Subgroups — #16016012

Order 3328.

598 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116643	C₂	1	4	yes	maximal, complemented
2	#13116635	C₂	1	4	yes	maximal, complemented
3	#13116595	C₂	1	2	yes	maximal, complemented
4	#13116450	C₂	1	2	yes	maximal
5	C₄×#88052	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₁₃×#131	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
9	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
10	C₄×C₁₃×#41	C₄	1	1	yes	characteristic
11	C₁₃×#405	C₄	1	2	yes	—
12	C₁₃×#405	C₄	1	1	yes	characteristic
13	C₁₃×#405	C₂²	1	1	yes	characteristic
14	C₁₃×#405	C₂²	1	2	yes	—
15	C₄×C₁₃×M₁₆	C₄	1	1	yes	characteristic
16	C₄×C₁₃×M₁₆	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
19	C₂×#88052	C₂²	1	2	yes	—
20	C₂²×#6090	C₂²	1	2	yes	—
21	C₂²⋊#6090	C₂²	1	8	yes	—
22	C₄⋊#6090	C₂²	1	8	yes	—
23	#11718053	C₂²	1	2	yes	—
24	C₄×#6090	C₂²	1	2	yes	—
25	C₄×#6090	C₂²	1	2	yes	—
26	C₄×#6090	C₂²	1	2	yes	—
27	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
28	C₁₃×#131	—	2	4	no	—
29	C₁₃×#131	—	2	4	no	—
30	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
32	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₄×Q₈×C₁₃	C₈	1	2	yes	—
39	C₄×D₄×C₁₃	C₈	1	4	yes	—
40	C₄×D₄×C₁₃	C₈	1	2	yes	—
41	C₁₃×#117	C₂×C₄	1	2	yes	—
42	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
45	C₁₃×#97	C₂×C₄	1	2	yes	—
46	C₁₃×#97	C₂×C₄	1	2	yes	—
47	C₁₃×#97	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₁₃	C₂³	1	2	yes	—
54	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
55	#88052	—	2	4	no	—
56	C₂×#6090	—	2	4	no	—
57	C₂×#6090	C₂³	1	4	yes	—
58	C₂×#6090	C₂³	1	4	yes	—
59	#33420	—	13	1	no	maximal, Sylow
60	C₁₃×#41	—	2	2	no	—
61	C₁₃×#41	—	2	2	no	—
62	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
63	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
64	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
66	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₁₃×M₁₆	—	2	2	no	—
70	C₁₃×M₁₆	—	2	2	no	—
71	C₁₃×M₁₆	—	2	4	no	—
72	C₁₃×M₁₆	—	2	4	no	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	—	2	4	no	—
75	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	#41	1	4	yes	—
82	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
83	C₁₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
84	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
85	C₁₃×C₂²⋊C₄	C₂×C₈	1	4	yes	—
86	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
87	C₄²×C₁₃	C₂×C₈	1	4	yes	—
88	C₄²×C₁₃	C₂×C₈	1	2	yes	—
89	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
92	#6090	—	2	4	no	—
93	#6090	—	2	4	no	—
94	C₄×#131	D₁₃	1	1	yes	characteristic
95	#2164	—	13	4	no	—
96	#2162	—	13	4	no	—
97	#2151	—	13	2	no	—
98	#2113	—	13	2	no	—
99	C₄×M₃₂	—	13	1	no	—
100	C₂×C₄×C₁₆	—	13	1	no	—
101	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
102	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
103	Q₈×C₁₃	—	2	2	no	—
104	D₄×C₁₃	—	2	4	no	—
105	D₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	—	2	8	no	—
110	C₂×C₄×C₁₃	#117	1	2	yes	—
111	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
112	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
113	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
114	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
117	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₈×C₁₃	—	2	2	no	—
119	C₈×C₁₃	—	2	2	no	—
120	C₈×C₁₃	#117	1	2	yes	—
121	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
122	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
123	C₈×C₁₃	C₂×#41	1	2	yes	—
124	C₂×#131	Dic₁₃	1	1	yes	characteristic
125	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
126	C₄×#41	Dic₁₃	1	1	yes	characteristic
127	C₂×M₃₂	—	13	2	no	—
128	C₂²×C₁₆	—	13	2	no	—
129	#405	Dic₁₃	1	2	yes	—
130	#405	Dic₁₃	1	1	yes	characteristic
131	#405	C₂×D₁₃	1	1	yes	characteristic
132	#405	C₂×D₁₃	1	2	yes	—
133	C₄×M₁₆	Dic₁₃	1	1	yes	characteristic
134	C₄×M₁₆	Dic₁₃	1	2	yes	—
135	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
136	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
137	C₄⋊C₁₆	—	13	8	no	—
138	C₂²⋊C₁₆	—	13	8	no	—
139	#346	—	13	2	no	—
140	C₄×C₁₆	—	13	2	no	—
141	C₄×C₁₆	—	13	2	no	—
142	C₄×C₁₆	—	13	2	no	—
143	C₂²×C₁₃	—	2	2	no	—
144	C₂²×C₁₃	—	2	4	no	—
145	C₂²×C₁₃	—	2	2	no	—
146	C₂²×C₁₃	#431	1	2	yes	—
147	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
148	C₄×C₁₃	—	2	2	no	—
149	C₄×C₁₃	—	2	4	no	—
150	C₄×C₁₃	—	2	2	no	—
151	C₄×C₁₃	#431	1	2	yes	—
