Subgroups — #15756048

Order 3328.

394 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	#13115008	C₂	1	1	yes	maximal, characteristic
2	#13113584	C₂	1	1	yes	maximal, characteristic
3	#13113265	C₂	1	2	yes	maximal
4	#13112733	C₂	1	2	yes	maximal
5	C₂×C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₈×Q₈×C₁₃	—	2	1	no	—
8	C₈×Q₈×C₁₃	—	2	1	no	—
9	C₈×Q₈×C₁₃	C₄	1	2	yes	—
10	C₈×Q₈×C₁₃	C₄	1	2	yes	—
11	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	—	2	1	no	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88059	—	2	1	no	—
16	C₂×#88059	—	2	1	no	—
17	#11718586	C₂²	1	2	yes	—
18	#11718585	C₂²	1	2	yes	—
19	Q₈⋊#1055	—	2	4	no	—
20	#11718050	—	2	1	no	—
21	#11718049	—	2	1	no	—
22	C₂²×Q₈×C₁₃	Q₈	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	—	2	1	no	—
24	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
26	C₄×Q₈×C₁₃	—	2	1	no	—
27	C₄×Q₈×C₁₃	—	2	1	no	—
28	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₄×Q₈×C₁₃	D₄	1	2	yes	—
30	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
31	C₂×C₁₃×C₄⋊C₄	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	—	2	1	no	—
33	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×#6099	—	2	2	no	—
35	C₁₃×C₄⋊C₈	—	2	2	no	—
36	C₁₃×C₄⋊C₈	—	2	2	no	—
37	C₁₃×C₄⋊C₈	—	2	1	no	—
38	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₃	—	2	2	no	—
40	C₄×C₈×C₁₃	—	2	2	no	—
41	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
45	C₂²⋊#1055	—	2	2	no	—
46	#88059	—	4	2	no	—
47	#88059	—	4	2	no	—
48	C₄⋊#1055	—	2	2	no	—
49	C₄⋊#1055	—	2	2	no	—
50	C₄⋊#1055	—	2	2	no	—
51	C₄×#1055	—	2	2	no	—
52	#8659	—	13	1	no	maximal, Sylow
53	C₂×Q₈×C₁₃	—	2	4	no	—
54	C₂×Q₈×C₁₃	—	2	1	no	—
55	C₂×Q₈×C₁₃	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	—	2	1	no	—
57	C₂²×C₄×C₁₃	—	2	1	no	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
61	C₂×C₈×C₁₃	—	2	1	no	—
62	C₂×C₈×C₁₃	—	2	1	no	—
63	C₂×C₈×C₁₃	—	2	2	no	—
64	C₂×C₈×C₁₃	—	4	1	no	—
65	C₂×C₈×C₁₃	—	4	1	no	—
66	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
71	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
72	C₁₃×C₄⋊C₄	—	2	2	no	—
73	C₁₃×C₄⋊C₄	—	2	2	no	—
74	C₁₃×C₄⋊C₄	—	2	1	no	—
75	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
76	C₄²×C₁₃	—	2	2	no	—
77	C₄²×C₁₃	—	2	2	no	—
78	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₁₃	#41	1	1	yes	characteristic
82	#6099	—	4	2	no	—
83	#6099	—	4	2	no	—
84	C₂×#1055	—	4	2	no	—
85	C₂×#1055	—	4	2	no	—
86	C₂×C₈×Q₈	D₁₃	1	1	yes	characteristic
87	#1848	—	13	1	no	—
88	#1564	—	13	1	no	—
89	#1487	—	13	2	no	—
90	#1301	—	13	2	no	—
91	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
92	Q₈×C₁₃	—	2	4	no	—
93	Q₈×C₁₃	—	2	2	no	—
94	Q₈×C₁₃	#108	1	4	yes	complemented
95	C₂×C₄×C₁₃	—	2	4	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	4	1	no	—
100	C₂×C₄×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
104	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
105	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
109	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
111	C₈×C₁₃	—	2	1	no	—
112	C₈×C₁₃	—	2	1	no	—
113	C₈×C₁₃	—	4	2	no	—
114	C₈×C₁₃	#103	1	2	yes	—
115	C₈×C₁₃	#103	1	2	yes	—
116	#1055	—	8	2	no	—
117	#1055	—	8	2	no	—
118	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
119	C₈×Q₈	—	2	1	no	—
120	C₈×Q₈	—	2	1	no	—
121	C₈×Q₈	Dic₁₃	1	2	yes	—
122	C₈×Q₈	Dic₁₃	1	2	yes	—
123	#431	—	13	2	no	—
124	C₂×#108	—	26	1	no	—
125	C₂×#108	—	26	1	no	—
126	#423	—	13	2	no	—
127	C₂×C₄⋊C₈	—	2	1	no	—
128	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
129	C₂×C₄×C₈	—	2	1	no	—
130	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
131	#335	—	26	1	no	—
132	#334	—	26	1	no	—
133	Q₈⋊C₈	—	26	4	no	—
134	C₂²×C₁₃	—	2	1	no	—
135	C₂²×C₁₃	—	2	1	no	—
136	C₂²×C₁₃	#380	1	1	yes	characteristic
137	C₂²×C₁₃	#475	1	1	yes	characteristic
138	C₂²×C₁₃	#477	1	1	yes	characteristic
139	C₄×C₁₃	—	2	4	no	—
140	C₄×C₁₃	—	2	1	no	—
141	C₄×C₁₃	—	2	1	no	—
142	C₄×C₁₃	—	2	1	no	—
143	C₄×C₁₃	—	4	2	no	—
