Subgroups — #29569494

Order 15616.

574 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839815	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839815	C₂	1	1	yes	maximal, characteristic
3	#21443653	C₂	1	4	yes	maximal, complemented
4	C₂×C₆₁×#432	C₂	1	1	yes	maximal, characteristic
5	#16839815	—	2	2	no	—
6	#16839815	—	2	2	no	—
7	#16839815	—	2	2	no	—
8	#16839815	C₄	1	4	yes	—
9	C₂×C₄⋊#92418	C₄	1	2	yes	—
10	C₂×C₄⋊#92418	C₂²	1	2	yes	—
11	C₂×C₂²⋊#92418	—	2	1	no	—
12	C₂×C₂²⋊#92418	C₄	1	2	yes	—
13	C₂×C₄×#92418	—	2	1	no	—
14	C₂×C₄×#92418	C₂²	1	2	yes	—
15	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#432	—	2	2	no	—
17	C₆₁×#432	C₄	1	4	yes	—
18	C₂×C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₈×C₆₁	—	2	1	no	—
20	C₂×#97×C₆₁	—	2	1	no	—
21	C₂²×#92418	—	2	2	no	—
22	C₂²×#92418	C₂×C₄	1	4	yes	—
23	C₄⋊#92418	—	2	4	no	—
24	C₄⋊#92418	—	2	2	no	—
25	C₄⋊#92418	C₂×C₄	1	4	yes	—
26	C₂²⋊#92418	—	2	4	no	—
27	C₂²⋊#92418	—	2	4	no	—
28	C₄×#92418	—	2	4	no	—
29	C₄×#92418	—	2	2	no	—
30	C₄×#92418	—	2	2	no	—
31	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₆₁	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₆₁	Q₈	1	1	yes	characteristic
34	C₂²×C₈×C₆₁	—	2	2	no	—
35	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₆₁	D₄	1	2	yes	—
39	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₁	D₄	1	2	yes	—
41	C₂×C₄²×C₆₁	Q₈	1	1	yes	characteristic
42	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₆₁	—	2	2	no	—
44	C₄⋊C₈×C₆₁	C₂×C₄	1	4	yes	—
45	C₂²⋊C₈×C₆₁	—	2	4	no	—
46	#97×C₆₁	—	2	4	no	—
47	C₂⁴×C₆₁	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₆₁	—	2	4	no	—
49	C₂²×C₄×C₆₁	—	2	4	no	—
50	C₂²×C₄×C₆₁	—	4	2	no	—
51	C₂²×C₄×C₆₁	C₄²	1	2	yes	—
52	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₆₁	—	2	4	no	—
64	C₂×C₈×C₆₁	—	4	4	no	—
65	C₄²×C₆₁	C₄²	1	2	yes	—
66	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
70	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
71	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
75	C₂×#92418	—	2	4	no	—
76	C₂×#92418	—	2	4	no	—
77	C₂×#92418	—	2	8	no	—
78	C₂×#92418	—	4	4	no	—
79	C₂³×C₆₁	—	2	2	no	—
80	C₂³×C₆₁	—	2	2	no	—
81	C₂³×C₆₁	—	2	2	no	—
82	C₂³×C₆₁	#95	1	2	yes	—
83	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₁	—	2	2	no	—
85	C₂×C₄×C₆₁	—	2	2	no	—
86	C₂×C₄×C₆₁	—	2	2	no	—
87	C₂×C₄×C₆₁	—	2	2	no	—
88	C₂×C₄×C₆₁	—	2	4	no	—
89	C₂×C₄×C₆₁	—	2	4	no	—
90	C₂×C₄×C₆₁	—	2	4	no	—
91	C₂×C₄×C₆₁	—	2	4	no	—
92	C₂×C₄×C₆₁	—	4	2	no	—
93	C₂×C₄×C₆₁	—	4	2	no	—
94	C₂×C₄×C₆₁	#95	1	2	yes	—
95	C₂×C₄×C₆₁	#95	1	2	yes	—
96	C₂×C₄×C₆₁	#95	1	2	yes	—
97	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₆₁	#131	1	4	yes	—
104	C₂×C₄×C₆₁	#135	1	4	yes	—
105	C₈×C₆₁	—	4	4	no	—
106	#92418	—	4	4	no	—
107	#92418	—	4	4	no	—
108	#7878	—	61	1	no	maximal, Sylow
109	C₂²×C₆₁	—	2	2	no	—
110	C₂²×C₆₁	—	2	2	no	—
111	C₂²×C₆₁	—	2	2	no	—
112	C₂²×C₆₁	—	4	4	no	—
113	C₂²×C₆₁	C₂×#95	1	1	yes	characteristic
114	C₂²×C₆₁	#424	1	2	yes	—
115	C₂²×C₆₁	#427	1	2	yes	—
116	C₂²×C₆₁	#430	1	2	yes	—
117	C₄×C₆₁	—	2	2	no	—
118	C₄×C₆₁	—	2	2	no	—
119	C₄×C₆₁	—	2	2	no	—
120	C₄×C₆₁	—	2	1	no	—
121	C₄×C₆₁	—	2	1	no	—
122	C₄×C₆₁	#405	1	2	yes	—
123	C₄×C₆₁	#408	1	2	yes	—
124	C₄×C₆₁	#416	1	2	yes	—
125	C₄×C₆₁	#416	1	2	yes	—
126	C₂×#432	D₆₁	1	1	yes	characteristic
127	C₂×#431	—	61	1	no	—
128	C₂×#431	—	61	1	no	—
129	#1277	—	61	4	no	—
130	C₂×C₆₁	—	4	2	no	—
131	C₂×C₆₁	#1572	1	4	yes	—
132	C₂×C₆₁	#1729	1	1	yes	characteristic
133	C₂×C₆₁	#1735	1	1	yes	characteristic
134	C₂×C₆₁	#1738	1	1	yes	characteristic
135	C₂²×C₄²	C₂×D₆₁	1	1	yes	characteristic
136	#432	—	2	2	no	—
137	#432	Dic₆₁	1	4	yes	—
138	#431	—	61	4	no	—
139	#431	—	122	2	no	—
