Subgroups — #29580139

Order 15616.

538 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445971	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#92418	C₂	1	1	yes	maximal, characteristic
3	#21443702	C₂	1	4	yes	maximal
4	C₆₁×#2868	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#92418	—	2	1	no	—
6	C₂×C₄⋊#92418	—	2	1	no	—
7	C₂×C₄⋊#92418	C₂²	1	2	yes	—
8	C₂×C₄⋊#92418	C₂²	1	2	yes	—
9	C₂×C₄×#92418	—	2	1	no	—
10	#16839517	—	2	1	no	—
11	C₂×C₄⋊Q₈×C₆₁	—	2	1	no	—
12	C₂×C₄⋊Q₈×C₆₁	—	2	1	no	—
13	C₂×C₄⋊Q₈×C₆₁	C₄	1	2	yes	—
14	C₆₁×#384	—	2	1	no	—
15	C₆₁×#384	—	2	1	no	—
16	C₆₁×#384	C₄	1	2	yes	—
17	C₆₁×#384	C₂²	1	2	yes	—
18	C₄³×C₆₁	C₂²	1	1	yes	characteristic
19	C₂²×#92418	—	2	2	no	—
20	C₂²×#92418	—	2	2	no	—
21	C₄⋊#92418	—	2	2	no	—
22	C₄⋊#92418	—	2	4	no	—
23	C₄⋊#92418	—	4	2	no	—
24	C₄⋊#92418	—	4	2	no	—
25	C₄⋊#92418	Q₈	1	4	yes	complemented
26	C₄×#92418	—	4	2	no	—
27	C₂²×Q₈×C₆₁	—	2	2	no	—
28	C₄⋊Q₈×C₆₁	—	4	4	no	—
29	C₄⋊Q₈×C₆₁	—	4	2	no	—
30	C₄⋊Q₈×C₆₁	—	4	2	no	—
31	C₄⋊Q₈×C₆₁	—	4	2	no	—
32	C₄⋊Q₈×C₆₁	—	4	2	no	—
33	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
46	C₂×Q₈×C₆₁	—	4	4	no	—
47	C₂×Q₈×C₆₁	—	4	4	no	—
48	C₂×Q₈×C₆₁	—	4	4	no	—
49	C₂²×C₄×C₆₁	—	2	4	no	—
50	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₄⋊C₄×C₆₁	—	4	4	no	—
58	C₄⋊C₄×C₆₁	—	4	4	no	—
59	C₄⋊C₄×C₆₁	—	4	4	no	—
60	C₄⋊C₄×C₆₁	—	4	4	no	—
61	C₄⋊C₄×C₆₁	—	4	2	no	—
62	C₄⋊C₄×C₆₁	—	4	2	no	—
63	C₄⋊C₄×C₆₁	—	4	2	no	—
64	C₄⋊C₄×C₆₁	—	4	2	no	—
65	C₄²×C₆₁	—	2	2	no	—
66	C₄²×C₆₁	—	2	2	no	—
67	C₄²×C₆₁	—	2	2	no	—
68	C₄²×C₆₁	D₈	1	2	yes	—
69	C₄²×C₆₁	D₈	1	2	yes	—
70	C₄²×C₆₁	SD₁₆	1	2	yes	—
71	C₄²×C₆₁	SD₁₆	1	2	yes	—
72	C₄²×C₆₁	SD₁₆	1	2	yes	—
73	C₄²×C₆₁	Q₁₆	1	2	yes	—
74	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
75	C₄²×C₆₁	#41	1	2	yes	—
76	C₂×#92418	—	2	4	no	—
77	C₂×#92418	—	4	2	no	—
78	C₂×#92418	—	4	2	no	—
79	C₂×#92418	—	4	2	no	—
80	C₂×#92418	—	4	2	no	—
81	C₂×#92418	—	4	2	no	—
82	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	Q₈×C₆₁	—	8	8	no	—
84	C₂×C₄×C₆₁	—	2	1	no	—
85	C₂×C₄×C₆₁	—	2	1	no	—
86	C₂×C₄×C₆₁	—	2	1	no	—
87	C₂×C₄×C₆₁	—	2	1	no	—
88	C₂×C₄×C₆₁	—	2	1	no	—
89	C₂×C₄×C₆₁	—	2	1	no	—
90	C₂×C₄×C₆₁	—	4	4	no	—
91	C₂×C₄×C₆₁	—	4	4	no	—
92	C₂×C₄×C₆₁	—	4	4	no	—
93	C₂×C₄×C₆₁	#100	1	2	yes	—
94	C₂×C₄×C₆₁	#101	1	2	yes	—
95	C₂×C₄×C₆₁	#102	1	2	yes	—
96	C₂×C₄×C₆₁	#102	1	2	yes	—
97	C₂×C₄×C₆₁	#102	1	2	yes	—
98	C₂×C₄×C₆₁	#102	1	2	yes	—
99	C₂×C₄×C₆₁	#103	1	2	yes	—
100	C₂×C₄×C₆₁	#103	1	2	yes	—
101	C₂×C₄×C₆₁	C₄×Q₈	1	2	yes	—
102	C₂×C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄×C₆₁	#124	1	1	yes	characteristic
104	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
111	C₂×C₄×C₆₁	#137	1	2	yes	—
112	#92418	—	4	4	no	—
113	#92418	—	8	2	no	—
114	#9898	—	61	1	no	maximal, Sylow
115	C₂²×C₆₁	#391	1	1	yes	characteristic
