Subgroups — C₄×#11718554 #15861023

Order 3328.

1034 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718554	C₂	1	2	yes	maximal
2	C₂×C₄×#6094	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆×D₁₃	C₂	1	2	yes	maximal
4	#13114744	C₂	1	2	yes	maximal, complemented
5	#13114741	C₂	1	2	yes	maximal, complemented
6	C₄×#88062	C₂	1	2	yes	maximal, complemented
7	C₄×#88059	C₂	1	1	yes	maximal, characteristic, complemented
8	#13114561	C₂	1	2	yes	maximal
9	C₄×C₁₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×M₁₆×D₁₃	C₂²	1	4	yes	—
11	C₂²×#6094	C₂²	1	1	yes	characteristic
12	C₂²×#6094	C₂²	1	1	yes	characteristic
13	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#108	C₂²	1	2	yes	—
15	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×#88062	C₂²	1	4	yes	—
18	M₁₆×Dic₁₃	C₂²	1	4	yes	—
19	C₂×#88059	C₂²	1	2	yes	—
20	C₂×#88055	C₂²	1	1	yes	characteristic
21	C₂×#88055	C₂²	1	1	yes	characteristic
22	C₄×#6099	C₂²	1	2	yes	—
23	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
24	#11718554	C₄	1	16	yes	direct product factor
25	#11718368	—	2	2	no	—
26	D₁₃×#97	—	2	2	no	—
27	C₄×#6094	—	2	2	no	—
28	C₄×#6094	C₄	1	8	yes	—
29	C₄×C₈×D₁₃	—	2	2	no	—
30	C₁₃×#341	C₂²	1	2	yes	—
31	#11718150	C₂²	1	4	yes	—
32	#11718144	C₂²	1	2	yes	—
33	C₂²×C₄×D₁₃	Q₈	1	2	yes	—
34	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
35	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×C₄×Dic₁₃	Q₈	1	2	yes	—
40	C₂×C₄×Dic₁₃	C₂³	1	1	yes	characteristic
41	C₂×#6099	C₂³	1	4	yes	—
42	C₂²×#1055	C₂³	1	1	yes	characteristic
43	C₂²×#1055	C₂³	1	1	yes	characteristic
44	M₁₆×D₁₃	—	2	8	no	—
45	M₁₆×D₁₃	C₂×C₄	1	16	yes	—
46	C₂×#6094	—	2	4	no	—
47	C₂×#6094	—	2	4	no	—
48	C₂×#6094	C₂×C₄	1	4	yes	—
49	C₂×#6094	C₂×C₄	1	8	yes	—
50	C₂×#6094	C₂×C₄	1	4	yes	—
51	C₂×C₈×D₁₃	—	2	4	no	—
52	C₄²×D₁₃	C₂×C₄	1	4	yes	—
53	C₄²×D₁₃	D₄	1	2	yes	—
54	C₄²×D₁₃	D₄	1	2	yes	—
55	C₁₃×#108	C₂×C₄	1	8	yes	complemented
56	C₁₃×#97	—	2	2	no	—
57	C₄×C₈×C₁₃	—	2	2	no	—
58	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
59	#88062	C₂×C₄	1	16	yes	complemented
60	#88059	C₂×C₄	1	8	yes	complemented
61	#88055	—	2	4	no	—
62	#88055	C₂×C₄	1	4	yes	—
63	#88055	C₂×C₄	1	4	yes	—
64	C₈×Dic₁₃	—	2	4	no	—
65	#88043	—	2	2	no	—
66	C₄×#1055	—	2	2	no	—
67	C₄×#1055	C₂×C₄	1	4	yes	—
68	C₄×#430	—	13	1	no	maximal, Sylow
69	C₂³×D₁₃	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
72	C₂²×Dic₁₃	C₂×Q₈	1	2	yes	—
