Subgroups — #15833580

Order 3328.

640 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	#13125415	C₂	1	1	yes	maximal, characteristic
2	#13123911	C₂	1	2	yes	maximal, complemented
3	#13113751	C₂	1	2	yes	maximal, complemented
4	#13113722	C₂	1	2	yes	maximal, complemented
5	#13113661	C₂	1	2	yes	maximal, complemented
6	C₈×D₁₀₄	C₂	1	2	yes	maximal, complemented
7	#13113599	C₂	1	1	yes	maximal, characteristic, complemented
8	#13113306	C₂	1	2	yes	maximal, complemented
9	C₁₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	#11719182	C₂²	1	2	yes	—
11	C₁₃×#431	C₂²	1	1	yes	characteristic
12	C₁₃×#431	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#11718776	C₄	1	2	yes	—
15	#11718776	C₄	1	2	yes	—
16	#11718587	C₂²	1	2	yes	—
17	#11718497	—	2	4	no	—
18	#11718388	—	2	4	no	—
19	#11718356	C₄	1	1	yes	characteristic
20	#11718355	C₄	1	2	yes	—
21	C₄×D₁₀₄	C₂²	1	2	yes	—
22	C₄×D₁₀₄	C₂²	1	2	yes	—
23	#11718349	C₂²	1	4	yes	—
24	C₈×D₅₂	C₂²	1	4	yes	—
25	#11718343	C₄	1	1	yes	characteristic
26	C₁₃×#322	C₂²	1	1	yes	characteristic
27	C₁₃×#322	C₂²	1	1	yes	characteristic
28	C₈²×C₁₃	C₂²	1	1	yes	characteristic
29	C₈²×C₁₃	C₂²	1	1	yes	characteristic
30	#11718051	C₂²	1	4	yes	—
31	#11718051	C₂²	1	4	yes	—
32	#11718050	C₂²	1	2	yes	—
33	#11718049	C₂²	1	2	yes	—
34	C₂×#6127	—	2	2	no	—
35	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	#88184	—	2	2	no	—
39	#88184	C₂×C₄	1	4	yes	—
40	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
41	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
42	C₂×#6094	—	2	4	no	—
43	C₂×C₈×D₁₃	—	2	4	no	—
44	#88129	—	2	2	no	—
45	#88128	C₂×C₄	1	2	yes	—
46	C₄×D₅₂	—	2	2	no	—
47	C₄×D₅₂	C₂³	1	4	yes	—
48	#88123	C₂×C₄	1	2	yes	—
49	C₁₃×C₄⋊C₈	—	2	1	no	—
50	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
51	C₁₃×C₂²⋊C₈	—	2	1	no	—
52	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
53	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₂²⋊#1055	—	2	2	no	—
62	#88061	C₂×C₄	1	4	yes	—
63	#88061	C₂×C₄	1	4	yes	—
64	#88060	—	2	4	no	—
65	#88060	—	2	4	no	—
66	#88058	C₂×C₄	1	2	yes	—
67	#88058	C₂×C₄	1	2	yes	—
68	C₄⋊#1055	C₂³	1	4	yes	—
69	#88043	—	2	2	no	—
70	#17970	—	13	1	no	maximal, Sylow
71	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
74	C₂×C₂²⋊D₁₃	—	2	4	no	—
75	#6127	—	8	4	no	—
76	C₂×D₅₂	—	2	2	no	—
77	C₂×D₅₂	—	2	2	no	—
78	C₂×D₅₂	C₂²×C₄	1	4	yes	—
79	C₂×C₄×D₁₃	—	2	4	no	—
80	C₂×Dic₂₆	—	2	2	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	1	no	—
84	C₂×C₈×C₁₃	—	2	1	no	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
91	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₃	#41	1	1	yes	characteristic
94	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
95	#6103	—	2	4	no	—
96	#6103	—	2	4	no	—
97	C₄⋊Dic₁₃	—	2	2	no	—
98	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
99	C₂×#1055	—	2	4	no	—
100	D₁₀₄	—	2	4	no	—
101	D₁₀₄	—	2	4	no	—
102	#6094	—	8	4	no	—
103	C₈×D₁₃	—	8	4	no	—
104	#3169	—	13	1	no	—
105	#2981	—	13	2	no	—
106	#1594	—	13	2	no	—
107	#1589	—	13	2	no	—
108	#1580	—	13	2	no	—
109	C₈×D₈	—	13	2	no	—
