Subgroups — #15863153

Order 3328.

778 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123399	C₂	1	2	yes	maximal
2	C₂×#11718762	C₂	1	1	yes	maximal, characteristic
3	#13114975	C₂	1	4	yes	maximal, complemented
4	#13114972	C₂	1	4	yes	maximal, complemented
5	#13114562	C₂	1	2	yes	maximal
6	C₂×#11718144	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×#341	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88180	C₂²	1	1	yes	characteristic
11	C₂²×#6099	C₂²	1	2	yes	—
12	C₂²×#6099	C₂²	1	1	yes	characteristic
13	#11718780	—	2	4	no	—
14	#11718779	C₂²	1	8	yes	—
15	#11718763	—	2	4	no	—
16	#11718762	C₄	1	4	yes	—
17	#11718762	C₄	1	4	yes	—
18	C₂×#88055	C₂²	1	2	yes	—
19	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
20	C₁₃×#341	C₂²	1	2	yes	—
21	C₁₃×#341	C₂²	1	2	yes	—
22	#11718150	C₂²	1	8	yes	—
23	#11718144	C₂²	1	2	yes	—
24	#11718144	C₂²	1	2	yes	—
25	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₁₃×M₁₆	—	2	2	no	—
27	C₂×C₁₃×M₁₆	—	2	4	no	—
28	C₂×C₁₃×M₁₆	—	2	4	no	—
29	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
30	C₂²×C₈×C₁₃	—	2	4	no	—
31	C₂²×C₈×C₁₃	—	2	2	no	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₂²⋊Dic₁₃	Q₈	1	2	yes	—
37	#88180	C₂×C₄	1	2	yes	—
38	#88180	C₂×C₄	1	2	yes	—
39	#88180	D₄	1	2	yes	—
40	#88180	D₄	1	2	yes	—
41	C₂×C₄⋊Dic₁₃	Q₈	1	2	yes	—
42	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
43	C₂×#6099	—	2	4	no	—
44	C₂×#6099	—	2	2	no	—
45	C₂×#6099	—	2	4	no	—
46	C₂×#6099	—	2	4	no	—
47	C₂×#6099	C₂×C₄	1	2	yes	—
48	C₂×#6099	C₂×C₄	1	2	yes	—
49	C₂×#6099	C₂³	1	4	yes	—
50	C₂²×#1055	—	2	2	no	—
51	C₂²×#1055	C₂³	1	2	yes	—
52	#88055	C₂×C₄	1	8	yes	—
53	#88054	—	2	16	no	—
54	C₈×Dic₁₃	C₂×C₄	1	8	yes	—
55	#20768	—	13	1	no	maximal, Sylow
56	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	#41	1	4	yes	—
63	C₂²×C₄×C₁₃	#41	1	4	yes	—
64	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
65	C₂²×Dic₁₃	C₂×Q₈	1	4	yes	—
66	C₁₃×M₁₆	—	2	4	no	—
67	C₁₃×M₁₆	—	2	4	no	—
68	C₁₃×M₁₆	—	4	4	no	—
69	C₂×C₈×C₁₃	—	2	4	no	—
70	C₂×C₈×C₁₃	—	2	4	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	—	4	4	no	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂²⋊Dic₁₃	C₄⋊C₄	1	4	yes	—
78	C₂²⋊Dic₁₃	C₄⋊C₄	1	4	yes	—
79	C₄⋊Dic₁₃	C₄⋊C₄	1	4	yes	—
80	C₄⋊Dic₁₃	C₄⋊C₄	1	4	yes	—
81	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
82	C₄×Dic₁₃	C₂×D₄	1	4	yes	—
83	#6099	—	2	4	no	—
84	#6099	—	2	4	no	—
85	#6099	—	2	4	no	—
86	#6099	—	2	4	no	—
87	#6099	—	4	4	no	—
88	C₂×#1055	—	2	4	no	—
89	C₂×#1055	—	2	4	no	—
90	C₂×#1055	—	4	4	no	—
91	C₂×#1055	C₂²×C₄	1	4	yes	—
92	#2905	—	13	2	no	—
93	C₂×#405	—	13	1	no	—
94	#1838	—	13	4	no	—
95	#1837	—	13	4	no	—
