Subgroups — #15881818

Order 3328.

638 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	Q₈×#6099	C₂	1	2	yes	maximal
2	#13115612	C₂	1	4	yes	maximal
3	#13115612	C₂	1	2	yes	maximal
4	#13115011	C₂	1	2	yes	maximal
5	#13115008	C₂	1	1	yes	maximal, characteristic
6	#13114742	C₂	1	2	yes	maximal
7	C₄×#88059	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×#446	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₁₃×#446	C₄	1	4	yes	—
11	C₁₃×#446	C₄	1	4	yes	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₁₃×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×#88059	C₂²	1	1	yes	characteristic
17	C₂×#88059	C₂²	1	2	yes	—
18	C₂×#88059	C₂²	1	2	yes	—
19	C₂×#88059	C₂²	1	4	yes	—
20	C₂×#88059	C₂²	1	1	yes	characteristic
21	#11718685	—	2	2	no	—
22	#11718685	—	2	4	no	—
23	Q₈×#1055	—	2	2	no	—
24	#11718585	C₂²	1	2	yes	—
25	#11718585	C₂²	1	4	yes	—
26	C₄×#6099	C₂²	1	2	yes	—
27	C₄×#6099	C₂²	1	4	yes	—
28	#11718144	C₂²	1	2	yes	—
29	#11718144	C₂²	1	4	yes	—
30	C₂²×Q₈×C₁₃	Q₈	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
35	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
36	C₄×Q₈×C₁₃	D₄	1	2	yes	—
37	C₄×Q₈×C₁₃	D₄	1	2	yes	—
38	C₂×C₁₃×C₄⋊C₄	Q₈	1	1	yes	characteristic
39	C₂×C₁₃×C₄⋊C₄	Q₈	1	2	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
42	C₂×#6099	C₂³	1	2	yes	—
43	C₂×#6099	C₂³	1	4	yes	—
44	C₂×#6099	C₂³	1	2	yes	—
45	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
46	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
47	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
48	C₁₃×#97	C₂×C₄	1	8	yes	—
49	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
51	#88059	—	2	2	no	—
52	#88059	—	2	4	no	—
53	#88059	—	4	2	no	—
54	#88059	—	4	4	no	—
55	#88059	Q₈	1	4	yes	complemented
56	C₄⋊#1055	—	2	2	no	—
57	C₄⋊#1055	—	2	4	no	—
58	C₄⋊#1055	—	2	4	no	—
59	C₄⋊#1055	—	2	2	no	—
60	#88043	—	2	4	no	—
61	#88043	—	2	2	no	—
62	C₄×#1055	—	2	2	no	—
63	C₄×#1055	—	2	4	no	—
64	#22733	—	13	1	no	maximal, Sylow
65	C₂×Q₈×C₁₃	—	4	2	no	—
66	C₂×Q₈×C₁₃	C₄⋊C₄	1	2	yes	—
67	C₂×Q₈×C₁₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₁₃	—	2	1	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	1	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
81	C₂×C₈×C₁₃	#41	1	2	yes	—
82	C₁₃×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
83	C₁₃×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
84	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
85	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
86	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
89	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₃	C₂×D₄	1	4	yes	—
91	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
92	#6099	—	2	2	no	—
93	#6099	—	2	2	no	—
94	#6099	—	2	4	no	—
95	#6099	—	2	2	no	—
96	#6099	—	2	4	no	—
97	#6099	C₂×Q₈	1	4	yes	—
98	C₂×#1055	—	2	2	no	—
99	C₂×#1055	—	2	2	no	—
100	C₂×#1055	—	2	4	no	—
101	Q₈×M₁₆	—	13	2	no	—
102	C₂×#446	D₁₃	1	1	yes	characteristic
103	#1950	—	13	4	no	—
104	#1950	—	13	2	no	—
105	#1849	—	13	2	no	—
106	#1848	—	13	1	no	—
107	#1778	—	13	2	no	—
108	C₄×#108	—	13	1	no	—
109	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
110	Q₈×C₁₃	—	2	4	no	—
111	Q₈×C₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	4	1	no	—
114	C₂×C₄×C₁₃	—	4	2	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
119	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
121	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
128	C₈×C₁₃	—	2	2	no	—
129	C₈×C₁₃	—	2	4	no	—
130	C₈×C₁₃	C₄×Q₈	1	4	yes	—
131	#1055	—	2	4	no	—
132	#1055	—	4	2	no	—
133	#1055	—	4	4	no	—
134	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
135	#446	—	26	2	no	—
136	#446	—	26	4	no	—
137	#446	Dic₁₃	1	4	yes	—
138	#446	Dic₁₃	1	4	yes	—
139	C₈×Q₈	—	26	2	no	—
140	C₂×#108	—	13	1	no	—
141	C₂×#108	—	13	2	no	—
142	C₂×#108	—	13	1	no	—
143	C₂×#108	—	13	2	no	—
144	C₂×#108	—	13	4	no	—
145	#423	—	13	2	no	—
146	#423	—	13	4	no	—
147	C₂×C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
148	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
149	C₄×M₁₆	—	13	2	no	—
150	C₄×M₁₆	—	13	4	no	—
151	C₂×#97	C₂×D₁₃	1	2	yes	—
152	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
153	#341	—	13	2	no	—
154	#341	—	13	4	no	—
155	C₂²×C₁₃	—	2	1	no	—
156	C₂²×C₁₃	—	2	1	no	—
157	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
158	C₂²×C₁₃	#557	1	1	yes	characteristic
159	C₂²×C₁₃	Q₈²	1	1	yes	characteristic
160	C₄×C₁₃	—	2	1	no	—
161	C₄×C₁₃	—	2	4	no	—
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	—	2	1	no	—
164	C₄×C₁₃	—	2	4	no	—
165	C₄×C₁₃	#430	1	2	yes	—
166	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