110	#1568	—	13	1	no	—
111	#1496	—	13	2	no	—
112	#1450	D₁₃	1	1	yes	characteristic, complemented
113	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
114	C₂²×D₁₃	—	2	4	no	—
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	—	4	1	no	—
119	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₄×D₄	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	C₂×#41	1	1	yes	characteristic
125	C₂²⋊D₁₃	—	8	4	no	—
126	C₂×Dic₁₃	—	2	4	no	—
127	D₅₂	—	2	4	no	—
128	D₅₂	—	2	4	no	—
129	D₅₂	—	8	2	no	—
130	D₅₂	—	8	2	no	—
131	C₄×D₁₃	—	4	4	no	—
132	C₄×D₁₃	—	4	4	no	—
133	Dic₂₆	—	8	2	no	—
134	C₈×C₁₃	—	2	1	no	—
135	C₈×C₁₃	—	2	1	no	—
136	C₈×C₁₃	—	2	2	no	—
137	C₈×C₁₃	—	2	2	no	—
138	C₈×C₁₃	—	4	1	no	—
139	C₈×C₁₃	—	4	1	no	—
140	C₈×C₁₃	#131	1	2	yes	—
141	C₈×C₁₃	#131	1	2	yes	—
142	#1055	—	4	4	no	—
143	#532	—	13	2	no	—
144	#500	—	13	1	no	—
145	#493	—	13	1	no	—
146	#486	—	13	2	no	—
147	#466	—	13	2	no	—
148	#466	—	13	2	no	—
149	C₄×D₈	—	13	2	no	—
150	C₄×D₈	—	13	2	no	—
151	#436	—	13	4	no	—
152	#435	—	26	4	no	—
153	C₈×D₄	—	13	4	no	—
154	#432	—	13	2	no	—
155	#431	C₂×D₁₃	1	1	yes	characteristic
156	#431	C₂×D₁₃	1	1	yes	characteristic
157	#408	—	26	4	no	—
158	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
159	#335	—	13	2	no	—
160	#334	—	13	2	no	—
161	D₄⋊C₈	—	13	4	no	—
162	D₄⋊C₈	—	13	4	no	—
163	#322	C₂×D₁₃	1	1	yes	characteristic
164	#322	C₂×D₁₃	1	1	yes	characteristic
165	C₈²	C₂×D₁₃	1	1	yes	characteristic
166	C₈²	C₂×D₁₃	1	1	yes	characteristic
167	C₂²×C₁₃	—	2	1	no	—
168	C₂²×C₁₃	—	2	1	no	—
169	C₂²×C₁₃	—	2	1	no	—
170	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂×D₁₃	—	4	4	no	—
172	C₂×D₁₃	—	4	4	no	—
173	C₂×D₁₃	—	4	4	no	—
174	C₄×C₁₃	—	2	1	no	—
175	C₄×C₁₃	—	2	1	no	—
176	C₄×C₁₃	—	2	1	no	—
177	C₄×C₁₃	—	2	1	no	—
178	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
179	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
180	C₄×C₁₃	#576	1	2	yes	—
181	Dic₁₃	—	4	4	no	—
182	C₂×#41	—	26	2	no	—
183	C₂×D₈	—	13	2	no	—
184	C₂×D₈	—	13	2	no	—
185	C₂×M₁₆	—	26	4	no	—
186	C₂²×C₈	—	26	4	no	—
187	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
188	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
189	C₄⋊Q₈	—	13	2	no	—
190	#127	—	13	2	no	—
191	#124	—	26	2	no	—
192	#121	—	13	4	no	—
193	#121	—	26	2	no	—
194	C₄×D₄	—	13	4	no	—
195	C₄×D₄	—	26	2	no	—
196	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
197	#107	—	13	2	no	—
198	#107	—	13	2	no	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	—	13	4	no	—
201	C₄⋊C₈	D₅₂	1	2	yes	—
202	#102	—	13	4	no	—
203	#102	—	13	4	no	—
204	C₂²⋊C₈	—	2	1	no	—
205	C₂²⋊C₈	—	26	2	no	—
206	C₂²⋊C₈	—	26	4	no	—
207	C₂²⋊C₈	—	26	4	no	—
208	C₂²⋊C₈	D₅₂	1	2	yes	—
209	#97	—	26	2	no	—
210	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
211	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
213	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
214	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
218	C₂×C₁₃	—	4	1	no	—
219	C₂×C₁₃	#2931	1	1	yes	characteristic
220	C₂×C₁₃	#2948	1	1	yes	characteristic
221	C₂×C₁₃	#2957	1	1	yes	characteristic
