Subgroups — #15833763

Order 3328.

624 conjugacy classes of subgroups.

The subgroup lattice has 286 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	#13113702	C₂	1	4	yes	maximal, complemented
4	#13113667	C₂	1	4	yes	maximal, complemented
5	#13113599	C₂	1	1	yes	maximal, characteristic, complemented
6	#13113306	C₂	1	2	yes	maximal, complemented
7	C₁₃×#1452	C₂	1	1	yes	maximal, characteristic, complemented
8	#11719182	C₂²	1	2	yes	—
9	C₁₃×#431	C₂²	1	1	yes	characteristic
10	C₁₃×#431	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	#11718776	C₄	1	2	yes	—
13	#11718776	C₄	1	2	yes	—
14	#11718587	C₂²	1	2	yes	—
15	#11718497	—	2	4	no	—
16	#11718388	—	2	4	no	—
17	#11718356	C₄	1	1	yes	characteristic
18	#11718355	C₄	1	2	yes	—
19	C₄×D₁₀₄	C₂²	1	2	yes	—
20	C₄×D₁₀₄	C₂²	1	2	yes	—
21	#11718349	C₂²	1	4	yes	—
22	C₈×D₅₂	C₂²	1	4	yes	—
23	#11718343	C₄	1	1	yes	characteristic
24	C₁₃×#322	C₂²	1	2	yes	—
25	C₁₃×#322	C₂²	1	2	yes	—
26	#11718051	C₂²	1	8	yes	—
27	#11718050	C₂²	1	2	yes	—
28	#11718049	C₂²	1	2	yes	—
29	C₂×#6127	—	2	2	no	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
33	#88184	—	2	2	no	—
34	#88184	C₂×C₄	1	4	yes	—
35	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
36	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
37	C₂×#6094	—	2	4	no	—
38	C₂×C₈×D₁₃	—	2	4	no	—
39	#88129	—	2	2	no	—
40	#88128	C₂×C₄	1	2	yes	—
41	C₄×D₅₂	—	2	2	no	—
42	C₄×D₅₂	C₂³	1	4	yes	—
43	#88123	C₂×C₄	1	2	yes	—
44	C₁₃×C₄⋊C₈	—	2	1	no	—
45	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
46	C₁₃×C₂²⋊C₈	—	2	1	no	—
47	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
48	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₁₃	C₂³	1	2	yes	—
51	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₁₃	C₂³	1	2	yes	—
54	C₂²⋊#1055	—	2	2	no	—
55	#88061	C₂×C₄	1	4	yes	—
56	#88061	C₂×C₄	1	4	yes	—
57	#88060	—	2	4	no	—
58	#88060	—	2	4	no	—
59	#88058	C₂×C₄	1	2	yes	—
60	#88058	C₂×C₄	1	2	yes	—
61	C₄⋊#1055	C₂³	1	4	yes	—
62	#88043	—	2	2	no	—
63	#17989	—	13	1	no	maximal, Sylow
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₃	—	2	4	no	—
68	#6127	—	8	4	no	—
69	C₂×D₅₂	—	2	2	no	—
70	C₂×D₅₂	—	2	2	no	—
71	C₂×D₅₂	C₂²×C₄	1	4	yes	—
72	C₂×C₄×D₁₃	—	2	4	no	—
73	C₂×Dic₂₆	—	2	2	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	4	1	no	—
76	C₂×C₈×C₁₃	—	4	1	no	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
82	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₃	#41	1	1	yes	characteristic
85	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
86	#6103	—	2	4	no	—
87	#6103	—	2	4	no	—
88	C₄⋊Dic₁₃	—	2	2	no	—
89	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
90	C₂×#1055	—	2	4	no	—
91	D₁₀₄	—	4	2	no	—
92	D₁₀₄	—	4	2	no	—
93	#6094	—	8	4	no	—
94	C₈×D₁₃	—	8	4	no	—
95	#3169	—	13	1	no	—
96	#2981	—	13	2	no	—
