Subgroups — #30613581

Order 15616.

626 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462722	C₂	1	2	yes	maximal
2	#21461459	C₂	1	2	yes	maximal, complemented
3	#21461444	C₂	1	4	yes	maximal, complemented
4	Q₁₆×#92420	C₂	1	2	yes	maximal, complemented
5	#21461381	C₂	1	1	yes	maximal, characteristic
6	#21461161	C₂	1	2	yes	maximal
7	C₈×#11758217	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic
9	#16840813	—	2	4	no	—
10	#16840813	C₂²	1	4	yes	—
11	Q₈×#92420	C₂²	1	4	yes	—
12	#16840783	—	2	2	no	—
13	#16840783	C₂²	1	2	yes	—
14	#16840779	—	2	2	no	—
15	#16840779	C₂²	1	1	yes	characteristic
16	C₄×#11758217	C₂²	1	1	yes	characteristic
17	#16840625	C₂²	1	2	yes	—
18	#16840624	C₂²	1	2	yes	—
19	Q₈⋊#92420	C₂²	1	4	yes	—
20	Q₈⋊#92420	C₂²	1	4	yes	—
21	#92420⋊C₈	C₂²	1	2	yes	—
22	C₈×#92420	C₂²	1	2	yes	—
23	C₂×Q₈×Dic₆₁	C₂²	1	2	yes	—
24	Q₁₆×Dic₆₁	C₄	1	2	yes	—
25	Q₁₆×Dic₆₁	C₄	1	1	yes	characteristic
26	Q₁₆×Dic₆₁	C₄	1	1	yes	characteristic
27	Q₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
28	Q₁₆×Dic₆₁	C₂²	1	2	yes	—
29	Q₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
30	C₂×#14519711	C₄	1	1	yes	characteristic
31	C₂×#14519701	C₄	1	2	yes	—
32	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
33	C₂²×Q₁₆×C₆₁	C₄	1	1	yes	characteristic
34	C₂×#11758217	—	2	1	no	—
35	#14519844	—	2	4	no	—
36	#14519844	D₄	1	2	yes	—
37	C₄⋊#92420	—	2	4	no	—
38	C₄⋊#92420	C₂³	1	4	yes	—
39	C₂²⋊#92420	—	2	2	no	—
40	C₂²⋊#92420	D₄	1	1	yes	characteristic
41	C₂²⋊#92420	D₄	1	1	yes	characteristic
42	#14519832	—	2	4	no	—
43	#14519832	—	2	2	no	—
44	#14519832	—	2	1	no	—
45	C₄×#92420	—	2	4	no	—
46	C₄×#92420	C₂³	1	2	yes	—
47	C₄×#92420	C₂³	1	2	yes	—
48	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
49	Q₈×Dic₆₁	—	2	2	no	—
50	Q₈×Dic₆₁	—	2	2	no	—
51	Q₈×Dic₆₁	C₂×C₄	1	2	yes	—
52	Q₈×Dic₆₁	C₂×C₄	1	2	yes	—
53	Q₈×Dic₆₁	C₂³	1	2	yes	—
54	Q₈×Dic₆₁	C₂³	1	2	yes	—
55	C₂×C₄⋊Dic₆₁	—	2	2	no	—
56	C₂×C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
57	C₂×C₄×Dic₆₁	—	2	2	no	—
58	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
59	#14519711	C₂×C₄	1	2	yes	—
60	#14519711	C₂×C₄	1	1	yes	characteristic
61	#14519711	C₂×C₄	1	1	yes	characteristic
62	#14519701	C₂×C₄	1	2	yes	—
63	#14519701	C₂×C₄	1	2	yes	—
64	#14519701	C₂×C₄	1	2	yes	—
65	#14519701	C₂×C₄	1	2	yes	—
66	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
67	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
69	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
70	C₂²×Q₈×C₆₁	C₂×C₄	1	2	yes	—
71	C₂×Q₁₆×C₆₁	—	2	4	no	—
72	C₂×Q₁₆×C₆₁	C₂×C₄	1	2	yes	—
73	C₂×Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
