Subgroups — #15781167

Order 3328.

510 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125264	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718110	C₂	1	4	yes	maximal, complemented
3	#13114993	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#88158	—	2	1	no	—
6	C₂×#88158	C₂²	1	2	yes	—
7	#11719182	—	2	1	no	—
8	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88057	—	2	1	no	—
11	C₂×#88057	C₂²	1	2	yes	—
12	#11718596	—	2	1	no	—
13	#11718360	C₄	1	1	yes	characteristic, complemented
14	#11718357	—	2	1	no	—
15	#11718357	C₄	1	2	yes	complemented
16	#11718356	C₄	1	1	yes	characteristic, complemented
17	#11718354	—	2	1	no	—
18	C₁₃×#346	—	2	1	no	—
19	C₄×C₁₃×C₁₆	—	2	1	no	—
20	#11718110	—	2	8	no	—
21	C₂×#6127	—	2	2	no	—
22	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
25	#88184	—	4	2	no	—
26	C₂×C₄⋊Dic₁₃	—	2	2	no	—
27	C₂×C₄⋊Dic₁₃	—	4	2	no	—
28	C₂×Dic₅₂	—	2	1	no	—
29	C₂×Dic₅₂	C₂×C₄	1	2	yes	—
30	#88158	—	2	4	no	—
31	#88158	—	2	4	no	—
32	#88158	—	4	2	no	—
33	#88158	—	4	2	no	—
34	C₂×D₁₀₄	—	2	1	no	—
35	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
36	C₂×#6095	—	2	2	no	—
37	C₂×#6095	—	2	2	no	—
38	#88129	—	2	2	no	—
39	#88128	—	2	1	no	—
40	C₄×D₅₂	—	4	2	no	—
41	#88123	—	2	1	no	—
42	C₁₃×M₃₂	—	2	2	no	—
43	C₁₃×M₃₂	D₄	1	4	yes	complemented
44	C₂×C₁₃×C₁₆	—	2	2	no	—
45	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
49	#88057	—	2	4	no	—
50	#88057	—	4	2	no	—
51	#13210	—	13	1	no	maximal, Sylow
52	C₂²×C₄×C₁₃	D₈	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₁₃	—	4	2	no	—
55	C₂²×Dic₁₃	—	4	2	no	—
56	#6127	—	4	2	no	—
57	#6127	—	4	2	no	—
58	#6127	—	4	2	no	—
59	#6127	—	4	2	no	—
60	C₂×D₅₂	—	2	2	no	—
61	C₂×D₅₂	—	4	2	no	—
62	C₂×C₄×D₁₃	—	4	2	no	—
63	C₂×Dic₂₆	—	2	2	no	—
64	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃	D₈	1	2	yes	—
67	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
68	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	#41	1	2	yes	—
72	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₁₃	SD₁₆	1	2	yes	—
75	#6103	—	4	4	no	—
76	C₄⋊Dic₁₃	—	4	2	no	—
77	C₄⋊Dic₁₃	—	4	2	no	—
78	C₄⋊Dic₁₃	—	4	2	no	—
79	C₄⋊Dic₁₃	—	8	4	no	—
80	Dic₅₂	—	2	4	no	—
81	Dic₅₂	—	4	2	no	—
82	D₁₀₄	—	2	4	no	—
83	D₁₀₄	—	4	2	no	—
84	#6095	—	4	4	no	—
85	#6095	—	4	4	no	—
86	C₁₃×C₁₆	—	2	4	no	—
87	#3147	—	13	1	no	—
88	C₂×#360	—	13	4	no	—
89	C₄×M₃₂	D₁₃	1	1	yes	characteristic, complemented
90	#1844	—	13	1	no	—
91	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
92	C₂²×D₁₃	—	4	2	no	—
93	C₂×C₄×C₁₃	—	2	4	no	—
94	C₂×C₄×C₁₃	#102	1	2	yes	—
95	C₂×C₄×C₁₃	#102	1	2	yes	—
96	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
102	C₂²⋊D₁₃	—	8	2	no	—
103	C₂²⋊D₁₃	—	8	2	no	—
104	C₂×Dic₁₃	—	4	2	no	—
