Subgroups — #16016139

Order 3328.

682 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718562	C₂	1	2	yes	maximal, complemented
2	#13123549	C₂	1	1	yes	maximal, characteristic
3	#13116667	C₂	1	2	yes	maximal, complemented
4	#13116660	C₂	1	4	yes	maximal, complemented
5	#13116655	C₂	1	2	yes	maximal, complemented
6	#13116449	C₂	1	2	yes	maximal, complemented
7	C₄×#88052	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#88155	C₂²	1	2	yes	—
10	#11719182	C₄	1	1	yes	characteristic
11	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	#11718771	C₄	1	2	yes	—
14	C₂×#88052	C₂²	1	1	yes	characteristic
15	C₂×#88052	C₂²	1	1	yes	characteristic
16	#11718585	C₄	1	1	yes	characteristic
17	#11718562	—	2	8	no	—
18	C₂×#88038	C₂²	1	4	yes	—
19	C₂×C₁₆×D₁₃	C₂²	1	4	yes	—
20	#11718349	C₄	1	2	yes	—
21	#11718349	C₄	1	2	yes	—
22	C₈×D₅₂	C₂²	1	2	yes	—
23	C₈×D₅₂	C₂²	1	2	yes	—
24	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
26	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
27	#11718099	C₂²	1	4	yes	—
28	#11718099	C₂²	1	4	yes	—
29	#11718053	C₂²	1	2	yes	—
30	C₄×#6090	C₂²	1	2	yes	—
31	C₂×#6127	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
35	#88184	C₈	1	2	yes	—
36	#88184	C₈	1	2	yes	—
37	C₂×#6099	C₂×C₄	1	2	yes	—
38	#88155	—	2	4	no	—
39	#88155	—	2	4	no	—
40	C₂×#6094	C₂×C₄	1	4	yes	—
41	C₂×C₈×D₁₃	C₂³	1	4	yes	—
42	#88129	C₈	1	2	yes	—
43	#88128	C₈	1	1	yes	characteristic
44	C₄×D₅₂	C₂×C₄	1	2	yes	—
45	C₄×D₅₂	C₂×C₄	1	2	yes	—
46	#88123	C₈	1	1	yes	characteristic
47	C₂×C₁₃×C₁₆	—	2	4	no	—
48	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
49	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
50	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
54	#88060	C₂×C₄	1	4	yes	—
55	#88060	C₂×C₄	1	4	yes	—
56	#88052	—	2	2	no	—
57	#88052	D₄	1	4	yes	complemented
58	C₂×#6090	C₂³	1	2	yes	—
59	C₂×#6090	C₂³	1	2	yes	—
60	C₄⋊#1055	C₂×C₄	1	2	yes	—
61	C₄⋊#1055	C₂×C₄	1	2	yes	—
62	#88038	—	2	8	no	—
63	C₁₆×D₁₃	—	2	8	no	—
64	#33438	—	13	1	no	maximal, Sylow
65	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₂²⋊D₁₃	C₂×C₈	1	4	yes	—
69	#6127	—	2	4	no	—
70	#6127	—	2	4	no	—
71	C₂×D₅₂	C₂×C₈	1	2	yes	—
72	C₂×D₅₂	C₂×C₈	1	2	yes	—
73	C₂×D₅₂	C₂×C₈	1	2	yes	—
74	C₂×C₄×D₁₃	C₂²×C₄	1	4	yes	—
75	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₁₃	#41	1	2	yes	—
82	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
83	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
84	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
85	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	#6103	C₂×C₈	1	4	yes	—
88	#6103	C₂×C₈	1	4	yes	—
89	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
90	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
91	#6099	—	2	4	no	—
92	C₂×#1055	C₂²×C₄	1	4	yes	—
93	#6094	—	2	8	no	—
94	C₈×D₁₃	—	2	4	no	—
95	C₈×D₁₃	—	2	4	no	—
96	C₁₃×C₁₆	—	2	2	no	—
97	C₁₃×C₁₆	—	2	2	no	—
98	#6090	—	2	2	no	—
99	#6090	C₂×D₄	1	4	yes	—
100	C₂×#504	—	13	2	no	—
101	#2931	—	13	1	no	—
102	#2169	—	13	2	no	—
103	#2168	—	13	4	no	—
104	D₄×C₁₆	—	13	2	no	—
105	C₂×C₄⋊C₁₆	D₁₃	1	1	yes	characteristic, complemented
106	#2113	—	13	2	no	—
107	C₄×M₃₂	—	13	1	no	—
