Subgroups — #15867318

Order 3328.

746 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	#13123480	C₂	1	2	yes	maximal, complemented
2	#13115730	C₂	1	2	yes	maximal
3	#13115590	C₂	1	2	yes	maximal, complemented
4	#13115549	C₂	1	4	yes	maximal, complemented
5	C₄×#88059	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#88077	C₂	1	2	yes	maximal, complemented
7	C₄×#88055	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₁₃×#135	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
10	C₄×C₁₃×#41	C₂²	1	2	yes	—
11	C₄×C₁₃×Q₁₆	C₄	1	2	yes	complemented
12	C₄×C₁₃×SD₁₆	C₄	1	4	yes	complemented
13	C₄×C₁₃×D₈	C₄	1	2	yes	complemented
14	C₁₃×#427	C₂²	1	1	yes	characteristic
15	C₁₃×#416	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×#88077	C₂²	1	4	yes	—
18	#11718895	C₂²	1	4	yes	—
19	C₂×#88059	C₂²	1	2	yes	—
20	C₂×#88055	C₂²	1	2	yes	—
21	C₂×#88055	C₂²	1	2	yes	—
22	#11718661	—	2	4	no	—
23	#11718660	C₂²	1	4	yes	—
24	C₄²×Dic₁₃	C₂²	1	1	yes	characteristic
25	#11718586	—	2	4	no	—
26	C₄×#6099	C₂²	1	2	yes	—
27	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
28	#11718144	C₂²	1	2	yes	—
29	#11718124	C₂²	1	4	yes	—
30	C₂×C₁₃×#41	C₂³	1	2	yes	—
31	C₁₃×#135	—	2	4	no	—
32	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₃×SD₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
38	C₄×D₄×C₁₃	—	2	4	no	—
39	C₄×D₄×C₁₃	C₂×C₄	1	4	yes	—
40	C₁₃×#117	—	2	2	no	—
41	C₁₃×#117	C₂³	1	2	yes	—
42	C₂×C₄²×C₁₃	—	2	2	no	—
43	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×C₄×Dic₁₃	—	2	2	no	—
45	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
46	C₂×#6099	C₂³	1	4	yes	—
47	C₂²×#1055	C₂³	1	2	yes	—
48	C₁₃×#107	C₂×C₄	1	2	yes	—
49	C₁₃×#106	C₂×C₄	1	2	yes	—
50	C₁₃×#103	C₂×C₄	1	4	yes	—
51	C₁₃×#102	C₂×C₄	1	4	yes	—
52	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
54	#88077	—	2	8	no	—
55	C₂²⋊#1055	—	2	8	no	—
56	#88059	—	2	4	no	—
57	#88055	—	2	4	no	—
58	#88055	—	2	4	no	—
59	C₄⋊#1055	—	2	8	no	—
60	#88043	—	2	2	no	—
61	C₄×#1055	—	2	2	no	—
62	C₄×#1055	D₄	1	2	yes	—
63	C₄×#1055	D₄	1	2	yes	—
64	#21293	—	13	1	no	maximal, Sylow
65	C₁₃×#41	—	2	4	no	—
66	C₁₃×#41	—	2	4	no	—
67	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂×D₄×C₁₃	—	2	2	no	—
69	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₁₃	—	2	4	no	—
71	C₂²×C₄×C₁₃	—	2	2	no	—
72	C₂²×C₄×C₁₃	#41	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₂²×Dic₁₃	—	2	2	no	—
75	C₁₃×Q₁₆	—	2	2	no	—
76	C₁₃×SD₁₆	—	2	4	no	—
77	C₁₃×D₈	—	2	2	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	#41	1	4	yes	—
83	C₁₃×C₄⋊C₄	—	2	4	no	—
84	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
85	C₁₃×C₂²⋊C₄	—	2	2	no	—
86	C₁₃×C₂²⋊C₄	—	2	4	no	—
87	C₄²×C₁₃	—	2	4	no	—
88	C₄²×C₁₃	—	2	2	no	—
89	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
90	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₄×Dic₁₃	—	2	2	no	—
93	C₄×Dic₁₃	—	4	4	no	—
94	C₄×Dic₁₃	#41	1	4	yes	—
95	#6099	—	2	4	no	—
96	#6099	—	2	4	no	—
97	C₂×#1055	—	2	2	no	—
98	C₂×#1055	—	2	2	no	—
