Subgroups — #15882727

Order 3328.

770 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13117793	C₂	1	2	yes	maximal
2	#13115708	C₂	1	2	yes	maximal
3	#13115472	C₂	1	4	yes	maximal
4	#13114993	C₂	1	1	yes	maximal, characteristic
5	#13114841	C₂	1	2	yes	maximal
6	#13114735	C₂	1	2	yes	maximal
7	#13114703	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
11	C₄×C₁₃×Q₁₆	C₄	1	8	yes	complemented
12	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#103	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	#11718891	C₂²	1	4	yes	—
16	C₂×#88075	C₂²	1	4	yes	—
17	C₂×#88057	C₂²	1	2	yes	—
18	C₂×#88057	C₂²	1	2	yes	—
19	C₂×#88055	C₂²	1	2	yes	—
20	#11718642	—	2	4	no	—
21	C₄⋊C₄×Dic₁₃	C₂²	1	4	yes	—
22	#11718597	—	2	2	no	—
23	#11718596	C₂²	1	2	yes	—
24	C₄×C₄⋊Dic₁₃	—	2	2	no	—
25	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
27	#11718142	C₂²	1	2	yes	—
28	#11718121	C₂²	1	4	yes	—
29	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
30	C₂×C₁₃×Q₁₆	—	2	2	no	—
31	C₂×C₁₃×Q₁₆	—	2	2	no	—
32	C₂×C₁₃×Q₁₆	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₄×Q₈×C₁₃	—	2	4	no	—
36	C₄×Q₈×C₁₃	C₂×C₄	1	8	yes	—
37	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
38	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
39	C₂×C₄²×C₁₃	—	2	2	no	—
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₄⋊Dic₁₃	—	2	2	no	—
42	C₂×C₄⋊Dic₁₃	—	2	2	no	—
43	C₂×C₄⋊Dic₁₃	D₄	1	2	yes	—
44	C₂×C₄⋊Dic₁₃	C₂³	1	4	yes	—
45	C₂×C₄×Dic₁₃	—	2	4	no	—
46	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
47	C₂²×#1055	C₂³	1	2	yes	—
48	C₁₃×#107	C₂×C₄	1	4	yes	—
49	C₁₃×#103	C₂×C₄	1	8	yes	—
50	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
51	#88075	—	4	4	no	—
52	#88071	—	2	4	no	—
53	#88071	—	2	4	no	—
54	#88071	—	2	4	no	—
55	#88071	D₄	1	2	yes	—
56	#88057	—	2	4	no	—
57	#88057	—	4	2	no	—
58	#88055	—	2	4	no	—
59	C₄⋊#1055	—	2	2	no	—
60	#22814	—	13	1	no	maximal, Sylow
61	C₂×Q₈×C₁₃	—	2	2	no	—
62	C₂×Q₈×C₁₃	—	2	2	no	—
63	C₂×Q₈×C₁₃	—	2	2	no	—
64	C₂×Q₈×C₁₃	—	2	2	no	—
65	C₂×Q₈×C₁₃	—	2	2	no	—
66	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₁₃	—	2	2	no	—
68	C₂²×C₄×C₁₃	—	2	2	no	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₁₃	—	2	4	no	—
73	C₂²×Dic₁₃	C₂×D₄	1	4	yes	—
74	C₁₃×Q₁₆	—	2	8	no	—
75	C₂×C₈×C₁₃	—	2	1	no	—
76	C₂×C₈×C₁₃	—	2	1	no	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₃	#41	1	2	yes	—
80	C₂×C₈×C₁₃	#41	1	2	yes	—
81	C₁₃×C₄⋊C₄	—	2	8	no	—
82	C₁₃×C₄⋊C₄	—	2	2	no	—
83	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
84	C₄²×C₁₃	—	2	8	no	—
85	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
86	C₄²×C₁₃	#41	1	2	yes	—
87	C₄⋊Dic₁₃	—	2	4	no	—
88	C₄⋊Dic₁₃	—	2	4	no	—
89	C₄⋊Dic₁₃	—	2	4	no	—
90	C₄⋊Dic₁₃	—	4	4	no	—
91	C₄⋊Dic₁₃	—	8	2	no	—
92	C₄×Dic₁₃	—	2	2	no	—
93	C₄×Dic₁₃	—	2	2	no	—
94	C₄×Dic₁₃	—	8	4	no	—
95	C₂×#1055	—	2	2	no	—
96	C₂×#1055	—	2	2	no	—
97	C₂×#1055	—	2	2	no	—
98	C₂×C₄×Q₁₆	D₁₃	1	1	yes	characteristic
99	#2355	—	13	2	no	—
100	#1971	—	13	2	no	—
