Subgroups — #15756040

Order 3328.

482 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₅₂	C₂	1	1	yes	maximal, characteristic
2	#13113584	C₂	1	1	yes	maximal, characteristic, complemented
3	#13113260	C₂	1	2	yes	maximal
4	#13112731	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₁₃	—	2	1	no	—
7	C₂×C₄×D₅₂	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#108	—	2	1	no	—
9	C₂×C₁₃×#108	—	2	1	no	—
10	C₁₃×#423	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×#88060	—	2	1	no	—
13	#11718586	C₂²	1	2	yes	—
14	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
15	C₈×D₅₂	—	2	1	no	—
16	C₈×D₅₂	—	2	1	no	—
17	C₈×D₅₂	C₄	1	2	yes	—
18	C₈×D₅₂	C₄	1	2	yes	—
19	#11718075	—	2	4	no	—
20	#11718050	—	2	1	no	—
21	#11718049	—	2	1	no	—
22	C₂²×D₅₂	Q₈	1	1	yes	characteristic
23	C₂²×C₄×D₁₃	—	2	1	no	—
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×#6103	—	2	1	no	—
29	C₂×C₄⋊Dic₁₃	Q₈	1	1	yes	characteristic
30	C₂²×#1055	—	2	1	no	—
31	C₂×C₈×D₁₃	—	2	2	no	—
32	C₂×C₈×D₁₃	—	2	2	no	—
33	C₂×C₈×D₁₃	—	4	2	no	—
34	C₂×C₈×D₁₃	—	4	2	no	—
35	C₄×D₅₂	—	2	1	no	—
36	C₄×D₅₂	—	2	1	no	—
37	C₄×D₅₂	C₂×C₄	1	2	yes	—
38	C₄×D₅₂	D₄	1	2	yes	—
39	C₁₃×#108	—	4	2	no	—
40	C₁₃×#108	—	4	2	no	—
41	C₁₃×C₄⋊C₈	—	2	2	no	—
42	C₁₃×C₄⋊C₈	—	2	2	no	—
43	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
47	C₂²⋊#1055	—	2	2	no	—
48	#88060	—	2	2	no	—
49	#88060	—	2	2	no	—
50	C₄⋊#1055	—	2	1	no	—
51	C₄⋊#1055	—	2	2	no	—
52	C₄⋊#1055	C₂×C₄	1	2	yes	—
53	C₄×#1055	—	2	2	no	—
54	#8658	—	13	1	no	maximal, Sylow
55	C₂³×D₁₃	—	2	1	no	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
59	C₂²×Dic₁₃	—	2	1	no	—
60	C₂×D₅₂	—	2	1	no	—
61	C₂×D₅₂	—	2	4	no	—
62	C₂×D₅₂	C₄⋊C₄	1	2	yes	—
63	C₂×C₄×D₁₃	—	2	2	no	—
64	C₂×C₄×D₁₃	—	2	2	no	—
65	C₂×C₄×D₁₃	—	4	1	no	—
66	C₂×C₄×D₁₃	—	4	1	no	—
67	C₂×C₄×D₁₃	—	4	1	no	—
68	C₂×C₄×D₁₃	—	4	1	no	—
69	C₁₃×M₁₆	—	4	2	no	—
70	C₁₃×M₁₆	—	4	2	no	—
71	C₂×C₈×C₁₃	—	2	1	no	—
72	C₂×C₈×C₁₃	—	2	1	no	—
73	C₂×C₈×C₁₃	—	4	2	no	—
74	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	D₈	1	2	yes	—
79	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
80	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₁₃	#41	1	1	yes	characteristic
84	#6103	—	2	2	no	—
85	#6103	—	2	2	no	—
86	C₄⋊Dic₁₃	—	2	1	no	—
87	C₄⋊Dic₁₃	C₄⋊C₄	1	2	yes	—
88	C₂×#1055	—	2	2	no	—
89	C₂×#1055	—	4	1	no	—
90	C₂×#1055	—	4	1	no	—
91	C₂×#1055	—	4	2	no	—
92	C₈×D₁₃	—	4	4	no	—
93	C₈×D₁₃	—	4	4	no	—
94	C₂×C₈×D₄	—	13	1	no	—
95	#1848	D₁₃	1	1	yes	characteristic, complemented
96	#1564	—	13	1	no	—
97	#1486	—	13	2	no	—
98	#1301	—	13	2	no	—
99	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂²×D₁₃	—	2	4	no	—
101	C₂²×D₁₃	—	2	2	no	—
102	C₂²×D₁₃	—	4	2	no	—
103	C₂²×D₁₃	—	4	1	no	—
104	C₂²×D₁₃	—	4	1	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	4	1	no	—
107	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
111	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
117	C₂×Dic₁₃	—	2	2	no	—