96	#1738	—	13	2	no	—
97	C₂×#341	—	13	1	no	—
98	C₂×#341	D₁₃	1	1	yes	characteristic
99	C₂³×C₁₃	—	4	2	no	—
100	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂³×C₁₃	#125	1	4	yes	—
103	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#117	1	2	yes	—
107	C₂×C₄×C₁₃	#117	1	2	yes	—
108	C₂×C₄×C₁₃	#117	1	2	yes	—
109	C₂×C₄×C₁₃	#117	1	2	yes	—
110	C₂×C₄×C₁₃	#124	1	4	yes	—
111	C₂×C₄×C₁₃	#124	1	4	yes	—
112	C₂×C₄×C₁₃	#125	1	4	yes	—
113	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
117	C₂×Dic₁₃	—	4	4	no	—
118	C₂×Dic₁₃	C₂×C₄⋊C₄	1	8	yes	—
119	C₈×C₁₃	—	2	2	no	—
120	C₈×C₁₃	—	2	2	no	—
121	C₈×C₁₃	—	4	4	no	—
122	#1055	—	2	4	no	—
123	#1055	—	4	4	no	—
124	C₂²×M₁₆	—	13	1	no	—
125	C₂²×M₁₆	—	13	2	no	—
126	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
127	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
128	C₂×#117	—	13	1	no	—
129	#424	—	26	4	no	—
130	#424	—	26	4	no	—
131	#423	—	13	8	no	—
132	#405	—	13	4	no	—
133	#405	—	13	4	no	—
134	C₂×#97	—	13	2	no	—
135	C₂×C₄×C₈	—	13	2	no	—
136	#341	—	13	8	no	—
137	#341	—	13	2	no	—
138	#341	—	13	2	no	—
139	#341	C₂×D₁₃	1	2	yes	—
140	#341	C₂×D₁₃	1	2	yes	—
141	C₂²×C₁₃	—	2	2	no	—
142	C₂²×C₁₃	—	2	2	no	—
143	C₂²×C₁₃	—	2	2	no	—
144	C₂²×C₁₃	—	4	4	no	—
145	C₂²×C₁₃	#382	1	2	yes	—
146	C₂²×C₁₃	#382	1	2	yes	—
147	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
148	C₂²×C₁₃	C₂×#125	1	2	yes	—
149	C₄×C₁₃	#382	1	2	yes	—
150	C₄×C₁₃	#382	1	2	yes	—
151	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
152	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
153	C₄×C₁₃	C₂×#124	1	2	yes	—
154	Dic₁₃	—	2	8	no	—
155	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
156	C₂×M₁₆	—	2	2	no	—
157	C₂×M₁₆	—	2	4	no	—
158	C₂×M₁₆	—	2	4	no	—
159	C₂×M₁₆	—	13	2	no	—
160	C₂×M₁₆	—	13	2	no	—
161	C₂×M₁₆	—	13	4	no	—
162	C₂×M₁₆	—	26	4	no	—
163	C₂×M₁₆	—	26	2	no	—
164	C₂×M₁₆	—	26	4	no	—
165	C₂×M₁₆	—	26	4	no	—
166	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
167	C₂²×C₈	—	2	4	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	13	2	no	—
170	C₂²×C₈	—	26	2	no	—
171	C₂²×C₈	C₄×D₁₃	1	2	yes	—
172	C₂²×C₈	C₄×D₁₃	1	2	yes	—
173	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
175	#117	—	13	2	no	—
176	#117	—	13	2	no	—
177	#117	—	13	2	no	—
178	#117	—	13	2	no	—
179	C₂×C₄⋊C₄	—	13	2	no	—
180	C₂×C₂²⋊C₄	—	13	2	no	—
181	C₂×C₄²	—	13	2	no	—
182	C₄⋊C₈	—	26	16	no	—
183	#97	—	13	8	no	—
184	C₄×C₈	—	13	8	no	—
185	C₂×C₁₃	—	2	2	no	—
186	C₂×C₁₃	—	2	2	no	—
187	C₂×C₁₃	—	2	2	no	—
188	C₂×C₁₃	C₂×#382	1	1	yes	characteristic
189	C₂×C₁₃	#2916	1	2	yes	—
190	C₂⁴	Q₈×D₁₃	1	1	yes	characteristic
191	C₂²×C₄	—	13	4	no	—
