Subgroups — #15867317

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	C₄×#88158	C₂	1	1	yes	maximal, characteristic
2	#13123478	C₂	1	2	yes	maximal, complemented
3	#13115729	C₂	1	2	yes	maximal
4	#13115589	C₂	1	2	yes	maximal, complemented
5	#13115547	C₂	1	4	yes	maximal, complemented
6	C₄×#88059	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#88045	C₂	1	2	yes	maximal, complemented
8	C₄×C₁₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#88158	C₂²	1	1	yes	characteristic
10	C₄×#6127	C₂²	1	2	yes	—
11	C₂×C₁₃×#97	C₂²	1	2	yes	—
12	C₂×C₁₃×#97	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₄³×C₁₃	C₂²	1	1	yes	characteristic
16	#11718774	C₂²	1	2	yes	—
17	#11718771	C₂²	1	4	yes	—
18	C₂×#88059	C₂²	1	2	yes	—
19	#11718767	C₂²	1	1	yes	characteristic
20	#11718763	C₂²	1	4	yes	—
21	C₂×#88045	C₂²	1	4	yes	—
22	C₄×#6099	C₂²	1	2	yes	—
23	C₄×Dic₅₂	C₄	1	2	yes	complemented
24	C₄×D₁₀₄	C₄	1	2	yes	complemented
25	C₄×#6095	C₄	1	4	yes	complemented
26	#11718351	—	2	4	no	—
27	#11718346	—	2	4	no	—
28	#11718144	C₂²	1	2	yes	—
29	#11718115	C₂²	1	4	yes	—
30	C₂×#6127	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	—	2	2	no	—
35	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₄×C₂²⋊D₁₃	—	2	4	no	—
38	#88180	C₂³	1	2	yes	—
39	C₂×#6099	C₂³	1	4	yes	—
40	C₂×Dic₅₂	C₂×C₄	1	1	yes	characteristic
41	#88158	—	2	4	no	—
42	C₂×D₁₀₄	C₂×C₄	1	1	yes	characteristic
43	C₂×#6095	C₂×C₄	1	2	yes	—
44	C₄×D₅₂	C₂×C₄	1	4	yes	—
45	#88126	—	2	2	no	—
46	C₄²×D₁₃	—	2	2	no	—
47	C₄×Dic₂₆	C₂×C₄	1	4	yes	—
48	C₁₃×#97	—	2	4	no	—
49	C₁₃×#97	—	2	4	no	—
50	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₁₃	D₄	1	2	yes	—
53	C₄×C₈×C₁₃	D₄	1	2	yes	—
54	#88061	C₂×C₄	1	4	yes	—
55	#88060	—	2	8	no	—
56	#88059	—	2	4	no	—
57	#88058	C₂×C₄	1	2	yes	—
58	#88057	C₂×C₄	1	2	yes	—
59	#88056	C₂×C₄	1	4	yes	—
60	#88054	—	2	8	no	—
61	#88045	—	2	8	no	—
62	#88043	—	2	2	no	—
63	C₄×#1055	—	2	2	no	—
64	#21293	—	13	1	no	maximal, Sylow
65	C₂²×C₄×C₁₃	—	2	2	no	—
66	C₂²×C₄×C₁₃	#41	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₂×C₂²⋊D₁₃	—	2	2	no	—
69	#6127	—	2	4	no	—
70	#6127	—	2	4	no	—
71	C₂×D₅₂	C₂²×C₄	1	2	yes	—
72	C₂×C₄×D₁₃	—	2	4	no	—
73	C₂×C₄×D₁₃	—	2	2	no	—
74	C₂×Dic₂₆	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
82	C₂×C₈×C₁₃	#41	1	4	yes	—
83	C₄²×C₁₃	—	2	2	no	—
84	C₄²×C₁₃	—	4	4	no	—
85	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
86	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₃	#41	1	4	yes	—
89	C₂²⋊Dic₁₃	—	2	2	no	—
90	#6103	—	2	4	no	—
91	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
92	#6101	—	2	4	no	—
93	C₄×Dic₁₃	—	2	4	no	—
94	C₄×Dic₁₃	—	2	2	no	—
95	#6099	—	2	4	no	—
96	#6099	—	2	4	no	—
97	C₂×#1055	—	2	4	no	—
98	Dic₅₂	—	2	2	no	—
99	D₁₀₄	—	2	2	no	—