101	#1927	—	13	4	no	—
102	#1844	—	13	1	no	—
103	#1803	—	13	2	no	—
104	#1776	—	13	2	no	—
105	#1770	—	13	1	no	—
106	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
107	Q₈×C₁₃	—	2	4	no	—
108	Q₈×C₁₃	—	2	4	no	—
109	Q₈×C₁₃	—	4	2	no	—
110	Q₈×C₁₃	—	4	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	4	2	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	—	4	2	no	—
117	C₂×C₄×C₁₃	—	4	2	no	—
118	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₁₃	#137	1	4	yes	—
121	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
127	C₂×Dic₁₃	—	2	4	no	—
128	C₂×Dic₁₃	—	2	4	no	—
129	C₂×Dic₁₃	—	2	4	no	—
130	C₂×Dic₁₃	—	4	4	no	—
131	C₂×Dic₁₃	—	4	4	no	—
132	C₂×Dic₁₃	—	4	4	no	—
133	C₈×C₁₃	—	2	2	no	—
134	C₈×C₁₃	#117	1	4	yes	—
135	#1055	—	4	2	no	—
136	C₂²×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
137	C₂×C₄×Q₈	C₂×D₁₃	1	2	yes	—
138	C₄×Q₁₆	Dic₁₃	1	8	yes	complemented
139	C₂×#107	C₂×D₁₃	1	1	yes	characteristic
140	C₂×#106	—	13	2	no	—
141	C₂×#106	—	13	2	no	—
142	C₂×C₄⋊C₈	—	13	1	no	—
143	C₂×#103	—	13	4	no	—
144	C₂×#103	C₂×D₁₃	1	2	yes	—
145	C₂×#97	—	13	2	no	—
146	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
147	#391	—	13	4	no	—
148	#387	—	26	4	no	—
149	#384	—	13	2	no	—
150	#382	—	26	2	no	—
151	C₄×C₄⋊C₄	—	13	1	no	—
152	C₄×C₄⋊C₄	—	13	4	no	—
153	C₄×C₄⋊C₄	—	26	2	no	—
154	#340	—	13	4	no	—
155	#336	—	13	2	no	—
156	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
157	C₂²×C₁₃	#547	1	1	yes	characteristic
158	C₂²×C₁₃	#550	1	1	yes	characteristic
159	C₂²×C₁₃	C₂×#137	1	2	yes	—
160	C₂²×C₁₃	#577	1	2	yes	—
161	C₄×C₁₃	—	2	2	no	—
162	C₄×C₁₃	—	4	4	no	—
163	C₄×C₁₃	—	4	4	no	—
164	C₄×C₁₃	#441	1	4	yes	—
165	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
166	C₄×C₁₃	#520	1	1	yes	characteristic
167	C₄×C₁₃	#555	1	1	yes	characteristic
168	C₄×C₁₃	#556	1	1	yes	characteristic
169	Dic₁₃	—	4	4	no	—
170	Dic₁₃	—	8	4	no	—
171	C₂²×Q₈	C₂²×D₁₃	1	2	yes	—
172	C₂×Q₁₆	—	2	2	no	—
173	C₂×Q₁₆	—	2	2	no	—
174	C₂×Q₁₆	C₂×Dic₁₃	1	4	yes	—
175	C₂²×C₈	—	13	2	no	—
176	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
177	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
178	C₄×Q₈	—	2	4	no	—
179	C₄×Q₈	C₂×Dic₁₃	1	8	yes	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	13	2	no	—
182	C₂×C₄⋊C₄	—	13	4	no	—
183	C₂×C₄⋊C₄	—	26	2	no	—
184	C₂×C₄⋊C₄	—	26	2	no	—
185	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	13	2	no	—
188	C₂×C₄²	—	26	4	no	—
189	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
190	#107	C₂×Dic₁₃	1	4	yes	—
191	#106	—	26	4	no	—
192	#106	—	52	2	no	—
193	C₄⋊C₈	—	26	2	no	—
194	#103	—	52	4	no	—
195	#103	C₂×Dic₁₃	1	8	yes	—
196	#97	—	26	4	no	—
197	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
198	#95	—	13	2	no	—
199	#95	—	26	4	no	—
200	#95	—	26	4	no	—
201	#95	—	26	4	no	—
202	C₂×C₁₃	#2354	1	1	yes	characteristic
203	C₂×C₁₃	C₂×#441	1	1	yes	characteristic
204	C₂×C₁₃	#2952	1	1	yes	characteristic
205	C₂×C₁₃	#3356	1	2	yes	—
