Subgroups — #16732561

Order 3328.

1770 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13128241	C₂	1	2	yes	maximal
2	#13125375	C₂	1	8	yes	maximal, complemented
3	#13125367	C₂	1	8	yes	maximal, complemented
4	#13125360	C₂	1	4	yes	maximal, complemented
5	#13125350	C₂	1	4	yes	maximal, complemented
6	#13123426	C₂	1	2	yes	maximal
7	C₂×#11718767	C₂	1	2	yes	maximal
8	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
12	C₂×#88180	C₂²	1	4	yes	—
13	#11719224	C₂²	1	16	yes	—
14	#11719222	—	2	8	no	—
15	#11719205	—	2	4	no	—
16	#11719188	—	2	4	no	—
17	#11719175	C₂²	1	8	yes	—
18	C₁₃×#405	C₂²	1	4	yes	—
19	C₁₃×#405	C₂²	1	4	yes	—
20	C₂×C₁₃×#97	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
22	#11718782	C₂²	1	16	yes	—
23	#11718767	C₂²	1	4	yes	—
24	#11718767	C₂²	1	4	yes	—
25	#11718767	C₂²	1	4	yes	—
26	#11718767	C₂²	1	4	yes	—
27	#11718767	C₂²	1	8	yes	—
28	C₂×#88058	C₂²	1	4	yes	—
29	C₂×#88057	C₂²	1	4	yes	—
30	#11718363	C₂²	1	32	yes	complemented
31	#11718344	C₂²	1	8	yes	complemented
32	#11718344	C₂²	1	8	yes	complemented
33	#11718343	C₂²	1	8	yes	complemented
34	#11718341	C₂²	1	8	yes	complemented
35	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₁₃×M₁₆	Q₈	1	8	yes	complemented
37	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
38	C₂²×C₈×C₁₃	Q₈	1	8	yes	complemented
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
43	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
44	C₁₃×#117	D₄	1	4	yes	—
45	C₁₃×#117	C₂³	1	4	yes	—
46	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
47	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
48	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
49	C₂×C₂²⋊Dic₁₃	—	2	4	no	—
50	#88180	—	2	4	no	—
51	#88180	—	2	4	no	—
52	#88180	C₂³	1	8	yes	—
53	#88180	C₂³	1	8	yes	—
54	C₂×C₄⋊Dic₁₃	C₂³	1	8	yes	—
55	#88178	—	2	16	no	—
56	C₂×C₄×Dic₁₃	—	2	4	no	—
57	#88144	—	2	16	no	—
58	#88142	—	2	16	no	—
59	#88132	—	2	32	no	—
60	#88124	C₂³	1	16	yes	—
61	#88123	C₂³	1	16	yes	—
62	C₄×Dic₂₆	—	2	16	no	—
63	C₁₃×#97	C₂³	1	8	yes	—
64	C₄×C₈×C₁₃	C₂³	1	4	yes	—
65	C₄×C₈×C₁₃	C₂³	1	4	yes	—
66	#88058	C₂³	1	8	yes	—
67	#88058	C₂³	1	8	yes	—
68	#88058	C₂³	1	16	yes	—
69	#88057	C₂³	1	8	yes	—
70	#88057	C₂³	1	8	yes	—
71	#88057	C₂³	1	16	yes	—
72	#60660	—	13	1	no	maximal, Sylow
73	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
78	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
79	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₂²×Dic₁₃	—	2	8	no	—
82	C₂×Dic₂₆	—	2	16	no	—
83	C₁₃×M₁₆	C₂×Q₈	1	16	yes	—
84	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
85	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
86	C₂×C₈×C₁₃	C₂×Q₈	1	8	yes	—
87	C₂×C₈×C₁₃	C₂×Q₈	1	8	yes	—
88	C₂×C₈×C₁₃	C₂×Q₈	1	8	yes	—
89	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
92	C₂×C₈×C₁₃	C₂⁴	1	2	yes	—
93	C₁₃×C₄⋊C₄	C₂×D₄	1	8	yes	—
94	C₁₃×C₂²⋊C₄	C₂×D₄	1	8	yes	—
95	C₄²×C₁₃	C₂×D₄	1	4	yes	—
96	C₄²×C₁₃	C₂⁴	1	4	yes	—
97	C₂²⋊Dic₁₃	—	2	8	no	—
98	C₂²⋊Dic₁₃	—	2	8	no	—
99	C₄⋊Dic₁₃	—	2	16	no	—
100	C₄⋊Dic₁₃	—	2	8	no	—
101	C₄⋊Dic₁₃	C₂⁴	1	16	yes	—
102	#6101	—	2	32	no	—
103	C₄×Dic₁₃	—	2	8	no	—
104	C₄×Dic₁₃	—	2	8	no	—
105	#3458	—	13	2	no	—
106	#3164	—	13	8	no	—
107	#3163	—	13	8	no	—
108	#3162	—	13	4	no	—
109	#3160	—	13	4	no	—
110	#2911	—	13	2	no	—
111	C₂×#427	—	13	2	no	—
112	C₂×#405	D₁₃	1	1	yes	characteristic
