Subgroups — #16807369

Order 3328.

494 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131253	C₂	1	4	yes	maximal, complemented
2	#13131251	C₂	1	2	yes	maximal, complemented
3	#13131250	C₂	1	2	yes	maximal, complemented
4	#13131200	C₂	1	1	yes	maximal, characteristic
5	#13131199	C₂	1	2	yes	maximal
6	#13131196	C₂	1	2	yes	maximal
7	#13131195	C₂	1	1	yes	maximal, characteristic
8	C₈×Q₈×D₁₃	C₂	1	1	yes	maximal, characteristic
9	C₄×Q₈×D₁₃	C₄	1	1	yes	characteristic
10	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic
11	Q₈×#1055	C₂²	1	1	yes	characteristic
12	D₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
13	D₁₃×C₄⋊C₈	C₄	1	2	yes	—
14	#11718491	C₂²	1	1	yes	characteristic
15	#11718491	C₂²	1	2	yes	—
16	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
17	C₄×C₈×D₁₃	C₂²	1	2	yes	—
18	C₈×Dic₂₆	C₄	1	1	yes	characteristic
19	C₈×Dic₂₆	C₄	1	2	yes	—
20	#11718260	C₂²	1	2	yes	—
21	#11718260	C₂²	1	4	yes	—
22	#11718259	C₂²	1	1	yes	characteristic
23	#11718259	C₂²	1	1	yes	characteristic
24	#11718259	C₂²	1	2	yes	—
25	#11718258	C₂²	1	2	yes	—
26	#11718257	C₂²	1	4	yes	—
27	#11718231	C₂²	1	4	yes	—
28	C₄⋊#6092	C₂²	1	4	yes	—
29	C₄⋊#6092	C₂²	1	2	yes	—
30	C₄×#6092	C₂²	1	2	yes	—
31	C₂×Q₈×D₁₃	C₈	1	1	yes	characteristic
32	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	Q₈×Dic₁₃	C₈	1	1	yes	characteristic
34	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
36	C₂×C₈×D₁₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×D₁₃	C₈	1	1	yes	characteristic
38	C₄⋊C₄×D₁₃	C₈	1	2	yes	—
39	#88141	C₈	1	2	yes	—
40	#88141	C₈	1	1	yes	characteristic
41	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
42	C₄²×D₁₃	C₂×C₄	1	2	yes	—
43	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
44	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
45	C₂×#6092	C₂³	1	2	yes	—
46	C₂×#6092	C₂³	1	2	yes	—
47	C₂×#6092	C₂³	1	4	yes	—
48	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
52	#88054	C₂×C₄	1	2	yes	—
53	#88054	C₂×C₄	1	2	yes	—
54	#88054	C₂×C₄	1	2	yes	—
55	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
56	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
57	C₈×Dic₁₃	C₂³	1	2	yes	—
58	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
59	C₄⋊#1055	C₂×C₄	1	2	yes	—
60	C₄⋊#1055	C₂×C₄	1	1	yes	characteristic
61	C₄×#1055	C₂³	1	1	yes	characteristic
62	C₄×#1055	C₂³	1	2	yes	—
63	#33498	—	13	1	no	maximal, Sylow
64	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
65	Q₈×D₁₃	—	2	4	no	—
66	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
67	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
68	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
69	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
70	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
72	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
77	C₁₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
78	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
79	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
81	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
82	#6101	C₂×C₈	1	2	yes	—