116	C₂²×C₆₁	C₂×#100	1	1	yes	characteristic
117	C₂²×C₆₁	C₂×#101	1	1	yes	characteristic
118	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
119	C₂²×C₆₁	C₂×#102	1	1	yes	characteristic
120	C₂²×C₆₁	C₂×#103	1	1	yes	characteristic
121	C₂²×C₆₁	#419	1	1	yes	characteristic
122	C₄×C₆₁	—	2	2	no	—
123	C₄×C₆₁	—	2	2	no	—
124	C₄×C₆₁	—	2	2	no	—
125	C₄×C₆₁	—	8	4	no	—
126	C₄×C₆₁	#331	1	4	yes	—
127	C₄×C₆₁	#332	1	4	yes	—
128	C₄×C₆₁	#474	1	2	yes	—
129	C₄×C₆₁	#475	1	2	yes	—
130	C₄×C₆₁	#486	1	1	yes	characteristic
131	C₄×C₆₁	#487	1	1	yes	characteristic
132	C₄×C₆₁	#498	1	1	yes	characteristic
133	C₄×C₆₁	#500	1	1	yes	characteristic
134	#2868	D₆₁	1	1	yes	characteristic
135	#1842	—	61	1	no	—
136	C₄×C₄⋊C₈	—	61	1	no	—
137	#1296	—	61	4	no	—
138	C₂×C₆₁	C₂×#331	1	1	yes	characteristic
139	C₂×C₆₁	C₂×#332	1	1	yes	characteristic
140	C₂×C₆₁	#1546	1	1	yes	characteristic
141	C₂×C₆₁	#1548	1	1	yes	characteristic
142	C₂×C₆₁	#1988	1	1	yes	characteristic
143	C₂×C₆₁	#1990	1	1	yes	characteristic
144	C₂×C₆₁	#1997	1	1	yes	characteristic
145	C₂×C₄⋊Q₈	—	2	1	no	—
146	C₂×C₄⋊Q₈	—	2	1	no	—
147	C₂×C₄⋊Q₈	Dic₆₁	1	2	yes	—
148	C₂×C₄⋊C₈	—	61	2	no	—
149	C₂×C₄⋊C₈	—	61	2	no	—
150	C₂×C₄⋊C₈	—	122	1	no	—
151	C₂×C₄⋊C₈	—	122	1	no	—
152	C₂×C₄×C₈	—	122	1	no	—
153	#384	—	2	1	no	—
154	#384	—	2	1	no	—
155	#384	Dic₆₁	1	2	yes	—
156	#384	C₂×D₆₁	1	2	yes	—
157	C₄³	C₂×D₆₁	1	1	yes	characteristic
158	#336	—	122	1	no	—
159	C₆₁	#9898	1	1	yes	characteristic, complemented, Sylow
160	C₂²×Q₈	—	2	2	no	—
161	C₂²×C₈	—	122	2	no	—
162	C₂²×C₈	—	122	2	no	—
163	C₄⋊Q₈	—	4	4	no	—
164	C₄⋊Q₈	—	4	2	no	—
165	C₄⋊Q₈	—	4	2	no	—
166	C₄⋊Q₈	—	4	2	no	—
167	C₄⋊Q₈	—	4	2	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	C₂×Dic₆₁	1	4	yes	—
174	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
175	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
176	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
177	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
178	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
179	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
180	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
181	C₄⋊C₈	—	61	4	no	—
182	C₄⋊C₈	—	122	2	no	—
183	C₄⋊C₈	—	122	4	no	—
184	C₄⋊C₈	—	244	2	no	—
185	C₄⋊C₈	—	244	2	no	—
186	C₄×C₈	—	244	2	no	—
187	C₂×Q₈	—	4	4	no	—
188	C₂×Q₈	—	4	4	no	—
189	C₂×Q₈	—	4	4	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
192	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
193	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
194	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
195	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
198	C₂×C₈	—	122	4	no	—
199	C₂×C₈	—	244	2	no	—
200	C₂×C₈	—	244	2	no	—
201	C₂×C₈	—	244	2	no	—
202	C₂×C₈	—	244	2	no	—
