Subgroups — #15828550

Order 3328.

874 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123469	C₂	1	2	yes	maximal
2	#13113128	C₂	1	2	yes	maximal, complemented
3	#13113125	C₂	1	4	yes	maximal, complemented
4	#13113119	C₂	1	2	yes	maximal, complemented
5	C₈×#6099	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₁₃	C₄	1	2	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₃	C₄	1	4	yes	—
10	C₂×C₄×C₈×C₁₃	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	#11718868	C₄	1	4	yes	—
16	#11718587	C₄	1	4	yes	—
17	#11718587	C₄	1	4	yes	—
18	#11718586	C₂²	1	4	yes	—
19	#11718586	C₂²	1	4	yes	—
20	#11718585	C₄	1	4	yes	—
21	C₄×#6099	C₂²	1	4	yes	—
22	#11718047	C₂²	1	4	yes	complemented
23	#11718046	C₂²	1	8	yes	complemented
24	C₈×#1055	C₂²	1	4	yes	complemented
25	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	—	2	8	no	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×#6099	C₂×C₄	1	8	yes	—
40	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₁₃	C₂³	1	2	yes	—
45	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₂²⋊#1055	C₂×C₄	1	8	yes	—
48	C₂²⋊#1055	C₂×C₄	1	8	yes	—
49	C₄⋊#1055	C₂×C₄	1	8	yes	—
50	C₄⋊#1055	C₂×C₄	1	8	yes	—
51	#88043	C₂×C₄	1	8	yes	—
52	C₄×#1055	C₂³	1	8	yes	—
53	#17323	—	13	1	no	maximal, Sylow
54	C₂⁴×C₁₃	C₄²	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	—	2	4	no	—
56	C₂²×C₄×C₁₃	—	2	8	no	—
57	C₂²×C₄×C₁₃	C₄²	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
59	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
60	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
61	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃	—	2	4	no	—
68	C₂×C₈×C₁₃	—	2	8	no	—
69	C₂×C₈×C₁₃	—	2	8	no	—
70	C₂×C₈×C₁₃	M₁₆	1	8	yes	complemented
71	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
74	C₄²×C₁₃	C₄²	1	2	yes	—
75	C₄²×C₁₃	C₄²	1	1	yes	characteristic
76	C₄²×C₁₃	C₄²	1	2	yes	—
77	C₄²×C₁₃	C₄²	1	1	yes	characteristic
78	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
79	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
80	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
81	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
83	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
84	#6099	—	2	16	no	—
85	C₂×#1055	C₂²×C₄	1	16	yes	—
86	#2920	—	13	2	no	—
87	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
88	#1452	—	13	2	no	—
89	#1451	—	13	4	no	—
90	#1450	—	13	2	no	—
91	C₈×M₁₆	—	13	4	no	—
92	C₂³×C₁₃	—	2	2	no	—
93	C₂³×C₁₃	—	2	4	no	—
94	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
95	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	4	no	—
98	C₂×C₄×C₁₃	—	2	4	no	—
99	C₂×C₄×C₁₃	—	2	4	no	—
100	C₂×C₄×C₁₃	—	2	4	no	—
101	C₂×C₄×C₁₃	—	2	4	no	—
102	C₂×C₄×C₁₃	—	2	4	no	—
103	C₂×C₄×C₁₃	—	2	4	no	—
104	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
110	C₂×C₄×C₁₃	#131	1	8	yes	—
111	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
113	C₈×C₁₃	—	2	4	no	—
114	C₈×C₁₃	C₂×M₁₆	1	4	yes	—
115	C₈×C₁₃	C₂×M₁₆	1	4	yes	—
116	#1055	—	2	16	no	—
117	C₂³×C₈	Dic₁₃	1	2	yes	—
118	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
119	#432	—	13	4	no	—
120	#432	—	13	4	no	—
121	#431	—	13	4	no	—
122	#431	—	13	4	no	—
123	#423	—	13	4	no	—
124	#407	—	13	4	no	—
125	C₄×M₁₆	—	13	4	no	—
126	C₂×C₄×C₈	Dic₁₃	1	4	yes	—
127	C₂×C₄×C₈	Dic₁₃	1	2	yes	—
128	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
129	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
131	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
132	#322	—	13	8	no	—
133	#322	—	13	4	no	—
134	C₈²	—	13	4	no	—
135	C₂²×C₁₃	—	2	8	no	—
136	C₂²×C₁₃	—	2	2	no	—
137	C₂²×C₁₃	—	2	2	no	—
138	C₂²×C₁₃	—	2	2	no	—
139	C₂²×C₁₃	C₄×M₁₆	1	2	yes	—
140	C₂²×C₁₃	#405	1	4	yes	—
141	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
142	C₄×C₁₃	—	2	2	no	—
143	C₄×C₁₃	—	2	2	no	—
144	C₄×C₁₃	—	2	2	no	—
145	C₄×C₁₃	C₄×M₁₆	1	2	yes	—
146	C₄×C₁₃	#405	1	4	yes	—
147	C₄×C₁₃	C₂²×M₁₆	1	2	yes	—
148	C₄×C₁₃	C₂×#131	1	4	yes	—
149	C₂³×C₄	C₄×D₁₃	1	2	yes	—
150	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
151	C₂×M₁₆	—	13	8	no	—
152	C₂²×C₈	—	2	8	no	—
153	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
154	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
155	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