108	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
109	C₂²×D₁₃	C₂²×C₈	1	4	yes	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	4	1	no	—
112	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#131	1	2	yes	—
119	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
120	C₂²⋊D₁₃	—	2	8	no	—
121	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
122	D₅₂	—	2	4	no	—
123	D₅₂	—	2	4	no	—
124	D₅₂	—	4	2	no	—
125	C₄×D₁₃	—	2	4	no	—
126	C₄×D₁₃	—	2	4	no	—
127	Dic₂₆	—	2	4	no	—
128	C₈×C₁₃	—	2	1	no	—
129	C₈×C₁₃	—	2	1	no	—
130	C₈×C₁₃	C₄×D₄	1	2	yes	—
131	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
132	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
133	#1055	—	2	4	no	—
134	C₂×#131	—	13	2	no	—
135	#532	—	13	1	no	—
136	#504	—	26	8	no	—
137	C₂×M₃₂	—	13	1	no	—
138	C₂×M₃₂	—	13	1	no	—
139	C₂×M₃₂	—	13	4	no	—
140	C₂²×C₁₆	—	13	4	no	—
141	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
142	#436	—	13	2	no	—
143	#436	—	13	2	no	—
144	C₈×D₄	—	13	2	no	—
145	C₈×D₄	—	13	2	no	—
146	#423	—	13	1	no	—
147	#408	—	13	2	no	—
148	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
149	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
150	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
151	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
152	C₂²⋊C₁₆	—	13	4	no	—
153	C₂²⋊C₁₆	—	13	4	no	—
154	#346	—	13	2	no	—
155	C₄×C₁₆	—	13	2	no	—
156	C₂²×C₁₃	—	2	1	no	—
157	C₂²×C₁₃	—	2	1	no	—
158	C₂²×C₁₃	C₈×D₄	1	2	yes	—
159	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
160	C₂×D₁₃	—	2	4	no	—
161	C₂×D₁₃	—	2	4	no	—
162	C₂×D₁₃	—	2	4	no	—
163	C₄×C₁₃	—	2	1	no	—
164	C₄×C₁₃	—	2	1	no	—
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	C₈×D₄	1	2	yes	—
167	C₄×C₁₃	#504	1	2	yes	complemented
168	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
169	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
170	Dic₁₃	—	2	4	no	—
171	C₂×#41	—	13	2	no	—
172	#131	—	26	4	no	—
173	#131	—	26	4	no	—
174	C₂×M₁₆	—	13	2	no	—
175	C₂×M₁₆	—	13	4	no	—
176	C₂²×C₈	—	13	4	no	—
177	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
178	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
179	C₄⋊Q₈	—	13	1	no	—
180	#127	—	13	1	no	—
181	#124	—	13	2	no	—
182	#121	—	13	2	no	—
183	#121	—	13	2	no	—
184	C₄×D₄	—	13	2	no	—
185	C₄×D₄	—	13	2	no	—
186	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
187	M₃₂	—	13	4	no	—
188	M₃₂	—	26	8	no	—
189	M₃₂	—	26	2	no	—
190	C₂×C₁₆	—	2	4	no	—
191	C₂×C₁₆	—	13	2	no	—
192	C₂×C₁₆	—	13	2	no	—
193	C₂×C₁₆	—	26	8	no	—
194	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
195	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
196	C₄⋊C₈	—	13	2	no	—
197	C₄⋊C₈	—	13	2	no	—
198	C₂²⋊C₈	—	13	4	no	—
199	C₂²⋊C₈	—	13	4	no	—
200	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
201	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
202	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
203	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
204	C₂×C₁₃	—	2	2	no	—
205	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
206	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
207	C₂×C₁₃	#3407	1	1	yes	characteristic