113	C₂³×C₁₃	—	4	2	no	—
114	C₂³×C₁₃	C₄⋊Q₈	1	4	yes	—
115	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
116	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
117	C₂×C₄×C₁₃	—	4	4	no	—
118	C₂×C₄×C₁₃	C₄⋊Q₈	1	8	yes	—
119	C₂×C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
120	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
121	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
122	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
124	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
125	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
126	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
127	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
128	C₂×Dic₁₃	—	2	16	no	—
129	C₂×Dic₁₃	—	8	8	no	—
130	Dic₂₆	—	8	16	no	—
131	C₈×C₁₃	C₂²×Q₈	1	2	yes	—
132	C₈×C₁₃	C₂²×Q₈	1	2	yes	—
133	C₈×C₁₃	C₂²×Q₈	1	4	yes	—
134	C₈×C₁₃	C₂²×Q₈	1	8	yes	—
135	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
136	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
137	#544	—	13	16	no	—
138	#544	—	26	4	no	—
139	#543	—	26	4	no	—
140	#533	—	13	8	no	—
141	#533	—	26	8	no	—
142	C₂×#117	—	13	4	no	—
143	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
144	#501	—	13	32	no	—
145	#500	—	13	8	no	—
146	#499	—	13	8	no	—
147	#499	—	13	8	no	—
148	#498	—	13	8	no	—
149	#428	—	13	16	no	—
150	#427	—	13	4	no	—
151	#427	—	13	8	no	—
152	#427	—	13	4	no	—
153	#427	—	13	4	no	—
154	#427	—	13	4	no	—
155	C₂×#107	—	13	4	no	—
156	C₂×#106	—	13	4	no	—
157	#405	C₂×D₁₃	1	4	yes	—
158	#405	C₂×D₁₃	1	4	yes	—
159	C₂×#97	C₂×D₁₃	1	2	yes	—
160	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
161	C₂²×C₁₃	—	2	4	no	—
162	C₂²×C₁₃	—	2	2	no	—
163	C₂²×C₁₃	—	4	4	no	—
164	C₂²×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
165	C₂²×C₁₃	C₂×C₄⋊Q₈	1	4	yes	—
166	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
167	C₄×C₁₃	—	2	8	no	—
168	C₄×C₁₃	C₂×C₄⋊Q₈	1	4	yes	—
169	C₄×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
170	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
171	C₄×C₁₃	C₂³×Q₈	1	1	yes	characteristic
172	Dic₁₃	—	4	16	no	—
173	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
174	C₂×M₁₆	Dic₂₆	1	8	yes	complemented
175	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
176	C₂²×C₈	Dic₂₆	1	8	yes	complemented
177	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
178	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
179	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
180	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
181	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
182	C₄⋊Q₈	—	13	16	no	—
183	C₄⋊Q₈	—	26	16	no	—
184	#125	—	13	16	no	—
185	#125	—	26	16	no	—
186	C₂²⋊Q₈	—	26	16	no	—
187	C₂²⋊Q₈	—	26	32	no	—
188	C₄×Q₈	—	26	16	no	—
189	#117	—	13	8	no	—
190	#117	—	13	8	no	—
191	#117	—	26	4	no	—
192	#117	—	26	4	no	—
193	#117	D₅₂	1	4	yes	—
194	#117	C₂²×D₁₃	1	4	yes	—
195	C₂×C₄⋊C₄	—	13	8	no	—
196	C₂×C₄⋊C₄	D₅₂	1	2	yes	—
197	C₂×C₂²⋊C₄	—	26	4	no	—
198	C₂×C₂²⋊C₄	D₅₂	1	2	yes	—
199	C₂×C₄²	—	26	4	no	—
200	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
201	#107	—	13	8	no	—
202	#107	—	13	8	no	—
203	#107	—	13	16	no	—
204	#106	—	13	8	no	—
205	#106	—	13	8	no	—
206	#106	—	13	16	no	—
207	#97	C₂²×D₁₃	1	8	yes	—
208	C₄×C₈	C₂²×D₁₃	1	4	yes	—