156	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
157	C₂×C₄²	C₄×D₁₃	1	2	yes	—
158	C₂×C₄²	C₄×D₁₃	1	2	yes	—
159	C₂×C₄²	C₄×D₁₃	1	2	yes	—
160	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
161	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
162	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
163	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
164	C₄⋊C₈	—	13	8	no	—
165	C₄⋊C₈	—	13	8	no	—
166	C₂²⋊C₈	—	13	8	no	—
167	C₂²⋊C₈	—	13	8	no	—
168	#97	—	13	8	no	—
169	C₄×C₈	—	13	8	no	—
170	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
171	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
172	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
173	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
174	C₄×C₈	C₂²×D₁₃	1	2	yes	—
175	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
176	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
177	C₂×C₁₃	—	2	2	no	—
178	C₂×C₁₃	—	2	4	no	—
179	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
180	C₂×C₁₃	C₂×#405	1	2	yes	—
181	C₂⁴	C₄×Dic₁₃	1	1	yes	characteristic
182	C₂²×C₄	—	2	8	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	C₄×Dic₁₃	1	1	yes	characteristic
185	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
186	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
187	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
188	C₂²×C₄	C₄×Dic₁₃	1	2	yes	—
189	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
190	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
191	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
192	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
193	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
194	M₁₆	—	26	16	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	8	no	—
197	C₂×C₈	—	2	8	no	—
198	C₂×C₈	—	13	16	no	—
199	C₂×C₈	#6099	1	8	yes	complemented
200	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
201	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
202	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
203	C₄²	C₄×Dic₁₃	1	2	yes	—
204	C₄²	C₄×Dic₁₃	1	2	yes	—
205	C₄²	C₄×Dic₁₃	1	1	yes	characteristic
206	C₄²	C₄×Dic₁₃	1	1	yes	characteristic
207	C₄²	C₂×C₄×D₁₃	1	2	yes	—
208	C₄²	C₂×C₄×D₁₃	1	2	yes	—
209	C₄²	C₂×C₄×D₁₃	1	2	yes	—
210	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
211	C₄²	C₂²×Dic₁₃	1	2	yes	—
212	C₄²	C₂³×D₁₃	1	1	yes	characteristic
213	C₁₃	#17323	1	1	yes	characteristic, complemented, Sylow
214	C₂³	—	2	4	no	—
215	C₂³	—	2	2	no	—
216	C₂³	C₂×C₄×Dic₁₃	1	2	yes	—
217	C₂³	C₂×C₄×Dic₁₃	1	1	yes	characteristic
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	#88155	1	8	yes	—
227	C₂×C₄	C₂×#6099	1	4	yes	—
228	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
229	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
230	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
231	C₂×C₄	C₂×C₄×Dic₁₃	1	2	yes	—
232	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
233	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
234	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
235	C₈	—	2	4	no	—
236	C₈	—	26	16	no	—
237	C₈	C₂×#6099	1	4	yes	—
238	C₈	C₂×#6099	1	4	yes	—
239	C₂²	—	2	8	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	C₄×#6099	1	2	yes	—
244	C₂²	#11718762	1	4	yes	—
245	C₂²	C₂²×C₄×Dic₁₃	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	C₄×#6099	1	2	yes	—
250	C₄	#11718762	1	4	yes	—
251	C₄	C₂×#88155	1	4	yes	—
252	C₄	C₂²×#6099	1	2	yes	—
253	C₂	—	2	4	no	—
254	C₂	—	2	2	no	—
255	C₂	C₂×C₄×#6099	1	1	yes	characteristic
256	C₂	C₂×#11718762	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₁₃
commutator, socle	C₂²×C₁₃	C₂²×C₄²
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
prime radical	C₂⁴×C₁₃	C₄²
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	#17323
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13113128	C₂	semidirect
3. #13113125	C₂	semidirect
4. #13113119	C₂	semidirect
5. C₈×#6099	C₂	semidirect
22. #11718047	C₂²	semidirect
23. #11718046	C₂²	semidirect
24. C₈×#1055	C₂²	semidirect
70. C₂×C₈×C₁₃	M₁₆	semidirect
199. C₂×C₈	#6099	semidirect
213. C₁₃	#17323	semidirect

Maximal subgroups

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

subgroup	index
#13123469	2
#13113128	2
#13113125	2
#13113119	2
C₈×#6099	2
C₂²×C₄×C₈×C₁₃	2
#17323	13