167	C₄×C₁₃	#520	1	1	yes	characteristic
168	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
169	C₄×C₁₃	#550	1	1	yes	characteristic
170	C₂²×Q₈	Dic₂₆	1	1	yes	characteristic
171	C₂×M₁₆	—	13	2	no	—
172	C₂×M₁₆	—	13	2	no	—
173	C₂×M₁₆	—	13	4	no	—
174	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
175	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
176	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
177	C₄×Q₈	D₅₂	1	2	yes	—
178	C₄×Q₈	D₅₂	1	2	yes	—
179	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
180	C₄×Q₈	C₂×Dic₁₃	1	2	yes	—
181	C₂×C₄⋊C₄	Dic₂₆	1	1	yes	characteristic
182	C₂×C₄⋊C₄	Dic₂₆	1	2	yes	—
183	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
184	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
185	#108	—	13	4	no	—
186	#108	—	26	4	no	—
187	#108	—	26	2	no	—
188	#108	—	52	2	no	—
189	#108	—	52	4	no	—
190	C₄⋊C₈	—	26	2	no	—
191	C₄⋊C₈	—	26	4	no	—
192	C₄⋊C₈	—	26	4	no	—
193	C₄⋊C₈	—	26	2	no	—
194	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
195	C₄⋊C₈	C₂×Dic₁₃	1	2	yes	—
196	C₄⋊C₈	C₂×Dic₁₃	1	8	yes	—
197	#97	—	26	4	no	—
198	#97	—	26	2	no	—
199	#97	C₂×Dic₁₃	1	8	yes	—
200	C₄×C₈	—	26	2	no	—
201	C₄×C₈	—	26	4	no	—
202	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
203	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
204	C₂×C₁₃	—	2	1	no	—
205	C₂×C₁₃	—	2	1	no	—
206	C₂×C₁₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
207	C₂×C₁₃	C₂×#430	1	1	yes	characteristic
208	C₂×C₁₃	#2916	1	1	yes	characteristic
209	C₂×Q₈	—	4	2	no	—
210	C₂×Q₈	C₄⋊Dic₁₃	1	2	yes	—
211	C₂×Q₈	C₄⋊Dic₁₃	1	2	yes	—
212	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
213	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
214	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
215	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
216	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
217	M₁₆	—	13	4	no	—
218	M₁₆	—	26	2	no	—
219	M₁₆	—	26	4	no	—
220	M₁₆	—	26	2	no	—
221	M₁₆	—	26	4	no	—
222	M₁₆	—	26	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	26	2	no	—
228	C₂×C₈	—	26	2	no	—
229	C₂×C₈	—	26	4	no	—
230	C₂×C₈	Q₈×D₁₃	1	2	yes	—
231	C₂×C₈	#6131	1	2	yes	—
232	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
233	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
234	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
235	C₄⋊C₄	C₄⋊Dic₁₃	1	4	yes	—
236	C₄⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
237	C₄⋊C₄	C₄⋊Dic₁₃	1	4	yes	—
238	C₄⋊C₄	C₄⋊Dic₁₃	1	2	yes	—
239	C₄²	C₂×D₅₂	1	1	yes	characteristic
240	C₄²	C₂×D₅₂	1	4	yes	—
241	C₄²	C₂×D₅₂	1	1	yes	characteristic
242	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
243	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
244	C₄²	C₂²×Dic₁₃	1	4	yes	—
245	C₁₃	#22733	1	1	yes	characteristic, complemented, Sylow
246	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
247	Q₈	—	2	4	no	—
248	Q₈	—	4	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
254	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
255	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
258	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
259	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
260	C₂×C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
261	C₂×C₄	C₂×#6131	1	1	yes	characteristic
262	C₂×C₄	#88256	1	1	yes	characteristic
263	C₂×C₄	#88257	1	1	yes	characteristic
264	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	2	4	no	—
267	C₈	—	26	4	no	—
268	C₈	—	52	2	no	—
269	C₈	—	52	4	no	—
270	C₈	Q₈×Dic₁₃	1	4	yes	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	Q₈×Dic₂₆	1	1	yes	characteristic
274	C₂²	#11719275	1	1	yes	characteristic
275	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	4	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	4	no	—
280	C₄	—	2	1	no	—
281	C₄	#11718786	1	2	yes	—
282	C₄	Q₈×D₅₂	1	1	yes	characteristic
283	C₄	#11719282	1	1	yes	characteristic
284	C₄	C₂×Q₈×Dic₁₃	1	1	yes	characteristic
285	C₄	#11719532	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	Q₈×C₄⋊Dic₁₃	1	1	yes	characteristic
289	C₂	C₂×#11718786	1	1	yes	characteristic
290	C₂	#13123447	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#446	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₁₃	C₂²×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#446	D₁₃
hypocenter	C₁₃	#22733
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
55. #88059	Q₈	semidirect
245. C₁₃	#22733	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#6099	2
#13115612	2
#13115011	2
#13115008	2
#13114742	2
C₄×#88059	2
C₂×C₁₃×#446	2
#22733	13