Subgroups — #15768839

Order 3328.

714 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88062	C₂	1	2	yes	maximal
2	C₂×#11718783	C₂	1	1	yes	maximal, characteristic
3	#13114574	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	2	yes	—
10	C₂³×#1055	C₂²	1	1	yes	characteristic
11	C₂×#88062	—	2	4	no	—
12	C₂×#88062	—	2	4	no	—
13	C₂×#88062	C₄	1	8	yes	complemented
14	#11718783	—	2	4	no	—
15	C₂×#88055	C₄	1	2	yes	—
16	C₂×C₈×Dic₁₃	C₄	1	2	yes	—
17	C₁₃×#338	—	2	2	no	—
18	#11718152	—	2	2	no	—
19	#11718146	—	2	4	no	—
20	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
27	C₂×C₁₃×M₁₆	—	2	2	no	—
28	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
29	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
30	C₂×C₁₃×M₁₆	Q₈	1	2	yes	—
31	C₂²×C₈×C₁₃	—	2	2	no	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
33	C₂×C₂²⋊Dic₁₃	Q₈	1	2	yes	—
34	#88180	—	2	2	no	—
35	#88180	—	2	2	no	—
36	C₂×C₄⋊Dic₁₃	D₄	1	2	yes	—
37	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
38	C₂×#6099	—	2	2	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	C₂×C₄	1	4	yes	—
44	C₂²×#1055	—	2	2	no	—
45	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
46	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
47	C₂²×#1055	C₂×C₄	1	4	yes	—
48	C₂²×#1055	D₄	1	4	yes	—
49	C₂²×#1055	D₄	1	2	yes	—
50	C₂²×#1055	Q₈	1	2	yes	—
51	#88062	—	2	16	no	—
52	#88055	—	2	4	no	—
53	C₈×Dic₁₃	—	2	4	no	—
54	#10867	—	13	1	no	maximal, Sylow
55	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
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×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	#41	1	2	yes	—
65	C₂²×C₄×C₁₃	#41	1	2	yes	—
66	C₂²×Dic₁₃	C₄⋊C₄	1	4	yes	—
67	C₁₃×M₁₆	—	2	4	no	—
68	C₁₃×M₁₆	—	2	4	no	—
69	C₁₃×M₁₆	—	2	4	no	—
70	C₁₃×M₁₆	—	2	4	no	—
71	C₁₃×M₁₆	—	4	2	no	—
72	C₁₃×M₁₆	—	4	2	no	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	—	4	2	no	—
77	C₂×C₈×C₁₃	—	4	2	no	—
78	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
79	C₂²⋊Dic₁₃	—	2	4	no	—
80	C₄⋊Dic₁₃	—	2	4	no	—
81	C₄×Dic₁₃	—	2	2	no	—
82	C₄×Dic₁₃	—	2	2	no	—
83	#6099	—	2	4	no	—
84	#6099	—	2	4	no	—
85	#6099	—	4	2	no	—
86	#6099	—	4	2	no	—
87	C₂×#1055	—	2	4	no	—
88	C₂×#1055	—	4	2	no	—
89	C₂×#1055	—	4	2	no	—
90	C₂×#1055	C₄²	1	2	yes	—
91	C₂×#1055	C₄²	1	2	yes	—
92	C₂×#1055	C₂²⋊C₄	1	2	yes	—
93	C₂×#1055	C₂²⋊C₄	1	2	yes	—
94	C₂×#1055	C₂²⋊C₄	1	8	yes	—
95	C₂×#1055	C₄⋊C₄	1	2	yes	—
96	C₂×#1055	C₄⋊C₄	1	2	yes	—
97	C₂×#1055	C₄⋊C₄	1	8	yes	—
98	C₂²×#108	—	13	2	no	—
99	C₂×#405	—	13	1	no	—
100	#1742	—	13	2	no	—
101	C₂×#338	—	13	1	no	—
102	C₂×#338	D₁₃	1	1	yes	characteristic
103	C₂³×C₁₃	—	4	2	no	—
104	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
106	C₂³×C₁₃	C₂²⋊Q₈	1	2	yes	—
107	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
108	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