118	C₂×Dic₁₃	—	4	1	no	—
119	C₂×Dic₁₃	—	4	1	no	—
120	D₅₂	—	2	4	no	—
121	D₅₂	—	2	2	no	—
122	D₅₂	#108	1	4	yes	complemented
123	C₄×D₁₃	—	4	2	no	—
124	C₄×D₁₃	—	4	2	no	—
125	C₄×D₁₃	—	4	2	no	—
126	C₄×D₁₃	—	4	2	no	—
127	C₈×C₁₃	—	2	1	no	—
128	C₈×C₁₃	—	2	1	no	—
129	C₈×C₁₃	—	8	2	no	—
130	C₈×C₁₃	#102	1	2	yes	—
131	C₈×C₁₃	#102	1	2	yes	—
132	#1055	—	4	2	no	—
133	#1055	—	8	2	no	—
134	C₂³×C₈	—	26	1	no	—
135	C₂×C₄×D₄	—	13	1	no	—
136	C₈×D₄	—	13	2	no	—
137	C₈×D₄	—	13	2	no	—
138	C₈×D₄	—	26	1	no	—
139	C₈×D₄	—	26	1	no	—
140	#431	—	13	2	no	—
141	C₂×#108	—	2	1	no	—
142	C₂×#108	—	2	1	no	—
143	#423	C₂×D₁₃	1	2	yes	—
144	C₂×C₄⋊C₈	—	13	1	no	—
145	C₂×C₂²⋊C₈	—	26	1	no	—
146	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
147	#335	—	26	1	no	—
148	#334	—	26	1	no	—
149	D₄⋊C₈	—	26	4	no	—
150	C₂²×C₁₃	—	2	1	no	—
151	C₂²×C₁₃	—	2	1	no	—
152	C₂²×C₁₃	#380	1	1	yes	characteristic
153	C₂²×C₁₃	#474	1	1	yes	characteristic
154	C₂²×C₁₃	#476	1	1	yes	characteristic
155	C₂×D₁₃	—	2	4	no	—
156	C₂×D₁₃	—	4	4	no	—
157	C₂×D₁₃	—	4	4	no	—
158	C₂×D₁₃	—	4	2	no	—
159	C₂×D₁₃	—	4	2	no	—
160	C₄×C₁₃	—	2	1	no	—
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
164	C₄×C₁₃	#417	1	1	yes	characteristic
165	C₄×C₁₃	C₂×#108	1	2	yes	—
166	C₄×C₁₃	#459	1	1	yes	characteristic
167	C₄×C₁₃	#461	1	1	yes	characteristic
168	Dic₁₃	—	4	2	no	—
169	C₂²×D₄	—	13	1	no	—
170	C₂³×C₄	—	26	1	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂²×C₈	—	26	1	no	—
173	C₂²×C₈	—	26	2	no	—
174	C₂²×C₈	—	26	2	no	—
175	C₂²×C₈	—	52	2	no	—
176	C₂²×C₈	—	52	2	no	—
177	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
178	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
179	C₄×D₄	—	13	2	no	—
180	C₄×D₄	—	13	2	no	—
181	C₄×D₄	—	26	1	no	—
182	C₄×D₄	—	26	1	no	—
183	C₂×C₄⋊C₄	—	13	1	no	—
184	C₂×C₂²⋊C₄	—	26	1	no	—
185	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
186	#108	—	4	2	no	—
187	#108	—	4	2	no	—
188	C₄⋊C₈	—	2	2	no	—
189	C₄⋊C₈	—	2	2	no	—
190	C₄⋊C₈	—	13	2	no	—
191	C₄⋊C₈	—	26	1	no	—
192	C₄⋊C₈	—	26	2	no	—
193	C₂²⋊C₈	—	26	2	no	—
194	C₂²⋊C₈	—	26	2	no	—
195	C₂²⋊C₈	—	26	2	no	—
196	C₄×C₈	—	26	2	no	—
197	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
198	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
199	C₄×C₈	D₅₂	1	1	yes	characteristic
200	C₄×C₈	D₅₂	1	1	yes	characteristic
201	C₂×C₁₃	—	2	1	no	—
202	C₂×C₁₃	—	2	1	no	—
203	C₂×C₁₃	#1802	1	1	yes	characteristic
204	C₂×C₁₃	#1810	1	1	yes	characteristic
205	C₂×C₁₃	#1814	1	1	yes	characteristic
206	D₁₃	—	4	4	no	—
207	C₂⁴	—	26	1	no	—
208	C₂×D₄	—	13	2	no	—
209	C₂×D₄	—	26	4	no	—
210	C₂×D₄	—	26	1	no	—
211	C₂²×C₄	—	26	1	no	—
212	C₂²×C₄	—	26	2	no	—
213	C₂²×C₄	—	26	2	no	—
214	C₂²×C₄	—	52	1	no	—
215	C₂²×C₄	—	52	1	no	—
216	C₂²×C₄	—	52	1	no	—
217	C₂²×C₄	—	52	1	no	—
218	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
219	C₂²×C₄	#6131	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
221	M₁₆	—	4	2	no	—
