Subgroups — #15762075

Order 3328.

430 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125422	C₂	1	1	yes	maximal, characteristic
2	#13115780	C₂	1	2	yes	maximal
3	C₄×#88058	C₂	1	1	yes	maximal, characteristic
4	#13113336	C₂	1	2	yes	maximal
5	C₈×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
6	#11719175	C₂²	1	2	yes	—
7	C₁₃×#431	C₂²	1	1	yes	characteristic
8	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	#11718777	—	2	1	no	—
11	#11718777	—	2	1	no	—
12	C₂×#88058	—	2	1	no	—
13	C₄×C₄⋊Dic₁₃	C₂²	1	2	yes	—
14	#11718360	C₄	1	1	yes	characteristic, complemented
15	C₄×Dic₅₂	—	2	1	no	—
16	C₄×Dic₅₂	—	2	1	no	—
17	#11718343	C₄	1	1	yes	characteristic, complemented
18	#11718342	C₄	1	2	yes	complemented
19	C₁₃×#322	—	2	1	no	—
20	C₈²×C₁₃	—	2	1	no	—
21	#11718147	—	2	4	no	—
22	#11718143	—	2	1	no	—
23	#11718059	—	2	4	no	—
24	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
25	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
28	#88180	—	2	2	no	—
29	C₂×C₄⋊Dic₁₃	—	2	2	no	—
30	#88178	—	2	2	no	—
31	#88178	—	2	2	no	—
32	C₂×C₄×Dic₁₃	—	2	2	no	—
33	C₂×Dic₅₂	—	2	1	no	—
34	C₂×Dic₅₂	—	2	1	no	—
35	#88124	—	2	2	no	—
36	#88123	C₂×C₄	1	2	yes	—
37	#88123	C₂×C₄	1	2	yes	—
38	C₄×Dic₂₆	—	2	2	no	—
39	C₄×Dic₂₆	—	2	2	no	—
40	C₁₃×C₄⋊C₈	—	2	1	no	—
41	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
42	C₁₃×#97	—	2	1	no	—
43	C₄×C₈×C₁₃	—	2	1	no	—
44	C₄×C₈×C₁₃	—	2	1	no	—
45	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
49	#88071	—	2	2	no	—
50	#88058	—	2	4	no	—
51	#88058	—	2	1	no	—
52	#88058	—	2	1	no	—
53	#88056	—	2	2	no	—
54	#88056	—	2	2	no	—
55	#9718	—	13	1	no	maximal, Sylow
56	C₂²×C₄×C₁₃	C₂×D₄	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	—	4	2	no	—
60	C₂×Dic₂₆	—	2	2	no	—
61	C₂×Dic₂₆	—	2	2	no	—
62	C₁₃×M₁₆	—	2	2	no	—
63	C₁₃×M₁₆	Q₁₆	1	4	yes	complemented
64	C₂×C₈×C₁₃	—	2	2	no	—
65	C₂×C₈×C₁₃	—	2	2	no	—
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	—	4	1	no	—
68	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₁₃	#41	1	1	yes	characteristic
75	C₄²×C₁₃	#41	1	1	yes	characteristic
76	C₂²⋊Dic₁₃	—	4	2	no	—
77	C₄⋊Dic₁₃	—	2	4	no	—
78	C₄⋊Dic₁₃	—	2	2	no	—
79	C₄⋊Dic₁₃	—	2	2	no	—
80	C₄⋊Dic₁₃	—	2	2	no	—
81	C₄⋊Dic₁₃	—	2	2	no	—
82	C₄⋊Dic₁₃	—	4	2	no	—
83	#6101	—	4	2	no	—
84	#6101	—	4	2	no	—
85	C₄×Dic₁₃	—	4	2	no	—
86	C₄×Dic₁₃	—	4	4	no	—
87	Dic₅₂	—	4	2	no	—
88	Dic₅₂	—	4	2	no	—
89	#3170	—	13	1	no	—
90	#1986	—	13	2	no	—
91	C₄×#107	—	13	1	no	—
92	#1503	—	13	2	no	—
93	C₈×M₁₆	D₁₃	1	1	yes	characteristic
94	C₂³×C₁₃	#127	1	1	yes	characteristic
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	4	1	no	—
