Subgroups — #15761502

Order 3328.

404 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114988	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718049	C₂	1	2	yes	maximal
4	#13112671	C₂	1	2	yes	maximal
5	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
9	C₄³×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊#1055	—	2	1	no	—
11	C₂×C₄⋊#1055	—	2	1	no	—
12	C₂×C₄⋊#1055	C₂²	1	2	yes	—
13	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
15	C₂×C₄×#1055	—	2	1	no	—
16	C₁₃×#336	C₄	1	2	yes	—
17	#11718116	—	2	1	no	—
18	#11718049	—	2	4	no	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	Q₈	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
28	C₂²×#1055	—	2	1	no	—
29	C₂²×#1055	—	2	1	no	—
30	C₂²×#1055	—	2	2	no	—
31	C₁₃×C₄⋊C₈	—	2	2	no	—
32	C₁₃×C₄⋊C₈	C₈	1	4	yes	complemented
33	C₄×C₈×C₁₃	—	2	2	no	—
34	C₄×C₈×C₁₃	—	2	2	no	—
35	C₄⋊#1055	—	2	1	no	—
36	C₄⋊#1055	—	2	1	no	—
37	C₄⋊#1055	—	2	2	no	—
38	C₄⋊#1055	—	2	2	no	—
39	C₄⋊#1055	—	4	2	no	—
40	C₄⋊#1055	—	4	2	no	—
41	C₄⋊#1055	D₄	1	2	yes	—
42	C₄⋊#1055	Q₈	1	2	yes	—
43	C₄×#1055	—	4	2	no	—
44	#9602	—	13	1	no	maximal, Sylow
45	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₁₃	—	2	2	no	—
53	C₂×C₈×C₁₃	—	2	2	no	—
54	C₂×C₈×C₁₃	—	2	2	no	—
55	C₂×C₈×C₁₃	—	2	2	no	—
56	C₂×C₈×C₁₃	—	2	2	no	—
57	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
58	C₄²×C₁₃	—	2	2	no	—
59	C₄²×C₁₃	—	2	1	no	—
60	C₄²×C₁₃	—	2	1	no	—
61	C₄²×C₁₃	—	2	1	no	—
62	C₄²×C₁₃	—	2	1	no	—
63	C₄²×C₁₃	—	4	1	no	—
64	C₄²×C₁₃	C₂×C₈	1	2	yes	—
65	C₄²×C₁₃	M₁₆	1	2	yes	—
66	C₄²×C₁₃	SD₁₆	1	2	yes	—
67	C₄²×C₁₃	SD₁₆	1	2	yes	—
68	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
70	C₄²×C₁₃	#41	1	1	yes	characteristic
71	C₄²×C₁₃	#41	1	1	yes	characteristic
72	C₂×#1055	—	2	2	no	—
73	C₂×#1055	—	2	2	no	—
74	C₂×#1055	—	4	1	no	—
75	C₂×#1055	—	4	1	no	—
76	C₂×#1055	—	4	1	no	—
77	C₂×#1055	—	4	1	no	—
78	C₂×#1055	—	4	2	no	—
79	C₂×#1055	—	4	2	no	—
80	C₂×#1055	—	4	2	no	—
81	#1842	—	13	1	no	—
82	C₄×C₄⋊C₈	—	13	1	no	—
83	C₄×C₄⋊C₈	D₁₃	1	1	yes	characteristic
84	C₂×#334	—	13	2	no	—
85	#1277	—	13	2	no	—
86	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
97	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
98	C₂×C₄×C₁₃	#106	1	2	yes	—
99	C₂×C₄×C₁₃	#106	1	2	yes	—
100	C₂×C₄×C₁₃	#108	1	2	yes	—
101	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	#125	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
115	C₈×C₁₃	—	2	4	no	—
116	C₈×C₁₃	—	4	2	no	—
117	#1055	—	4	2	no	—
118	#1055	—	4	2	no	—
119	#1055	—	8	2	no	—
120	C₂×C₄⋊C₈	—	13	1	no	—
121	C₂×C₄⋊C₈	—	13	1	no	—
122	C₂×C₄⋊C₈	—	13	2	no	—
123	C₂×C₄⋊C₈	—	26	1	no	—
124	C₂×C₄⋊C₈	—	26	1	no	—
125	C₂×C₄⋊C₈	Dic₁₃	1	1	yes	characteristic
126	C₂×C₄⋊C₈	Dic₁₃	1	1	yes	characteristic
