Subgroups — #16015513

Order 3328.

894 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718517	C₂	1	4	yes	maximal
2	#13123233	C₂	1	1	yes	maximal, characteristic
3	#13123197	C₂	1	1	yes	maximal, characteristic
4	#13115001	C₂	1	4	yes	maximal
5	#13114996	C₂	1	1	yes	maximal, characteristic
6	#13114995	C₂	1	1	yes	maximal, characteristic
7	#13114992	C₂	1	1	yes	maximal, characteristic
8	C₄×#88055	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#1844	C₂	1	1	yes	maximal, characteristic
10	C₂×#88144	C₂²	1	2	yes	—
11	C₂×#88142	—	2	2	no	—
12	C₂×#88142	C₂²	1	2	yes	—
13	C₂×C₁₃×#106	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×#384	C₂²	1	1	yes	characteristic
16	C₁₃×#384	C₂²	1	1	yes	characteristic
17	C₂×#88058	C₂²	1	4	yes	—
18	C₂×#88055	C₂²	1	2	yes	—
19	C₂×#88055	C₂²	1	2	yes	—
20	#11718646	—	2	2	no	—
21	#11718638	C₂²	1	2	yes	—
22	#11718637	—	2	2	no	—
23	#11718637	C₂²	1	2	yes	—
24	#11718634	—	2	2	no	—
25	C₂×#88048	C₂²	1	4	yes	—
26	C₂×#88047	C₂²	1	4	yes	—
27	#11718596	C₂²	1	1	yes	characteristic
28	#11718596	C₂²	1	1	yes	characteristic
29	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
31	#11718517	—	2	16	no	—
32	C₂²×Dic₂₆	—	2	2	no	—
33	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
34	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
36	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
37	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊Dic₁₃	—	2	2	no	—
40	C₂×C₄⋊Dic₁₃	—	2	2	no	—
41	C₂×C₄⋊Dic₁₃	—	2	2	no	—
42	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
43	C₂×C₄⋊Dic₁₃	C₂³	1	2	yes	—
44	C₂×#6101	—	2	4	no	—
45	C₂×#6101	—	2	2	no	—
46	C₂×#6101	—	2	2	no	—
47	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
48	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
49	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
51	C₂²×#1055	C₂³	1	2	yes	—
52	#88144	—	2	8	no	—
53	#88142	—	2	8	no	—
54	#88142	—	4	8	no	—
55	C₁₃×#106	—	2	8	no	—
56	C₄×C₈×C₁₃	Q₈	1	4	yes	complemented
57	#88071	—	2	4	no	—
58	#88058	—	2	8	no	—
59	#88055	D₄	1	8	yes	—
60	#88055	Q₈	1	8	yes	complemented
61	#88048	—	2	8	no	—
62	#88047	—	2	8	no	—
63	C₄×#1055	Q₈	1	4	yes	complemented
64	#33316	—	13	1	no	maximal, Sylow
65	C₂²×C₄×C₁₃	—	2	2	no	—
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
69	C₂²×Dic₁₃	—	2	2	no	—
70	C₂²×Dic₁₃	—	2	2	no	—
71	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×Dic₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×Dic₁₃	C₂×D₄	1	2	yes	—
74	C₂×Dic₂₆	—	4	8	no	—
75	C₂×Dic₂₆	—	4	4	no	—
76	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
77	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
78	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
79	C₁₃×C₄⋊C₄	—	2	4	no	—
80	C₁₃×C₄⋊C₄	—	2	4	no	—
81	C₁₃×C₄⋊C₄	—	4	4	no	—
82	C₁₃×C₄⋊C₄	—	4	4	no	—
83	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
84	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
85	C₄⋊Dic₁₃	—	2	4	no	—
86	C₄⋊Dic₁₃	—	2	4	no	—
87	C₄⋊Dic₁₃	—	4	4	no	—
88	C₄⋊Dic₁₃	—	4	4	no	—
89	C₄⋊Dic₁₃	—	4	4	no	—
90	#6101	—	4	8	no	—
91	#6101	—	4	4	no	—
92	#6101	—	4	4	no	—
93	C₄×Dic₁₃	—	2	4	no	—
94	C₄×Dic₁₃	—	2	4	no	—
95	C₄×Dic₁₃	C₂×D₄	1	4	yes	—