222	M₁₆	—	4	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	26	2	no	—
227	C₂×C₈	—	52	1	no	—
228	C₂×C₈	—	52	1	no	—
229	C₂×C₈	—	52	4	no	—
230	C₂×C₈	—	52	4	no	—
231	C₂×C₈	—	52	2	no	—
232	C₂×C₈	#6103	1	1	yes	characteristic
233	C₂×C₈	#6103	1	1	yes	characteristic
234	C₂×C₈	#6103	1	1	yes	characteristic
235	C₂×C₈	#6103	1	1	yes	characteristic
236	C₂×C₈	#6104	1	2	yes	—
237	C₂×C₈	#6106	1	2	yes	—
238	C₄⋊C₄	—	13	2	no	—
239	C₄⋊C₄	—	26	1	no	—
240	C₂²⋊C₄	—	26	2	no	—
241	C₂²⋊C₄	—	26	2	no	—
242	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
243	C₄²	C₂×D₅₂	1	1	yes	characteristic
244	C₄²	D₄×D₁₃	1	1	yes	characteristic
245	C₄²	#6129	1	1	yes	characteristic
246	C₁₃	#8658	1	1	yes	characteristic, complemented, Sylow
247	C₂³	—	26	4	no	—
248	C₂³	—	26	2	no	—
249	C₂³	—	52	2	no	—
250	C₂³	—	52	1	no	—
251	C₂³	—	52	1	no	—
252	C₂³	#88200	1	1	yes	characteristic
253	D₄	—	13	4	no	—
254	D₄	—	26	4	no	—
255	D₄	—	26	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	26	2	no	—
259	C₂×C₄	—	52	1	no	—
260	C₂×C₄	—	52	1	no	—
261	C₂×C₄	—	52	2	no	—
262	C₂×C₄	—	52	2	no	—
263	C₂×C₄	—	52	2	no	—
264	C₂×C₄	—	52	2	no	—
265	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
266	C₂×C₄	#88146	1	1	yes	characteristic
267	C₂×C₄	#88149	1	1	yes	characteristic
268	C₂×C₄	#88151	1	1	yes	characteristic
269	C₂×C₄	C₂×#6103	1	1	yes	characteristic
270	C₂×C₄	C₂×#6104	1	1	yes	characteristic
271	C₂×C₄	#88186	1	1	yes	characteristic
272	C₂×C₄	#88192	1	1	yes	characteristic
273	C₂×C₄	C₂×#6106	1	1	yes	characteristic
274	C₂×C₄	#88196	1	1	yes	characteristic
275	C₈	—	2	1	no	—
276	C₈	—	2	1	no	—
277	C₈	—	8	2	no	—
278	C₈	—	52	2	no	—
279	C₈	—	104	2	no	—
280	C₈	#88049	1	2	yes	—
281	C₈	#88049	1	2	yes	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	26	4	no	—
285	C₂²	—	52	4	no	—
286	C₂²	—	52	4	no	—
287	C₂²	—	52	2	no	—
288	C₂²	—	52	2	no	—
289	C₂²	#11718648	1	1	yes	characteristic
290	C₂²	#11718745	1	1	yes	characteristic
291	C₂²	#11718748	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	52	2	no	—
296	C₄	D₁₃×#108	1	2	yes	—
297	C₄	C₂×#88049	1	1	yes	characteristic
298	C₄	#11718627	1	1	yes	characteristic
299	C₄	#11718733	1	1	yes	characteristic
300	C₄	#11718744	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	52	4	no	—
304	C₂	#13114833	1	1	yes	characteristic
305	C₂	#13114871	1	1	yes	characteristic
306	C₂	#13114891	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88192
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1848	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂²×D₅₂	Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1848	D₁₃
hypocenter	C₁₃	#8658
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
2. #13113584	C₂	semidirect
4. #13112731	C₂	semidirect
5. C₁₃×#1848	C₂	semidirect
95. #1848	D₁₃	semidirect
122. D₅₂	#108	semidirect
246. C₁₃	#8658	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₅₂	2
#13113584	2
#13113260	2
#13112731	2
C₁₃×#1848	2
#8658	13