Subgroups — #16807397

Order 3328.

536 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131252	C₂	1	2	yes	maximal, complemented
2	#13131217	C₂	1	4	yes	maximal, complemented
3	#13131215	C₂	1	2	yes	maximal, complemented
4	#13131200	C₂	1	2	yes	maximal, complemented
5	#13131193	C₂	1	1	yes	maximal, characteristic, complemented
6	#13131168	C₂	1	1	yes	maximal, characteristic, complemented
7	D₁₃⋊(C₄⋊C₁₆)	C₂	1	2	yes	maximal, complemented
8	#13123792	C₂	1	1	yes	maximal, characteristic
9	C₄×#6127	C₄	1	1	yes	characteristic
10	C₂×#88106	C₂²	1	2	yes	—
11	C₂×D₁₃⋊C₁₆	C₂²	1	4	yes	—
12	C₁₃×#431	C₄	1	1	yes	characteristic
13	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
14	#11718495	C₄	1	1	yes	characteristic
15	#11718495	C₂²	1	1	yes	characteristic
16	#11718491	C₄	1	1	yes	characteristic
17	#11718491	C₂²	1	1	yes	characteristic
18	#11718386	C₄	1	2	yes	—
19	#11718386	C₂²	1	2	yes	—
20	#11718346	C₄	1	2	yes	—
21	C₄×C₈×D₁₃	C₂²	1	2	yes	—
22	C₂²⋊#6092	C₂²	1	2	yes	—
23	#11718260	C₂²	1	4	yes	—
24	#11718259	C₂²	1	4	yes	—
25	#11718258	C₂²	1	4	yes	—
26	C₄⋊#6092	C₂²	1	1	yes	characteristic
27	C₄⋊#6092	C₂²	1	2	yes	—
28	C₄⋊#6092	C₂²	1	2	yes	—
29	C₄⋊#6092	C₂²	1	1	yes	characteristic
30	C₄×#6092	C₂²	1	2	yes	—
31	C₂×#6127	C₈	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₂²⋊D₁₃	C₈	1	2	yes	—
34	C₄×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
35	#88180	C₈	1	1	yes	characteristic
36	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
37	C₂²×#1055	C₂³	1	1	yes	characteristic
38	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₁₃	C₂³	1	2	yes	—
40	C₄×D₅₂	C₈	1	1	yes	characteristic
41	C₄×D₅₂	C₂×C₄	1	1	yes	characteristic
42	#88126	C₈	1	2	yes	—
43	C₄²×D₁₃	C₂×C₄	1	2	yes	—
44	C₄×Dic₂₆	C₈	1	1	yes	characteristic
45	C₄×Dic₂₆	C₂×C₄	1	1	yes	characteristic
46	#88106	—	4	2	no	—
47	C₂×#6092	C₂³	1	2	yes	—
48	C₂×#6092	C₂³	1	2	yes	—
49	C₂×#6092	C₂³	1	4	yes	—
50	D₁₃⋊C₁₆	—	4	4	no	—
51	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₁₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
55	#88060	C₂×C₄	1	2	yes	—
56	#88060	C₂×C₄	1	2	yes	—
57	#88054	C₂×C₄	1	2	yes	—
58	#88054	C₂×C₄	1	2	yes	—
59	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
60	C₈×Dic₁₃	C₂³	1	2	yes	—
61	C₄×#1055	C₂×C₄	1	1	yes	characteristic
62	C₄×#1055	C₂×C₄	1	1	yes	characteristic
63	C₄×#1055	C₂³	1	1	yes	characteristic
64	C₄×#1055	C₂³	1	1	yes	characteristic
65	#33504	—	13	1	no	maximal, Sylow
66	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
70	#6127	—	2	4	no	—
71	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
73	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
74	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
75	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
78	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
79	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
80	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²⋊Dic₁₃	C₂×C₈	1	2	yes	—
83	#6103	C₂×C₈	1	2	yes	—
84	#6103	C₂×C₈	1	2	yes	—
85	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
86	C₄⋊Dic₁₃	C₂×C₈	1	1	yes	characteristic
87	#6101	C₂×C₈	1	2	yes	—
88	#6101	C₂×C₈	1	2	yes	—
89	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
90	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
