Subgroups — #15834696

Order 3328.

586 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123785	C₂	1	2	yes	maximal
2	#13113733	C₂	1	2	yes	maximal, complemented
3	#13113687	C₂	1	4	yes	maximal, complemented
4	Q₁₆×#1055	C₂	1	2	yes	maximal, complemented
5	C₂×#11718050	C₂	1	1	yes	maximal, characteristic
6	#13113239	C₂	1	2	yes	maximal
7	C₈×#6099	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#3170	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#533	C₂²	1	2	yes	—
10	C₁₃×#500	C₄	1	1	yes	characteristic
11	C₁₃×#494	C₄	1	1	yes	characteristic
12	C₁₃×#487	C₄	1	2	yes	—
13	C₁₃×#467	C₄	1	2	yes	—
14	C₁₃×#467	C₄	1	2	yes	—
15	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
16	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
18	#11718685	—	2	4	no	—
19	#11718685	C₂²	1	4	yes	—
20	Q₈×#1055	C₂²	1	4	yes	—
21	#11718660	—	2	2	no	—
22	#11718658	—	2	2	no	—
23	#11718586	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊#1055	C₂²	1	2	yes	—
25	C₄×#6099	C₂²	1	1	yes	characteristic
26	Q₈⋊#1055	C₂²	1	4	yes	—
27	Q₈⋊#1055	C₂²	1	4	yes	—
28	#11718050	C₂²	1	1	yes	characteristic
29	#11718050	C₂²	1	2	yes	—
30	#11718050	C₂²	1	1	yes	characteristic
31	#11718046	C₂²	1	2	yes	—
32	C₈×#1055	C₂²	1	2	yes	—
33	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
38	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
39	C₁₃×#125	—	2	2	no	—
40	C₁₃×C₂²⋊Q₈	—	2	2	no	—
41	C₁₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
42	C₄×Q₈×C₁₃	—	2	2	no	—
43	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
44	C₁₃×#117	—	2	2	no	—
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×#6099	D₄	1	1	yes	characteristic
47	C₂×#6099	D₄	1	1	yes	characteristic
48	C₂²×#1055	—	2	2	no	—
49	C₁₃×#107	C₂×C₄	1	2	yes	—
50	C₁₃×#107	C₂×C₄	1	2	yes	—
51	C₁₃×#103	C₂×C₄	1	4	yes	—
52	C₁₃×#103	C₂×C₄	1	4	yes	—
53	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
57	C₂²⋊#1055	—	2	1	no	—
58	C₄⋊#1055	—	2	2	no	—
59	C₄⋊#1055	—	2	1	no	—
60	C₄⋊#1055	—	2	4	no	—
61	C₄⋊#1055	—	2	4	no	—
62	C₄⋊#1055	C₂³	1	2	yes	—
63	C₄⋊#1055	C₂³	1	2	yes	—
64	#88043	—	2	4	no	—
65	#88043	D₄	1	2	yes	—
66	C₄×#1055	—	2	4	no	—
67	C₄×#1055	C₂³	1	2	yes	—
68	C₄×#1055	C₂³	1	2	yes	—
69	#18068	—	13	1	no	maximal, Sylow
70	C₂×Q₈×C₁₃	—	2	2	no	—
71	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
75	C₁₃×Q₁₆	—	2	4	no	—
76	C₁₃×Q₁₆	—	4	2	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	—	4	1	no	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₁₃×C₄⋊C₄	—	2	2	no	—
84	C₁₃×C₄⋊C₄	—	2	2	no	—
85	C₁₃×C₄⋊C₄	—	2	2	no	—
86	C₁₃×C₄⋊C₄	—	2	4	no	—
87	C₁₃×C₄⋊C₄	—	2	4	no	—
88	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
89	C₁₃×C₂²⋊C₄	—	2	4	no	—
90	C₄²×C₁₃	—	2	4	no	—
91	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₃	#41	1	1	yes	characteristic
94	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
95	#6099	—	2	2	no	—
96	#6099	Q₁₆	1	4	yes	complemented
97	C₂×#1055	—	2	2	no	—
98	C₂×#1055	—	2	2	no	—
99	C₂×#1055	—	2	2	no	—
100	C₂×#1055	—	4	4	no	—