105	C₂×Dic₁₃	—	8	4	no	—
106	C₂×Dic₁₃	—	8	2	no	—
107	D₅₂	—	4	2	no	—
108	D₅₂	—	4	2	no	—
109	D₅₂	—	8	4	no	—
110	C₄×D₁₃	—	8	2	no	—
111	C₄×D₁₃	—	8	2	no	—
112	Dic₂₆	—	4	2	no	—
113	Dic₂₆	—	4	2	no	—
114	C₈×C₁₃	#102	1	2	yes	—
115	C₈×C₁₃	#102	1	2	yes	—
116	C₈×C₁₃	C₄×D₄	1	2	yes	—
117	C₈×C₁₃	#121	1	2	yes	—
118	C₂×#135	—	13	2	no	—
119	C₂×#135	—	26	1	no	—
120	#532	—	26	1	no	—
121	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
122	#495	—	13	2	no	—
123	#495	—	26	1	no	—
124	#494	—	13	1	no	—
125	#493	—	13	1	no	—
126	#492	—	26	1	no	—
127	C₂×#106	—	13	2	no	—
128	C₂×#106	—	26	1	no	—
129	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
130	#384	—	26	1	no	—
131	#360	—	26	8	no	—
132	#346	—	2	1	no	—
133	C₄×C₁₆	—	2	1	no	—
134	C₂²×C₁₃	—	2	2	no	—
135	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
136	C₂²×C₁₃	#486	1	1	yes	characteristic
137	C₂²×C₁₃	#493	1	1	yes	characteristic
138	C₂×D₁₃	—	8	4	no	—
139	C₂×D₁₃	—	8	2	no	—
140	C₄×C₁₃	—	2	2	no	—
141	C₄×C₁₃	#360	1	4	yes	complemented
142	C₄×C₁₃	#390	1	1	yes	characteristic
143	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
144	C₄×C₁₃	#486	1	1	yes	characteristic
145	C₄×C₁₃	#492	1	1	yes	characteristic
146	Dic₁₃	—	8	2	no	—
147	Dic₁₃	—	16	2	no	—
148	C₂×#41	—	26	2	no	—
149	#135	—	26	4	no	—
150	#135	—	26	4	no	—
151	#135	—	52	2	no	—
152	#135	—	52	2	no	—
153	C₂×Q₁₆	—	13	2	no	—
154	C₂×Q₁₆	—	26	1	no	—
155	C₂×SD₁₆	—	26	2	no	—
156	C₂×SD₁₆	—	26	2	no	—
157	C₂×D₈	—	13	2	no	—
158	C₂×D₈	—	26	1	no	—
159	C₂²×C₈	D₅₂	1	1	yes	characteristic
160	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
161	C₄⋊Q₈	—	26	1	no	—
162	#127	—	26	1	no	—
163	#124	—	26	2	no	—
164	#121	—	52	2	no	—
165	C₄×D₄	—	52	2	no	—
166	C₂×C₄⋊C₄	—	26	2	no	—
167	C₂×C₄⋊C₄	—	52	2	no	—
168	C₂×C₄²	D₅₂	1	1	yes	characteristic
169	M₃₂	—	2	2	no	—
170	M₃₂	D₅₂	1	4	yes	complemented
171	C₂×C₁₆	—	2	2	no	—
172	#106	—	26	4	no	—
173	#106	—	52	2	no	—
174	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
175	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
176	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
177	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
178	C₂×C₁₃	—	2	1	no	—
179	C₂×C₁₃	—	2	1	no	—
180	C₂×C₁₃	#1957	1	1	yes	characteristic
181	C₂×C₁₃	C₂×#360	1	2	yes	—
182	D₁₃	—	16	2	no	—
183	#41	—	52	2	no	—
184	#41	—	52	2	no	—
185	#41	—	52	2	no	—
186	#41	—	52	2	no	—
187	C₂×Q₈	—	26	2	no	—
188	C₂×D₄	—	26	2	no	—
189	C₂×D₄	—	52	2	no	—
190	C₂×D₄	—	52	2	no	—
191	C₂²×C₄	—	52	2	no	—
192	C₂²×C₄	—	52	2	no	—
193	C₂²×C₄	D₁₀₄	1	2	yes	—
194	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
195	Q₁₆	—	26	4	no	—
196	Q₁₆	—	52	2	no	—
197	SD₁₆	—	52	4	no	—
198	SD₁₆	—	52	4	no	—
199	D₈	—	26	4	no	—
200	D₈	—	52	2	no	—