91	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
92	C₂×#1055	—	2	1	no	—
93	C₂×#1055	—	2	1	no	—
94	C₂×#1055	—	4	1	no	—
95	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
96	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
97	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
98	C₂×#1055	C₂⁴	1	1	yes	characteristic
99	C₈×D₁₃	—	2	2	no	—
100	C₈×D₁₃	—	2	2	no	—
101	C₈×D₁₃	—	4	2	no	—
102	#6092	—	2	2	no	—
103	#6092	—	2	4	no	—
104	#6092	—	2	2	no	—
105	C₈×#41	—	13	1	no	—
106	#2183	—	13	2	no	—
107	#2168	—	13	4	no	—
108	D₄×C₁₆	—	13	2	no	—
109	#2163	—	13	2	no	—
110	#2162	—	13	1	no	—
111	#2150	—	13	1	no	—
112	C₂×C₄⋊C₁₆	—	13	2	no	—
113	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
114	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
125	C₂²⋊D₁₃	—	2	4	no	—
126	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
127	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
128	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
129	D₅₂	M₃₂	1	4	yes	complemented
130	C₄×D₁₃	—	2	4	no	—
131	C₄×D₁₃	—	2	2	no	—
132	C₄×D₁₃	—	2	2	no	—
133	C₄×D₁₃	—	2	4	no	—
134	Dic₂₆	—	2	2	no	—
135	C₈×C₁₃	—	2	2	no	—
136	C₈×C₁₃	—	2	2	no	—
137	#1055	—	2	1	no	—
138	#1055	—	2	1	no	—
139	#1055	—	2	1	no	—
140	#1055	H(5,𝔽₂)	1	1	yes	characteristic
141	#1055	2₋¹⁺⁴	1	1	yes	characteristic
142	C₄×#41	—	13	1	no	—
143	C₂×M₃₂	—	13	2	no	—
144	C₂²×C₁₆	—	13	4	no	—
145	C₈×Q₈	—	13	1	no	—
146	C₈×Q₈	—	13	1	no	—
147	C₈×D₄	—	13	1	no	—
148	C₈×D₄	—	13	1	no	—
149	C₈×D₄	—	13	2	no	—
150	C₈×D₄	—	13	2	no	—
151	#431	—	13	2	no	—
152	#431	#261	1	1	yes	characteristic
153	C₂×C₄×C₈	—	13	1	no	—
154	C₂×C₄×C₈	—	13	2	no	—
155	C₄⋊C₁₆	—	13	1	no	—
156	C₄⋊C₁₆	—	13	2	no	—
157	C₄⋊C₁₆	—	13	2	no	—
158	C₄⋊C₁₆	—	13	1	no	—
159	C₄⋊C₁₆	—	13	4	no	—
160	C₂²⋊C₁₆	—	13	2	no	—
161	C₂²⋊C₁₆	—	13	4	no	—
162	#346	—	13	4	no	—
163	C₄×C₁₆	—	13	2	no	—
164	C₂²×C₁₃	—	2	1	no	—
165	C₂²×C₁₃	—	2	1	no	—
166	C₂²×C₁₃	—	2	1	no	—
167	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
168	C₂×D₁₃	—	2	2	no	—
169	C₂×D₁₃	—	2	2	no	—
170	C₂×D₁₃	C₂×M₃₂	1	4	yes	—
171	C₄×C₁₃	—	2	1	no	—
172	C₄×C₁₃	—	2	1	no	—
173	C₄×C₁₃	—	2	1	no	—
174	C₄×C₁₃	C₂×M₃₂	1	2	yes	—
175	C₄×C₁₃	#504	1	2	yes	—
176	C₄×C₁₃	#520	1	1	yes	characteristic
177	C₄×C₁₃	#568	1	1	yes	characteristic
178	Dic₁₃	—	2	2	no	—
179	Dic₁₃	—	2	2	no	—
180	Dic₁₃	—	2	2	no	—
181	C₂×#41	—	13	1	no	—
182	C₂²×C₈	—	13	1	no	—
183	C₂²×C₈	—	13	1	no	—
184	C₂²×C₈	—	13	2	no	—
185	C₂²×C₈	—	13	2	no	—
186	C₄×Q₈	—	13	1	no	—
187	C₄×Q₈	—	13	1	no	—
188	C₄×D₄	—	13	2	no	—
189	C₄×D₄	—	13	2	no	—
190	C₄×D₄	—	13	1	no	—
191	C₄×D₄	—	13	1	no	—
192	#117	—	13	1	no	—
193	#117	—	13	2	no	—
194	C₂×C₄²	—	13	2	no	—
195	C₂×C₄²	C₂×#261	1	1	yes	characteristic
196	M₃₂	—	52	2	no	—
197	C₂×C₁₆	—	13	2	no	—
198	C₂×C₁₆	—	13	2	no	—
199	C₂×C₁₆	—	13	4	no	—
200	C₂×C₁₆	—	52	4	no	—
201	C₄⋊C₈	—	13	2	no	—
202	C₄⋊C₈	—	13	2	no	—
203	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
204	C₄⋊C₈	C₂×#261	1	1	yes	characteristic
205	C₂²⋊C₈	—	13	2	no	—
206	C₂²⋊C₈	—	13	2	no	—
207	C₂²⋊C₈	C₂×#261	1	2	yes	—
208	C₄×C₈	—	13	2	no	—
209	C₄×C₈	—	13	2	no	—
210	C₄×C₈	—	13	1	no	—
211	C₄×C₈	—	13	1	no	—
212	C₄×C₈	—	13	1	no	—
213	C₄×C₈	—	13	1	no	—
214	C₄×C₈	C₂×#261	1	2	yes	—
215	C₂×C₁₃	—	4	1	no	—
216	C₂×C₁₃	#2988	1	1	yes	characteristic