96	C₄×Dic₁₃	C₂×Q₈	1	4	yes	—
97	C₂×#1055	C₂×D₄	1	4	yes	—
98	C₂×#1055	C₂×Q₈	1	4	yes	—
99	C₂×#1055	C₂×Q₈	1	4	yes	—
100	C₂×#501	—	13	4	no	—
101	#2868	—	13	1	no	—
102	#2859	—	13	1	no	—
103	#1847	—	13	4	no	—
104	#1845	—	13	1	no	—
105	#1845	—	13	1	no	—
106	#1844	—	13	1	no	—
107	#1844	D₁₃	1	1	yes	characteristic
108	C₄×#97	—	13	1	no	—
109	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	—	4	2	no	—
111	C₂×C₄×C₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	—	4	4	no	—
113	C₂×C₄×C₁₃	—	4	4	no	—
114	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
115	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
116	C₂×C₄×C₁₃	#137	1	4	yes	—
117	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
118	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
121	C₂×Dic₁₃	—	2	2	no	—
122	C₂×Dic₁₃	—	2	2	no	—
123	C₂×Dic₁₃	—	2	2	no	—
124	C₂×Dic₁₃	—	2	2	no	—
125	C₂×Dic₁₃	—	4	2	no	—
126	C₂×Dic₁₃	—	4	2	no	—
127	C₂×Dic₁₃	—	4	4	no	—
128	C₂×Dic₁₃	—	4	4	no	—
129	C₂×Dic₁₃	#127	1	2	yes	—
130	C₂×Dic₁₃	#127	1	2	yes	—
131	C₂×Dic₁₃	C₄⋊Q₈	1	4	yes	—
132	Dic₂₆	—	8	8	no	—
133	C₈×C₁₃	C₄⋊Q₈	1	8	yes	complemented
134	#1055	C₄⋊Q₈	1	8	yes	complemented
135	C₂×C₄⋊Q₈	—	13	2	no	—
136	C₂×C₄⋊Q₈	—	26	2	no	—
137	C₂×#125	—	13	2	no	—
138	#501	—	26	16	no	—
139	C₂×#107	—	13	4	no	—
140	C₂×#107	—	13	4	no	—
141	C₂×#106	—	13	4	no	—
142	C₂×#106	C₂×D₁₃	1	4	yes	—
143	C₂×#97	—	13	2	no	—
144	C₂×#97	—	13	2	no	—
145	C₂×C₄×C₈	—	13	1	no	—
146	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
147	#389	—	26	2	no	—
148	#389	—	26	2	no	—
149	#384	—	13	2	no	—
150	#384	—	13	1	no	—
151	#384	—	13	1	no	—
152	#384	—	26	2	no	—
153	#384	C₂×D₁₃	1	1	yes	characteristic
154	#384	C₂×D₁₃	1	1	yes	characteristic
155	#382	—	13	2	no	—
156	C₄³	—	13	1	no	—
157	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
158	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
159	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
160	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
161	C₂²×C₁₃	C₂×#137	1	2	yes	—
162	C₄×C₁₃	—	8	2	no	—
163	C₄×C₁₃	—	8	2	no	—
164	C₄×C₁₃	#501	1	8	yes	complemented
165	C₄×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
166	C₄×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
167	Dic₁₃	—	2	2	no	—
168	Dic₁₃	—	2	2	no	—
169	Dic₁₃	—	2	4	no	—
170	Dic₁₃	—	8	2	no	—
171	Dic₁₃	—	8	2	no	—
172	C₂²×Q₈	—	26	2	no	—
173	C₂²×C₈	—	13	2	no	—
174	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
175	C₄⋊Q₈	—	26	8	no	—
176	C₄⋊Q₈	—	52	8	no	—
177	#125	—	26	8	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	13	2	no	—
181	C₂×C₄⋊C₄	—	13	2	no	—
182	C₂×C₄⋊C₄	—	26	2	no	—
183	C₂×C₄⋊C₄	—	26	2	no	—
184	C₂×C₄⋊C₄	—	26	2	no	—
185	C₂×C₄⋊C₄	—	26	4	no	—
186	C₂×C₄⋊C₄	—	26	2	no	—
187	C₂×C₄⋊C₄	—	26	2	no	—
188	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
189	C₂×C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
190	C₂×C₄²	—	13	1	no	—
191	C₂×C₄²	—	13	1	no	—
192	C₂×C₄²	—	13	2	no	—
193	C₂×C₄²	—	13	2	no	—
194	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
195	#107	—	26	8	no	—
196	#107	—	26	8	no	—
197	#106	—	2	8	no	—
198	#106	—	26	8	no	—
199	#97	—	13	8	no	—
200	#97	—	13	8	no	—