208	D₁₃	—	4	4	no	—
209	#41	—	26	4	no	—
210	#41	—	26	4	no	—
211	C₂×Q₈	—	13	2	no	—
212	C₂×D₄	—	13	4	no	—
213	C₂×D₄	—	13	2	no	—
214	C₂×D₄	—	13	2	no	—
215	C₂×D₄	—	13	2	no	—
216	C₂²×C₄	—	13	4	no	—
217	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
218	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
220	M₁₆	—	26	4	no	—
221	M₁₆	—	26	8	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	13	4	no	—
224	C₂×C₈	—	26	4	no	—
225	C₂×C₈	—	26	4	no	—
226	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
229	C₂×C₈	D₄×D₁₃	1	2	yes	—
230	C₂×C₈	#6131	1	2	yes	—
231	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
232	C₄⋊C₄	—	13	2	no	—
233	C₄⋊C₄	—	13	2	no	—
234	C₂²⋊C₄	—	13	4	no	—
235	C₂²⋊C₄	—	13	4	no	—
236	C₄²	C₈×D₁₃	1	1	yes	characteristic
237	C₄²	C₈×D₁₃	1	1	yes	characteristic
238	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
239	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
240	C₁₆	—	2	2	no	—
241	C₁₆	—	2	2	no	—
242	C₁₆	—	13	4	no	—
243	C₁₆	—	26	2	no	—
244	C₁₃	#33438	1	1	yes	characteristic, complemented, Sylow
245	C₂³	—	13	4	no	—
246	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
247	Q₈	—	26	4	no	—
248	D₄	—	26	8	no	—
249	D₄	—	26	4	no	—
250	D₄	—	26	4	no	—
251	D₄	—	52	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	13	4	no	—
255	C₂×C₄	—	26	4	no	—
256	C₂×C₄	—	26	4	no	—
257	C₂×C₄	#88147	1	2	yes	—
258	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
259	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
260	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
261	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
262	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
263	C₂×C₄	#88161	1	2	yes	—
264	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	26	4	no	—
268	C₈	#88147	1	2	yes	—
269	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
270	C₈	C₂×#6131	1	1	yes	characteristic
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	26	4	no	—
274	C₂²	—	26	4	no	—
275	C₂²	—	26	4	no	—
276	C₂²	#11718495	1	2	yes	—
277	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	26	4	no	—
282	C₄	#11718495	1	2	yes	—
283	C₄	#11718562	1	2	yes	complemented
284	C₄	C₂×#88147	1	1	yes	characteristic
285	C₄	C₂×#88161	1	1	yes	characteristic
286	C₂	—	2	2	no	—
287	C₂	—	52	4	no	—
288	C₂	C₂×#11718495	1	1	yes	characteristic
289	C₂	C₂×#11718562	1	1	yes	characteristic
290	C₂	#13127938	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Fitting	C₂×C₁₃×C₄⋊C₁₆	C₂
prime radical	#11719182	C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₁₃
hypocenter	C₁₃	#33438
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#11718562	C₂	semidirect
3. #13116667	C₂	semidirect
4. #13116660	C₂	semidirect
5. #13116655	C₂	semidirect
6. #13116449	C₂	semidirect
7. C₄×#88052	C₂	semidirect
8. C₂×C₁₃×C₄⋊C₁₆	C₂	semidirect
57. #88052	D₄	semidirect
105. C₂×C₄⋊C₁₆	D₁₃	semidirect
167. C₄×C₁₃	#504	semidirect
244. C₁₃	#33438	semidirect
283. C₄	#11718562	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718562	2
#13123549	2
#13116667	2
#13116660	2
#13116655	2
#13116449	2
C₄×#88052	2
C₂×C₁₃×C₄⋊C₁₆	2
#33438	13