99	C₂×#1055	—	2	4	no	—
100	C₂×#1055	C₂×D₄	1	4	yes	—
101	C₄×#135	D₁₃	1	1	yes	characteristic
102	#2922	—	13	2	no	—
103	#1975	—	13	2	no	—
104	#1946	—	13	2	no	—
105	#1938	—	13	4	no	—
106	C₄×#108	—	13	1	no	—
107	C₄×#104	—	13	2	no	—
108	C₄×#97	—	13	1	no	—
109	C₂³×C₁₃	—	2	2	no	—
110	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
111	Q₈×C₁₃	—	2	2	no	—
112	Q₈×C₁₃	—	2	2	no	—
113	D₄×C₁₃	—	2	2	no	—
114	D₄×C₁₃	—	2	2	no	—
115	D₄×C₁₃	—	4	4	no	—
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	—	2	4	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	4	4	no	—
120	C₂×C₄×C₁₃	—	4	8	no	—
121	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
127	C₂×Dic₁₃	—	2	4	no	—
128	C₂×Dic₁₃	—	4	4	no	—
129	C₈×C₁₃	—	2	2	no	—
130	C₈×C₁₃	#117	1	2	yes	—
131	C₈×C₁₃	#117	1	2	yes	—
132	#1055	—	2	4	no	—
133	#1055	—	4	4	no	—
134	C₂×#135	C₂×D₁₃	1	1	yes	characteristic
135	C₄×#41	C₂×D₁₃	1	2	yes	—
136	C₄×Q₁₆	Dic₁₃	1	2	yes	complemented
137	C₄×SD₁₆	Dic₁₃	1	4	yes	complemented
138	C₄×D₈	Dic₁₃	1	2	yes	complemented
139	#433	—	26	4	no	—
140	#431	—	26	4	no	—
141	C₂×#108	—	13	2	no	—
142	#427	C₂×D₁₃	1	1	yes	characteristic
143	#424	—	13	4	no	—
144	C₂×#104	—	13	4	no	—
145	#416	C₂×D₁₃	1	2	yes	—
146	#408	—	13	4	no	—
147	C₄×M₁₆	—	13	2	no	—
148	C₂×#97	—	13	2	no	—
149	C₂×#97	—	13	2	no	—
150	C₂×C₄×C₈	—	13	1	no	—
151	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
152	C₄³	—	13	1	no	—
153	#341	—	13	2	no	—
154	#339	—	13	4	no	—
155	C₂²×C₁₃	—	2	2	no	—
156	C₂²×C₁₃	—	4	4	no	—
157	C₂²×C₁₃	—	4	2	no	—
158	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
159	C₂²×C₁₃	#529	1	2	yes	—
160	C₄×C₁₃	—	2	2	no	—
161	C₄×C₁₃	—	4	4	no	—
162	C₄×C₁₃	—	4	2	no	—
163	C₄×C₁₃	#444	1	4	yes	—
164	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
165	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
166	C₄×C₁₃	#532	1	2	yes	—
167	Dic₁₃	—	4	4	no	—
168	C₂×#41	C₂²×D₁₃	1	2	yes	—
169	#135	—	2	4	no	—
170	C₂×Q₁₆	C₂×Dic₁₃	1	1	yes	characteristic
171	C₂×SD₁₆	C₂×Dic₁₃	1	2	yes	—
172	C₂×D₈	C₂×Dic₁₃	1	1	yes	characteristic
173	C₂×M₁₆	—	13	4	no	—
174	C₂²×C₈	—	13	2	no	—
175	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
177	C₄×Q₈	C₂×Dic₁₃	1	4	yes	—
178	C₄×D₄	—	2	4	no	—
179	C₄×D₄	C₂×Dic₁₃	1	4	yes	—
180	#117	—	2	2	no	—
181	#117	C₂²×D₁₃	1	2	yes	—
182	C₂×C₄²	—	2	2	no	—
183	C₂×C₄²	—	13	2	no	—
184	C₂×C₄²	—	26	2	no	—
185	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
186	#108	—	26	4	no	—
187	#107	C₂×Dic₁₃	1	2	yes	—
188	#106	C₂×Dic₁₃	1	2	yes	—
189	C₄⋊C₈	—	26	8	no	—
190	#104	—	26	8	no	—
191	#103	C₂×Dic₁₃	1	4	yes	—
192	#102	C₂×Dic₁₃	1	4	yes	—
193	C₂²⋊C₈	—	26	8	no	—
194	#97	—	26	4	no	—
195	#97	—	26	4	no	—
196	#97	—	26	2	no	—
197	C₄×C₈	—	13	2	no	—
198	C₄×C₈	—	13	2	no	—
199	C₄×C₈	—	26	2	no	—
200	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
201	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
202	C₂×C₁₃	—	2	2	no	—
203	C₂×C₁₃	—	8	2	no	—
204	C₂×C₁₃	#2293	1	1	yes	characteristic
205	C₂×C₁₃	C₂×#444	1	2	yes	—
206	#41	—	2	4	no	—