201	C₄×C₈	—	13	4	no	—
202	C₄×C₈	Dic₂₆	1	4	yes	complemented
203	#95	—	26	4	no	—
204	C₂×C₁₃	#2868	1	1	yes	characteristic
205	C₂×C₁₃	#3151	1	1	yes	characteristic
206	C₂×C₁₃	#3152	1	1	yes	characteristic
207	C₂×C₁₃	C₂×#501	1	2	yes	—
208	C₂×C₁₃	#3165	1	2	yes	—
209	C₂×Q₈	—	52	4	no	—
210	C₂×Q₈	—	52	8	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	13	2	no	—
214	C₂²×C₄	—	13	1	no	—
215	C₂²×C₄	—	13	1	no	—
216	C₂²×C₄	—	26	2	no	—
217	C₂²×C₄	—	26	2	no	—
218	C₂²×C₄	D₄×D₁₃	1	2	yes	—
219	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
220	C₂×C₈	—	13	4	no	—
221	C₂×C₈	—	13	4	no	—
222	C₂×C₈	—	13	4	no	—
223	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
224	C₂×C₈	D₄×D₁₃	1	4	yes	—
225	C₂×C₈	Q₈×D₁₃	1	4	yes	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₄⋊C₄	—	26	4	no	—
231	C₄⋊C₄	—	26	4	no	—
232	C₄⋊C₄	—	52	4	no	—
233	C₄⋊C₄	—	52	4	no	—
234	C₄⋊C₄	—	52	4	no	—
235	C₄⋊C₄	—	52	8	no	—
236	C₄⋊C₄	—	52	4	no	—
237	C₄⋊C₄	—	52	4	no	—
238	C₄²	—	13	4	no	—
239	C₄²	—	13	4	no	—
240	C₄²	—	26	4	no	—
241	C₄²	—	26	4	no	—
242	C₄²	C₂×Dic₂₆	1	2	yes	—
243	C₄²	Q₈×D₁₃	1	2	yes	—
244	C₁₃	#33316	1	1	yes	characteristic, complemented, Sylow
245	C₂³	C₂×D₄×D₁₃	1	1	yes	characteristic
246	Q₈	—	104	8	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	13	4	no	—
252	C₂×C₄	—	13	2	no	—
253	C₂×C₄	—	13	2	no	—
254	C₂×C₄	—	26	2	no	—
255	C₂×C₄	—	26	2	no	—
256	C₂×C₄	—	26	2	no	—
257	C₂×C₄	—	26	2	no	—
258	C₂×C₄	—	52	4	no	—
259	C₂×C₄	—	52	4	no	—
260	C₂×C₄	—	52	2	no	—
261	C₂×C₄	—	52	2	no	—
262	C₂×C₄	#88142	1	4	yes	—
263	C₂×C₄	#88168	1	4	yes	—
264	C₂×C₄	#88169	1	4	yes	—
265	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
266	C₂×C₄	C₂×D₄×D₁₃	1	2	yes	—
267	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
268	C₂×C₄	C₂×Q₈×D₁₃	1	2	yes	—
269	C₈	—	13	8	no	—
270	C₈	#88142	1	8	yes	complemented
271	C₂²	C₂×#88142	1	1	yes	characteristic
272	C₂²	D₁₃×#127	1	1	yes	characteristic
273	C₂²	D₁₃×C₄⋊Q₈	1	1	yes	characteristic
274	C₂²	C₂×#88168	1	2	yes	—
275	C₂²	C₂×#88169	1	2	yes	—
276	C₄	—	8	2	no	—
277	C₄	—	8	2	no	—
278	C₄	—	26	4	no	—
279	C₄	—	26	2	no	—
280	C₄	—	26	2	no	—
281	C₄	—	104	2	no	—
282	C₄	—	104	2	no	—
283	C₄	#11718517	1	8	yes	complemented
284	C₄	C₂×#88142	1	2	yes	—
285	C₄	D₁₃×C₄⋊Q₈	1	2	yes	—
286	C₂	#13123233	1	1	yes	characteristic
287	C₂	#13125283	1	1	yes	characteristic
288	C₂	#13125290	1	1	yes	characteristic
289	C₂	C₂×#11718517	1	2	yes	—
290	C₂	#13125377	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₁₃
commutator, cosocle	C₂²×C₄×C₁₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₁₃×#1844	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
hypercenter	#1844	D₁₃
hypocenter	C₁₃	#33316
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
56. C₄×C₈×C₁₃	Q₈	semidirect
60. #88055	Q₈	semidirect
63. C₄×#1055	Q₈	semidirect
133. C₈×C₁₃	C₄⋊Q₈	semidirect
134. #1055	C₄⋊Q₈	semidirect
164. C₄×C₁₃	#501	semidirect
202. C₄×C₈	Dic₂₆	semidirect
244. C₁₃	#33316	semidirect
270. C₈	#88142	semidirect
283. C₄	#11718517	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718517	2
#13123233	2
#13123197	2
#13115001	2
#13114996	2
#13114995	2
#13114992	2
C₄×#88055	2
C₁₃×#1844	2
#33316	13