83	#6101	C₂×C₈	1	2	yes	—
84	#6101	C₂×C₈	1	2	yes	—
85	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
86	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
87	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
88	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
89	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
91	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
92	C₂×#1055	C₂²×C₄	1	2	yes	—
93	C₂×#1055	C₂⁴	1	1	yes	characteristic
94	C₈×D₁₃	—	2	2	no	—
95	C₈×D₁₃	—	4	1	no	—
96	C₈×D₁₃	—	4	2	no	—
97	#6092	—	2	2	no	—
98	#6092	—	2	2	no	—
99	#6092	—	2	4	no	—
100	C₂×C₈×Q₈	—	13	1	no	—
101	#2183	—	13	2	no	—
102	#2183	—	13	4	no	—
103	Q₈×C₁₆	—	13	2	no	—
104	#2163	—	13	2	no	—
105	#2163	—	13	1	no	—
106	#2162	—	13	1	no	—
107	#2162	—	13	2	no	—
108	C₂²×D₁₃	C₂²×C₈	1	1	yes	characteristic
109	Q₈×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
112	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
116	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
117	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
119	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
120	C₄×D₁₃	—	2	2	no	—
121	C₄×D₁₃	—	2	4	no	—
122	C₄×D₁₃	—	2	1	no	—
123	C₄×D₁₃	—	2	1	no	—
124	C₄×D₁₃	—	2	2	no	—
125	C₄×D₁₃	—	2	4	no	—
126	Dic₂₆	—	2	4	no	—
127	Dic₂₆	M₃₂	1	4	yes	—
128	C₈×C₁₃	—	2	1	no	—
129	C₈×C₁₃	—	2	2	no	—
130	C₈×C₁₃	—	2	1	no	—
131	#1055	—	2	1	no	—
132	#1055	—	2	2	no	—
133	#1055	2₋¹⁺⁴	1	2	yes	—
134	C₂×C₄×Q₈	—	13	1	no	—
135	C₈×Q₈	—	13	1	no	—
136	C₈×Q₈	—	13	1	no	—
137	C₈×Q₈	—	13	2	no	—
138	C₈×Q₈	—	13	2	no	—
139	C₈×Q₈	—	13	1	no	—
140	C₈×Q₈	#261	1	1	yes	characteristic
141	C₂×C₄⋊C₈	—	13	2	no	—
142	C₂×C₄⋊C₈	—	13	1	no	—
143	C₂×C₄×C₈	—	13	2	no	—
144	C₂×C₄×C₈	—	13	1	no	—
145	C₄⋊C₁₆	—	13	4	no	—
146	C₄⋊C₁₆	—	13	2	no	—
147	C₄⋊C₁₆	—	13	2	no	—
148	C₄⋊C₁₆	—	13	4	no	—
149	C₂²⋊C₁₆	—	13	1	no	—
150	C₂²⋊C₁₆	—	13	1	no	—
151	C₂²⋊C₁₆	—	13	2	no	—
152	#346	—	13	4	no	—
153	#346	—	13	2	no	—
154	C₄×C₁₆	—	13	2	no	—
155	C₄×C₁₆	—	13	4	no	—
156	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
157	C₂×D₁₃	—	2	1	no	—
158	C₂×D₁₃	—	2	1	no	—
159	C₂×D₁₃	#504	1	2	yes	—
160	C₄×C₁₃	—	2	2	no	—
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	—	2	2	no	—
163	C₄×C₁₃	C₂×M₃₂	1	2	yes	—
164	C₄×C₁₃	#519	1	1	yes	characteristic
165	C₄×C₁₃	#568	1	1	yes	characteristic
166	Dic₁₃	—	2	1	no	—
167	Dic₁₃	—	2	2	no	—
168	Dic₁₃	—	2	1	no	—
169	Dic₁₃	—	2	1	no	—
170	Dic₁₃	C₂×M₃₂	1	4	yes	—
171	C₂²×Q₈	—	13	1	no	—
172	C₂²×C₈	—	13	1	no	—
173	C₂²×C₈	—	13	1	no	—
174	C₂²×C₈	—	13	2	no	—
175	C₄×Q₈	—	13	1	no	—
176	C₄×Q₈	—	13	1	no	—
177	C₄×Q₈	—	13	2	no	—
178	C₄×Q₈	—	13	1	no	—
179	C₄×Q₈	—	13	2	no	—
180	C₄×Q₈	C₂×#261	1	1	yes	characteristic
181	C₂×C₄⋊C₄	—	13	2	no	—
182	C₂×C₄⋊C₄	—	13	1	no	—
183	C₂×C₄²	—	13	2	no	—
184	C₂×C₄²	—	13	1	no	—
185	C₂×C₁₆	—	13	2	no	—
186	C₂×C₁₆	—	13	2	no	—
187	C₂×C₁₆	—	13	4	no	—
188	C₄⋊C₈	—	13	2	no	—
189	C₄⋊C₈	—	13	2	no	—
190	C₄⋊C₈	—	13	2	no	—
191	C₄⋊C₈	—	13	2	no	—
192	C₄⋊C₈	—	13	1	no	—
193	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
194	C₄⋊C₈	C₂×#261	1	2	yes	—