206	C₂×C₁₃	#3400	1	2	yes	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	C₂²×Dic₁₃	1	4	yes	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	13	4	no	—
216	C₂²×C₄	—	26	4	no	—
217	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
218	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
219	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
220	Q₁₆	—	2	8	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	26	2	no	—
224	C₂×C₈	—	26	2	no	—
225	C₂×C₈	—	26	2	no	—
226	C₂×C₈	#6127	1	2	yes	—
227	C₂×C₈	#6127	1	2	yes	—
228	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
229	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
230	C₄⋊C₄	—	2	8	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	26	4	no	—
233	C₄⋊C₄	—	26	4	no	—
234	C₄⋊C₄	—	26	4	no	—
235	C₄⋊C₄	—	52	4	no	—
236	C₄⋊C₄	—	104	2	no	—
237	C₄⋊C₄	C₂²×Dic₁₃	1	4	yes	—
238	C₄²	—	2	8	no	—
239	C₄²	—	26	2	no	—
240	C₄²	—	26	2	no	—
241	C₄²	—	104	4	no	—
242	C₄²	#6129	1	2	yes	—
243	C₄²	C₂²×Dic₁₃	1	2	yes	—
244	C₁₃	#22814	1	1	yes	characteristic, complemented, Sylow
245	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
246	Q₈	—	2	4	no	—
247	Q₈	—	2	4	no	—
248	Q₈	—	4	2	no	—
249	Q₈	—	4	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	26	4	no	—
258	C₂×C₄	—	26	4	no	—
259	C₂×C₄	—	26	4	no	—
260	C₂×C₄	—	52	4	no	—
261	C₂×C₄	—	52	4	no	—
262	C₂×C₄	—	52	4	no	—
263	C₂×C₄	#88172	1	4	yes	—
264	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
265	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
266	C₂×C₄	C₂×#6127	1	1	yes	characteristic
267	C₂×C₄	C₂×#6127	1	1	yes	characteristic
268	C₂×C₄	C₂×#6129	1	1	yes	characteristic
269	C₂×C₄	#88256	1	1	yes	characteristic
270	C₂×C₄	#88257	1	1	yes	characteristic
271	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
272	C₈	—	2	2	no	—
273	C₈	—	52	2	no	—
274	C₈	#88180	1	4	yes	—
275	C₂²	#11719256	1	1	yes	characteristic
276	C₂²	#11719258	1	1	yes	characteristic
277	C₂²	C₂×#88172	1	2	yes	—
278	C₂²	#11719475	1	2	yes	—
279	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	4	4	no	—
282	C₄	—	4	4	no	—
283	C₄	—	52	4	no	—
284	C₄	—	104	4	no	—
285	C₄	#11718845	1	4	yes	—
286	C₄	#11719247	1	1	yes	characteristic
287	C₄	#11719268	1	1	yes	characteristic
288	C₄	C₂×#88180	1	1	yes	characteristic
289	C₄	#11719532	1	1	yes	characteristic
290	C₂	#13117785	1	1	yes	characteristic
291	C₂	C₂×#11718845	1	1	yes	characteristic
292	C₂	#13123720	1	1	yes	characteristic
293	C₂	#13127453	1	2	yes	—
294	C₂	#13127887	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×Q₁₆	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₁₃
hypocenter	C₁₃	#22814
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
11. C₄×C₁₃×Q₁₆	C₄	semidirect
138. C₄×Q₁₆	Dic₁₃	semidirect
244. C₁₃	#22814	semidirect

Maximal subgroups

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

subgroup	index
#13117793	2
#13115708	2
#13115472	2
#13114993	2
#13114841	2
#13114735	2
#13114703	2
C₂×C₄×C₁₃×Q₁₆	2
#22814	13