Subgroups — #16018356

Order 3328.

814 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718570	C₂	1	4	yes	maximal, complemented
2	#13125264	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718360	C₂	1	1	yes	maximal, characteristic, complemented
4	#13117107	C₂	1	4	yes	maximal, complemented
5	#13117092	C₂	1	2	yes	maximal, complemented
6	#13117089	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₅₂	—	2	1	no	—
9	C₂²×Dic₅₂	C₂²	1	2	yes	—
10	C₂×#88158	—	2	1	no	—
11	C₂×#88158	C₂²	1	2	yes	—
12	#11719182	—	2	1	no	—
13	C₂×#88123	—	2	1	no	—
14	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	#11718570	—	2	16	no	—
17	C₂×Dic₁₀₄	C₂²	1	8	yes	—
18	C₂×#88040	C₂²	1	8	yes	—
19	#11718360	—	2	1	no	—
20	#11718360	—	2	1	no	—
21	#11718360	C₂²	1	2	yes	—
22	#11718360	C₂²	1	2	yes	—
23	#11718360	C₂²	1	1	yes	characteristic
24	#11718357	—	2	1	no	—
25	#11718357	C₂²	1	2	yes	—
26	#11718356	C₂²	1	1	yes	characteristic
27	#11718354	—	2	1	no	—
28	C₁₃×#346	C₂²	1	2	yes	—
29	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
30	C₂×#6127	—	2	2	no	—
31	C₂²×Dic₂₆	—	2	2	no	—
32	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
35	#88184	—	4	2	no	—
36	C₂×C₄⋊Dic₁₃	—	4	2	no	—
37	C₂×Dic₅₂	—	2	8	no	—
38	C₂×Dic₅₂	—	2	1	no	—
39	C₂×Dic₅₂	—	2	2	no	—
40	C₂×Dic₅₂	—	2	2	no	—
41	C₂×Dic₅₂	—	2	2	no	—
42	C₂×Dic₅₂	—	4	4	no	—
43	C₂×Dic₅₂	C₂³	1	2	yes	—
44	C₂×Dic₅₂	C₂³	1	4	yes	—
45	#88158	—	2	8	no	—
46	#88158	—	4	4	no	—
47	C₂×D₁₀₄	—	2	1	no	—
48	C₂×D₁₀₄	C₂³	1	2	yes	—
49	C₂×#6095	—	2	2	no	—
50	C₂×#6095	—	2	2	no	—
51	#88129	—	2	2	no	—
52	#88128	—	2	1	no	—
53	C₄×D₅₂	—	4	2	no	—
54	#88123	—	2	2	no	—
55	#88123	—	2	2	no	—
56	#88123	—	2	1	no	—
57	#88123	—	4	2	no	—
58	C₁₃×M₃₂	D₄	1	8	yes	complemented
59	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
60	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	Dic₁₀₄	—	2	16	no	—
65	#88040	—	2	16	no	—
66	#33718	—	13	1	no	maximal, Sylow
67	C₂²×C₄×C₁₃	D₈	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₁₃	—	4	2	no	—
70	C₂²×Dic₁₃	—	4	2	no	—
71	#6127	—	4	4	no	—
72	#6127	—	4	4	no	—
73	C₂×D₅₂	—	2	2	no	—
74	C₂×D₅₂	—	4	2	no	—
75	C₂×C₄×D₁₃	—	4	2	no	—
76	C₂×Dic₂₆	—	2	4	no	—
77	C₂×Dic₂₆	—	2	2	no	—
78	C₂×Dic₂₆	—	4	4	no	—
79	C₂×Dic₂₆	—	4	4	no	—
80	C₂×Dic₂₆	—	4	2	no	—
81	C₂×C₈×C₁₃	D₈	1	4	yes	—
82	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
84	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₄²×C₁₃	D₈	1	2	yes	—
88	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
90	#6103	—	4	4	no	—
91	C₄⋊Dic₁₃	—	4	4	no	—
92	C₄⋊Dic₁₃	—	4	4	no	—
93	C₄⋊Dic₁₃	—	4	2	no	—
94	Dic₅₂	—	2	4	no	—
95	Dic₅₂	—	2	8	no	—
96	Dic₅₂	—	4	4	no	—
97	Dic₅₂	—	4	2	no	—
98	Dic₅₂	—	4	4	no	—
99	Dic₅₂	—	4	4	no	—
100	D₁₀₄	—	2	4	no	—
101	D₁₀₄	—	4	2	no	—
102	#6095	—	4	4	no	—
103	#6095	—	4	4	no	—
104	C₁₃×C₁₆	C₂×D₄	1	8	yes	—
105	C₂×#510	—	13	4	no	—
106	#3147	—	13	1	no	—
107	C₂×#494	—	13	1	no	—
108	#2251	—	13	4	no	—
109	#2248	—	13	2	no	—
110	#2247	—	13	2	no	—