127	C₂×C₄×C₈	—	26	1	no	—
128	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
129	C₄³	C₂×D₁₃	1	1	yes	characteristic
130	#336	—	26	1	no	—
131	#336	Dic₁₃	1	2	yes	—
132	#334	—	26	4	no	—
133	C₂²×C₁₃	#389	1	1	yes	characteristic
134	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
135	C₂²×C₁₃	#424	1	1	yes	characteristic
136	C₂²×C₁₃	C₂×#106	1	1	yes	characteristic
137	C₂²×C₁₃	#428	1	1	yes	characteristic
138	C₂²×C₁₃	C₂×#108	1	1	yes	characteristic
139	C₂²×C₁₃	#430	1	1	yes	characteristic
140	C₄×C₁₃	—	2	2	no	—
141	C₄×C₁₃	—	2	2	no	—
142	C₄×C₁₃	—	2	1	no	—
143	C₄×C₁₃	—	2	1	no	—
144	C₄×C₁₃	—	4	1	no	—
145	C₄×C₁₃	#334	1	4	yes	complemented
146	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
147	C₄×C₁₃	#436	1	1	yes	characteristic
148	C₄×C₁₃	C₈×Q₈	1	1	yes	characteristic
149	C₄×C₁₃	#446	1	1	yes	characteristic
150	C₄×C₁₃	#460	1	1	yes	characteristic
151	C₄×C₁₃	#461	1	1	yes	characteristic
152	C₄×C₁₃	#475	1	1	yes	characteristic
153	C₄×C₁₃	#476	1	1	yes	characteristic
154	C₂²×C₈	—	26	1	no	—
155	C₂²×C₈	—	26	1	no	—
156	C₂²×C₈	—	26	2	no	—
157	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
158	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
159	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
160	C₂×C₄²	Dic₂₆	1	1	yes	characteristic
161	C₂×C₄²	D₅₂	1	1	yes	characteristic
162	C₂×C₄²	D₅₂	1	1	yes	characteristic
163	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
164	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
165	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	13	2	no	—
168	C₄⋊C₈	—	13	2	no	—
169	C₄⋊C₈	—	26	2	no	—
170	C₄⋊C₈	—	26	1	no	—
171	C₄⋊C₈	—	26	1	no	—
172	C₄⋊C₈	—	26	2	no	—
173	C₄⋊C₈	—	52	2	no	—
174	C₄⋊C₈	—	52	2	no	—
175	C₄⋊C₈	#1055	1	4	yes	complemented
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	2	2	no	—
178	C₄×C₈	—	52	2	no	—
179	C₂×C₁₃	C₂×#334	1	1	yes	characteristic
180	C₂×C₁₃	#1565	1	1	yes	characteristic
181	C₂×C₁₃	#1570	1	1	yes	characteristic
182	C₂×C₁₃	#1574	1	1	yes	characteristic
183	C₂×C₁₃	#1939	1	1	yes	characteristic
184	C₂×C₁₃	#1948	1	1	yes	characteristic
185	C₂×C₁₃	#1950	1	1	yes	characteristic
186	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
187	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
188	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
189	C₂²×C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
190	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
192	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	26	2	no	—
199	C₂×C₈	—	26	2	no	—
200	C₂×C₈	—	52	1	no	—
201	C₂×C₈	—	52	1	no	—
202	C₂×C₈	—	52	1	no	—
203	C₂×C₈	—	52	1	no	—
204	C₂×C₈	—	52	2	no	—
205	C₂×C₈	—	52	2	no	—
206	C₂×C₈	—	52	2	no	—