98	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
103	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
105	C₂×C₄×C₁₃	#135	1	2	yes	—
106	C₂×Dic₁₃	—	4	4	no	—
107	C₂×Dic₁₃	—	4	2	no	—
108	C₂×Dic₁₃	—	4	2	no	—
109	C₂×Dic₁₃	—	8	2	no	—
110	C₂×Dic₁₃	—	8	2	no	—
111	Dic₂₆	—	4	2	no	—
112	Dic₂₆	—	4	2	no	—
113	Dic₂₆	—	8	2	no	—
114	C₈×C₁₃	—	2	1	no	—
115	C₈×C₁₃	—	2	1	no	—
116	C₈×C₁₃	—	2	1	no	—
117	C₈×C₁₃	—	2	4	no	—
118	C₈×C₁₃	—	4	2	no	—
119	C₈×C₁₃	#104	1	2	yes	—
120	#533	—	13	2	no	—
121	#500	—	13	1	no	—
122	#494	—	13	1	no	—
123	#487	—	13	2	no	—
124	#467	—	26	1	no	—
125	#467	—	26	1	no	—
126	C₄×Q₁₆	—	26	1	no	—
127	C₄×Q₁₆	—	26	1	no	—
128	#431	C₂×D₁₃	1	1	yes	characteristic
129	C₂×#107	—	26	1	no	—
130	C₄×M₁₆	C₂×D₁₃	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
132	C₄×C₄⋊C₄	—	13	2	no	—
133	#340	—	26	1	no	—
134	#339	—	26	4	no	—
135	#328	—	26	4	no	—
136	#322	—	2	1	no	—
137	C₈²	—	2	1	no	—
138	C₂²×C₁₃	—	2	1	no	—
139	C₂²×C₁₃	—	2	1	no	—
140	C₂²×C₁₃	#390	1	1	yes	characteristic
141	C₂²×C₁₃	#494	1	1	yes	characteristic
142	C₂²×C₁₃	#495	1	1	yes	characteristic
143	C₄×C₁₃	—	2	1	no	—
144	C₄×C₁₃	—	2	1	no	—
145	C₄×C₁₃	—	2	1	no	—
146	C₄×C₁₃	C₂×#104	1	1	yes	characteristic
147	C₄×C₁₃	#421	1	1	yes	characteristic
148	C₄×C₁₃	C₄×Q₁₆	1	2	yes	—
149	C₄×C₁₃	#467	1	2	yes	—
150	Dic₁₃	—	8	4	no	—
151	Dic₁₃	—	8	2	no	—
152	Dic₁₃	—	8	2	no	—
153	C₂×Q₁₆	—	26	1	no	—
154	C₂×Q₁₆	—	26	1	no	—
155	C₂×M₁₆	D₅₂	1	2	yes	—
156	C₂²×C₈	D₅₂	1	1	yes	characteristic
157	C₂²×C₈	D₅₂	1	1	yes	characteristic
158	C₄⋊Q₈	—	13	2	no	—
159	C₄⋊Q₈	—	13	2	no	—
160	#125	—	26	2	no	—
161	C₂²⋊Q₈	—	26	2	no	—
162	C₂²⋊Q₈	—	26	2	no	—
163	C₄×Q₈	—	26	2	no	—
164	C₄×Q₈	—	26	2	no	—
165	#117	—	26	2	no	—
166	C₂×C₄⋊C₄	—	26	2	no	—
167	C₂×C₄²	—	26	2	no	—
168	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
169	#107	—	26	4	no	—
170	#107	—	26	1	no	—
171	#107	—	26	1	no	—
172	C₄⋊C₈	—	2	1	no	—
173	#103	—	26	2	no	—
174	#103	—	26	2	no	—
175	C₂²⋊C₈	D₅₂	1	2	yes	—
176	#97	—	2	1	no	—
177	C₄×C₈	—	2	1	no	—
178	C₄×C₈	—	2	1	no	—
179	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
180	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
183	#95	—	26	2	no	—
184	C₂×C₁₃	—	2	1	no	—
185	C₂×C₁₃	—	2	1	no	—
186	C₂×C₁₃	#1969	1	1	yes	characteristic
187	C₂×C₁₃	#1976	1	1	yes	characteristic
188	C₂×C₁₃	#1980	1	1	yes	characteristic
189	C₂×Q₈	—	26	2	no	—
190	C₂×Q₈	—	26	2	no	—
191	C₂²×C₄	—	52	2	no	—
192	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
195	Q₁₆	—	52	2	no	—
196	Q₁₆	—	52	2	no	—
197	M₁₆	—	2	2	no	—
198	M₁₆	Dic₅₂	1	4	yes	complemented
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	#6103	1	1	yes	characteristic