100	#6095	—	2	4	no	—
101	C₄×#135	—	13	1	no	—
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	D₁₃	1	1	yes	characteristic, complemented
109	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
110	C₂²×D₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	—	4	4	no	—
114	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
120	C₂²⋊D₁₃	—	4	4	no	—
121	C₂×Dic₁₃	—	2	2	no	—
122	C₂×Dic₁₃	—	2	4	no	—
123	D₅₂	—	2	2	no	—
124	D₅₂	—	2	2	no	—
125	C₄×D₁₃	—	4	4	no	—
126	C₄×D₁₃	—	4	8	no	—
127	Dic₂₆	—	2	2	no	—
128	Dic₂₆	—	2	2	no	—
129	C₈×C₁₃	—	2	2	no	—
130	C₈×C₁₃	—	2	4	no	—
131	C₈×C₁₃	#117	1	2	yes	—
132	C₈×C₁₃	#117	1	2	yes	—
133	#1055	—	4	4	no	—
134	C₂×#135	—	13	1	no	—
135	C₄×#41	—	13	2	no	—
136	C₄×Q₁₆	—	13	2	no	—
137	C₄×SD₁₆	—	13	4	no	—
138	C₄×D₈	—	13	2	no	—
139	#433	—	26	4	no	—
140	#431	—	26	4	no	—
141	C₂×#108	—	13	2	no	—
142	#427	—	13	1	no	—
143	#424	—	13	4	no	—
144	C₂×#104	—	13	4	no	—
145	#416	—	13	2	no	—
146	#408	—	13	4	no	—
147	C₄×M₁₆	—	13	2	no	—
148	C₂×#97	C₂×D₁₃	1	2	yes	—
149	C₂×#97	C₂×D₁₃	1	2	yes	—
150	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
151	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
152	C₄³	C₂×D₁₃	1	1	yes	characteristic
153	#341	—	13	2	no	—
154	#339	—	13	4	no	—
155	C₂²×C₁₃	—	2	2	no	—
156	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
157	C₂²×C₁₃	#529	1	2	yes	—
158	C₂×D₁₃	—	4	4	no	—
159	C₂×D₁₃	—	4	2	no	—
160	C₄×C₁₃	—	2	2	no	—
161	C₄×C₁₃	—	4	4	no	—
162	C₄×C₁₃	#444	1	4	yes	—
163	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
164	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
165	C₄×C₁₃	#532	1	2	yes	—
166	Dic₁₃	—	4	4	no	—
167	Dic₁₃	—	4	2	no	—
168	C₂×#41	—	13	2	no	—
169	#135	—	26	4	no	—
170	C₂×Q₁₆	—	13	1	no	—
171	C₂×SD₁₆	—	13	2	no	—
172	C₂×D₈	—	13	1	no	—
173	C₂×M₁₆	—	13	4	no	—
174	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
175	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
176	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
177	C₄×Q₈	—	13	4	no	—
178	C₄×D₄	—	13	4	no	—
179	C₄×D₄	—	26	4	no	—
180	#117	—	13	2	no	—
181	#117	—	26	2	no	—
182	C₂×C₄²	—	2	2	no	—
183	C₂×C₄²	—	26	2	no	—
184	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
185	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
186	#108	—	26	4	no	—
187	#107	—	13	2	no	—
188	#106	—	13	2	no	—
189	C₄⋊C₈	—	26	8	no	—
190	#104	—	26	8	no	—
191	#103	—	13	4	no	—
192	#102	—	13	4	no	—
193	C₂²⋊C₈	—	26	8	no	—
194	#97	—	2	4	no	—
195	#97	—	2	4	no	—
196	#97	—	26	2	no	—
197	C₄×C₈	—	26	2	no	—
198	C₄×C₈	C₄×D₁₃	1	2	yes	—
199	C₄×C₈	C₄×D₁₃	1	2	yes	—
200	C₄×C₈	D₅₂	1	2	yes	—
201	C₄×C₈	D₅₂	1	2	yes	—
202	C₂×C₁₃	—	2	2	no	—
203	C₂×C₁₃	#2293	1	1	yes	characteristic
204	C₂×C₁₃	C₂×#444	1	2	yes	—
205	D₁₃	—	8	2	no	—