195	C₄×C₈	—	13	2	no	—
196	C₄×C₈	—	13	1	no	—
197	C₄×C₈	—	13	2	no	—
198	C₄×C₈	—	13	1	no	—
199	C₄×C₈	—	13	1	no	—
200	C₄×C₈	—	13	2	no	—
201	C₄×C₈	C₂×#261	1	2	yes	—
202	C₄×C₈	C₂×#261	1	1	yes	characteristic
203	C₂×C₁₃	#2982	1	1	yes	characteristic
204	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
205	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
206	D₁₃	—	2	2	no	—
207	C₂×Q₈	—	13	1	no	—
208	C₂×Q₈	—	13	2	no	—
209	C₂×Q₈	—	26	4	no	—
210	C₂×Q₈	D₁₃⋊C₈	1	1	yes	characteristic
211	C₂²×C₄	—	13	2	no	—
212	C₂²×C₄	—	13	1	no	—
213	C₂²×C₄	—	13	2	no	—
214	C₂²×C₄	—	13	1	no	—
215	C₂²×C₄	—	13	1	no	—
216	C₂×C₈	—	13	1	no	—
217	C₂×C₈	—	13	1	no	—
218	C₂×C₈	—	13	2	no	—
219	C₂×C₈	—	26	2	no	—
220	C₂×C₈	—	52	2	no	—
221	C₂×C₈	—	52	1	no	—
222	C₂×C₈	C₂²×#261	1	1	yes	characteristic
223	C₂×C₈	C₂²×#261	1	2	yes	—
224	C₂×C₈	C₂²×#261	1	1	yes	characteristic
225	C₄⋊C₄	—	13	2	no	—
226	C₄⋊C₄	—	13	2	no	—
227	C₄⋊C₄	—	13	2	no	—
228	C₄⋊C₄	—	13	2	no	—
229	C₄⋊C₄	—	13	1	no	—
230	C₄⋊C₄	D₁₃⋊C₈	1	2	yes	—
231	C₄⋊C₄	D₁₃⋊C₈	1	1	yes	characteristic
232	C₄²	—	13	2	no	—
233	C₄²	—	13	2	no	—
234	C₄²	—	13	1	no	—
235	C₄²	—	13	1	no	—
236	C₄²	—	13	2	no	—
237	C₄²	—	13	1	no	—
238	C₄²	C₂²×#261	1	2	yes	—
239	C₄²	C₂²×#261	1	1	yes	characteristic
240	C₁₆	—	26	2	no	—
241	C₁₆	—	26	2	no	—
242	C₁₆	—	26	4	no	—
243	C₁₃	#33498	1	1	yes	characteristic, complemented, Sylow
244	C₂³	—	13	1	no	—
245	Q₈	—	2	2	no	—
246	Q₈	—	13	4	no	—
247	Q₈	—	26	4	no	—
248	C₂×C₄	—	13	2	no	—
249	C₂×C₄	—	13	1	no	—
250	C₂×C₄	—	13	2	no	—
251	C₂×C₄	—	13	1	no	—
252	C₂×C₄	—	13	1	no	—
253	C₂×C₄	—	26	4	no	—
254	C₂×C₄	—	26	4	no	—
255	C₂×C₄	—	26	2	no	—
256	C₂×C₄	—	26	2	no	—
257	C₂×C₄	—	26	1	no	—
258	C₂×C₄	—	26	1	no	—
259	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
260	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
261	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
262	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
263	C₂×C₄	C₂³×#261	1	1	yes	characteristic
264	C₈	—	2	1	no	—
265	C₈	—	2	2	no	—
266	C₈	—	2	1	no	—
267	C₈	—	13	2	no	—
268	C₈	—	26	1	no	—
269	C₈	—	26	2	no	—
270	C₂²	—	13	2	no	—
271	C₂²	—	26	1	no	—
272	C₂²	—	26	1	no	—
273	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	1	no	—
277	C₄	—	13	4	no	—
278	C₄	—	26	1	no	—
279	C₄	—	26	2	no	—
280	C₄	—	26	1	no	—
281	C₄	—	26	1	no	—
282	C₄	D₁₃⋊M₃₂	1	2	yes	—
283	C₄	#11719627	1	1	yes	characteristic
284	C₄	#11719629	1	1	yes	characteristic
285	C₂	—	26	2	no	—
286	C₂	#13131730	1	1	yes	characteristic
287	C₂	C₂×D₁₃⋊M₃₂	1	1	yes	characteristic
288	C₂	D₁₃⋊#504	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₈×Q₈×C₁₃	C₄
prime radical	C₂²×D₁₃	C₂²×C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	C₈×Q₈	#261
hypocenter	C₁₃	#33498
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. #13131253	C₂	semidirect
2. #13131251	C₂	semidirect
3. #13131250	C₂	semidirect
243. C₁₃	#33498	semidirect

Maximal subgroups

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

subgroup	index
#13131253	2
#13131251	2
#13131250	2
#13131200	2
#13131199	2
#13131196	2
#13131195	2
C₈×Q₈×D₁₃	2
#33498	13