74	C₂×Q₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
75	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
76	C₂×Q₈×C₆₁	—	2	2	no	—
77	C₂×Q₈×C₆₁	—	2	4	no	—
78	C₂×Q₈×C₆₁	—	4	2	no	—
79	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
80	C₂×Q₈×C₆₁	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₆₁	—	2	2	no	—
82	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
83	C₂²×Dic₆₁	—	2	2	no	—
84	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
85	C₂²×Dic₆₁	#41	1	1	yes	characteristic
86	#11758217	—	2	4	no	—
87	C₂×#92420	—	2	2	no	—
88	C₂×#92420	—	2	4	no	—
89	C₂×#92420	C₂×D₄	1	2	yes	—
90	C₂×#92420	C₂×D₄	1	2	yes	—
91	Q₁₆×C₆₁	—	2	4	no	—
92	Q₁₆×C₆₁	—	2	2	no	—
93	Q₁₆×C₆₁	M₁₆	1	4	yes	complemented
94	C₂×C₈×C₆₁	—	2	2	no	—
95	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
97	C₄⋊Dic₆₁	—	2	2	no	—
98	C₄⋊Dic₆₁	—	2	2	no	—
99	C₄⋊Dic₆₁	—	2	2	no	—
100	C₄⋊Dic₆₁	—	2	2	no	—
101	C₄⋊Dic₆₁	—	2	2	no	—
102	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
103	C₄⋊Dic₆₁	C₂²×C₄	1	2	yes	—
104	C₄×Dic₆₁	—	2	2	no	—
105	C₄×Dic₆₁	—	2	2	no	—
106	C₄×Dic₆₁	—	2	2	no	—
107	C₄×Dic₆₁	—	2	2	no	—
108	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
109	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
110	C₄×Dic₆₁	#41	1	1	yes	characteristic
111	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
112	C₂×#92418	—	4	1	no	—
113	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
114	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
115	C₂³×C₆₁	C₄×D₄	1	1	yes	characteristic
116	Q₈×C₆₁	—	2	2	no	—
117	Q₈×C₆₁	—	2	2	no	—
118	Q₈×C₆₁	—	2	2	no	—
119	Q₈×C₆₁	—	4	4	no	—
120	Q₈×C₆₁	C₂×M₁₆	1	4	yes	—
121	C₂×C₄×C₆₁	—	2	2	no	—
122	C₂×C₄×C₆₁	—	2	1	no	—
123	C₂×C₄×C₆₁	—	2	1	no	—
124	C₂×C₄×C₆₁	—	2	2	no	—
125	C₂×C₄×C₆₁	—	4	4	no	—
126	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
128	C₂×Dic₆₁	—	2	2	no	—
129	C₂×Dic₆₁	—	2	2	no	—
130	C₂×Dic₆₁	—	2	2	no	—
131	C₂×Dic₆₁	—	2	2	no	—
132	C₂×Dic₆₁	—	4	4	no	—
133	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
134	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
135	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₆₁	C₂×#41	1	1	yes	characteristic
137	#92420	—	2	4	no	—
138	#92420	—	2	2	no	—
139	#92420	—	2	2	no	—
140	#92420	—	4	4	no	—
141	C₈×C₆₁	—	2	1	no	—
142	C₈×C₆₁	C₂×M₁₆	1	2	yes	—
143	#92418	—	2	1	no	—
144	#92418	—	2	1	no	—
145	#18058	—	61	1	no	maximal, Sylow
146	C₂²×C₆₁	—	2	1	no	—
147	C₂²×C₆₁	—	2	1	no	—
148	C₂²×C₆₁	#443	1	2	yes	—
149	C₂²×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
150	C₄×C₆₁	—	2	1	no	—
151	C₄×C₆₁	—	2	4	no	—
152	C₄×C₆₁	—	4	2	no	—