73	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₄×D₁₃	—	4	4	no	—
75	C₂×C₄×D₁₃	C₄⋊C₄	1	4	yes	—
76	C₂×C₄×D₁₃	C₄⋊C₄	1	4	yes	—
77	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×D₁₃	C₂²×C₄	1	4	yes	—
79	C₂×C₄×D₁₃	C₂×D₄	1	2	yes	—
80	C₂×C₄×D₁₃	#41	1	4	yes	—
81	C₁₃×M₁₆	—	2	4	no	—
82	C₁₃×M₁₆	C₂²×C₄	1	8	yes	—
83	C₂×C₈×C₁₃	—	2	4	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
88	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₄×Dic₁₃	C₄⋊C₄	1	4	yes	—
92	C₄×Dic₁₃	C₄⋊C₄	1	4	yes	—
93	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
94	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
95	#6099	—	2	4	no	—
96	#6099	C₂²×C₄	1	8	yes	—
97	C₂×#1055	—	2	4	no	—
98	C₂×#1055	—	2	2	no	—
99	C₂×#1055	C₂²×C₄	1	2	yes	—
100	C₂×#1055	C₂²×C₄	1	2	yes	—
101	C₂×#1055	C₂²×C₄	1	4	yes	—
102	#6094	—	2	8	no	—
103	#6094	—	2	16	no	—
104	#6094	C₄²	1	16	yes	—
105	C₈×D₁₃	—	2	16	no	—
106	C₂×#430	—	13	2	no	—
107	C₂×C₄×M₁₆	—	13	1	no	—
108	C₂×C₄×M₁₆	—	13	2	no	—
109	#1778	—	13	2	no	—
110	#1778	—	13	2	no	—
111	C₄×#108	—	13	2	no	—
112	C₄×#108	—	13	1	no	—
113	C₄×#108	D₁₃	1	1	yes	characteristic, complemented
114	#1738	—	13	2	no	—
115	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
116	C₂²×D₁₃	—	4	2	no	—
117	C₂²×D₁₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂²×D₁₃	C₄×Q₈	1	4	yes	—
119	C₂×C₄×C₁₃	—	2	4	no	—
120	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
124	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
126	C₂×Dic₁₃	—	4	2	no	—
127	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
128	C₂×Dic₁₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×Dic₁₃	C₄×Q₈	1	4	yes	—
130	C₄×D₁₃	—	2	8	no	—
131	C₄×D₁₃	—	4	4	no	—
132	C₄×D₁₃	C₂×C₄²	1	4	yes	—
133	C₄×D₁₃	#117	1	4	yes	—
134	C₄×D₁₃	C₄×D₄	1	8	yes	—
135	C₈×C₁₃	—	2	2	no	—
136	C₈×C₁₃	—	2	8	no	—
137	C₈×C₁₃	C₂×C₄²	1	4	yes	—
138	#1055	—	2	2	no	—
139	#1055	—	2	8	no	—
140	#1055	C₂×C₄²	1	4	yes	—
141	C₂²×M₁₆	—	13	1	no	—
142	C₂²×M₁₆	—	13	1	no	—
143	C₂²×M₁₆	—	13	4	no	—
144	C₂²×C₄²	—	13	1	no	—
145	#430	—	13	16	no	—
146	C₂×#108	—	13	4	no	—
147	C₂×#108	—	13	2	no	—
148	C₂×#108	C₂×D₁₃	1	2	yes	—
149	C₄×M₁₆	—	13	8	no	—
150	C₄×M₁₆	—	13	4	no	—
151	C₄×M₁₆	—	13	2	no	—
152	C₄×M₁₆	—	26	2	no	—
153	C₄×M₁₆	—	26	2	no	—
154	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
155	C₂×#97	—	13	1	no	—
156	C₂×#97	—	13	1	no	—
157	C₂×#97	—	26	2	no	—
158	C₂×C₄×C₈	—	13	1	no	—
159	C₂×C₄×C₈	—	26	2	no	—
160	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