204	C₂×C₈	#6103	1	1	yes	characteristic
205	C₂×C₈	#6103	1	1	yes	characteristic
206	C₂×C₈	#6103	1	1	yes	characteristic
207	C₄⋊C₄	—	26	4	no	—
208	C₄⋊C₄	—	26	2	no	—
209	C₄⋊C₄	—	26	2	no	—
210	C₄⋊C₄	—	26	2	no	—
211	C₄⋊C₄	—	26	2	no	—
212	C₄⋊C₄	—	52	2	no	—
213	C₄⋊C₄	—	52	2	no	—
214	C₄⋊C₄	—	52	2	no	—
215	C₂²⋊C₄	—	52	2	no	—
216	C₄²	—	52	2	no	—
217	C₄²	—	52	4	no	—
218	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
219	C₄²	#6127	1	1	yes	characteristic
220	C₄²	#6127	1	1	yes	characteristic
221	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
222	C₁₃	#9718	1	1	yes	characteristic, complemented, Sylow
223	C₂³	#88128	1	1	yes	characteristic
224	Q₈	—	52	2	no	—
225	Q₈	—	52	2	no	—
226	Q₈	—	104	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	52	4	no	—
231	C₂×C₄	—	52	2	no	—
232	C₂×C₄	—	52	2	no	—
233	C₂×C₄	—	104	2	no	—
234	C₂×C₄	—	104	2	no	—
235	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
236	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
237	C₂×C₄	#88129	1	1	yes	characteristic
238	C₂×C₄	#88158	1	2	yes	—
239	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
240	C₂×C₄	C₂×#6103	1	1	yes	characteristic
241	C₂×C₄	#88184	1	1	yes	characteristic
242	C₂×C₄	#88184	1	1	yes	characteristic
243	C₈	—	2	4	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	4	2	no	—
248	C₈	#88065	1	2	yes	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	#11718357	1	1	yes	characteristic
252	C₂²	#11718360	1	1	yes	characteristic
253	C₂²	#11718600	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	104	4	no	—
258	C₄	—	104	2	no	—
259	C₄	—	104	2	no	—
260	C₄	C₄×Dic₅₂	1	2	yes	—
261	C₄	#11718777	1	2	yes	—
262	C₄	C₂×#88065	1	1	yes	characteristic
263	C₄	#11718800	1	1	yes	characteristic
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	#13115697	1	1	yes	characteristic
267	C₂	#13115734	1	1	yes	characteristic
268	C₂	#13115755	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88184
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₈×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	#390
prime radical	C₂³×C₁₃	#127
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₈×M₁₆	D₁₃
hypocenter	C₁₃	#9718
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
14. #11718360	C₄	semidirect
17. #11718343	C₄	semidirect
18. #11718342	C₄	semidirect
63. C₁₃×M₁₆	Q₁₆	semidirect
198. M₁₆	Dic₅₂	semidirect
222. C₁₃	#9718	semidirect

Maximal subgroups

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

subgroup	index
#13125422	2
#13115780	2
C₄×#88058	2
#13113336	2
C₈×C₁₃×M₁₆	2
#9718	13