207	#41	—	2	4	no	—
208	C₂×Q₈	C₂²×Dic₁₃	1	2	yes	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	C₂²×Dic₁₃	1	2	yes	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	26	2	no	—
214	C₂²×C₄	#6129	1	2	yes	—
215	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
216	Q₁₆	—	2	2	no	—
217	SD₁₆	—	2	4	no	—
218	D₈	—	2	2	no	—
219	M₁₆	—	26	4	no	—
220	M₁₆	—	26	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	13	4	no	—
223	C₂×C₈	—	26	2	no	—
224	C₂×C₈	—	26	2	no	—
225	C₂×C₈	—	26	4	no	—
226	C₂×C₈	#6127	1	4	yes	—
227	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
228	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
229	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	C₂²×Dic₁₃	1	4	yes	—
232	C₂²⋊C₄	—	2	2	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	13	4	no	—
237	C₄²	—	26	2	no	—
238	C₄²	—	52	4	no	—
239	C₄²	D₄×D₁₃	1	1	yes	characteristic
240	C₄²	D₄×D₁₃	1	1	yes	characteristic
241	C₄²	C₂²×Dic₁₃	1	2	yes	—
242	C₁₃	#21293	1	1	yes	characteristic, complemented, Sylow
243	C₂³	—	2	2	no	—
244	C₂³	C₂×#6129	1	1	yes	characteristic
245	Q₈	—	2	2	no	—
246	Q₈	—	2	2	no	—
247	D₄	—	2	2	no	—
248	D₄	—	2	2	no	—
249	D₄	—	4	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	8	no	—
255	C₂×C₄	—	26	4	no	—
256	C₂×C₄	—	52	4	no	—
257	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
258	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
259	C₂×C₄	C₂×#6127	1	2	yes	—
260	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
261	C₂×C₄	#41×D₁₃	1	2	yes	—
262	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	26	4	no	—
265	C₈	—	52	4	no	—
266	C₈	#88180	1	2	yes	—
267	C₈	#88180	1	2	yes	—
268	C₂²	—	2	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	4	no	—
271	C₂²	#11719262	1	2	yes	—
272	C₂²	C₂×D₄×Dic₁₃	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	4	2	no	—
275	C₄	—	4	4	no	—
276	C₄	—	52	4	no	—
277	C₄	#11718858	1	4	yes	—
278	C₄	#11719253	1	2	yes	—
279	C₄	C₂×#88180	1	1	yes	characteristic
280	C₄	#41×Dic₁₃	1	1	yes	characteristic
281	C₂	—	2	2	no	—
282	C₂	—	8	2	no	—
283	C₂	#13117301	1	1	yes	characteristic
284	C₂	C₂×#11718858	1	2	yes	—

	subgroup	G/H
center	C₄²	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×#135	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂×C₁₃×#135	C₂²
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#135	D₁₃
hypocenter	C₁₃	#21293
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #13123480	C₂	semidirect
3. #13115590	C₂	semidirect
4. #13115549	C₂	semidirect
5. C₄×#88059	C₂	semidirect
6. C₄×#88077	C₂	semidirect
7. C₄×#88055	C₂	semidirect
11. C₄×C₁₃×Q₁₆	C₄	semidirect
12. C₄×C₁₃×SD₁₆	C₄	semidirect
13. C₄×C₁₃×D₈	C₄	semidirect
136. C₄×Q₁₆	Dic₁₃	semidirect
137. C₄×SD₁₆	Dic₁₃	semidirect
138. C₄×D₈	Dic₁₃	semidirect
242. C₁₃	#21293	semidirect

Maximal subgroups

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

subgroup	index
#13123480	2
#13115730	2
#13115590	2
#13115549	2
C₄×#88059	2
C₄×#88077	2
C₄×#88055	2
C₄×C₁₃×#135	2
#21293	13