101	C₂×#1055	C₂×D₄	1	2	yes	—
102	C₂×#1055	C₂×D₄	1	2	yes	—
103	#3170	D₁₃	1	1	yes	characteristic
104	#2962	—	13	2	no	—
105	#1591	—	13	2	no	—
106	#1584	—	13	4	no	—
107	C₈×Q₁₆	—	13	2	no	—
108	C₂×#335	—	13	1	no	—
109	#1481	—	13	2	no	—
110	C₈×M₁₆	—	13	1	no	—
111	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
112	Q₈×C₁₃	—	2	4	no	—
113	Q₈×C₁₃	—	2	4	no	—
114	Q₈×C₁₃	—	8	2	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	4	no	—
118	C₂×C₄×C₁₃	—	2	4	no	—
119	C₂×C₄×C₁₃	—	4	1	no	—
120	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
124	C₂×C₄×C₁₃	#135	1	2	yes	—
125	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
128	C₈×C₁₃	—	2	1	no	—
129	C₈×C₁₃	—	2	1	no	—
130	C₈×C₁₃	—	2	1	no	—
131	C₈×C₁₃	#131	1	2	yes	—
132	#1055	—	2	2	no	—
133	#1055	—	4	2	no	—
134	#1055	—	4	2	no	—
135	#1055	—	4	2	no	—
136	#1055	C₂×Q₁₆	1	4	yes	—
137	#533	C₂×D₁₃	1	2	yes	—
138	#500	Dic₁₃	1	1	yes	characteristic
139	#494	Dic₁₃	1	1	yes	characteristic
140	#487	Dic₁₃	1	2	yes	—
141	#467	Dic₁₃	1	2	yes	—
142	#467	Dic₁₃	1	2	yes	—
143	#446	—	13	4	no	—
144	#446	—	26	4	no	—
145	C₈×Q₈	—	13	4	no	—
146	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
147	C₄×Q₁₆	C₂×D₁₃	1	2	yes	—
148	#431	—	13	1	no	—
149	#424	—	26	2	no	—
150	C₂×C₄⋊C₈	—	13	2	no	—
151	#405	—	26	2	no	—
152	C₄×M₁₆	—	13	1	no	—
153	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
154	#335	—	13	1	no	—
155	#335	—	13	2	no	—
156	#335	—	13	1	no	—
157	Q₈⋊C₈	—	13	4	no	—
158	Q₈⋊C₈	—	13	4	no	—
159	#322	—	13	2	no	—
160	C₈²	—	13	2	no	—
161	C₂²×C₁₃	—	2	1	no	—
162	C₂²×C₁₃	—	2	1	no	—
163	C₂²×C₁₃	C₄×Q₁₆	1	2	yes	—
164	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	2	1	no	—
168	C₄×C₁₃	—	4	4	no	—
169	C₄×C₁₃	—	4	4	no	—
170	C₄×C₁₃	C₄×Q₁₆	1	2	yes	—
171	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
172	C₄×C₁₃	#568	1	1	yes	characteristic
173	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
174	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
175	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
176	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
177	C₂×M₁₆	—	13	1	no	—
178	C₂×M₁₆	—	13	1	no	—
179	C₂²×C₈	—	26	2	no	—
180	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
181	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
182	C₄⋊Q₈	C₂×Dic₁₃	1	2	yes	—
183	C₄⋊Q₈	C₂×Dic₁₃	1	2	yes	—
184	#125	—	2	2	no	—
185	C₂²⋊Q₈	—	2	2	no	—
186	C₂²⋊Q₈	C₂×Dic₁₃	1	4	yes	—
187	C₄×Q₈	—	2	2	no	—
188	C₄×Q₈	C₂²×D₁₃	1	4	yes	—
189	#117	—	2	2	no	—
190	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
191	#107	C₂×Dic₁₃	1	2	yes	—
192	#107	C₂×Dic₁₃	1	2	yes	—
193	C₄⋊C₈	—	13	2	no	—
194	C₄⋊C₈	—	13	2	no	—
195	C₄⋊C₈	—	26	4	no	—
196	C₄⋊C₈	—	26	4	no	—
197	C₄⋊C₈	—	26	2	no	—
198	C₄⋊C₈	—	26	1	no	—
199	#103	C₂×Dic₁₃	1	4	yes	—
200	#103	C₂×Dic₁₃	1	4	yes	—
201	C₂²⋊C₈	—	26	1	no	—
202	#97	—	13	2	no	—
203	#97	—	26	4	no	—
204	C₄×C₈	—	13	2	no	—
205	C₄×C₈	—	13	2	no	—
206	C₄×C₈	—	26	4	no	—
207	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
208	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