222	D₁₃	—	8	4	no	—
223	#41	—	104	4	no	—
224	C₂×Q₈	—	26	2	no	—
225	C₂×D₄	—	13	4	no	—
226	C₂×D₄	—	26	4	no	—
227	C₂×D₄	—	26	2	no	—
228	C₂×D₄	—	26	2	no	—
229	C₂²×C₄	—	26	4	no	—
230	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
232	C₂²×C₄	#6127	1	1	yes	characteristic
233	D₈	—	26	4	no	—
234	D₈	—	26	4	no	—
235	M₁₆	—	104	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	26	4	no	—
241	C₂×C₈	—	104	4	no	—
242	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
243	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
244	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
245	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
246	C₂×C₈	C₂×D₅₂	1	2	yes	—
247	C₂×C₈	C₂×D₅₂	1	2	yes	—
248	C₄⋊C₄	—	13	4	no	—
249	C₄⋊C₄	—	26	2	no	—
250	C₂²⋊C₄	—	26	4	no	—
251	C₂²⋊C₄	—	26	4	no	—
252	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
253	C₄²	C₂×D₅₂	1	1	yes	characteristic
254	C₄²	#6127	1	1	yes	characteristic
255	C₄²	C₂³×D₁₃	1	1	yes	characteristic
256	C₁₃	#17970	1	1	yes	characteristic, complemented, Sylow
257	C₂³	—	26	4	no	—
258	C₂³	C₄×D₅₂	1	1	yes	characteristic
259	Q₈	—	104	2	no	—
260	D₄	—	26	4	no	—
261	D₄	—	26	4	no	—
262	D₄	—	104	2	no	—
263	D₄	—	104	2	no	—
264	D₄	—	104	4	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	26	4	no	—
270	C₂×C₄	—	52	4	no	—
271	C₂×C₄	—	52	4	no	—
272	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
273	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
274	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
275	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
276	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
277	C₂×C₄	C₂×#6127	1	1	yes	characteristic
278	C₈	—	2	1	no	—
279	C₈	—	2	1	no	—
280	C₈	—	2	2	no	—
281	C₈	—	2	2	no	—
282	C₈	—	4	1	no	—
283	C₈	—	4	1	no	—
284	C₈	—	52	4	no	—
285	C₈	#88155	1	2	yes	—
286	C₈	#88155	1	2	yes	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	52	4	no	—
291	C₂²	—	52	4	no	—
292	C₂²	—	52	4	no	—
293	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	52	4	no	—
299	C₄	C₂×#88155	1	1	yes	characteristic
300	C₄	C₂×#88155	1	1	yes	characteristic
301	C₄	#11719457	1	2	yes	—
302	C₂	—	4	1	no	—
303	C₂	—	104	4	no	—
304	C₂	#13123549	1	1	yes	characteristic
305	C₂	#13123689	1	1	yes	characteristic
306	C₂	#13123754	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1450	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#13125415	C₂
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1450	D₁₃
hypocenter	C₁₃	#17970
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
2. #13123911	C₂	semidirect
3. #13113751	C₂	semidirect
4. #13113722	C₂	semidirect
5. #13113661	C₂	semidirect
6. C₈×D₁₀₄	C₂	semidirect
7. #13113599	C₂	semidirect
8. #13113306	C₂	semidirect
9. C₁₃×#1450	C₂	semidirect
112. #1450	D₁₃	semidirect
256. C₁₃	#17970	semidirect

Maximal subgroups

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

subgroup	index
#13125415	2
#13123911	2
#13113751	2
#13113722	2
#13113661	2
C₈×D₁₀₄	2
#13113599	2
#13113306	2
C₁₃×#1450	2
#17970	13