Subgroups — #15768606

Order 3328.

612 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123254	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718151	C₂	1	2	yes	maximal
3	#13114562	C₂	1	2	yes	maximal
4	C₂×#11718152	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#338	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂×#88180	C₂²	1	1	yes	characteristic
9	C₂²×#6099	C₂²	1	1	yes	characteristic
10	C₂²×#6099	C₂²	1	2	yes	—
11	#11718783	—	2	2	no	—
12	M₁₆×Dic₁₃	—	2	2	no	—
13	#11718762	C₄	1	4	yes	—
14	C₁₃×#338	—	2	2	no	—
15	#11718152	—	2	2	no	—
16	#11718151	—	2	4	no	—
17	#11718150	C₄	1	8	yes	complemented
18	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×C₁₃×M₁₆	—	2	2	no	—
20	C₂×C₁₃×M₁₆	—	2	2	no	—
21	C₂×C₁₃×M₁₆	—	2	2	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	2	no	—
26	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
27	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
28	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
29	C₂²×C₈×C₁₃	—	2	2	no	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₂×C₂²⋊Dic₁₃	Q₈	1	2	yes	—
32	#88180	—	2	1	no	—
33	#88180	—	2	1	no	—
34	#88180	C₂×C₄	1	2	yes	—
35	#88180	D₄	1	2	yes	—
36	C₂×C₄⋊Dic₁₃	—	2	1	no	—
37	C₂×C₄×Dic₁₃	—	2	1	no	—
38	C₂×#6099	—	2	1	no	—
39	C₂×#6099	—	2	2	no	—
40	C₂×#6099	—	2	2	no	—
41	C₂×#6099	—	2	2	no	—
42	C₂×#6099	—	2	2	no	—
43	C₂×#6099	—	2	2	no	—
44	C₂×#6099	—	2	2	no	—
45	C₂×#6099	—	2	2	no	—
46	C₂×#6099	C₂×C₄	1	4	yes	—
47	C₂×#6099	C₂×C₄	1	2	yes	—
48	C₂²×#1055	—	2	1	no	—
49	C₂²×#1055	C₂×C₄	1	4	yes	—
50	#88055	—	2	4	no	—
51	C₈×Dic₁₃	—	2	4	no	—
52	#10812	—	13	1	no	maximal, Sylow
53	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	#41	1	2	yes	—
60	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	#41	1	2	yes	—
62	C₂²×C₄×C₁₃	#41	1	2	yes	—
63	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
64	C₂²×Dic₁₃	—	2	2	no	—
65	C₁₃×M₁₆	—	2	4	no	—
66	C₁₃×M₁₆	—	2	4	no	—
67	C₁₃×M₁₆	—	2	4	no	—
68	C₁₃×M₁₆	—	2	4	no	—
69	C₁₃×M₁₆	—	4	2	no	—
70	C₁₃×M₁₆	—	4	2	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	2	no	—
75	C₂×C₈×C₁₃	—	4	2	no	—
76	C₂×C₈×C₁₃	C₂²⋊C₄	1	4	yes	—
77	C₂²⋊Dic₁₃	—	2	2	no	—
78	C₂²⋊Dic₁₃	C₄⋊C₄	1	4	yes	—
79	C₄⋊Dic₁₃	—	2	2	no	—
80	C₄⋊Dic₁₃	—	2	2	no	—
81	C₄×Dic₁₃	—	2	2	no	—
82	C₄×Dic₁₃	—	2	2	no	—
83	#6099	—	2	2	no	—
84	#6099	—	2	4	no	—
85	#6099	—	4	2	no	—
86	#6099	—	4	2	no	—
87	#6099	—	4	2	no	—
88	#6099	—	4	2	no	—
89	#6099	C₄²	1	4	yes	—
90	C₂×#1055	—	2	2	no	—
91	C₂×#1055	—	2	4	no	—
92	C₂×#1055	—	2	4	no	—
93	C₂×#1055	—	2	4	no	—
94	C₂×#1055	—	2	4	no	—
95	#2873	—	13	1	no	—
96	C₂×#344	—	13	2	no	—
97	#1738	—	13	2	no	—
98	C₂×#338	—	13	1	no	—
99	C₂×#338	D₁₃	1	1	yes	characteristic
100	C₂³×C₁₃	—	8	1	no	—