209	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
210	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
211	C₂×C₁₃	—	2	2	no	—
212	C₂×C₁₃	#2935	1	1	yes	characteristic
213	C₂×C₁₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
214	C₂×C₁₃	#2957	1	1	yes	characteristic
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	C₂²×Dic₁₃	1	4	yes	—
217	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
218	C₂²×C₄	#6129	1	1	yes	characteristic
219	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
220	Q₁₆	—	2	4	no	—
221	Q₁₆	—	4	2	no	—
222	M₁₆	—	13	4	no	—
223	M₁₆	—	26	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	13	2	no	—
227	C₂×C₈	—	13	2	no	—
228	C₂×C₈	—	26	2	no	—
229	C₂×C₈	—	26	2	no	—
230	C₂×C₈	—	26	2	no	—
231	C₂×C₈	—	52	4	no	—
232	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
233	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
234	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
235	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	C₂²×Dic₁₃	1	4	yes	—
242	C₂²⋊C₄	—	2	4	no	—
243	C₄²	—	2	4	no	—
244	C₄²	D₄×D₁₃	1	1	yes	characteristic
245	C₄²	#6129	1	1	yes	characteristic
246	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
247	C₄²	C₂³×D₁₃	1	1	yes	characteristic
248	C₁₃	#18068	1	1	yes	characteristic, complemented, Sylow
249	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
250	Q₈	—	2	4	no	—
251	Q₈	—	2	4	no	—
252	Q₈	—	8	2	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	Q₁₆×D₁₃	1	2	yes	—
259	C₂×C₄	#88173	1	2	yes	—
260	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
261	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
262	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
263	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
264	C₂×C₄	C₂×#6129	1	1	yes	characteristic
265	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	13	4	no	—
270	C₈	—	26	2	no	—
271	C₈	—	52	2	no	—
272	C₈	—	52	2	no	—
273	C₈	—	52	2	no	—
274	C₈	#88202	1	2	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	Q₁₆×Dic₁₃	1	2	yes	—
278	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	4	4	no	—
283	C₄	—	4	4	no	—
284	C₄	Q₁₆×Dic₁₃	1	2	yes	—
285	C₄	C₂×Q₁₆×D₁₃	1	1	yes	characteristic
286	C₄	C₂×#88173	1	1	yes	characteristic
287	C₄	C₂×#88202	1	1	yes	characteristic
288	C₄	#11719524	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	#13123580	1	1	yes	characteristic
291	C₂	C₂×Q₁₆×Dic₁₃	1	1	yes	characteristic
292	C₂	#13123753	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#3170	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#3170	D₁₃
hypocenter	C₁₃	#18068
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #13113733	C₂	semidirect
3. #13113687	C₂	semidirect
4. Q₁₆×#1055	C₂	semidirect
96. #6099	Q₁₆	semidirect
248. C₁₃	#18068	semidirect

Maximal subgroups

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

subgroup	index
#13123785	2
#13113733	2
#13113687	2
Q₁₆×#1055	2
C₂×#11718050	2
#13113239	2
C₈×#6099	2
C₁₃×#3170	2
#18068	13