201	C₂×C₈	#6095	1	2	yes	—
202	C₂×C₈	D₁₀₄	1	2	yes	—
203	C₂×C₈	#6103	1	1	yes	characteristic
204	C₂×C₈	#6103	1	1	yes	characteristic
205	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
206	C₂×C₈	C₂×D₅₂	1	2	yes	—
207	C₂×C₈	#6127	1	2	yes	—
208	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
209	C₄⋊C₄	—	52	2	no	—
210	C₄⋊C₄	—	52	2	no	—
211	C₄⋊C₄	—	52	2	no	—
212	C₄⋊C₄	—	104	4	no	—
213	C₂²⋊C₄	—	52	4	no	—
214	C₄²	#6095	1	2	yes	—
215	C₄²	#6103	1	1	yes	characteristic
216	C₄²	#6103	1	1	yes	characteristic
217	C₁₆	—	2	4	no	—
218	C₁₃	#13210	1	1	yes	characteristic, complemented, Sylow
219	C₂³	—	52	2	no	—
220	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
221	Q₈	—	52	2	no	—
222	Q₈	—	52	2	no	—
223	D₄	—	52	2	no	—
224	D₄	—	52	2	no	—
225	D₄	—	104	2	no	—
226	D₄	—	104	2	no	—
227	D₄	—	104	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	52	2	no	—
230	C₂×C₄	—	104	2	no	—
231	C₂×C₄	—	104	2	no	—
232	C₂×C₄	—	104	4	no	—
233	C₂×C₄	—	104	2	no	—
234	C₂×C₄	#88061	1	2	yes	—
235	C₂×C₄	#88061	1	2	yes	—
236	C₂×C₄	#88128	1	1	yes	characteristic
237	C₂×C₄	#88129	1	1	yes	characteristic
238	C₂×C₄	C₂×#6095	1	1	yes	characteristic
239	C₂×C₄	C₂×#6095	1	1	yes	characteristic
240	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
241	C₂×C₄	C₂×#6103	1	1	yes	characteristic
242	C₈	#88061	1	2	yes	—
243	C₈	#88061	1	2	yes	—
244	C₈	C₄×D₅₂	1	2	yes	—
245	C₈	#88184	1	2	yes	—
246	C₂²	—	2	2	no	—
247	C₂²	—	104	4	no	—
248	C₂²	—	104	2	no	—
249	C₂²	#11718355	1	1	yes	characteristic
250	C₂²	#11718356	1	1	yes	characteristic
251	C₂²	C₂×#88061	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	104	2	no	—
254	C₄	—	208	2	no	—
255	C₄	#11718110	1	4	yes	complemented
256	C₄	#11718354	1	1	yes	characteristic
257	C₄	#11718355	1	1	yes	characteristic
258	C₄	#11718600	1	1	yes	characteristic
259	C₄	C₂×#88061	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	208	2	no	—
263	C₂	#13115642	1	1	yes	characteristic
264	C₂	C₂×#11718110	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#88061
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	#13125264	C₂
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#13210
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×#11718110	C₂	semidirect
3. #13114993	C₂	semidirect
4. C₄×C₁₃×M₃₂	C₂	semidirect
13. #11718360	C₄	semidirect
15. #11718357	C₄	semidirect
16. #11718356	C₄	semidirect
43. C₁₃×M₃₂	D₄	semidirect
89. C₄×M₃₂	D₁₃	semidirect
141. C₄×C₁₃	#360	semidirect
170. M₃₂	D₅₂	semidirect
218. C₁₃	#13210	semidirect
255. C₄	#11718110	semidirect

Maximal subgroups

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

subgroup	index
#13125264	2
C₂×#11718110	2
#13114993	2
C₄×C₁₃×M₃₂	2
#13210	13