153	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
154	C₄×C₆₁	#568	1	1	yes	characteristic
155	Dic₆₁	—	2	1	no	—
156	Dic₆₁	—	2	1	no	—
157	Dic₆₁	—	2	1	no	—
158	Dic₆₁	—	4	2	no	—
159	Dic₆₁	—	4	2	no	—
160	Dic₆₁	#578	1	2	yes	—
161	#2984	—	61	2	no	—
162	C₂×C₄×Q₁₆	—	61	1	no	—
163	#1591	—	61	2	no	—
164	#1587	—	61	4	no	—
165	C₈×Q₁₆	—	61	2	no	—
166	#1568	—	61	1	no	—
167	#1491	—	61	2	no	—
168	C₈×M₁₆	—	61	1	no	—
169	C₂×C₆₁	—	2	2	no	—
170	C₂×C₆₁	D₄×M₁₆	1	1	yes	characteristic
171	C₂×C₆₁	#2950	1	1	yes	characteristic
172	C₂×C₆₁	C₂×#443	1	1	yes	characteristic
173	C₂²×Q₁₆	#6923	1	1	yes	characteristic
174	C₂×C₄×Q₈	—	61	2	no	—
175	#446	—	61	4	no	—
176	#446	—	122	4	no	—
177	C₈×Q₈	—	61	4	no	—
178	C₄×Q₁₆	—	61	1	no	—
179	C₄×Q₁₆	—	61	2	no	—
180	C₄×Q₁₆	—	61	1	no	—
181	C₄×Q₁₆	—	61	2	no	—
182	C₄×Q₁₆	—	61	1	no	—
183	C₄×Q₁₆	—	61	1	no	—
184	#432	—	61	2	no	—
185	#432	—	122	2	no	—
186	#431	—	61	1	no	—
187	#431	—	122	2	no	—
188	C₂×#107	—	61	1	no	—
189	C₂×#103	—	61	2	no	—
190	C₄×M₁₆	—	61	1	no	—
191	C₂×C₄×C₈	—	61	1	no	—
192	#335	—	61	2	no	—
193	#334	—	61	2	no	—
194	Q₈⋊C₈	—	61	4	no	—
195	Q₈⋊C₈	—	61	4	no	—
196	#322	—	61	2	no	—
197	C₈²	—	61	2	no	—
198	C₆₁	#18058	1	1	yes	characteristic, complemented, Sylow
199	C₂²×Q₈	C₂×#6923	1	2	yes	—
200	C₂×Q₁₆	—	2	4	no	—
201	C₂×Q₁₆	C₂×#6923	1	2	yes	—
202	C₂×Q₁₆	C₂×#6923	1	1	yes	characteristic
203	C₂×Q₁₆	C₂×#6923	1	1	yes	characteristic
204	C₂×M₁₆	—	122	1	no	—
205	C₂²×C₈	—	61	1	no	—
206	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
207	C₄×Q₈	—	61	2	no	—
208	C₄×Q₈	—	61	2	no	—
209	C₄×Q₈	—	61	2	no	—
210	C₄×Q₈	—	61	2	no	—
211	C₄×Q₈	—	122	2	no	—
212	C₄×Q₈	—	122	2	no	—
213	C₂×C₄⋊C₄	—	61	2	no	—
214	C₂×C₄⋊C₄	—	122	2	no	—
215	C₂×C₄²	—	61	1	no	—
216	C₂×C₄²	—	122	2	no	—
217	#107	—	61	2	no	—
218	#107	—	61	1	no	—
219	#107	—	61	1	no	—
220	C₄⋊C₈	—	61	2	no	—
221	C₄⋊C₈	—	61	4	no	—
222	C₄⋊C₈	—	122	4	no	—
223	C₄⋊C₈	—	122	4	no	—
224	#103	—	61	2	no	—
225	#103	—	61	2	no	—
226	#103	—	61	2	no	—
227	#103	—	61	2	no	—
228	C₂²⋊C₈	—	61	1	no	—
229	C₂²⋊C₈	—	61	1	no	—
230	C₂²⋊C₈	—	122	2	no	—
231	#97	—	122	1	no	—
232	#97	—	122	2	no	—
233	#97	—	122	4	no	—
234	C₄×C₈	—	61	2	no	—
235	C₄×C₈	—	61	1	no	—
236	C₄×C₈	—	61	1	no	—
237	C₄×C₈	—	61	1	no	—
238	C₄×C₈	—	61	1	no	—
239	C₄×C₈	—	61	2	no	—
240	C₄×C₈	—	122	4	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	4	no	—
243	C₂×Q₈	—	4	2	no	—
244	C₂×Q₈	C₂²×#6923	1	2	yes	—
245	C₂×Q₈	C₂²×#6923	1	2	yes	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	61	1	no	—
248	C₂²×C₄	—	61	1	no	—
249	C₂²×C₄	—	122	2	no	—
250	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
251	Q₁₆	—	2	4	no	—