192	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
194	C₂²×C₄	#6127	1	4	yes	—
195	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
196	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
197	C₂²×C₄	#6129	1	4	yes	—
198	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
199	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	—	4	4	no	—
203	M₁₆	—	26	4	no	—
204	M₁₆	—	26	4	no	—
205	M₁₆	—	26	4	no	—
206	M₁₆	—	26	4	no	—
207	M₁₆	—	52	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	4	4	no	—
214	C₂×C₈	—	13	4	no	—
215	C₂×C₈	—	26	4	no	—
216	C₂×C₈	—	26	4	no	—
217	C₂×C₈	—	52	4	no	—
218	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
219	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
220	C₄⋊C₄	—	13	4	no	—
221	C₄⋊C₄	—	13	4	no	—
222	C₂²⋊C₄	—	13	4	no	—
223	C₂²⋊C₄	—	13	4	no	—
224	C₄²	—	13	4	no	—
225	C₄²	—	13	4	no	—
226	C₁₃	#20768	1	1	yes	characteristic, complemented, Sylow
227	C₂³	—	4	2	no	—
228	C₂³	#88143	1	4	yes	—
229	C₂³	C₄⋊C₄×D₁₃	1	1	yes	characteristic
230	C₂³	C₄⋊C₄×D₁₃	1	1	yes	characteristic
231	C₂³	C₂×Q₈×D₁₃	1	1	yes	characteristic
232	C₂×C₄	—	13	8	no	—
233	C₂×C₄	—	52	4	no	—
234	C₂×C₄	#88126	1	2	yes	—
235	C₂×C₄	#88126	1	2	yes	—
236	C₂×C₄	#88131	1	2	yes	—
237	C₂×C₄	#88131	1	2	yes	—
238	C₂×C₄	#88139	1	4	yes	—
239	C₂×C₄	#88139	1	4	yes	—
240	C₂×C₄	#88143	1	4	yes	—
241	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
242	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
244	C₂×C₄	C₂×#6127	1	2	yes	—
245	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
246	C₂×C₄	C₂×#6129	1	2	yes	—
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	4	4	no	—
250	C₈	—	26	4	no	—
251	C₈	—	52	4	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	4	4	no	—
256	C₂²	#11718311	1	2	yes	—
257	C₂²	#11718311	1	2	yes	—
258	C₂²	C₂×#88143	1	2	yes	—
259	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
260	C₄	—	26	8	no	—
261	C₄	#11718311	1	2	yes	—
262	C₄	#11718311	1	2	yes	—
263	C₄	C₂×#88126	1	1	yes	characteristic
264	C₄	C₂×#88131	1	1	yes	characteristic
265	C₄	C₂×#88139	1	2	yes	—
266	C₂	—	2	2	no	—
267	C₂	—	2	2	no	—
268	C₂	—	2	2	no	—
269	C₂	C₂×#11718311	1	1	yes	characteristic
270	C₂	#13123444	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#341	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#341	D₁₃
hypocenter	C₁₃	#20768
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
3. #13114975	C₂	semidirect
4. #13114972	C₂	semidirect
226. C₁₃	#20768	semidirect

Maximal subgroups

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

subgroup	index
#13123399	2
C₂×#11718762	2
#13114975	2
#13114972	2
#13114562	2
C₂×#11718144	2
C₂×C₁₃×#341	2
#20768	13