161	#341	—	13	4	no	—
162	#341	—	13	2	no	—
163	#341	C₂×D₁₃	1	2	yes	—
164	C₂²×C₁₃	—	2	2	no	—
165	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
166	C₂²×C₁₃	C₂×C₄×Q₈	1	2	yes	—
167	C₂×D₁₃	—	2	4	no	—
168	C₂×D₁₃	—	4	4	no	—
169	C₂×D₁₃	C₄×C₄⋊C₄	1	4	yes	—
170	C₄×C₁₃	—	2	2	no	—
171	C₄×C₁₃	#430	1	4	yes	complemented
172	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
173	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
174	C₄×C₁₃	C₂×C₄×D₄	1	2	yes	—
175	Dic₁₃	—	2	4	no	—
176	Dic₁₃	C₄×C₄⋊C₄	1	4	yes	—
177	C₂³×C₄	—	13	1	no	—
178	C₂³×C₄	—	13	2	no	—
179	C₂×M₁₆	—	13	4	no	—
180	C₂×M₁₆	—	13	8	no	—
181	C₂×M₁₆	—	13	4	no	—
182	C₂×M₁₆	—	13	4	no	—
183	C₂×M₁₆	—	13	16	no	—
184	C₂×M₁₆	—	26	4	no	—
185	C₂×M₁₆	—	26	4	no	—
186	C₂×M₁₆	—	26	8	no	—
187	C₂×M₁₆	C₂²×D₁₃	1	4	yes	—
188	C₂²×C₈	—	13	1	no	—
189	C₂²×C₈	—	13	1	no	—
190	C₂²×C₈	—	26	4	no	—
191	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
192	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
193	C₂×C₄²	—	13	2	no	—
194	C₂×C₄²	—	13	1	no	—
195	C₂×C₄²	—	13	4	no	—
196	C₂×C₄²	—	13	2	no	—
197	C₂×C₄²	—	13	2	no	—
198	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
199	#108	—	13	16	no	—
200	#108	—	13	8	no	—
201	#108	C₄×D₁₃	1	8	yes	complemented
202	#97	—	2	2	no	—
203	#97	—	13	4	no	—
204	#97	—	13	4	no	—
205	#97	—	26	4	no	—
206	#97	—	26	2	no	—
207	C₄×C₈	—	2	2	no	—
208	C₄×C₈	—	13	4	no	—
209	C₄×C₈	—	26	4	no	—
210	C₄×C₈	—	26	2	no	—
211	C₄×C₈	C₄×D₁₃	1	4	yes	—
212	C₂×C₁₃	—	2	2	no	—
213	C₂×C₁₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
214	C₂×C₁₃	C₂×#430	1	2	yes	—
215	D₁₃	—	2	4	no	—
216	C₂⁴	—	13	1	no	—
217	C₂²×C₄	—	13	1	no	—
218	C₂²×C₄	—	13	2	no	—
219	C₂²×C₄	—	13	4	no	—
220	C₂²×C₄	—	13	2	no	—
221	C₂²×C₄	—	13	4	no	—
222	C₂²×C₄	—	13	2	no	—
223	C₂²×C₄	—	13	4	no	—
224	C₂²×C₄	—	13	4	no	—
225	C₂²×C₄	—	52	4	no	—
226	C₂²×C₄	Q₈×D₁₃	1	2	yes	—
227	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
228	M₁₆	—	2	4	no	—
229	M₁₆	—	13	16	no	—
230	M₁₆	—	13	8	no	—
231	M₁₆	—	26	8	no	—
232	M₁₆	—	26	16	no	—
233	M₁₆	—	26	4	no	—
234	M₁₆	C₂×C₄×D₁₃	1	8	yes	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	13	2	no	—
238	C₂×C₈	—	13	4	no	—
239	C₂×C₈	—	13	2	no	—
240	C₂×C₈	—	26	16	no	—
241	C₂×C₈	—	26	4	no	—
242	C₂×C₈	—	26	2	no	—
243	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
244	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
245	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