206	#41	—	26	4	no	—
207	#41	—	26	4	no	—
208	C₂×Q₈	—	13	2	no	—
209	C₂×D₄	—	13	2	no	—
210	C₂×D₄	—	26	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	26	4	no	—
213	C₂²×C₄	—	26	2	no	—
214	C₂²×C₄	#6127	1	2	yes	—
215	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
216	Q₁₆	—	26	2	no	—
217	SD₁₆	—	26	4	no	—
218	D₈	—	26	2	no	—
219	M₁₆	—	26	4	no	—
220	M₁₆	—	26	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	26	4	no	—
225	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
226	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
228	C₂×C₈	C₂×D₅₂	1	4	yes	—
229	C₂×C₈	#6127	1	4	yes	—
230	C₄⋊C₄	—	13	4	no	—
231	C₄⋊C₄	—	26	4	no	—
232	C₂²⋊C₄	—	26	2	no	—
233	C₂²⋊C₄	—	26	4	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	4	4	no	—
236	C₄²	—	26	4	no	—
237	C₄²	—	26	2	no	—
238	C₄²	C₂×C₄×D₁₃	1	2	yes	—
239	C₄²	C₂×D₅₂	1	1	yes	characteristic
240	C₄²	C₂×D₅₂	1	1	yes	characteristic
241	C₄²	#6127	1	4	yes	—
242	C₁₃	#21293	1	1	yes	characteristic, complemented, Sylow
243	C₂³	—	26	2	no	—
244	C₂³	C₂×#6127	1	1	yes	characteristic
245	Q₈	—	26	2	no	—
246	Q₈	—	26	2	no	—
247	D₄	—	26	2	no	—
248	D₄	—	26	2	no	—
249	D₄	—	52	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	26	4	no	—
254	C₂×C₄	—	26	2	no	—
255	C₂×C₄	—	52	4	no	—
256	C₂×C₄	—	52	8	no	—
257	C₂×C₄	C₄×D₅₂	1	2	yes	—
258	C₂×C₄	C₄×D₅₂	1	2	yes	—
259	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
261	C₂×C₄	C₂×#6127	1	2	yes	—
262	C₂×C₄	C₂×#6127	1	2	yes	—
263	C₈	—	2	2	no	—
264	C₈	—	2	4	no	—
265	C₈	—	52	4	no	—
266	C₈	#88126	1	2	yes	—
267	C₈	#88126	1	2	yes	—
268	C₂²	—	2	2	no	—
269	C₂²	—	52	2	no	—
270	C₂²	—	52	4	no	—
271	C₂²	C₂×C₄×D₅₂	1	1	yes	characteristic
272	C₂²	#11719184	1	2	yes	—
273	C₄	—	2	2	no	—
274	C₄	—	4	4	no	—
275	C₄	—	52	2	no	—
276	C₄	—	52	4	no	—
277	C₄	#11718378	1	4	yes	—
278	C₄	C₂×#88126	1	1	yes	characteristic
279	C₄	C₄×#6127	1	1	yes	characteristic
280	C₄	#11719182	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	104	2	no	—
283	C₂	#13117300	1	1	yes	characteristic
284	C₂	C₂×#11718378	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×#97	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂×#88158	C₂²
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#97	D₁₃
hypocenter	C₁₃	#21293
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
2. #13123478	C₂	semidirect
4. #13115589	C₂	semidirect
5. #13115547	C₂	semidirect
6. C₄×#88059	C₂	semidirect
7. C₄×#88045	C₂	semidirect
8. C₄×C₁₃×#97	C₂	semidirect
23. C₄×Dic₅₂	C₄	semidirect
24. C₄×D₁₀₄	C₄	semidirect
25. C₄×#6095	C₄	semidirect
108. C₄×#97	D₁₃	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
C₄×#88158	2
#13123478	2
#13115729	2
#13115589	2
#13115547	2
C₄×#88059	2
C₄×#88045	2
C₄×C₁₃×#97	2
#21293	13