Subgroups — #15768833

Order 3328.

746 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718659	C₂	1	2	yes	maximal
2	C₂×#11718658	C₂	1	1	yes	maximal, characteristic
3	#13114569	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718123	C₂	1	2	yes	maximal
5	C₁₃×#1740	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#117	C₂²	1	2	yes	—
7	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
8	C₂×#88180	C₂²	1	2	yes	—
9	C₂²×#6099	C₂²	1	1	yes	characteristic
10	C₂³×#1055	C₂²	1	1	yes	characteristic
11	#11718659	—	2	4	no	—
12	#11718659	C₄	1	8	yes	complemented
13	#11718658	—	2	2	no	—
14	#11718658	—	2	2	no	—
15	C₂×#88048	—	2	2	no	—
16	C₂×#88047	—	2	2	no	—
17	C₂×#88043	C₄	1	2	yes	complemented
18	C₂×C₄×#1055	C₄	1	2	yes	complemented
19	C₁₃×#342	—	2	2	no	—
20	#11718123	—	2	4	no	—
21	#11718118	—	2	2	no	—
22	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₁₃×#117	—	2	2	no	—
24	C₁₃×#117	—	2	2	no	—
25	C₁₃×#117	—	2	2	no	—
26	C₁₃×#117	—	2	2	no	—
27	C₁₃×#117	—	2	2	no	—
28	C₁₃×#117	C₂×C₄	1	4	yes	—
29	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
30	C₂×C₁₃×C₄⋊C₄	Q₈	1	2	yes	—
31	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
32	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
33	C₂×C₄²×C₁₃	—	2	2	no	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₂²⋊Dic₁₃	—	2	2	no	—
36	#88180	—	2	2	no	—
37	#88180	—	2	2	no	—
38	#88180	—	2	2	no	—
39	#88180	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊Dic₁₃	—	2	2	no	—
41	C₂×C₄×Dic₁₃	—	2	2	no	—
42	C₂×#6099	—	2	2	no	—
43	C₂×#6099	—	2	2	no	—
44	C₂×#6099	D₄	1	2	yes	—
45	C₂×#6099	Q₈	1	2	yes	—
46	C₂²×#1055	C₂×C₄	1	2	yes	—
47	C₂²×#1055	C₂×C₄	1	4	yes	—
48	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
49	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
50	C₂²×#1055	D₄	1	2	yes	—
51	C₂²×#1055	D₄	1	2	yes	—
52	C₂²×#1055	D₄	1	2	yes	—
53	C₂²×#1055	Q₈	1	2	yes	—
54	#88048	—	2	8	no	—
55	#88047	—	2	8	no	—
56	#88043	—	2	4	no	—
57	C₄×#1055	—	2	2	no	—
58	C₄×#1055	—	2	2	no	—
59	#10866	—	13	1	no	maximal, Sylow
60	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	—	4	2	no	—
62	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	#41	1	2	yes	—
72	C₂²×C₄×C₁₃	#41	1	2	yes	—
73	C₂²×Dic₁₃	—	4	2	no	—
74	C₁₃×C₄⋊C₄	—	2	4	no	—
75	C₁₃×C₄⋊C₄	—	2	4	no	—
76	C₁₃×C₄⋊C₄	—	4	2	no	—
77	C₁₃×C₂²⋊C₄	—	2	4	no	—
78	C₁₃×C₂²⋊C₄	—	2	4	no	—
79	C₁₃×C₂²⋊C₄	—	4	2	no	—
80	C₄²×C₁₃	—	2	2	no	—
81	C₄²×C₁₃	—	2	2	no	—
82	C₄²×C₁₃	—	2	4	no	—
83	C₄²×C₁₃	—	4	2	no	—
84	C₂²⋊Dic₁₃	—	2	4	no	—
85	C₂²⋊Dic₁₃	—	4	2	no	—
86	C₄⋊Dic₁₃	—	2	4	no	—
87	C₄⋊Dic₁₃	—	4	2	no	—
88	C₄×Dic₁₃	—	2	4	no	—
89	C₄×Dic₁₃	—	4	2	no	—
90	#6099	—	2	4	no	—
91	#6099	—	2	4	no	—
92	C₂×#1055	—	2	2	no	—
93	C₂×#1055	—	2	2	no	—
94	C₂×#1055	C₄²	1	2	yes	—
95	C₂×#1055	C₄²	1	2	yes	—
96	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×#1055	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×#1055	C₂²⋊C₄	1	2	yes	—
99	C₂×#1055	C₂²⋊C₄	1	4	yes	—
100	C₂×#1055	C₂²⋊C₄	1	4	yes	—
101	C₂×#1055	C₄⋊C₄	1	4	yes	—
102	C₂×#1055	C₄⋊C₄	1	2	yes	—
103	C₂×#1055	C₄⋊C₄	1	2	yes	—
104	C₂×#1055	C₄⋊C₄	1	4	yes	—
105	C₂×#1055	C₄⋊C₄	1	4	yes	—
106	C₂×#427	—	13	2	no	—
107	C₂×#405	—	13	1	no	—
108	#1740	—	13	1	no	—
109	#1740	D₁₃	1	1	yes	characteristic
110	C₂×H(3,ℤ/4)	—	13	2	no	—
111	C₂³×C₁₃	—	4	2	no	—
112	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂³×C₁₃	C₄×D₄	1	2	yes	—