97	#1586	—	13	4	no	—
98	#1581	—	13	4	no	—
99	#1568	—	13	1	no	—
100	#1496	—	13	2	no	—
101	#1452	D₁₃	1	1	yes	characteristic, complemented
102	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
103	C₂²×D₁₃	—	2	4	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	4	1	no	—
108	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
114	C₂²⋊D₁₃	—	8	4	no	—
115	C₂×Dic₁₃	—	2	4	no	—
116	D₅₂	—	2	4	no	—
117	D₅₂	—	2	4	no	—
118	D₅₂	—	8	2	no	—
119	D₅₂	—	8	2	no	—
120	C₄×D₁₃	—	4	4	no	—
121	C₄×D₁₃	—	4	4	no	—
122	Dic₂₆	—	8	2	no	—
123	C₈×C₁₃	—	2	1	no	—
124	C₈×C₁₃	—	2	1	no	—
125	C₈×C₁₃	—	2	1	no	—
126	C₈×C₁₃	—	2	1	no	—
127	C₈×C₁₃	—	2	4	no	—
128	C₈×C₁₃	—	4	2	no	—
129	#1055	—	4	4	no	—
130	#532	—	13	2	no	—
131	#500	—	13	1	no	—
132	#493	—	13	1	no	—
133	#486	—	13	2	no	—
134	#466	—	13	2	no	—
135	#466	—	13	2	no	—
136	C₄×D₈	—	13	2	no	—
137	C₄×D₈	—	13	2	no	—
138	#436	—	13	4	no	—
139	#435	—	26	4	no	—
140	C₈×D₄	—	13	4	no	—
141	#432	—	13	2	no	—
142	#431	C₂×D₁₃	1	1	yes	characteristic
143	#431	C₂×D₁₃	1	1	yes	characteristic
144	#408	—	26	4	no	—
145	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
146	#335	—	13	2	no	—
147	#334	—	13	2	no	—
148	D₄⋊C₈	—	13	8	no	—
149	#322	C₂×D₁₃	1	2	yes	—
150	#322	C₂×D₁₃	1	2	yes	—
151	C₂²×C₁₃	—	2	1	no	—
152	C₂²×C₁₃	—	2	1	no	—
153	C₂²×C₁₃	—	2	1	no	—
154	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
155	C₂×D₁₃	—	4	4	no	—
156	C₂×D₁₃	—	4	4	no	—
157	C₂×D₁₃	—	4	4	no	—
158	C₄×C₁₃	—	2	1	no	—
159	C₄×C₁₃	—	2	1	no	—
160	C₄×C₁₃	—	2	1	no	—
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	#568	1	1	yes	characteristic
163	C₄×C₁₃	#568	1	1	yes	characteristic
164	C₄×C₁₃	#576	1	2	yes	—
165	Dic₁₃	—	4	4	no	—
166	C₂×#41	—	26	2	no	—
167	C₂×D₈	—	13	2	no	—
168	C₂×D₈	—	13	2	no	—
169	C₂×M₁₆	—	26	4	no	—
170	C₂²×C₈	—	26	4	no	—
171	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
172	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
173	C₄⋊Q₈	—	13	2	no	—
174	#127	—	13	2	no	—
175	#124	—	26	2	no	—
176	#121	—	13	4	no	—
177	#121	—	26	2	no	—
178	C₄×D₄	—	13	4	no	—
179	C₄×D₄	—	26	2	no	—
180	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
181	#107	—	13	2	no	—
182	#107	—	13	2	no	—
183	C₄⋊C₈	—	2	1	no	—
184	C₄⋊C₈	—	13	4	no	—
185	C₄⋊C₈	D₅₂	1	2	yes	—
186	#102	—	13	4	no	—
187	#102	—	13	4	no	—
188	C₂²⋊C₈	—	2	1	no	—
189	C₂²⋊C₈	—	26	2	no	—
190	C₂²⋊C₈	—	26	4	no	—
191	C₂²⋊C₈	—	26	4	no	—
192	C₂²⋊C₈	D₅₂	1	2	yes	—
193	#97	—	26	2	no	—
194	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
195	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
196	C₄×C₈	C₂²×D₁₃	1	2	yes	—
197	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
198	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
199	C₄×C₈	C₂²×D₁₃	1	2	yes	—
200	C₂×C₁₃	—	4	1	no	—
201	C₂×C₁₃	#2932	1	1	yes	characteristic
202	C₂×C₁₃	#2948	1	1	yes	characteristic
