Subgroups — #16806737

Order 3328.

834 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719096	C₂	1	2	yes	maximal
2	#13131046	C₂	1	4	yes	maximal, complemented
3	#13131045	C₂	1	2	yes	maximal, complemented
4	#13131043	C₂	1	2	yes	maximal, complemented
5	#13131042	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718257	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×D₁₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
8	#13123258	C₂	1	1	yes	maximal, characteristic
9	C₂×#88161	C₂²	1	1	yes	characteristic
10	C₂×#88155	C₄	1	1	yes	characteristic
11	#11719227	C₄	1	1	yes	characteristic
12	C₂×#88106	C₂²	1	2	yes	—
13	C₂²×#6092	C₂²	1	2	yes	—
14	#11719096	C₄	1	16	yes	complemented
15	C₂×#88098	C₂²	1	4	yes	—
16	C₂×D₁₃⋊C₁₆	C₂²	1	4	yes	—
17	C₁₃×#405	C₄	1	1	yes	characteristic
18	M₁₆×Dic₁₃	C₄	1	1	yes	characteristic
19	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
20	#11718658	C₂²	1	1	yes	characteristic
21	#11718369	C₂²	1	2	yes	—
22	#11718368	C₄	1	2	yes	—
23	#11718350	C₄	1	2	yes	—
24	C₄×C₈×D₁₃	C₂²	1	2	yes	—
25	#11718258	C₂²	1	2	yes	—
26	#11718257	C₂²	1	1	yes	characteristic
27	#11718257	C₂²	1	2	yes	—
28	#11718257	C₂²	1	1	yes	characteristic
29	#11718257	C₂²	1	2	yes	—
30	#11718231	C₂²	1	4	yes	—
31	C₄×#6092	C₂²	1	4	yes	—
32	C₂×#6127	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×#6099	C₂×C₄	1	1	yes	characteristic
38	C₂²×#1055	C₂³	1	1	yes	characteristic
39	#88161	C₂×C₄	1	8	yes	—
40	#88155	C₈	1	8	yes	—
41	C₂×#6094	C₂×C₄	1	2	yes	—
42	C₂×#6094	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
44	C₂×C₈×D₁₃	C₂³	1	2	yes	—
45	#88147	C₈	1	2	yes	—
46	#88141	C₈	1	2	yes	—
47	#88135	C₈	1	4	yes	—
48	#88126	C₂×C₄	1	2	yes	—
49	C₄²×D₁₃	C₂×C₄	1	2	yes	—
50	#88106	C₂×C₄	1	8	yes	complemented
51	C₂×#6092	C₂×C₄	1	8	yes	complemented
52	C₂×#6092	C₂³	1	2	yes	—
53	C₂×#6092	C₂³	1	4	yes	—
54	C₂×#6092	C₂³	1	2	yes	—
55	#88098	C₂×C₄	1	16	yes	complemented
56	D₁₃⋊C₁₆	C₂×C₄	1	16	yes	complemented
57	C₁₃×#97	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
59	#88055	C₂×C₄	1	2	yes	—
60	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
61	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
62	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
63	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
64	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
65	#88043	C₂×C₄	1	2	yes	—
66	C₄×#1055	C₂³	1	2	yes	—
67	C₄×#504	—	13	1	no	maximal, Sylow
68	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₁₃	C₂×C₈	1	2	yes	—
70	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
71	#6127	C₄²	1	4	yes	—
72	#6127	C₂×C₈	1	4	yes	—
73	C₂×D₅₂	C₂×C₈	1	1	yes	characteristic
74	C₂×C₄×D₁₃	C₂×C₈	1	4	yes	—
75	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
76	C₂×Dic₂₆	C₂×C₈	1	1	yes	characteristic
77	C₁₃×M₁₆	C₄²	1	4	yes	—
78	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
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	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
83	C₁₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
84	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
85	#6103	C₂×C₈	1	4	yes	—
86	#6101	C₂×C₈	1	4	yes	—