101	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
103	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂³×C₁₃	#125	1	2	yes	—
106	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#117	1	2	yes	—
111	C₂×C₄×C₁₃	#117	1	2	yes	—
112	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#123	1	2	yes	—
120	C₂×C₄×C₁₃	#123	1	2	yes	—
121	C₂×C₄×C₁₃	#124	1	2	yes	—
122	C₂×C₄×C₁₃	#126	1	2	yes	—
123	C₂×C₄×C₁₃	#126	1	2	yes	—
124	C₂×Dic₁₃	—	2	4	no	—
125	C₂×Dic₁₃	—	8	2	no	—
126	C₈×C₁₃	—	2	4	no	—
127	C₈×C₁₃	—	4	4	no	—
128	#1055	—	2	4	no	—
129	#1055	—	4	4	no	—
130	C₂²×M₁₆	—	13	2	no	—
131	C₂²×M₁₆	—	13	1	no	—
132	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
133	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
134	C₂×#117	—	13	1	no	—
135	#405	—	13	4	no	—
136	#405	—	26	2	no	—
137	C₄×M₁₆	—	26	2	no	—
138	#344	—	26	4	no	—
139	#341	—	13	8	no	—
140	#338	—	2	2	no	—
141	#338	—	26	2	no	—
142	C₂²×C₁₃	—	4	1	no	—
143	C₂²×C₁₃	—	4	1	no	—
144	C₂²×C₁₃	—	4	2	no	—
145	C₂²×C₁₃	—	4	1	no	—
146	C₂²×C₁₃	—	4	2	no	—
147	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
148	C₂²×C₁₃	#380	1	1	yes	characteristic
149	C₂²×C₁₃	#382	1	1	yes	characteristic
150	C₂²×C₁₃	#387	1	2	yes	—
151	C₂²×C₁₃	#387	1	2	yes	—
152	C₄×C₁₃	#376	1	1	yes	characteristic
153	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
154	C₄×C₁₃	#381	1	1	yes	characteristic
155	C₄×C₁₃	#383	1	1	yes	characteristic
156	C₄×C₁₃	#388	1	2	yes	—
157	C₄×C₁₃	#388	1	2	yes	—
158	Dic₁₃	—	4	4	no	—
159	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	13	4	no	—
168	C₂×M₁₆	—	13	2	no	—
169	C₂×M₁₆	—	26	1	no	—
170	C₂×M₁₆	—	26	2	no	—
171	C₂×M₁₆	—	26	2	no	—
172	C₂×M₁₆	—	26	2	no	—
173	C₂×M₁₆	—	26	2	no	—
174	C₂×M₁₆	—	26	2	no	—
175	C₂×M₁₆	—	26	2	no	—
176	C₂×M₁₆	—	26	2	no	—
177	C₂×M₁₆	C₄×D₁₃	1	4	yes	—
178	C₂×M₁₆	D₅₂	1	2	yes	—
179	C₂×M₁₆	C₂²⋊D₁₃	1	2	yes	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	13	4	no	—
182	C₂²×C₈	—	26	1	no	—
183	C₂²×C₈	C₄×D₁₃	1	2	yes	—
184	#117	—	13	2	no	—
185	#117	—	13	2	no	—
186	#117	—	26	1	no	—
187	#117	—	26	1	no	—
188	C₂×C₄⋊C₄	—	26	1	no	—
189	C₂×C₂²⋊C₄	—	13	2	no	—
190	C₂×C₄²	—	26	1	no	—
191	#97	—	26	4	no	—
192	C₄×C₈	—	26	4	no	—
193	C₂×C₁₃	—	2	1	no	—
194	C₂×C₁₃	—	2	1	no	—
195	C₂×C₁₃	—	2	1	no	—
196	C₂×C₁₃	—	2	2	no	—
197	C₂×C₁₃	—	2	2	no	—
198	C₂×C₁₃	#1434	1	1	yes	characteristic
199	C₂⁴	Q₈×D₁₃	1	1	yes	characteristic
200	C₂²×C₄	—	26	2	no	—
201	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
203	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
204	C₂²×C₄	#6127	1	2	yes	—
205	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
206	C₂²×C₄	#6129	1	1	yes	characteristic
207	C₂²×C₄	#6129	1	2	yes	—
208	C₂²×C₄	#6129	1	2	yes	—
