Subgroups — #15866491

Order 3328.

844 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718786	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718785	C₂	1	2	yes	maximal
3	#13123254	C₂	1	1	yes	maximal, characteristic
4	#13115624	C₂	1	8	yes	maximal, complemented
5	C₂×#11718148	C₂	1	2	yes	maximal
6	C₂×C₁₃×#430	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×#88180	C₂²	1	1	yes	characteristic
10	C₂²×#6099	C₂²	1	2	yes	—
11	C₂²×#6099	C₂²	1	1	yes	characteristic
12	C₁₃×#430	—	2	2	no	—
13	C₁₃×#430	—	2	2	no	—
14	C₂×C₁₃×#108	C₂²	1	4	yes	—
15	#11718786	C₄	1	8	yes	complemented
16	#11718785	—	2	4	no	—
17	#11718785	C₂²	1	8	yes	—
18	C₂×#88062	—	2	4	no	—
19	#11718783	—	2	2	no	—
20	M₁₆×Dic₁₃	—	2	2	no	—
21	C₂×#88059	C₂²	1	4	yes	—
22	#11718762	C₂²	1	4	yes	—
23	#11718148	C₂²	1	8	yes	—
24	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
29	C₂×C₁₃×M₁₆	—	2	2	no	—
30	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
31	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
32	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
33	C₂×C₁₃×M₁₆	Q₈	1	4	yes	—
34	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
35	C₂²×C₈×C₁₃	—	2	2	no	—
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₂×C₂²⋊Dic₁₃	Q₈	1	2	yes	—
38	#88180	—	2	1	no	—
39	#88180	—	2	1	no	—
40	#88180	Q₈	1	2	yes	—
41	#88180	C₂³	1	2	yes	—
42	C₂×C₄⋊Dic₁₃	—	2	1	no	—
43	C₂×C₄×Dic₁₃	—	2	1	no	—
44	C₂×#6099	—	2	1	no	—
45	C₂×#6099	—	2	2	no	—
46	C₂×#6099	—	2	2	no	—
47	C₂×#6099	—	2	2	no	—
48	C₂×#6099	—	2	2	no	—
49	C₂×#6099	C₂×C₄	1	4	yes	—
50	C₂×#6099	C₂×C₄	1	4	yes	—
51	C₂×#6099	D₄	1	4	yes	—
52	C₂×#6099	C₂³	1	4	yes	—
53	C₂×#6099	C₂³	1	2	yes	—
54	C₂²×#1055	—	2	1	no	—
55	C₂²×#1055	—	2	2	no	—
56	C₁₃×#108	—	2	8	no	—
57	#88062	—	2	16	no	—
58	#88059	C₂×C₄	1	16	yes	complemented
59	#88055	—	2	4	no	—
60	C₈×Dic₁₃	—	2	4	no	—
61	#21184	—	13	1	no	maximal, Sylow
62	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	#41	1	2	yes	—
70	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₁₃	—	2	2	no	—
74	C₁₃×M₁₆	—	2	4	no	—
75	C₁₃×M₁₆	—	2	4	no	—
76	C₁₃×M₁₆	—	4	2	no	—
77	C₁₃×M₁₆	—	4	2	no	—
78	C₁₃×M₁₆	C₄⋊C₄	1	8	yes	—
79	C₁₃×M₁₆	C₄⋊C₄	1	8	yes	—
80	C₂×C₈×C₁₃	—	2	4	no	—
81	C₂×C₈×C₁₃	—	4	2	no	—
82	C₂×C₈×C₁₃	—	4	2	no	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
85	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
86	C₂²⋊Dic₁₃	—	2	2	no	—
87	C₂²⋊Dic₁₃	C₂×Q₈	1	4	yes	—
88	C₄⋊Dic₁₃	—	2	2	no	—
89	C₄⋊Dic₁₃	—	2	2	no	—
90	C₄×Dic₁₃	—	2	2	no	—
91	C₄×Dic₁₃	—	2	2	no	—
92	#6099	—	2	4	no	—
93	#6099	—	2	4	no	—
94	#6099	—	2	4	no	—
95	#6099	—	2	4	no	—
96	#6099	—	2	2	no	—
97	#6099	C₂²×C₄	1	8	yes	—
98	#6099	C₂×D₄	1	4	yes	—
99	C₂×#1055	—	2	4	no	—
100	C₂×#1055	—	2	4	no	—
101	C₂×#1055	—	2	4	no	—
102	C₂×#1055	—	2	2	no	—
103	C₂×#1055	—	2	4	no	—
104	C₂×#430	—	13	1	no	—
105	C₂×#430	—	13	2	no	—
106	C₂×#430	D₁₃	1	1	yes	characteristic
107	#2873	—	13	1	no	—
108	#1953	—	13	8	no	—
109	C₂×#344	—	13	2	no	—
110	C₂³×C₁₃	—	8	1	no	—
111	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