144	C₄×C₁₃	C₂×#103	1	1	yes	characteristic
145	C₄×C₁₃	#417	1	1	yes	characteristic
146	C₄×C₁₃	C₂×#108	1	2	yes	—
147	C₄×C₁₃	#460	1	1	yes	characteristic
148	C₄×C₁₃	#462	1	1	yes	characteristic
149	C₂²×Q₈	Dic₂₆	1	1	yes	characteristic
150	C₂×M₁₆	—	26	2	no	—
151	C₂²×C₈	—	2	1	no	—
152	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
153	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	—	2	1	no	—
156	C₄×Q₈	D₅₂	1	2	yes	—
157	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	Dic₂₆	1	1	yes	characteristic
160	C₂×C₄²	—	2	1	no	—
161	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
162	#108	—	52	2	no	—
163	#108	—	52	2	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	2	1	no	—
167	C₄⋊C₈	—	26	2	no	—
168	C₄⋊C₈	—	26	2	no	—
169	C₄⋊C₈	—	26	2	no	—
170	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
171	C₂²⋊C₈	—	26	2	no	—
172	C₄×C₈	—	2	2	no	—
173	C₄×C₈	—	2	2	no	—
174	C₄×C₈	—	26	2	no	—
175	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
176	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
177	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
178	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
179	C₂×C₁₃	—	2	1	no	—
180	C₂×C₁₃	—	2	1	no	—
181	C₂×C₁₃	#1803	1	1	yes	characteristic
182	C₂×C₁₃	#1810	1	1	yes	characteristic
183	C₂×C₁₃	#1814	1	1	yes	characteristic
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	—	2	1	no	—
186	C₂×Q₈	C₄⋊Dic₁₃	1	2	yes	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
190	C₂²×C₄	#6127	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
192	M₁₆	—	52	2	no	—
193	M₁₆	—	52	2	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	52	2	no	—
200	C₂×C₈	—	52	2	no	—
201	C₂×C₈	#6106	1	2	yes	—
202	C₂×C₈	#6107	1	2	yes	—
203	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
205	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	C₂×D₅₂	1	1	yes	characteristic
214	C₄²	#6127	1	1	yes	characteristic
215	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
216	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
217	C₁₃	#8659	1	1	yes	characteristic, complemented, Sylow
218	C₂³	#88178	1	1	yes	characteristic
219	Q₈	—	2	4	no	—
220	Q₈	—	2	2	no	—
221	Q₈	#88059	1	4	yes	complemented
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	#88178	1	1	yes	characteristic
229	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
230	C₂×C₄	#88180	1	1	yes	characteristic
231	C₂×C₄	#88184	1	1	yes	characteristic
232	C₂×C₄	#88184	1	1	yes	characteristic
233	C₂×C₄	C₂×#6106	1	1	yes	characteristic
234	C₂×C₄	C₂×#6107	1	1	yes	characteristic
235	C₂×C₄	#88203	1	1	yes	characteristic
236	C₂×C₄	#88205	1	1	yes	characteristic
237	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	4	2	no	—
241	C₈	—	104	2	no	—
242	C₈	—	104	2	no	—
243	C₈	#88075	1	2	yes	—
244	C₈	#88075	1	2	yes	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	#11718681	1	1	yes	characteristic
248	C₂²	#11718682	1	1	yes	characteristic
249	C₂²	#11718873	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	4	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	2	no	—
255	C₄	#11718688	1	1	yes	characteristic
256	C₄	#11718692	1	1	yes	characteristic
257	C₄	C₂×#88059	1	2	yes	—
258	C₄	C₂×#88075	1	1	yes	characteristic
259	C₄	#11718893	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	#13114841	1	1	yes	characteristic
263	C₂	#13114874	1	1	yes	characteristic
264	C₂	#13114890	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88184
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₈×Q₈×C₁₃	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂³×C₁₃	C₂²⋊Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×C₈×Q₈	D₁₃
hypocenter	C₁₃	#8659
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
94. Q₈×C₁₃	#108	semidirect
217. C₁₃	#8659	semidirect
221. Q₈	#88059	semidirect

Maximal subgroups

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

subgroup	index
#13115008	2
#13113584	2
#13113265	2
#13112733	2
C₂×C₈×Q₈×C₁₃	2
#8659	13