217	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
218	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
219	D₁₃	—	2	4	no	—
220	#41	—	26	4	no	—
221	C₂×Q₈	—	13	1	no	—
222	C₂×D₄	—	13	2	no	—
223	C₂×D₄	—	13	1	no	—
224	C₂²×C₄	—	13	2	no	—
225	C₂²×C₄	—	13	2	no	—
226	C₂²×C₄	—	13	2	no	—
227	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
228	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
229	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
230	C₂×C₈	—	13	1	no	—
231	C₂×C₈	—	13	1	no	—
232	C₂×C₈	—	13	1	no	—
233	C₂×C₈	—	13	1	no	—
234	C₂×C₈	—	26	2	no	—
235	C₂×C₈	—	26	2	no	—
236	C₂×C₈	—	26	1	no	—
237	C₂×C₈	—	26	1	no	—
238	C₂×C₈	—	52	2	no	—
239	C₂×C₈	—	52	1	no	—
240	C₂×C₈	C₂²×#261	1	2	yes	—
241	C₂×C₈	C₂²×#261	1	2	yes	—
242	C₄⋊C₄	—	13	2	no	—
243	C₄⋊C₄	—	13	2	no	—
244	C₄⋊C₄	—	13	1	no	—
245	C₄⋊C₄	—	13	1	no	—
246	C₂²⋊C₄	—	13	2	no	—
247	C₂²⋊C₄	—	13	2	no	—
248	C₂²⋊C₄	—	13	2	no	—
249	C₄²	—	13	2	no	—
250	C₄²	—	13	2	no	—
251	C₄²	—	13	2	no	—
252	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
253	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
254	C₄²	C₂²×#261	1	1	yes	characteristic
255	C₄²	C₂²×#261	1	1	yes	characteristic
256	C₁₆	—	26	2	no	—
257	C₁₆	—	26	4	no	—
258	C₁₆	—	26	2	no	—
259	C₁₃	#33504	1	1	yes	characteristic, complemented, Sylow
260	C₂³	—	13	2	no	—
261	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
262	Q₈	—	26	2	no	—
263	D₄	—	13	4	no	—
264	D₄	—	26	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	13	2	no	—
270	C₂×C₄	—	13	2	no	—
271	C₂×C₄	—	13	2	no	—
272	C₂×C₄	—	26	4	no	—
273	C₂×C₄	—	26	4	no	—
274	C₂×C₄	—	26	2	no	—
275	C₂×C₄	—	26	2	no	—
276	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
277	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
278	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
279	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
280	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
281	C₂×C₄	C₂³×#261	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	2	2	no	—
284	C₈	—	13	1	no	—
285	C₈	—	13	1	no	—
286	C₈	—	26	1	no	—
287	C₈	—	26	1	no	—
288	C₈	—	26	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	13	4	no	—
293	C₂²	—	26	2	no	—
294	C₂²	—	26	2	no	—
295	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	26	2	no	—
300	C₄	—	26	2	no	—
301	C₄	—	26	2	no	—
302	C₄	D₁₃⋊M₃₂	1	2	yes	—
303	C₄	#11719096	1	2	yes	—
304	C₄	#11719629	1	1	yes	characteristic
305	C₄	#11719631	1	1	yes	characteristic
306	C₂	—	4	1	no	—
307	C₂	—	26	4	no	—
308	C₂	#13131753	1	1	yes	characteristic
309	C₂	C₂×D₁₃⋊M₃₂	1	1	yes	characteristic
310	C₂	C₂×#11719096	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₁₃×#431	C₄
prime radical	C₂×#6127	C₈
cosocle	C₂×#1055	C₂⁴
hypercenter	#431	#261
hypocenter	C₁₃	#33504
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
1. #13131252	C₂	semidirect
2. #13131217	C₂	semidirect
3. #13131215	C₂	semidirect
4. #13131200	C₂	semidirect
5. #13131193	C₂	semidirect
6. #13131168	C₂	semidirect
7. D₁₃⋊(C₄⋊C₁₆)	C₂	semidirect
129. D₅₂	M₃₂	semidirect
259. C₁₃	#33504	semidirect

Maximal subgroups

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

subgroup	index
#13131252	2
#13131217	2
#13131215	2
#13131200	2
#13131193	2
#13131168	2
D₁₃⋊(C₄⋊C₁₆)	2
#13123792	2
#33504	13