116	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
120	C₂×C₄×C₁₃	#121	1	2	yes	—
121	C₂×C₄×C₁₃	#121	1	2	yes	—
122	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₁₃	#124	1	2	yes	—
124	C₂×C₄×C₁₃	#125	1	2	yes	—
125	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
127	C₂×Dic₁₃	—	2	4	no	—
128	C₂×Dic₁₃	—	4	4	no	—
129	C₈×C₁₃	—	2	4	no	—
130	C₈×C₁₃	—	4	4	no	—
131	#1055	—	4	4	no	—
132	#1055	#95	1	4	yes	—
133	#1055	#95	1	4	yes	—
134	C₂²×M₁₆	—	13	2	no	—
135	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
136	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
137	C₂³×C₈	—	13	1	no	—
138	C₂×#117	—	13	1	no	—
139	C₂×#108	—	13	8	no	—
140	C₂×#108	—	26	4	no	—
141	C₂×#108	—	26	4	no	—
142	#405	—	26	4	no	—
143	C₂×#97	—	13	2	no	—
144	C₂×C₄×C₈	—	13	2	no	—
145	#343	—	26	4	no	—
146	#338	—	2	2	no	—
147	#338	—	26	2	no	—
148	C₂²×C₁₃	—	2	2	no	—
149	C₂²×C₁₃	—	2	4	no	—
150	C₂²×C₁₃	—	4	2	no	—
151	C₂²×C₁₃	—	4	1	no	—
152	C₂²×C₁₃	—	4	1	no	—
153	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
154	C₂²×C₁₃	#380	1	1	yes	characteristic
155	C₂²×C₁₃	#384	1	1	yes	characteristic
156	C₂²×C₁₃	#389	1	2	yes	—
157	C₂²×C₁₃	#391	1	2	yes	—
158	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
159	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
160	C₄×C₁₃	#380	1	1	yes	characteristic
161	C₄×C₁₃	#382	1	1	yes	characteristic
162	C₄×C₁₃	#389	1	2	yes	—
163	C₄×C₁₃	#390	1	2	yes	—
164	Dic₁₃	—	4	4	no	—
165	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	13	4	no	—
174	C₂×M₁₆	—	26	2	no	—
175	C₂×M₁₆	—	26	2	no	—
176	C₂×M₁₆	—	26	2	no	—
177	C₂×M₁₆	—	26	2	no	—
178	C₂×M₁₆	—	26	2	no	—
179	C₂×M₁₆	Dic₂₆	1	2	yes	—
180	C₂×M₁₆	C₄×D₁₃	1	4	yes	—
181	C₂×M₁₆	C₂²⋊D₁₃	1	2	yes	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	13	1	no	—
184	C₂²×C₈	—	13	1	no	—
185	C₂²×C₈	—	13	4	no	—
186	C₂²×C₈	—	13	4	no	—
187	C₂²×C₈	—	13	2	no	—
188	C₂²×C₈	—	13	2	no	—
189	C₂²×C₈	—	26	2	no	—
190	C₂²×C₈	C₄×D₁₃	1	2	yes	—
191	#117	—	26	2	no	—
192	#117	—	26	2	no	—
193	C₂×C₄⋊C₄	—	13	2	no	—
194	C₂×C₂²⋊C₄	—	13	2	no	—
195	C₂×C₄²	—	13	2	no	—
196	#108	—	26	16	no	—
197	#97	—	26	4	no	—
198	C₄×C₈	—	26	4	no	—
199	C₂×C₁₃	—	2	1	no	—
200	C₂×C₁₃	—	2	1	no	—
201	C₂×C₁₃	—	2	2	no	—
202	C₂×C₁₃	—	2	2	no	—
203	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
204	C₂×C₁₃	#1813	1	2	yes	—
205	C₂⁴	Q₈×D₁₃	1	1	yes	characteristic
206	C₂²×C₄	—	13	4	no	—
207	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
209	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
210	C₂²×C₄	#6127	1	2	yes	—
211	C₂²×C₄	D₄×D₁₃	1	2	yes	—
212	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
213	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
214	C₂²×C₄	#6129	1	2	yes	—
215	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	4	2	no	—