87	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
88	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
89	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
91	#6099	C₂×C₈	1	4	yes	—
92	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
93	C₂×#1055	C₂²×C₄	1	4	yes	—
94	C₂×#1055	C₂²×C₄	1	2	yes	—
95	C₂×#1055	C₂⁴	1	1	yes	characteristic
96	#6094	C₄²	1	8	yes	—
97	#6094	C₂×C₈	1	8	yes	—
98	C₈×D₁₃	C₂×C₈	1	8	yes	—
99	C₈×D₁₃	C₂²×C₄	1	8	yes	—
100	#6092	C₂²×C₄	1	4	yes	—
101	#6092	C₂²×C₄	1	8	yes	—
102	#6092	C₂²×C₄	1	4	yes	—
103	C₂×#504	—	13	2	no	—
104	C₄×#131	—	13	1	no	—
105	#2108	—	13	2	no	—
106	#2108	—	13	4	no	—
107	C₄×M₃₂	—	13	1	no	—
108	C₄×M₃₂	—	13	2	no	—
109	C₂×C₄×C₁₆	—	13	2	no	—
110	C₂×C₄×C₁₆	—	13	1	no	—
111	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
112	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
115	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₂²⋊D₁₃	C₄×C₈	1	8	yes	—
119	C₂×Dic₁₃	—	2	4	no	—
120	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
121	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
122	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
123	D₅₂	C₄×C₈	1	4	yes	—
124	C₄×D₁₃	—	2	8	no	—
125	C₄×D₁₃	C₂×C₄²	1	4	yes	—
126	C₄×D₁₃	C₂²×C₈	1	4	yes	—
127	Dic₂₆	C₄×C₈	1	4	yes	—
128	C₈×C₁₃	C₂×C₄²	1	4	yes	—
129	C₈×C₁₃	C₂²×C₈	1	2	yes	—
130	C₈×C₁₃	C₂²×C₈	1	2	yes	—
131	#1055	C₂×C₄²	1	2	yes	—
132	#1055	C₂²×C₈	1	4	yes	—
133	#1055	C₂³×C₄	1	2	yes	—
134	C₂×#131	—	13	1	no	—
135	C₂×#131	—	13	1	no	—
136	C₄×#41	—	13	1	no	—
137	#504	—	13	16	no	—
138	C₂×M₃₂	—	13	4	no	—
139	C₂×M₃₂	—	13	2	no	—
140	C₂²×C₁₆	—	13	4	no	—
141	C₂²×C₁₆	—	13	2	no	—
142	#405	—	13	1	no	—
143	#405	—	13	2	no	—
144	#405	—	13	2	no	—
145	#405	#261	1	1	yes	characteristic
146	C₄×M₁₆	—	13	1	no	—
147	C₄×M₁₆	—	13	2	no	—
148	C₂×C₄×C₈	—	13	1	no	—
149	C₂×C₄×C₈	—	13	2	no	—
150	#346	—	13	4	no	—
151	#346	—	13	2	no	—
152	C₄×C₁₆	—	13	1	no	—
153	C₄×C₁₆	—	13	2	no	—
154	C₄×C₁₆	—	13	1	no	—
155	C₄×C₁₆	—	13	4	no	—
156	C₄×C₁₆	—	13	2	no	—
157	C₂²×C₁₃	—	2	2	no	—
158	C₂²×C₁₃	C₂×C₄×C₈	1	2	yes	—
159	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
160	C₂×D₁₃	—	2	4	no	—
161	C₂×D₁₃	C₂×C₄×C₈	1	4	yes	—
162	C₄×C₁₃	—	2	4	no	—
163	C₄×C₁₃	C₂×C₄×C₈	1	2	yes	—
164	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
165	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
166	Dic₁₃	—	2	2	no	—
167	Dic₁₃	C₂×C₄×C₈	1	4	yes	—
168	Dic₁₃	#504	1	4	yes	complemented
169	C₂×#41	—	13	1	no	—
170	#131	—	13	8	no	—
171	#131	—	13	8	no	—
172	C₂×M₁₆	—	13	1	no	—
173	C₂×M₁₆	—	13	2	no	—
174	C₂×M₁₆	—	13	2	no	—
175	C₂×M₁₆	C₂×#261	1	1	yes	characteristic
176	C₂²×C₈	—	13	1	no	—
177	C₂²×C₈	—	13	2	no	—
178	C₂²×C₈	—	13	2	no	—
179	C₂²×C₈	C₂×#261	1	1	yes	characteristic
180	C₄×Q₈	—	13	2	no	—
181	C₄×D₄	—	13	2	no	—
182	C₄×D₄	—	13	4	no	—
183	#117	—	13	2	no	—
184	#117	C₂×#261	1	1	yes	characteristic
185	C₂×C₄²	—	13	1	no	—
186	C₂×C₄²	—	13	2	no	—
187	M₃₂	—	13	16	no	—
188	M₃₂	—	13	8	no	—
189	C₂×C₁₆	—	13	16	no	—
190	C₂×C₁₆	—	13	8	no	—
191	C₂×C₁₆	—	13	2	no	—
192	C₂×C₁₆	—	13	4	no	—
193	C₂×C₁₆	—	13	2	no	—
194	#97	—	13	2	no	—
195	#97	—	13	2	no	—
196	#97	C₂×#261	1	2	yes	—
197	C₄×C₈	—	13	1	no	—
198	C₄×C₈	—	13	1	no	—