203	C₂×C₈	—	244	2	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄⋊C₄	—	4	4	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	#11758199	1	2	yes	—
216	C₄²	#11758199	1	2	yes	—
217	C₄²	#11758200	1	2	yes	—
218	C₄²	#11758200	1	2	yes	—
219	C₄²	#11758201	1	2	yes	—
220	C₄²	#11758202	1	2	yes	—
221	C₄²	Q₈×D₆₁	1	2	yes	—
222	C₄²	#11758226	1	2	yes	—
223	C₂³	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
224	Q₈	—	8	8	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	#14519694	1	2	yes	—
235	C₂×C₄	#14519696	1	2	yes	—
236	C₂×C₄	#14519698	1	2	yes	—
237	C₂×C₄	#14519698	1	2	yes	—
238	C₂×C₄	#14519698	1	2	yes	—
239	C₂×C₄	#14519698	1	2	yes	—
240	C₂×C₄	#14519701	1	2	yes	—
241	C₂×C₄	#14519701	1	2	yes	—
242	C₂×C₄	Q₈×Dic₆₁	1	2	yes	—
243	C₂×C₄	#14519753	1	2	yes	—
244	C₂×C₄	#14519766	1	1	yes	characteristic
245	C₂×C₄	#14519775	1	1	yes	characteristic
246	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
247	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
248	C₂×C₄	C₂×#11758201	1	1	yes	characteristic
249	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
250	C₂×C₄	C₂×#11758200	1	1	yes	characteristic
251	C₂×C₄	C₂×#11758202	1	1	yes	characteristic
252	C₂×C₄	#14519808	1	2	yes	—
253	C₈	—	488	2	no	—
254	C₈	—	244	4	no	—
255	C₂²	#16839669	1	1	yes	characteristic
256	C₂²	C₂×#14519694	1	1	yes	characteristic
257	C₂²	C₂×#14519696	1	1	yes	characteristic
258	C₂²	C₂×#14519698	1	1	yes	characteristic
259	C₂²	C₂×#14519698	1	1	yes	characteristic
260	C₂²	C₂×#14519701	1	1	yes	characteristic
261	C₂²	#16839771	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	8	4	no	—
266	C₄	#16839506	1	4	yes	—
267	C₄	#16839508	1	4	yes	—
268	C₄	#16840039	1	2	yes	—
269	C₄	#16840044	1	2	yes	—
270	C₄	#16840110	1	1	yes	characteristic
271	C₄	#16840114	1	1	yes	characteristic
272	C₄	#16840158	1	1	yes	characteristic
273	C₄	#16840165	1	1	yes	characteristic
274	C₂	C₂×#16839506	1	1	yes	characteristic
275	C₂	C₂×#16839508	1	1	yes	characteristic
276	C₂	#21444491	1	1	yes	characteristic
277	C₂	#21444500	1	1	yes	characteristic
278	C₂	#21446772	1	1	yes	characteristic
279	C₂	#21446781	1	1	yes	characteristic
280	C₂	#21446809	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₁
commutator	C₂²×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#2868	C₂
socle, prime radical	C₂³×C₆₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#2868	D₆₁
hypocenter	C₆₁	#9898
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2868

Complements

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

N	H ≅ G/N	type
25. C₄⋊#92418	Q₈	semidirect
159. C₆₁	#9898	semidirect

Maximal subgroups

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

subgroup	index
#21445971	2
C₄×C₄⋊#92418	2
#21443702	2
C₆₁×#2868	2
#9898	61