112	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂³×C₁₃	C₄⋊Q₈	1	2	yes	—
115	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
121	C₂×C₄×C₁₃	#121	1	2	yes	—
122	C₂×C₄×C₁₃	#121	1	2	yes	—
123	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₁₃	#125	1	2	yes	—
126	C₂×C₄×C₁₃	#125	1	2	yes	—
127	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
128	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
134	C₂×Dic₁₃	—	2	4	no	—
135	C₂×Dic₁₃	—	8	2	no	—
136	C₈×C₁₃	—	4	4	no	—
137	C₈×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
138	#1055	—	2	8	no	—
139	#1055	—	2	4	no	—
140	C₂²×M₁₆	—	13	1	no	—
141	C₂²×M₁₆	—	13	2	no	—
142	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
143	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
144	C₂×#117	—	13	1	no	—
145	#430	—	2	2	no	—
146	#430	—	2	2	no	—
147	#430	—	13	8	no	—
148	#430	—	13	8	no	—
149	#430	—	26	4	no	—
150	C₂×#108	—	13	4	no	—
151	C₂×#108	—	26	4	no	—
152	C₂×#108	C₂×D₁₃	1	4	yes	—
153	#405	—	13	4	no	—
154	#405	—	26	2	no	—
155	C₄×M₁₆	—	26	2	no	—
156	#344	—	13	8	no	—
157	C₂²×C₁₃	—	4	1	no	—
158	C₂²×C₁₃	—	4	2	no	—
159	C₂²×C₁₃	—	4	1	no	—
160	C₂²×C₁₃	—	4	2	no	—
161	C₂²×C₁₃	—	4	1	no	—
162	C₂²×C₁₃	#389	1	2	yes	—
163	C₂²×C₁₃	#389	1	2	yes	—
164	C₂²×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
165	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
166	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
167	C₄×C₁₃	#389	1	2	yes	—
168	C₄×C₁₃	#389	1	2	yes	—
169	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
170	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
171	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
172	C₄×C₁₃	C₂×#125	1	1	yes	characteristic
173	Dic₁₃	—	4	4	no	—
174	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	13	4	no	—
181	C₂×M₁₆	—	13	4	no	—
182	C₂×M₁₆	—	13	4	no	—
183	C₂×M₁₆	—	13	4	no	—
184	C₂×M₁₆	—	13	2	no	—
185	C₂×M₁₆	—	26	2	no	—
186	C₂×M₁₆	—	26	2	no	—
187	C₂×M₁₆	—	26	2	no	—
188	C₂×M₁₆	—	26	2	no	—
189	C₂×M₁₆	—	26	1	no	—
190	C₂×M₁₆	Dic₂₆	1	4	yes	—
191	C₂×M₁₆	C₄×D₁₃	1	4	yes	—
192	C₂×M₁₆	C₂²⋊D₁₃	1	2	yes	—
193	C₂×M₁₆	C₂²⋊D₁₃	1	2	yes	—
194	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	26	2	no	—
197	C₂²×C₈	—	26	1	no	—
198	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
199	#117	—	13	2	no	—
200	#117	—	13	2	no	—
201	#117	—	26	1	no	—
202	#117	—	26	1	no	—
203	C₂×C₄⋊C₄	—	26	1	no	—
204	C₂×C₂²⋊C₄	—	13	2	no	—
205	C₂×C₄²	—	26	1	no	—
206	#108	—	2	8	no	—
207	#108	—	13	16	no	—
208	#108	—	26	16	no	—
209	#97	—	26	4	no	—
210	C₄×C₈	—	26	4	no	—
211	C₂×C₁₃	—	2	1	no	—
212	C₂×C₁₃	—	2	1	no	—
213	C₂×C₁₃	—	2	1	no	—
214	C₂×C₁₃	—	2	2	no	—
215	C₂×C₁₃	—	2	2	no	—
216	C₂×C₁₃	C₂×#389	1	1	yes	characteristic
217	C₂⁴	Q₈×D₁₃	1	1	yes	characteristic
218	C₂²×C₄	—	26	2	no	—
219	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
220	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
221	C₂²×C₄	D₄×D₁₃	1	2	yes	—
222	C₂²×C₄	#6129	1	2	yes	—
223	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
224	C₂²×C₄	#6131	1	1	yes	characteristic