152	C₄×C₁₃	#504	1	4	yes	—
153	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
154	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
155	C₂×#41	C₂×Dic₁₃	1	1	yes	characteristic
156	#131	—	2	4	no	—
157	#131	—	2	4	no	—
158	C₂×M₁₆	C₂×Dic₁₃	1	1	yes	characteristic
159	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
160	C₂×M₁₆	C₂×Dic₁₃	1	1	yes	characteristic
161	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
162	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
163	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
165	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
166	C₄×Q₈	#1055	1	2	yes	—
167	C₄×D₄	#1055	1	4	yes	—
168	C₄×D₄	#1055	1	2	yes	—
169	#117	C₂×Dic₁₃	1	2	yes	—
170	#117	C₂×Dic₁₃	1	1	yes	characteristic
171	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
172	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
173	M₃₂	—	26	4	no	—
174	C₂×C₁₆	—	13	4	no	—
175	C₂×C₁₆	—	13	4	no	—
176	C₂×C₁₆	—	26	4	no	—
177	#97	C₂×Dic₁₃	1	2	yes	—
178	#97	C₂×Dic₁₃	1	2	yes	—
179	#97	C₂×Dic₁₃	1	2	yes	—
180	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
181	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
182	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
183	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
184	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
185	C₄×C₈	C₂²×D₁₃	1	2	yes	—
186	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
187	C₂×C₁₃	—	2	2	no	—
188	C₂×C₁₃	—	4	2	no	—
189	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
190	C₂×C₁₃	C₂×#504	1	2	yes	—
191	#41	—	2	2	no	—
192	#41	—	2	2	no	—
193	C₂×Q₈	C₂×#1055	1	1	yes	characteristic
194	C₂×D₄	C₂×#1055	1	2	yes	—
195	C₂×D₄	C₂×#1055	1	1	yes	characteristic
196	C₂²×C₄	C₂×#1055	1	4	yes	—
197	C₂²×C₄	C₂×#1055	1	2	yes	—
198	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
199	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
200	M₁₆	—	2	2	no	—
201	M₁₆	—	2	2	no	—
202	M₁₆	—	2	4	no	—
203	M₁₆	—	2	4	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	#6099	1	4	yes	—
207	C₂×C₈	#6127	1	4	yes	—
208	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
209	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
210	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
211	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
212	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
213	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
214	C₄⋊C₄	C₂×#1055	1	4	yes	—
215	C₄⋊C₄	C₂×#1055	1	2	yes	—
216	C₂²⋊C₄	C₂×#1055	1	4	yes	—
217	C₂²⋊C₄	C₂×#1055	1	2	yes	—
218	C₄²	C₂×#1055	1	4	yes	—
219	C₄²	C₂×#1055	1	2	yes	—
220	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
221	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
222	C₄²	C₂²×Dic₁₃	1	2	yes	—
223	C₁₆	—	26	4	no	—
224	C₁₆	—	26	4	no	—
225	C₁₃	#33420	1	1	yes	characteristic, complemented, Sylow
226	C₂³	C₂²×#1055	1	2	yes	—
227	C₂³	C₂²×#1055	1	1	yes	characteristic
228	Q₈	—	2	2	no	—
229	D₄	—	2	4	no	—
230	D₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	C₂²×#1055	1	2	yes	—
236	C₂×C₄	C₂²×#1055	1	4	yes	—
237	C₂×C₄	C₂²×#1055	1	2	yes	—
238	C₂×C₄	C₂²×#1055	1	2	yes	—
239	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
240	C₂×C₄	C₂×#6099	1	2	yes	—
241	C₂×C₄	#88180	1	2	yes	—
242	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₈	C₂×#6099	1	2	yes	—
246	C₈	C₂×#6099	1	2	yes	—
247	C₈	#88180	1	2	yes	—
248	C₈	C₂×#6127	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	4	no	—
252	C₂²	#11718586	1	2	yes	—
253	C₂²	C₂³×#1055	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	4	no	—
257	C₄	#11718586	1	2	yes	—
258	C₄	#11718801	1	4	yes	—
259	C₄	C₂²×#6099	1	1	yes	characteristic
260	C₄	C₂×#88180	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	4	2	no	—
263	C₂	C₂×#11718586	1	1	yes	characteristic
264	C₂	C₂×#11718801	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₄×C₁₃×#131	C₂
prime radical	C₂×C₁₃×#41	C₂×C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₄×#131	D₁₃
hypocenter	C₁₃	#33420
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. #13116643	C₂	semidirect
2. #13116635	C₂	semidirect
3. #13116595	C₂	semidirect
5. C₄×#88052	C₂	semidirect
6. C₂×C₄×#6090	C₂	semidirect
225. C₁₃	#33420	semidirect

Maximal subgroups

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

subgroup	index
#13116643	2
#13116635	2
#13116595	2
#13116450	2
C₄×#88052	2
C₂×C₄×#6090	2
C₄×C₁₃×#131	2
#33420	13