140	#431	—	122	2	no	—
141	#431	—	122	2	no	—
142	C₂×C₄⋊C₈	—	61	2	no	—
143	C₂×C₄⋊C₈	—	61	2	no	—
144	C₂×C₄⋊C₈	C₂×D₆₁	1	2	yes	—
145	C₂×C₂²⋊C₈	—	2	1	no	—
146	C₂×C₂²⋊C₈	—	61	2	no	—
147	C₂×C₂²⋊C₈	—	122	1	no	—
148	C₂×#97	—	2	1	no	—
149	C₂×C₄×C₈	—	61	2	no	—
150	C₂×C₄×C₈	—	122	1	no	—
151	C₆₁	#7878	1	1	yes	characteristic, complemented, Sylow
152	C₂³×C₄	Dic₁₂₂	1	1	yes	characteristic
153	C₂³×C₄	C₄×D₆₁	1	1	yes	characteristic
154	C₂³×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	—	61	4	no	—
157	C₂²×C₈	—	122	2	no	—
158	C₂×C₄²	Dic₁₂₂	1	1	yes	characteristic
159	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
160	C₂×C₄²	C₄×D₆₁	1	2	yes	—
161	C₂×C₄²	D₂₄₄	1	2	yes	—
162	C₂×C₄²	C₂×Dic₆₁	1	2	yes	—
163	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
164	C₂×C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
165	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	61	4	no	—
168	C₄⋊C₈	—	122	4	no	—
169	C₄⋊C₈	—	122	2	no	—
170	C₄⋊C₈	C₂×Dic₆₁	1	4	yes	—
171	C₂²⋊C₈	—	2	4	no	—
172	C₂²⋊C₈	—	122	4	no	—
173	C₂²⋊C₈	—	122	4	no	—
174	#97	—	2	4	no	—
175	C₄×C₈	—	122	4	no	—
176	C₄×C₈	—	122	2	no	—
177	C₄×C₈	—	122	2	no	—
178	C₂⁴	#11758196	1	1	yes	characteristic
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	C₄×Dic₆₁	1	2	yes	—
183	C₂²×C₄	#11758196	1	1	yes	characteristic
184	C₂²×C₄	#11758196	1	1	yes	characteristic
185	C₂²×C₄	#11758196	1	1	yes	characteristic
186	C₂²×C₄	#11758196	1	2	yes	—
187	C₂²×C₄	C₄⋊Dic₆₁	1	2	yes	—
188	C₂²×C₄	#11758198	1	2	yes	—
189	C₂²×C₄	#11758198	1	2	yes	—
190	C₂²×C₄	C₂²⋊Dic₆₁	1	2	yes	—
191	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	4	4	no	—
196	C₂×C₈	—	122	4	no	—
197	C₂×C₈	—	122	4	no	—
198	C₂×C₈	—	122	8	no	—
199	C₂×C₈	—	244	4	no	—
200	C₄²	C₄×Dic₆₁	1	2	yes	—
201	C₄²	#11758196	1	2	yes	—
202	C₄²	C₄⋊Dic₆₁	1	2	yes	—
203	C₄²	#11758198	1	2	yes	—
204	C₄²	#11758198	1	2	yes	—
205	C₄²	C₂²⋊Dic₆₁	1	2	yes	—
206	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
207	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
208	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
209	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	#14519684	1	2	yes	—
214	C₂³	C₂×#11758196	1	1	yes	characteristic
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	#14519684	1	2	yes	—
226	C₂×C₄	#14519684	1	2	yes	—
227	C₂×C₄	#14519684	1	2	yes	—
228	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
229	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
230	C₂×C₄	C₂×#11758198	1	1	yes	characteristic
231	C₂×C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
232	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
234	C₂×C₄	#14519785	1	4	yes	—
235	C₂×C₄	#14519797	1	4	yes	—
236	C₈	—	4	4	no	—
237	C₈	—	244	4	no	—
238	C₈	—	244	4	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	4	4	no	—
243	C₂²	C₂×#14519684	1	1	yes	characteristic
244	C₂²	#16839790	1	2	yes	—
245	C₂²	#16839798	1	2	yes	—
246	C₂²	#16839810	1	2	yes	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	#16839720	1	2	yes	—
253	C₄	#16839730	1	2	yes	—
254	C₄	#16839756	1	2	yes	—
255	C₄	#16839758	1	2	yes	—
256	C₂	—	4	2	no	—
257	C₂	#21444600	1	4	yes	—
258	C₂	#21445517	1	1	yes	characteristic
259	C₂	#21445535	1	1	yes	characteristic
260	C₂	#21445545	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₆₁×#432	C₂
socle	C₂³×C₆₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₆₁	C₄⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×#432	D₆₁
hypocenter	C₆₁	#7878
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #21443653	C₂	semidirect
151. C₆₁	#7878	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839815	2
#21443653	2
C₂×C₆₁×#432	2
#7878	61