114	C₂³×C₁₃	#121	1	2	yes	—
115	C₂³×C₁₃	#127	1	1	yes	characteristic
116	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃	—	2	4	no	—
118	C₂×C₄×C₁₃	—	4	4	no	—
119	C₂×C₄×C₁₃	—	4	4	no	—
120	C₂×C₄×C₁₃	—	8	4	no	—
121	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
124	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
128	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
130	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
131	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
132	C₂×C₄×C₁₃	#121	1	2	yes	—
133	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₁₃	#124	1	2	yes	—
136	C₂×C₄×C₁₃	#125	1	2	yes	—
137	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
139	C₂×Dic₁₃	—	4	4	no	—
140	C₂×Dic₁₃	—	8	4	no	—
141	#1055	—	2	4	no	—
142	#1055	#95	1	2	yes	—
143	#1055	#95	1	2	yes	—
144	#1055	#95	1	4	yes	—
145	C₂²×M₁₆	—	13	1	no	—
146	C₂³×C₈	—	13	1	no	—
147	C₂×#117	—	13	2	no	—
148	C₂×#117	C₂×D₁₃	1	2	yes	—
149	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
150	#427	—	13	8	no	—
151	#427	—	26	4	no	—
152	C₂×#107	—	26	2	no	—
153	C₂×#106	—	26	2	no	—
154	#405	—	26	2	no	—
155	#405	—	26	2	no	—
156	C₂×#97	—	13	2	no	—
157	C₂×C₄×C₈	—	13	2	no	—
158	#342	—	2	2	no	—
159	#342	—	26	2	no	—
160	H(3,ℤ/4)	—	26	4	no	—
161	C₂²×C₁₃	—	2	4	no	—
162	C₂²×C₁₃	—	2	2	no	—
163	C₂²×C₁₃	—	4	2	no	—
164	C₂²×C₁₃	—	4	1	no	—
165	C₂²×C₁₃	—	4	1	no	—
166	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
167	C₂²×C₁₃	#380	1	1	yes	characteristic
168	C₂²×C₁₃	#384	1	1	yes	characteristic
169	C₂²×C₁₃	#389	1	2	yes	—
170	C₂²×C₁₃	#390	1	2	yes	—
171	C₄×C₁₃	—	4	4	no	—
172	C₄×C₁₃	—	8	4	no	—
173	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
174	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
175	C₄×C₁₃	#380	1	1	yes	characteristic
176	C₄×C₁₃	#382	1	1	yes	characteristic
177	C₄×C₁₃	#389	1	2	yes	—
178	C₄×C₁₃	#391	1	2	yes	—
179	Dic₁₃	—	8	4	no	—
180	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
181	C₂×M₁₆	—	13	2	no	—
182	C₂×M₁₆	—	13	2	no	—
183	C₂×M₁₆	—	26	2	no	—
184	C₂×M₁₆	—	26	2	no	—
185	C₂²×C₈	—	13	1	no	—
186	C₂²×C₈	—	13	1	no	—
187	C₂²×C₈	—	13	4	no	—
188	C₂²×C₈	—	13	2	no	—
189	C₂²×C₈	—	13	2	no	—
190	C₂²×C₈	—	13	2	no	—
191	C₂²×C₈	—	13	2	no	—
192	C₂²×C₈	—	13	2	no	—
193	#117	—	2	2	no	—
194	#117	—	2	2	no	—
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	—	2	2	no	—
198	#117	—	13	4	no	—
199	#117	—	26	2	no	—
200	#117	—	26	2	no	—
201	#117	—	26	2	no	—
202	#117	C₄×D₁₃	1	4	yes	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	26	2	no	—
205	C₂×C₄⋊C₄	Dic₂₆	1	2	yes	—
206	C₂×C₂²⋊C₄	—	2	2	no	—
207	C₂×C₂²⋊C₄	—	26	2	no	—
208	C₂×C₂²⋊C₄	C₂²⋊D₁₃	1	2	yes	—
209	C₂×C₄²	—	2	2	no	—
210	C₂×C₄²	—	26	2	no	—
211	C₂×C₄²	C₄×D₁₃	1	2	yes	—
212	#107	—	26	8	no	—
213	#106	—	26	8	no	—
214	#97	—	26	4	no	—
215	C₄×C₈	—	26	2	no	—
216	C₄×C₈	—	26	2	no	—
217	C₂×C₁₃	—	2	2	no	—
218	C₂×C₁₃	—	2	1	no	—
219	C₂×C₁₃	—	2	1	no	—
220	C₂×C₁₃	—	2	2	no	—
221	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
222	C₂×C₁₃	#1812	1	2	yes	—
223	C₂⁴	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	52	2	no	—
226	C₂²×C₄	#6101	1	4	yes	—
227	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
229	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
230	C₂²×C₄	#6127	1	2	yes	—
231	C₂²×C₄	D₄×D₁₃	1	2	yes	—