222	M₁₆	—	4	2	no	—
223	M₁₆	—	26	4	no	—
224	M₁₆	—	26	4	no	—
225	M₁₆	—	52	2	no	—
226	M₁₆	—	52	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	13	2	no	—
233	C₂×C₈	—	13	2	no	—
234	C₂×C₈	—	13	2	no	—
235	C₂×C₈	—	13	2	no	—
236	C₂×C₈	—	13	8	no	—
237	C₂×C₈	—	13	8	no	—
238	C₂×C₈	—	13	2	no	—
239	C₂×C₈	—	13	2	no	—
240	C₂×C₈	—	26	4	no	—
241	C₂×C₈	—	52	2	no	—
242	C₂×C₈	—	52	2	no	—
243	C₂×C₈	#6101	1	4	yes	—
244	C₄⋊C₄	—	26	4	no	—
245	C₂²⋊C₄	—	26	4	no	—
246	C₄²	—	26	2	no	—
247	C₄²	—	26	2	no	—
248	C₁₃	#10867	1	1	yes	characteristic, complemented, Sylow
249	C₂³	—	4	2	no	—
250	C₂³	#88141	1	2	yes	—
251	C₂³	#88142	1	1	yes	characteristic
252	C₂³	C₄⋊C₄×D₁₃	1	1	yes	characteristic
253	C₂³	#88150	1	2	yes	—
254	C₂³	#88198	1	1	yes	characteristic
255	C₂×C₄	—	26	4	no	—
256	C₂×C₄	—	52	4	no	—
257	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
258	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
259	C₂×C₄	#88126	1	1	yes	characteristic
260	C₂×C₄	#88131	1	1	yes	characteristic
261	C₂×C₄	#88132	1	2	yes	—
262	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
263	C₂×C₄	C₂²⋊C₄×D₁₃	1	2	yes	—
264	C₂×C₄	#88135	1	2	yes	—
265	C₂×C₄	#88138	1	2	yes	—
266	C₂×C₄	#88139	1	2	yes	—
267	C₂×C₄	#88142	1	1	yes	characteristic
268	C₂×C₄	#88143	1	2	yes	—
269	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
270	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
271	C₂×C₄	C₂×#6101	1	1	yes	characteristic
272	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
273	C₂×C₄	#88193	1	2	yes	—
274	C₂×C₄	#88194	1	1	yes	characteristic
275	C₈	—	2	4	no	—
276	C₈	—	4	4	no	—
277	C₈	—	13	4	no	—
278	C₈	—	13	4	no	—
279	C₈	—	52	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	#11718307	1	2	yes	—
286	C₂²	#11718310	1	1	yes	characteristic
287	C₂²	#11718328	1	1	yes	characteristic
288	C₂²	#11718633	1	2	yes	—
289	C₂²	#11718637	1	1	yes	characteristic
290	C₄	—	52	4	no	—
291	C₄	#11718308	1	2	yes	—
292	C₄	#11718311	1	1	yes	characteristic
293	C₄	D₁₃×#95	1	1	yes	characteristic
294	C₄	C₄×#6101	1	1	yes	characteristic
295	C₄	#11718603	1	1	yes	characteristic
296	C₄	#11718618	1	2	yes	—
297	C₂	—	2	2	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	—	2	1	no	—
301	C₂	#13113064	1	1	yes	characteristic
302	C₂	#13114885	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#88126
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×#338	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×#338	D₁₃
hypocenter	C₁₃	#10867
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#338

Complements

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

N	H ≅ G/N	type
13. C₂×#88062	C₄	semidirect
248. C₁₃	#10867	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88062	2
C₂×#11718783	2
#13114574	2
C₂×#11718152	2
C₂×C₁₃×#338	2
#10867	13