203	C₂×C₁₃	#2957	1	1	yes	characteristic
204	D₁₃	—	8	4	no	—
205	#41	—	104	4	no	—
206	C₂×Q₈	—	26	2	no	—
207	C₂×D₄	—	13	4	no	—
208	C₂×D₄	—	26	4	no	—
209	C₂×D₄	—	26	2	no	—
210	C₂×D₄	—	26	2	no	—
211	C₂²×C₄	—	26	4	no	—
212	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
214	C₂²×C₄	#6127	1	1	yes	characteristic
215	D₈	—	52	2	no	—
216	D₈	—	52	2	no	—
217	M₁₆	—	104	4	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	26	4	no	—
222	C₂×C₈	—	104	4	no	—
223	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
224	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
227	C₂×C₈	C₂×D₅₂	1	4	yes	—
228	C₄⋊C₄	—	13	4	no	—
229	C₄⋊C₄	—	26	2	no	—
230	C₂²⋊C₄	—	26	4	no	—
231	C₂²⋊C₄	—	26	4	no	—
232	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
233	C₄²	C₂×D₅₂	1	1	yes	characteristic
234	C₄²	#6127	1	1	yes	characteristic
235	C₄²	C₂³×D₁₃	1	1	yes	characteristic
236	C₁₃	#17989	1	1	yes	characteristic, complemented, Sylow
237	C₂³	—	26	4	no	—
238	C₂³	C₄×D₅₂	1	1	yes	characteristic
239	Q₈	—	104	2	no	—
240	D₄	—	26	4	no	—
241	D₄	—	26	4	no	—
242	D₄	—	104	2	no	—
243	D₄	—	104	2	no	—
244	D₄	—	104	4	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	26	4	no	—
250	C₂×C₄	—	52	4	no	—
251	C₂×C₄	—	52	4	no	—
252	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
253	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
254	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
255	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
256	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
257	C₂×C₄	C₂×#6127	1	1	yes	characteristic
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	4	no	—
263	C₈	—	4	2	no	—
264	C₈	—	52	4	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	52	4	no	—
269	C₂²	—	52	4	no	—
270	C₂²	—	52	4	no	—
271	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	52	4	no	—
277	C₄	#11719450	1	1	yes	characteristic
278	C₄	#11719450	1	1	yes	characteristic
279	C₄	#11719457	1	2	yes	—
280	C₂	—	4	1	no	—
281	C₂	—	104	4	no	—
282	C₂	#13123555	1	1	yes	characteristic
283	C₂	#13123689	1	1	yes	characteristic
284	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×#1452	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	#13125415	C₂
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1452	D₁₃
hypocenter	C₁₃	#17989
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1452

Complements

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

N	H ≅ G/N	type
2. #13123911	C₂	semidirect
3. #13113702	C₂	semidirect
4. #13113667	C₂	semidirect
5. #13113599	C₂	semidirect
6. #13113306	C₂	semidirect
7. C₁₃×#1452	C₂	semidirect
101. #1452	D₁₃	semidirect
236. C₁₃	#17989	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
#13113702	2
#13113667	2
#13113599	2
#13113306	2
C₁₃×#1452	2
#17989	13