Subgroups — #30598010

Order 15616.

438 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21461605	C₂	1	2	yes	maximal
2	C₂×#16840623	C₂	1	2	yes	maximal
3	#21460649	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#92420	C₂²	1	4	yes	—
6	C₂×#14519832	—	2	2	no	—
7	#16840628	—	2	2	no	—
8	#16840623	C₄	1	8	yes	complemented
9	C₂×C₄×#92418	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₆₁	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
14	C₄²×Dic₆₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
16	C₂²×#92420	—	2	4	no	—
17	C₄⋊#92420	—	2	4	no	—
18	C₄⋊#92420	C₂×C₄	1	8	yes	—
19	#14519832	—	4	4	no	—
20	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
21	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
24	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
25	C₂×C₄×Dic₆₁	Q₈	1	1	yes	characteristic
26	C₂×C₄×Dic₆₁	Q₈	1	1	yes	characteristic
27	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
28	C₄×#92418	—	2	2	no	—
29	C₈×Dic₆₁	—	2	2	no	—
30	C₈×Dic₆₁	—	2	2	no	—
31	C₈×Dic₆₁	—	2	2	no	—
32	C₈×Dic₆₁	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₆₁	—	2	2	no	—
37	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₂²×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×Dic₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×#92420	—	2	8	no	—
45	C₂×#92420	—	4	4	no	—
46	C₂×#92420	—	4	4	no	—
47	C₂×C₈×C₆₁	—	2	1	no	—
48	C₂×C₈×C₆₁	—	2	1	no	—
49	C₂×C₈×C₆₁	C₄²	1	2	yes	—
50	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
51	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
52	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
53	C₄²×C₆₁	—	2	1	no	—
54	C₄²×C₆₁	—	2	1	no	—
55	C₄×Dic₆₁	—	2	2	no	—
56	C₄×Dic₆₁	—	2	1	no	—
57	C₄×Dic₆₁	—	2	1	no	—
58	C₄×Dic₆₁	—	2	2	no	—
59	C₄×Dic₆₁	—	2	2	no	—
60	C₄×Dic₆₁	SD₁₆	1	4	yes	—
61	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
62	C₄×Dic₆₁	#41	1	2	yes	—
63	C₂×#92418	—	2	1	no	—
64	C₂×#92418	—	2	1	no	—
65	C₂×#92418	—	2	1	no	—
66	C₂×#92418	—	2	1	no	—
67	C₂×#92418	C₄²	1	2	yes	—
68	C₂×#92418	M₁₆	1	2	yes	—
69	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₆₁	—	2	1	no	—
71	C₂×C₄×C₆₁	—	2	1	no	—
72	C₂×C₄×C₆₁	—	2	1	no	—
73	C₂×C₄×C₆₁	—	2	1	no	—
74	C₂×C₄×C₆₁	#106	1	2	yes	—
75	C₂×C₄×C₆₁	#108	1	2	yes	—
76	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
77	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
82	C₂×Dic₆₁	—	2	1	no	—
83	C₂×Dic₆₁	—	2	1	no	—
84	C₂×Dic₆₁	—	2	1	no	—
85	C₂×Dic₆₁	—	2	1	no	—
86	C₂×Dic₆₁	—	2	1	no	—
87	C₂×Dic₆₁	—	2	1	no	—
88	C₂×Dic₆₁	#106	1	2	yes	—
89	C₂×Dic₆₁	#108	1	2	yes	—
90	C₂×Dic₆₁	C₄×D₄	1	2	yes	—
91	C₂×Dic₆₁	C₄×Q₈	1	2	yes	—
92	C₂×Dic₆₁	C₂×SD₁₆	1	2	yes	—
93	C₂×Dic₆₁	#135	1	2	yes	—
94	#92420	—	4	8	no	—
95	C₈×C₆₁	—	2	2	no	—
96	C₈×C₆₁	#97	1	4	yes	complemented
97	#92418	—	2	2	no	—
98	#92418	—	2	2	no	—
99	#8273	—	61	1	no	maximal, Sylow
100	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₆₁	#423	1	1	yes	characteristic
102	C₂²×C₆₁	#423	1	1	yes	characteristic
103	C₂²×C₆₁	C₂×#106	1	1	yes	characteristic
104	C₂²×C₆₁	#427	1	1	yes	characteristic
105	C₂²×C₆₁	C₂×#108	1	1	yes	characteristic
106	C₂²×C₆₁	C₂×#108	1	1	yes	characteristic
107	C₄×C₆₁	—	2	2	no	—
108	C₄×C₆₁	—	2	2	no	—
109	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
110	C₄×C₆₁	C₄×M₁₆	1	1	yes	characteristic
111	C₄×C₆₁	#432	1	1	yes	characteristic
112	C₄×C₆₁	#432	1	1	yes	characteristic
113	Dic₆₁	—	2	2	no	—
114	Dic₆₁	—	2	2	no	—
115	Dic₆₁	—	2	2	no	—
116	Dic₆₁	C₄×SD₁₆	1	4	yes	—
117	#1767	—	61	2	no	—
118	C₄²×C₈	—	61	1	no	—
119	C₂×#334	—	61	2	no	—
120	#1277	—	61	2	no	—
121	C₂×C₆₁	#1566	1	1	yes	characteristic
122	C₂×C₆₁	#1566	1	1	yes	characteristic
123	C₂×C₆₁	#1568	1	1	yes	characteristic
124	C₂×C₆₁	#1568	1	1	yes	characteristic
125	C₂×C₆₁	#1767	1	1	yes	characteristic
126	C₂×C₆₁	C₄×#106	1	1	yes	characteristic
127	C₂×C₆₁	C₄×#108	1	1	yes	characteristic
128	C₂×C₄⋊C₈	—	61	4	no	—
129	C₂×#97	—	122	2	no	—
130	C₂×C₄×C₈	—	61	1	no	—
131	C₂×C₄×C₈	—	61	1	no	—
132	C₂×C₄×C₈	—	61	1	no	—