209	C₂²×C₄	#6131	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	13	4	no	—
218	M₁₆	—	26	2	no	—
219	M₁₆	—	26	4	no	—
220	M₁₆	—	52	2	no	—
221	M₁₆	—	52	2	no	—
222	M₁₆	—	52	2	no	—
223	M₁₆	—	52	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	26	2	no	—
230	C₂×C₈	—	26	4	no	—
231	C₂×C₈	—	26	4	no	—
232	C₂×C₈	—	26	4	no	—
233	C₂×C₈	—	26	4	no	—
234	C₂×C₈	#6103	1	4	yes	—
235	C₄⋊C₄	—	26	2	no	—
236	C₄⋊C₄	—	26	2	no	—
237	C₂²⋊C₄	—	13	4	no	—
238	C₂²⋊C₄	—	26	2	no	—
239	C₄²	—	26	2	no	—
240	C₄²	—	26	2	no	—
241	C₁₃	#10812	1	1	yes	characteristic, complemented, Sylow
242	C₂³	—	8	1	no	—
243	C₂³	#88141	1	2	yes	—
244	C₂³	#88143	1	2	yes	—
245	C₂³	C₄⋊C₄×D₁₃	1	1	yes	characteristic
246	C₂³	#88151	1	1	yes	characteristic
247	C₂³	#88200	1	1	yes	characteristic
248	C₂×C₄	—	26	4	no	—
249	C₂×C₄	—	104	2	no	—
250	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
251	C₂×C₄	#88126	1	1	yes	characteristic
252	C₂×C₄	C₄×D₅₂	1	2	yes	—
253	C₂×C₄	#88131	1	1	yes	characteristic
254	C₂×C₄	#88131	1	2	yes	—
255	C₂×C₄	#88131	1	1	yes	characteristic
256	C₂×C₄	#88133	1	2	yes	—
257	C₂×C₄	#88135	1	2	yes	—
258	C₂×C₄	#88139	1	2	yes	—
259	C₂×C₄	#88140	1	2	yes	—
260	C₂×C₄	#88146	1	1	yes	characteristic
261	C₂×C₄	#88146	1	2	yes	—
262	C₂×C₄	#88149	1	1	yes	characteristic
263	C₂×C₄	#88152	1	2	yes	—
264	C₂×C₄	C₂×#6103	1	1	yes	characteristic
265	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
266	C₂×C₄	#88189	1	2	yes	—
267	C₂×C₄	#88192	1	1	yes	characteristic
268	C₈	—	2	4	no	—
269	C₈	—	4	4	no	—
270	C₈	—	26	4	no	—
271	C₈	—	52	4	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	1	no	—
277	C₂²	#11718310	1	1	yes	characteristic
278	C₂²	#11718311	1	1	yes	characteristic
279	C₂²	#11718314	1	2	yes	—
280	C₂²	#11718639	1	2	yes	—
281	C₂²	#11718648	1	1	yes	characteristic
282	C₄	—	52	4	no	—
283	C₄	#11718309	1	1	yes	characteristic
284	C₄	#11718312	1	1	yes	characteristic
285	C₄	#11718331	1	2	yes	—
286	C₄	C₄×#6103	1	1	yes	characteristic
287	C₄	#11718604	1	1	yes	characteristic
288	C₄	#11718605	1	2	yes	—
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	C₂	#13113060	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×#6103
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×#338	C₂
socle	C₂×C₁₃	#1434
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×#338	D₁₃
hypocenter	C₁₃	#10812
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ C₂×#338

Complements

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

N	H ≅ G/N	type
17. #11718150	C₄	semidirect
241. C₁₃	#10812	semidirect

Maximal subgroups

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

subgroup	index
#13123254	2
C₂×#11718151	2
#13114562	2
C₂×#11718152	2
C₂×C₁₃×#338	2
#10812	13