111	C₄×M₃₂	D₁₃	1	1	yes	characteristic, complemented
112	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
113	C₂²×D₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	#127	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
121	C₂²⋊D₁₃	—	8	4	no	—
122	C₂×Dic₁₃	—	4	4	no	—
123	C₂×Dic₁₃	—	4	2	no	—
124	C₂×Dic₁₃	—	8	4	no	—
125	D₅₂	—	4	2	no	—
126	D₅₂	—	4	2	no	—
127	D₅₂	—	8	4	no	—
128	C₄×D₁₃	—	8	4	no	—
129	Dic₂₆	—	4	2	no	—
130	Dic₂₆	—	4	4	no	—
131	Dic₂₆	—	4	2	no	—
132	Dic₂₆	—	4	4	no	—
133	Dic₂₆	—	8	4	no	—
134	C₈×C₁₃	#127	1	2	yes	—
135	C₈×C₁₃	C₂×D₈	1	2	yes	—
136	C₈×C₁₃	C₂×D₈	1	2	yes	—
137	C₈×C₁₃	C₂²×D₄	1	2	yes	—
138	C₂×#135	—	13	2	no	—
139	C₂×#135	—	26	1	no	—
140	C₂²×Q₁₆	—	13	2	no	—
141	C₂²×Q₁₆	—	26	1	no	—
142	#532	—	26	1	no	—
143	C₂×C₄⋊Q₈	—	26	1	no	—
144	#510	—	26	16	no	—
145	C₂×Q₃₂	—	13	8	no	—
146	C₂×SD₃₂	—	13	8	no	—
147	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
148	#495	—	13	2	no	—
149	#495	—	26	1	no	—
150	#494	—	13	2	no	—
151	#494	—	13	2	no	—
152	#494	—	13	1	no	—
153	#494	—	26	1	no	—
154	#494	—	26	1	no	—
155	#493	—	13	1	no	—
156	#492	—	26	1	no	—
157	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
158	#346	C₂×D₁₃	1	2	yes	—
159	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
160	C₂²×C₁₃	—	2	2	no	—
161	C₂²×C₁₃	#493	1	2	yes	—
162	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
163	C₂×D₁₃	—	8	2	no	—
164	C₂×D₁₃	—	8	4	no	—
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	#493	1	2	yes	—
167	C₄×C₁₃	#510	1	4	yes	complemented
168	C₄×C₁₃	C₂×#127	1	1	yes	characteristic
169	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
170	Dic₁₃	—	8	4	no	—
171	Dic₁₃	—	8	2	no	—
172	C₂×#41	—	26	2	no	—
173	C₂²×Q₈	—	26	2	no	—
174	#135	—	26	8	no	—
175	#135	—	52	4	no	—
176	C₂×Q₁₆	—	13	4	no	—
177	C₂×Q₁₆	—	13	2	no	—
178	C₂×Q₁₆	—	26	8	no	—
179	C₂×Q₁₆	—	26	2	no	—
180	C₂×Q₁₆	—	26	2	no	—
181	C₂×Q₁₆	—	26	2	no	—
182	C₂×Q₁₆	—	26	1	no	—
183	C₂×Q₁₆	—	52	4	no	—
184	C₂×SD₁₆	—	26	2	no	—
185	C₂×SD₁₆	—	26	2	no	—
186	C₂×D₈	—	13	2	no	—
187	C₂×D₈	—	26	1	no	—
188	C₂²×C₈	D₅₂	1	1	yes	characteristic
189	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
190	C₄⋊Q₈	—	26	2	no	—
191	C₄⋊Q₈	—	26	2	no	—
192	C₄⋊Q₈	—	26	1	no	—
193	C₄⋊Q₈	—	52	2	no	—
194	#127	—	26	1	no	—
195	#124	—	26	2	no	—
196	#121	—	52	2	no	—
197	C₄×D₄	—	52	2	no	—
198	C₂×C₄⋊C₄	—	52	2	no	—
199	C₂×C₄²	D₅₂	1	1	yes	characteristic
200	Q₃₂	—	26	16	no	—
201	SD₃₂	—	26	16	no	—
202	M₃₂	D₅₂	1	8	yes	complemented
203	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
204	C₄×C₈	D₅₂	1	1	yes	characteristic
205	C₄×C₈	D₅₂	1	1	yes	characteristic
206	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
208	C₂×C₁₃	—	2	2	no	—
209	C₂×C₁₃	C₂×#493	1	1	yes	characteristic
210	C₂×C₁₃	C₂×#510	1	2	yes	—
211	D₁₃	—	16	2	no	—
212	#41	—	52	4	no	—
213	#41	—	52	4	no	—
214	C₂×Q₈	—	26	2	no	—
215	C₂×Q₈	—	26	4	no	—
216	C₂×Q₈	—	52	2	no	—
217	C₂×Q₈	—	52	4	no	—
218	C₂×Q₈	—	52	4	no	—