199	C₄×C₈	—	13	2	no	—
200	C₄×C₈	—	13	1	no	—
201	C₄×C₈	—	13	1	no	—
202	C₄×C₈	—	13	2	no	—
203	C₄×C₈	C₂×#261	1	2	yes	—
204	C₂×C₁₃	—	2	2	no	—
205	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
206	C₂×C₁₃	C₂×#504	1	2	yes	—
207	D₁₃	—	2	4	no	—
208	#41	—	13	4	no	—
209	#41	—	13	4	no	—
210	C₂×Q₈	—	13	1	no	—
211	C₂×D₄	—	13	2	no	—
212	C₂×D₄	—	13	1	no	—
213	C₂²×C₄	—	13	2	no	—
214	C₂²×C₄	—	13	2	no	—
215	C₂²×C₄	—	13	4	no	—
216	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
217	M₁₆	—	13	4	no	—
218	M₁₆	—	13	8	no	—
219	M₁₆	—	13	8	no	—
220	M₁₆	C₄×#261	1	4	yes	—
221	C₂×C₈	—	13	4	no	—
222	C₂×C₈	—	13	1	no	—
223	C₂×C₈	—	13	2	no	—
224	C₂×C₈	—	13	1	no	—
225	C₂×C₈	—	13	8	no	—
226	C₂×C₈	—	13	8	no	—
227	C₂×C₈	D₁₃⋊C₈	1	4	yes	—
228	C₂×C₈	C₂²×#261	1	1	yes	characteristic
229	C₂×C₈	C₂²×#261	1	2	yes	—
230	C₂×C₈	C₂²×#261	1	1	yes	characteristic
231	C₄⋊C₄	—	13	4	no	—
232	C₄⋊C₄	D₁₃⋊C₈	1	2	yes	—
233	C₂²⋊C₄	—	13	4	no	—
234	C₂²⋊C₄	D₁₃⋊C₈	1	2	yes	—
235	C₄²	—	13	4	no	—
236	C₄²	—	13	2	no	—
237	C₄²	—	13	1	no	—
238	C₄²	—	13	1	no	—
239	C₄²	C₂²×#261	1	2	yes	—
240	C₁₆	—	13	4	no	—
241	C₁₆	—	13	8	no	—
242	C₁₆	—	13	4	no	—
243	C₁₃	C₄×#504	1	1	yes	characteristic, complemented, Sylow
244	C₂³	—	13	2	no	—
245	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
246	Q₈	—	13	4	no	—
247	D₄	—	13	8	no	—
248	D₄	—	13	4	no	—
249	C₂×C₄	—	13	2	no	—
250	C₂×C₄	—	13	2	no	—
251	C₂×C₄	—	13	4	no	—
252	C₂×C₄	—	13	4	no	—
253	C₂×C₄	—	13	2	no	—
254	C₂×C₄	—	26	8	no	—
255	C₂×C₄	—	26	4	no	—
256	C₂×C₄	C₂×D₁₃⋊C₈	1	4	yes	—
257	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
258	C₂×C₄	C₂×D₁₃⋊C₈	1	2	yes	—
259	C₂×C₄	C₂×C₄×#261	1	2	yes	—
260	C₂×C₄	C₂³×#261	1	1	yes	characteristic
261	C₈	—	13	4	no	—
262	C₈	—	13	2	no	—
263	C₈	—	13	2	no	—
264	C₈	C₂×D₁₃⋊C₈	1	2	yes	—
265	C₈	C₂×D₁₃⋊C₈	1	2	yes	—
266	C₈	C₂×C₄×#261	1	4	yes	—
267	C₂²	—	2	2	no	—
268	C₂²	—	13	4	no	—
269	C₂²	—	26	4	no	—
270	C₂²	C₄×D₁₃⋊C₈	1	2	yes	—
271	C₂²	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
272	C₄	—	2	4	no	—
273	C₄	—	13	4	no	—
274	C₄	—	13	4	no	—
275	C₄	—	26	2	no	—
276	C₄	C₄×D₁₃⋊C₈	1	2	yes	—
277	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
278	C₄	C₂²×C₄×#261	1	1	yes	characteristic
279	C₂	—	2	2	no	—
280	C₂	—	26	4	no	—
281	C₂	C₂×C₄×D₁₃⋊C₈	1	1	yes	characteristic
282	C₂	D₁₃⋊#504	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×#261
commutator	C₂×C₁₃	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₁₃×#405	C₄
socle	C₂²×C₁₃	C₂³×C₈
prime radical	C₂×#6127	C₂×C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	#405	#261
hypocenter	C₁₃	C₄×#504
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #13131046	C₂	semidirect
3. #13131045	C₂	semidirect
4. #13131043	C₂	semidirect
5. #13131042	C₂	semidirect
6. C₂×#11718257	C₂	semidirect
7. C₄×D₁₃⋊C₁₆	C₂	semidirect
14. #11719096	C₄	semidirect
50. #88106	C₂×C₄	semidirect
51. C₂×#6092	C₂×C₄	semidirect
55. #88098	C₂×C₄	semidirect
56. D₁₃⋊C₁₆	C₂×C₄	semidirect
168. Dic₁₃	#504	semidirect
243. C₁₃	C₄×#504	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11719096	2
#13131046	2
#13131045	2
#13131043	2
#13131042	2
C₂×#11718257	2
C₄×D₁₃⋊C₁₆	2
#13123258	2
C₄×#504	13