246	C₄²	—	13	4	no	—
247	C₄²	—	13	4	no	—
248	C₄²	—	13	2	no	—
249	C₄²	—	13	2	no	—
250	C₄²	C₂×C₄×D₁₃	1	2	yes	—
251	C₄²	D₄×D₁₃	1	1	yes	characteristic
252	C₄²	D₄×D₁₃	1	1	yes	characteristic
253	C₁₃	C₄×#430	1	1	yes	characteristic, complemented, Sylow
254	C₂³	—	13	2	no	—
255	C₂³	—	13	4	no	—
256	C₂³	—	52	2	no	—
257	C₂³	C₂×Q₈×D₁₃	1	1	yes	characteristic
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	13	2	no	—
260	C₂×C₄	—	13	4	no	—
261	C₂×C₄	—	13	4	no	—
262	C₂×C₄	—	13	4	no	—
263	C₂×C₄	—	13	8	no	—
264	C₂×C₄	—	13	4	no	—
265	C₂×C₄	—	26	8	no	—
266	C₂×C₄	—	52	4	no	—
267	C₂×C₄	—	52	2	no	—
268	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
269	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
270	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
272	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
273	C₂×C₄	#41×D₁₃	1	2	yes	—
274	C₈	—	2	2	no	—
275	C₈	—	2	8	no	—
276	C₈	—	13	4	no	—
277	C₈	—	26	2	no	—
278	C₈	—	26	8	no	—
279	C₈	C₄²×D₁₃	1	4	yes	—
280	C₂²	—	2	2	no	—
281	C₂²	—	13	4	no	—
282	C₂²	—	26	4	no	—
283	C₂²	—	52	4	no	—
284	C₂²	C₂×C₄⋊C₄×D₁₃	1	1	yes	characteristic
285	C₂²	C₄×Q₈×D₁₃	1	2	yes	—
286	C₄	—	2	2	no	—
287	C₄	—	13	4	no	—
288	C₄	—	26	4	no	—
289	C₄	#11718554	1	4	yes	direct product factor
290	C₄	C₂×C₄²×D₁₃	1	1	yes	characteristic
291	C₄	D₁₃×#117	1	1	yes	characteristic
292	C₄	C₄×D₄×D₁₃	1	2	yes	—
293	C₂	—	2	2	no	—
294	C₂	—	26	4	no	—
295	C₂	C₄×C₄⋊C₄×D₁₃	1	1	yes	characteristic
296	C₂	C₂×#11718554	1	2	yes	—

	subgroup	G/H
center	C₄²	D₄×D₁₃
commutator	C₄×C₁₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×#108	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#108	D₁₃
hypocenter	C₁₃	C₄×#430
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
4. #13114744	C₂	semidirect
5. #13114741	C₂	semidirect
6. C₄×#88062	C₂	semidirect
7. C₄×#88059	C₂	semidirect
9. C₄×C₁₃×#108	C₂	semidirect
24. #11718554	C₄	direct (N × H)
55. C₁₃×#108	C₂×C₄	semidirect
59. #88062	C₂×C₄	semidirect
60. #88059	C₂×C₄	semidirect
113. C₄×#108	D₁₃	semidirect
171. C₄×C₁₃	#430	semidirect
201. #108	C₄×D₁₃	semidirect
253. C₁₃	C₄×#430	semidirect
289. C₄	#11718554	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11718554	2
C₂×C₄×#6094	2
C₄×M₁₆×D₁₃	2
#13114744	2
#13114741	2
C₄×#88062	2
C₄×#88059	2
#13114561	2
C₄×C₁₃×#108	2
C₄×#430	13

Subgroups — C4×#11718554 #15861023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#11718554 #15861023