219	C₂×D₄	—	26	2	no	—
220	C₂×D₄	—	52	2	no	—
221	C₂×D₄	—	52	2	no	—
222	C₂²×C₄	—	52	2	no	—
223	C₂²×C₄	—	52	2	no	—
224	C₂²×C₄	D₁₀₄	1	2	yes	—
225	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
226	Q₁₆	—	26	8	no	—
227	Q₁₆	—	26	4	no	—
228	Q₁₆	—	52	4	no	—
229	Q₁₆	—	52	4	no	—
230	Q₁₆	—	52	4	no	—
231	Q₁₆	—	52	2	no	—
232	SD₁₆	—	52	4	no	—
233	SD₁₆	—	52	4	no	—
234	D₈	—	26	4	no	—
235	D₈	—	52	2	no	—
236	C₂×C₈	D₁₀₄	1	4	yes	—
237	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
238	C₂×C₈	C₂×D₅₂	1	4	yes	—
239	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
240	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
241	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
242	C₄⋊C₄	—	52	4	no	—
243	C₄⋊C₄	—	52	4	no	—
244	C₄⋊C₄	—	52	2	no	—
245	C₂²⋊C₄	—	52	4	no	—
246	C₄²	D₁₀₄	1	2	yes	—
247	C₄²	C₂×D₅₂	1	1	yes	characteristic
248	C₄²	C₂×D₅₂	1	1	yes	characteristic
249	C₁₆	C₂×D₅₂	1	8	yes	—
250	C₁₃	#33718	1	1	yes	characteristic, complemented, Sylow
251	C₂³	—	52	2	no	—
252	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
253	Q₈	—	52	2	no	—
254	Q₈	—	52	4	no	—
255	Q₈	—	52	2	no	—
256	Q₈	—	52	4	no	—
257	Q₈	—	104	4	no	—
258	D₄	—	52	2	no	—
259	D₄	—	52	2	no	—
260	D₄	—	104	4	no	—
261	D₄	—	104	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	52	2	no	—
264	C₂×C₄	—	52	4	no	—
265	C₂×C₄	—	104	4	no	—
266	C₂×C₄	—	104	4	no	—
267	C₂×C₄	#88128	1	2	yes	—
268	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
269	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
270	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
271	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
272	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
273	C₈	#88128	1	2	yes	—
274	C₈	C₂×D₁₀₄	1	2	yes	—
275	C₈	C₂×D₁₀₄	1	2	yes	—
276	C₈	C₂²×D₅₂	1	2	yes	—
277	C₂²	—	2	2	no	—
278	C₂²	—	104	4	no	—
279	C₂²	—	104	2	no	—
280	C₂²	#11718356	1	2	yes	—
281	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	104	2	no	—
284	C₄	—	104	4	no	—
285	C₄	#11718356	1	2	yes	—
286	C₄	#11718570	1	4	yes	complemented
287	C₄	C₂×#88128	1	1	yes	characteristic
288	C₄	C₂²×D₁₀₄	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	—	208	2	no	—
291	C₂	C₂×#11718356	1	1	yes	characteristic
292	C₂	C₂×#11718570	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₄×C₁₃×M₃₂	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	#13125264	C₂
hypercenter	C₄×M₃₂	D₁₃
hypocenter	C₁₃	#33718
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
1. C₂×#11718570	C₂	semidirect
3. C₂×#11718360	C₂	semidirect
4. #13117107	C₂	semidirect
5. #13117092	C₂	semidirect
6. #13117089	C₂	semidirect
7. C₄×C₁₃×M₃₂	C₂	semidirect
58. C₁₃×M₃₂	D₄	semidirect
111. C₄×M₃₂	D₁₃	semidirect
167. C₄×C₁₃	#510	semidirect
202. M₃₂	D₅₂	semidirect
250. C₁₃	#33718	semidirect
286. C₄	#11718570	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718570	2
#13125264	2
C₂×#11718360	2
#13117107	2
#13117092	2
#13117089	2
C₄×C₁₃×M₃₂	2
#33718	13