225	C₂²×C₄	#6131	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
228	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
229	M₁₆	—	2	4	no	—
230	M₁₆	—	2	4	no	—
231	M₁₆	—	4	2	no	—
232	M₁₆	—	4	2	no	—
233	M₁₆	—	13	8	no	—
234	M₁₆	—	13	4	no	—
235	M₁₆	—	26	4	no	—
236	M₁₆	—	26	4	no	—
237	M₁₆	—	26	4	no	—
238	M₁₆	—	26	2	no	—
239	M₁₆	—	26	4	no	—
240	M₁₆	#6101	1	8	yes	—
241	M₁₆	#6101	1	8	yes	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	26	4	no	—
246	C₂×C₈	—	26	4	no	—
247	C₂×C₈	—	26	4	no	—
248	C₂×C₈	—	26	4	no	—
249	C₂×C₈	—	26	2	no	—
250	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
251	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
252	C₂×C₈	C₂×C₂²⋊D₁₃	1	4	yes	—
253	C₄⋊C₄	—	26	2	no	—
254	C₄⋊C₄	—	26	2	no	—
255	C₂²⋊C₄	—	13	4	no	—
256	C₂²⋊C₄	—	26	2	no	—
257	C₄²	—	26	2	no	—
258	C₄²	—	26	2	no	—
259	C₁₃	#21184	1	1	yes	characteristic, complemented, Sylow
260	C₂³	—	8	1	no	—
261	C₂³	#88141	1	2	yes	—
262	C₂³	#88142	1	2	yes	—
263	C₂³	#88200	1	1	yes	characteristic
264	C₂³	#88200	1	1	yes	characteristic
265	C₂³	C₂×Q₈×D₁₃	1	1	yes	characteristic
266	C₂×C₄	—	26	4	no	—
267	C₂×C₄	—	104	2	no	—
268	C₂×C₄	#88141	1	2	yes	—
269	C₂×C₄	#88142	1	2	yes	—
270	C₂×C₄	#88144	1	2	yes	—
271	C₂×C₄	#88144	1	2	yes	—
272	C₂×C₄	#88147	1	2	yes	—
273	C₂×C₄	#88147	1	2	yes	—
274	C₂×C₄	C₂×#6101	1	2	yes	—
275	C₂×C₄	C₂×#6101	1	2	yes	—
276	C₂×C₄	#88192	1	2	yes	—
277	C₂×C₄	#88192	1	2	yes	—
278	C₂×C₄	#88200	1	1	yes	characteristic
279	C₂×C₄	#88200	1	1	yes	characteristic
280	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
282	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
283	C₂×C₄	C₂×#6129	1	1	yes	characteristic
284	C₂×C₄	C₂×#6131	1	1	yes	characteristic
285	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
286	C₈	—	4	4	no	—
287	C₈	—	26	8	no	—
288	C₈	—	26	4	no	—
289	C₈	C₂×#6101	1	8	yes	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	#11718634	1	2	yes	—
296	C₂²	#11718634	1	2	yes	—
297	C₂²	C₂×#88141	1	1	yes	characteristic
298	C₂²	C₂×#88142	1	1	yes	characteristic
299	C₂²	C₂×#88200	1	1	yes	characteristic
300	C₄	—	52	4	no	—
301	C₄	#11718634	1	2	yes	—
302	C₄	#11718634	1	2	yes	—
303	C₄	C₂×#88144	1	1	yes	characteristic
304	C₄	C₂×#88147	1	1	yes	characteristic
305	C₄	C₂²×#6101	1	1	yes	characteristic
306	C₄	C₂×#88192	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	2	1	no	—
310	C₂	—	2	2	no	—
311	C₂	—	2	2	no	—
312	C₂	C₂×#11718634	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#88192
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#430	C₂
socle	C₂×C₁₃	C₂×#389
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#430	D₁₃
hypocenter	C₁₃	#21184
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#430

Complements

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

N	H ≅ G/N	type
4. #13115624	C₂	semidirect
15. #11718786	C₄	semidirect
58. #88059	C₂×C₄	semidirect
259. C₁₃	#21184	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718786	2
C₂×#11718785	2
#13123254	2
#13115624	2
C₂×#11718148	2
C₂×C₁₃×#430	2
#21184	13