252	Q₁₆	—	2	2	no	—
253	Q₁₆	#11758217	1	4	yes	complemented
254	M₁₆	—	122	4	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	61	2	no	—
257	C₂×C₈	—	61	2	no	—
258	C₂×C₈	—	61	1	no	—
259	C₂×C₈	—	61	1	no	—
260	C₂×C₈	—	122	2	no	—
261	C₂×C₈	—	122	4	no	—
262	C₂×C₈	—	244	1	no	—
263	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
264	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
265	C₄⋊C₄	—	61	2	no	—
266	C₄⋊C₄	—	61	2	no	—
267	C₄⋊C₄	—	122	2	no	—
268	C₄⋊C₄	—	122	2	no	—
269	C₄⋊C₄	—	122	2	no	—
270	C₄⋊C₄	—	122	2	no	—
271	C₄⋊C₄	—	122	2	no	—
272	C₄²	—	61	1	no	—
273	C₄²	—	61	1	no	—
274	C₄²	—	61	1	no	—
275	C₄²	—	61	1	no	—
276	C₄²	—	122	2	no	—
277	C₄²	—	122	2	no	—
278	C₄²	—	122	2	no	—
279	C₄²	—	122	2	no	—
280	C₂³	D₄×#6923	1	1	yes	characteristic
281	Q₈	—	2	2	no	—
282	Q₈	—	2	2	no	—
283	Q₈	—	2	2	no	—
284	Q₈	—	4	4	no	—
285	Q₈	C₂×#11758217	1	4	yes	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	61	1	no	—
292	C₂×C₄	—	61	1	no	—
293	C₂×C₄	—	61	1	no	—
294	C₂×C₄	—	61	1	no	—
295	C₂×C₄	—	122	2	no	—
296	C₂×C₄	—	122	2	no	—
297	C₂×C₄	—	122	2	no	—
298	C₂×C₄	—	122	2	no	—
299	C₂×C₄	—	244	4	no	—
300	C₂×C₄	D₄×#6923	1	1	yes	characteristic
301	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	122	2	no	—
304	C₈	—	122	2	no	—
305	C₈	—	122	4	no	—
306	C₈	—	122	1	no	—
307	C₈	—	122	1	no	—
308	C₈	—	244	4	no	—
309	C₈	C₂×#11758217	1	2	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	#16840806	1	2	yes	—
313	C₂²	C₂×D₄×#6923	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	4	no	—
316	C₄	—	4	2	no	—
317	C₄	—	61	2	no	—
318	C₄	—	122	1	no	—
319	C₄	—	122	1	no	—
320	C₄	—	122	1	no	—
321	C₄	—	244	2	no	—
322	C₄	—	244	2	no	—
323	C₄	C₂²×#11758217	1	1	yes	characteristic
324	C₄	#16840840	1	1	yes	characteristic
325	C₂	—	2	2	no	—
326	C₂	D₄×#11758217	1	1	yes	characteristic
327	C₂	#21462580	1	1	yes	characteristic
328	C₂	C₂×#16840806	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#6923
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₂²×Q₁₆×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₄×D₄
prime radical	C₂³×C₆₁	C₄×D₄
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	C₂²×Q₁₆	#6923
hypocenter	C₆₁	#18058
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #21461459	C₂	semidirect
3. #21461444	C₂	semidirect
4. Q₁₆×#92420	C₂	semidirect
93. Q₁₆×C₆₁	M₁₆	semidirect
198. C₆₁	#18058	semidirect
253. Q₁₆	#11758217	semidirect

Maximal subgroups

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

subgroup	index
#21462722	2
#21461459	2
#21461444	2
Q₁₆×#92420	2
#21461381	2
#21461161	2
C₈×#11758217	2
C₂×Q₁₆×Dic₆₁	2
#18058	61