207	C₂×C₈	C₂×#1055	1	4	yes	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	4	1	no	—
214	C₄²	#6095	1	2	yes	—
215	C₄²	#6095	1	2	yes	—
216	C₄²	C₂×#1055	1	2	yes	—
217	C₄²	#6099	1	2	yes	—
218	C₄²	D₄×D₁₃	1	1	yes	characteristic
219	C₄²	#6129	1	1	yes	characteristic
220	C₄²	Q₈×D₁₃	1	1	yes	characteristic
221	C₄²	#6131	1	1	yes	characteristic
222	C₁₃	#9602	1	1	yes	characteristic, complemented, Sylow
223	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	C₄⋊#1055	1	2	yes	—
234	C₂×C₄	C₄⋊#1055	1	2	yes	—
235	C₂×C₄	#88057	1	2	yes	—
236	C₂×C₄	#88057	1	2	yes	—
237	C₂×C₄	#88059	1	2	yes	—
238	C₂×C₄	#88142	1	1	yes	characteristic
239	C₂×C₄	#88144	1	1	yes	characteristic
240	C₂×C₄	#88149	1	1	yes	characteristic
241	C₂×C₄	#88151	1	1	yes	characteristic
242	C₂×C₄	C₂×#6095	1	1	yes	characteristic
243	C₂×C₄	C₂×#6095	1	1	yes	characteristic
244	C₂×C₄	#88162	1	1	yes	characteristic
245	C₂×C₄	#88163	1	1	yes	characteristic
246	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
247	C₂×C₄	C₂×#6099	1	1	yes	characteristic
248	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
249	C₂×C₄	Q₈×Dic₁₃	1	1	yes	characteristic
250	C₂×C₄	#88202	1	1	yes	characteristic
251	C₂×C₄	#88202	1	1	yes	characteristic
252	C₈	—	2	4	no	—
253	C₈	—	4	2	no	—
254	C₈	—	52	2	no	—
255	C₈	—	52	2	no	—
256	C₈	—	104	2	no	—
257	C₂²	C₂×C₄⋊#1055	1	1	yes	characteristic
258	C₂²	#11718646	1	1	yes	characteristic
259	C₂²	#11718660	1	1	yes	characteristic
260	C₂²	C₂×#88057	1	1	yes	characteristic
261	C₂²	C₂×#88059	1	1	yes	characteristic
262	C₂²	#11718782	1	1	yes	characteristic
263	C₂²	#11718786	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	2	no	—
268	C₄	—	4	1	no	—
269	C₄	#11718049	1	4	yes	complemented
270	C₄	#11718502	1	1	yes	characteristic
271	C₄	#11718503	1	1	yes	characteristic
272	C₄	#11718509	1	1	yes	characteristic
273	C₄	#11718512	1	1	yes	characteristic
274	C₄	D₄×#1055	1	1	yes	characteristic
275	C₄	#11718673	1	1	yes	characteristic
276	C₄	Q₈×#1055	1	1	yes	characteristic
277	C₄	#11718685	1	1	yes	characteristic
278	C₂	C₂×#11718049	1	1	yes	characteristic
279	C₂	#13113588	1	1	yes	characteristic
280	C₂	#13113607	1	1	yes	characteristic
281	C₂	#13113625	1	1	yes	characteristic
282	C₂	#13115555	1	1	yes	characteristic
283	C₂	#13115600	1	1	yes	characteristic
284	C₂	#13115612	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₁₃
hypocenter	C₁₃	#9602
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
32. C₁₃×C₄⋊C₈	C₈	semidirect
145. C₄×C₁₃	#334	semidirect
175. C₄⋊C₈	#1055	semidirect
222. C₁₃	#9602	semidirect
269. C₄	#11718049	semidirect

Maximal subgroups

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

subgroup	index
#13114988	2
C₄×C₄⋊#1055	2
C₂×#11718049	2
#13112671	2
C₄×C₁₃×C₄⋊C₈	2
#9602	13