133	C₂×C₄×C₈	—	61	1	no	—
134	C₂×C₄×C₈	—	61	1	no	—
135	C₂×C₄×C₈	#6923	1	1	yes	characteristic
136	C₄³	—	61	1	no	—
137	#336	—	122	2	no	—
138	#334	—	61	8	no	—
139	C₆₁	#8273	1	1	yes	characteristic, complemented, Sylow
140	C₂²×C₈	—	61	1	no	—
141	C₂²×C₈	—	61	1	no	—
142	C₂²×C₈	—	122	4	no	—
143	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
144	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
145	C₂×C₄²	—	61	1	no	—
146	C₂×C₄²	—	61	1	no	—
147	C₂×C₄²	—	61	1	no	—
148	C₂×C₄²	—	61	1	no	—
149	C₂×C₄²	—	61	1	no	—
150	C₂×C₄²	—	61	1	no	—
151	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
152	C₄⋊C₈	—	61	8	no	—
153	C₄⋊C₈	—	122	4	no	—
154	#97	—	244	4	no	—
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	—	61	4	no	—
157	C₄×C₈	—	122	2	no	—
158	C₄×C₈	—	122	2	no	—
159	C₄×C₈	—	122	2	no	—
160	C₄×C₈	—	122	2	no	—
161	C₂²×C₄	—	61	1	no	—
162	C₂²×C₄	—	61	1	no	—
163	C₂²×C₄	—	61	1	no	—
164	C₂²×C₄	—	61	1	no	—
165	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
166	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
167	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	61	2	no	—
171	C₂×C₈	—	61	2	no	—
172	C₂×C₈	—	122	1	no	—
173	C₂×C₈	—	122	1	no	—
174	C₂×C₈	—	122	1	no	—
175	C₂×C₈	—	122	1	no	—
176	C₂×C₈	—	122	8	no	—
177	C₂×C₈	—	244	4	no	—
178	C₂×C₈	—	244	4	no	—
179	C₂×C₈	#11758213	1	2	yes	—
180	C₂×C₈	#11758213	1	2	yes	—
181	C₂×C₈	C₄×#6923	1	2	yes	—
182	C₂×C₈	#11758217	1	2	yes	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	61	4	no	—
186	C₄²	—	61	2	no	—
187	C₄²	—	61	2	no	—
188	C₄²	—	122	2	no	—
189	C₄²	—	122	2	no	—
190	C₄²	—	122	2	no	—
191	C₄²	—	122	1	no	—
192	C₄²	—	122	1	no	—
193	C₂³	C₂×C₄⋊#6923	1	1	yes	characteristic
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	61	2	no	—
199	C₂×C₄	—	61	2	no	—
200	C₂×C₄	—	61	2	no	—
201	C₂×C₄	—	61	2	no	—
202	C₂×C₄	—	61	2	no	—
203	C₂×C₄	—	61	2	no	—
204	C₂×C₄	—	122	1	no	—
205	C₂×C₄	—	122	1	no	—
206	C₂×C₄	—	122	1	no	—
207	C₂×C₄	—	122	1	no	—
208	C₂×C₄	—	122	1	no	—
209	C₂×C₄	—	122	1	no	—
210	C₂×C₄	#14519845	1	2	yes	—
211	C₂×C₄	#14519847	1	2	yes	—
212	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
213	C₂×C₄	#14519856	1	1	yes	characteristic
214	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
215	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
216	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
217	C₂×C₄	C₂×#11758213	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	122	2	no	—
220	C₈	—	122	2	no	—
221	C₈	—	244	8	no	—
222	C₈	#14519832	1	4	yes	complemented
223	C₂²	C₄×C₄⋊#6923	1	1	yes	characteristic
224	C₂²	#16840765	1	1	yes	characteristic
225	C₂²	#16840766	1	1	yes	characteristic
226	C₂²	C₂×#14519845	1	1	yes	characteristic
227	C₂²	#16840772	1	1	yes	characteristic
228	C₂²	C₂×#14519847	1	1	yes	characteristic
229	C₂²	D₆₁⋊#108	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	—	61	4	no	—
233	C₄	—	122	2	no	—
234	C₄	—	122	2	no	—
235	C₄	—	122	2	no	—
236	C₄	C₂×#14519832	1	1	yes	characteristic
237	C₄	C₄×#11758213	1	1	yes	characteristic
238	C₄	#16840783	1	1	yes	characteristic
239	C₄	#16840784	1	1	yes	characteristic
240	C₂	#21461373	1	1	yes	characteristic
241	C₂	#21461374	1	1	yes	characteristic
242	C₂	#21461379	1	1	yes	characteristic
243	C₂	#21461380	1	1	yes	characteristic
244	C₂	#21461605	1	1	yes	characteristic
245	C₂	C₄×#14519845	1	1	yes	characteristic
246	C₂	C₄×#14519847	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#6923
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×C₄×C₈×C₆₁	C₄
socle, prime radical	C₂³×C₆₁	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₆₁	C₂³
hypercenter	C₂×C₄×C₈	#6923
hypocenter	C₆₁	#8273
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
8. #16840623	C₄	semidirect
96. C₈×C₆₁	#97	semidirect
139. C₆₁	#8273	semidirect
222. C₈	#14519832	semidirect

Maximal subgroups

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

subgroup	index
#21461605	2
C₂×#16840623	2
#21460649	2
C₄×C₈×Dic₆₁	2
#8273	61