232	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
233	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
234	C₂²×C₄	#6129	1	2	yes	—
235	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
236	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
237	M₁₆	—	26	4	no	—
238	M₁₆	—	26	4	no	—
239	C₂×C₈	—	13	2	no	—
240	C₂×C₈	—	13	2	no	—
241	C₂×C₈	—	13	1	no	—
242	C₂×C₈	—	13	1	no	—
243	C₂×C₈	—	13	2	no	—
244	C₂×C₈	—	13	4	no	—
245	C₂×C₈	—	13	4	no	—
246	C₂×C₈	—	13	4	no	—
247	C₂×C₈	—	13	4	no	—
248	C₂×C₈	—	13	2	no	—
249	C₂×C₈	—	13	2	no	—
250	C₂×C₈	—	13	4	no	—
251	C₂×C₈	—	26	2	no	—
252	C₂×C₈	—	26	2	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	26	4	no	—
257	C₄⋊C₄	—	52	2	no	—
258	C₂²⋊C₄	—	2	4	no	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₂²⋊C₄	—	4	2	no	—
261	C₂²⋊C₄	—	26	4	no	—
262	C₂²⋊C₄	—	52	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	4	2	no	—
267	C₄²	—	26	4	no	—
268	C₄²	—	52	2	no	—
269	C₁₃	#10866	1	1	yes	characteristic, complemented, Sylow
270	C₂³	—	4	2	no	—
271	C₂³	C₂×#6101	1	1	yes	characteristic
272	C₂³	C₄×C₂²⋊D₁₃	1	2	yes	—
273	C₂³	#88193	1	2	yes	—
274	C₂³	#88194	1	1	yes	characteristic
275	C₂³	#88198	1	1	yes	characteristic
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	8	4	no	—
280	C₂×C₄	—	52	4	no	—
281	C₂×C₄	—	104	4	no	—
282	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
283	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
284	C₂×C₄	#88126	1	1	yes	characteristic
285	C₂×C₄	#88131	1	1	yes	characteristic
286	C₂×C₄	#88132	1	2	yes	—
287	C₂×C₄	C₂²⋊C₄×D₁₃	1	2	yes	—
288	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
289	C₂×C₄	#88135	1	2	yes	—
290	C₂×C₄	#88138	1	2	yes	—
291	C₂×C₄	#88139	1	2	yes	—
292	C₂×C₄	#88141	1	2	yes	—
293	C₂×C₄	#88142	1	1	yes	characteristic
294	C₂×C₄	#88142	1	1	yes	characteristic
295	C₂×C₄	#88143	1	2	yes	—
296	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
297	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
298	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
299	C₂×C₄	#88150	1	2	yes	—
300	C₈	—	13	2	no	—
301	C₈	—	13	2	no	—
302	C₈	—	13	4	no	—
303	C₈	—	26	4	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	C₄×#6101	1	1	yes	characteristic
310	C₂²	#11718603	1	1	yes	characteristic
311	C₂²	#11718618	1	2	yes	—
312	C₂²	#11718633	1	2	yes	—
313	C₂²	#11718637	1	1	yes	characteristic
314	C₄	—	4	4	no	—
315	C₄	—	8	4	no	—
316	C₄	—	104	4	no	—
317	C₄	#11718307	1	2	yes	—
318	C₄	#11718308	1	2	yes	—
319	C₄	#11718310	1	1	yes	characteristic
320	C₄	#11718311	1	1	yes	characteristic
321	C₄	D₁₃×#95	1	1	yes	characteristic
322	C₄	#11718328	1	1	yes	characteristic
323	C₂	—	2	2	no	—
324	C₂	—	2	1	no	—
325	C₂	—	2	2	no	—
326	C₂	—	2	1	no	—
327	C₂	#13113064	1	1	yes	characteristic
328	C₂	#13114880	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#88131
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₁₃×#1740	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	#1740	D₁₃
hypocenter	C₁₃	#10866
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1740

Complements

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

N	H ≅ G/N	type
12. #11718659	C₄	semidirect
17. C₂×#88043	C₄	semidirect
18. C₂×C₄×#1055	C₄	semidirect
269. C₁₃	#10866	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718659	2
C₂×#11718658	2
#13114569	2
C₂×#11718123	2
C₁₃×#1740	2
#10866	13