129	1	1	yes	characteristic
267	C₄²	Q₈×D₁₃	1	1	yes	characteristic
268	C₄²	#6131	1	1	yes	characteristic
269	C₁₃	#8570	1	1	yes	characteristic, complemented, Sylow
270	C₂³	C₂×#6103	1	1	yes	characteristic
271	Q₈	—	26	4	no	—
272	Q₈	—	52	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	26	2	no	—
287	C₂×C₄	—	26	2	no	—
288	C₂×C₄	—	52	1	no	—
289	C₂×C₄	—	52	1	no	—
290	C₂×C₄	—	52	1	no	—
291	C₂×C₄	—	52	1	no	—
292	C₂×C₄	—	52	1	no	—
293	C₂×C₄	—	52	1	no	—
294	C₂×C₄	—	52	1	no	—
295	C₂×C₄	—	52	1	no	—
296	C₂×C₄	—	52	1	no	—
297	C₂×C₄	—	52	1	no	—
298	C₂×C₄	#88065	1	2	yes	—
299	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
300	C₂×C₄	#88146	1	1	yes	characteristic
301	C₂×C₄	#88149	1	1	yes	characteristic
302	C₂×C₄	#88151	1	1	yes	characteristic
303	C₂×C₄	#88155	1	2	yes	—
304	C₂×C₄	#88161	1	2	yes	—
305	C₂×C₄	#88192	1	1	yes	characteristic
306	C₂×C₄	#88200	1	1	yes	characteristic
307	C₂×C₄	#88204	1	2	yes	—
308	C₂×C₄	#88205	1	2	yes	—
309	C₈	—	8	2	no	—
310	C₈	—	52	4	no	—
311	C₂²	#11718648	1	1	yes	characteristic
312	C₂²	#11718663	1	1	yes	characteristic
313	C₂²	#11718665	1	1	yes	characteristic
314	C₂²	#11718771	1	1	yes	characteristic
315	C₂²	#11718790	1	1	yes	characteristic
316	C₂²	C₂×#88065	1	1	yes	characteristic
317	C₂²	#11718800	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	52	2	no	—
325	C₄	—	52	2	no	—
326	C₄	—	104	1	no	—
327	C₄	—	104	1	no	—
328	C₄	#11718500	1	2	yes	—
329	C₄	#11718501	1	2	yes	—
330	C₄	#11718715	1	2	yes	—
331	C₄	#11718724	1	2	yes	—
332	C₂	#13113242	1	1	yes	characteristic
333	C₂	#13113256	1	1	yes	characteristic
334	C₂	#13113294	1	1	yes	characteristic
335	C₂	#13113308	1	1	yes	characteristic
336	C₂	#13114817	1	1	yes	characteristic
337	C₂	#13114848	1	1	yes	characteristic
338	C₂	#13114853	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1831	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1831	D₁₃
hypocenter	C₁₃	#8570
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
269. C₁₃	#8570	semidirect

Maximal subgroups

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

subgroup	index
#13117313	2
C₄×C₄⋊#1055	2
C₂×#11718076	2
#13112675	2
C₁₃×#1831	2
#8570	13