209	C₄×C₈	C₂²×D₁₃	1	4	yes	—
210	C₂×C₁₃	—	2	4	no	—
211	C₂×C₁₃	—	2	2	no	—
212	C₂×C₁₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
213	C₂×C₁₃	#3584	1	2	yes	—
214	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
215	C₂×Q₈	—	26	16	no	—
216	C₂²×C₄	—	26	8	no	—
217	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
218	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
219	C₂²×C₄	C₂×D₅₂	1	4	yes	—
220	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₅₂	1	2	yes	—
222	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
223	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
224	M₁₆	C₂×Dic₂₆	1	16	yes	—
225	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
226	C₂×C₈	C₂×Dic₂₆	1	4	yes	—
227	C₂×C₈	C₂×Dic₂₆	1	8	yes	—
228	C₂×C₈	C₂×Dic₂₆	1	8	yes	—
229	C₂×C₈	C₂×Dic₂₆	1	8	yes	—
230	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
231	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
232	C₂×C₈	C₂³×D₁₃	1	4	yes	—
233	C₂×C₈	C₂³×D₁₃	1	2	yes	—
234	C₄⋊C₄	—	13	16	no	—
235	C₄⋊C₄	—	26	8	no	—
236	C₄⋊C₄	—	26	32	no	—
237	C₄⋊C₄	—	26	16	no	—
238	C₄⋊C₄	C₂×D₅₂	1	8	yes	—
239	C₂²⋊C₄	—	26	8	no	—
240	C₂²⋊C₄	—	26	8	no	—
241	C₂²⋊C₄	C₂×D₅₂	1	8	yes	—
242	C₄²	—	26	8	no	—
243	C₄²	—	26	8	no	—
244	C₄²	C₂×D₅₂	1	4	yes	—
245	C₄²	C₂³×D₁₃	1	4	yes	—
246	C₁₃	#60660	1	1	yes	characteristic, complemented, Sylow
247	C₂³	—	4	2	no	—
248	C₂³	#88123	1	4	yes	—
249	C₂³	C₂²×D₅₂	1	1	yes	characteristic
250	C₂³	C₂²×D₅₂	1	2	yes	—
251	Q₈	—	104	16	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	26	16	no	—
254	C₂×C₄	—	104	8	no	—
255	C₂×C₄	#88123	1	4	yes	—
256	C₂×C₄	#88123	1	8	yes	—
257	C₂×C₄	C₂²×Dic₂₆	1	4	yes	—
258	C₂×C₄	C₂²×Dic₂₆	1	2	yes	—
259	C₂×C₄	C₂²×Dic₂₆	1	2	yes	—
260	C₂×C₄	C₂²×Dic₂₆	1	4	yes	—
261	C₂×C₄	C₂²×D₅₂	1	8	yes	—
262	C₂×C₄	C₂²×D₅₂	1	2	yes	—
263	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
264	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
265	C₈	C₂²×Dic₂₆	1	2	yes	—
266	C₈	C₂²×Dic₂₆	1	2	yes	—
267	C₈	C₂²×Dic₂₆	1	4	yes	—
268	C₈	C₂²×Dic₂₆	1	8	yes	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	4	4	no	—
272	C₂²	C₂×#88123	1	2	yes	—
273	C₂²	C₂×#88123	1	4	yes	—
274	C₂²	C₂³×D₅₂	1	1	yes	characteristic
275	C₄	—	2	8	no	—
276	C₄	—	52	16	no	—
277	C₄	C₂×#88123	1	4	yes	—
278	C₄	C₂×#88123	1	2	yes	—
279	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
280	C₄	C₂³×Dic₂₆	1	1	yes	characteristic
281	C₂	—	2	2	no	—
282	C₂	—	2	4	no	—
283	C₂	C₂²×#88123	1	1	yes	characteristic
284	C₂	#13129610	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂×C₁₃×#405	C₂
socle	C₂²×C₁₃	C₂³×D₄
prime radical	C₂⁴×C₁₃	C₂×D₄
hypercenter	C₂×#405	D₁₃
hypocenter	C₁₃	#60660
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #13125375	C₂	semidirect
3. #13125367	C₂	semidirect
4. #13125360	C₂	semidirect
5. #13125350	C₂	semidirect
30. #11718363	C₂²	semidirect
31. #11718344	C₂²	semidirect
32. #11718344	C₂²	semidirect
33. #11718343	C₂²	semidirect
34. #11718341	C₂²	semidirect
36. C₂×C₁₃×M₁₆	Q₈	semidirect
38. C₂²×C₈×C₁₃	Q₈	semidirect
174. C₂×M₁₆	Dic₂₆	semidirect
176. C₂²×C₈	Dic₂₆	semidirect
246. C₁₃	#60660	semidirect

Maximal subgroups

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

subgroup	index
#13128241	2
#13125375	2
#13125367	2
#13125360	2
#13125350	2
#13123426	2
C₂×#11718767	2
C₂×C₁₃×#405	2
#60660	13