Subgroups — #30598012

Order 15616.

438 conjugacy classes of subgroups.

The subgroup lattice has 254 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×#16840626	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	#16840626	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	—	2	1	no	—
50	C₂×C₈×C₆₁	—	2	1	no	—
51	C₂×C₈×C₆₁	C₄²	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	D₈	1	2	yes	—
61	C₄×Dic₆₁	Q₁₆	1	2	yes	—
62	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
63	C₄×Dic₆₁	#41	1	2	yes	—
64	C₂×#92418	—	2	1	no	—
65	C₂×#92418	—	2	1	no	—
66	C₂×#92418	C₄²	1	2	yes	—
67	C₂×#92418	M₁₆	1	2	yes	—
68	C₂×#92418	M₁₆	1	2	yes	—
69	C₂×#92418	M₁₆	1	2	yes	—
70	C₂³×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
75	C₂×C₄×C₆₁	#107	1	2	yes	—
76	C₂×C₄×C₆₁	#108	1	2	yes	—
77	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₆₁	#117	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×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
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	—	2	1	no	—
89	C₂×Dic₆₁	#107	1	2	yes	—
90	C₂×Dic₆₁	#108	1	2	yes	—
91	C₂×Dic₆₁	C₄×D₄	1	2	yes	—
92	C₂×Dic₆₁	C₄×Q₈	1	2	yes	—
93	C₂×Dic₆₁	C₂×D₈	1	1	yes	characteristic
94	C₂×Dic₆₁	C₂×Q₁₆	1	1	yes	characteristic
95	C₂×Dic₆₁	#135	1	2	yes	—
96	#92420	—	4	8	no	—
97	C₈×C₆₁	—	2	2	no	—
98	C₈×C₆₁	—	2	2	no	—
99	#92418	—	2	2	no	—
100	#92418	#97	1	4	yes	complemented
101	#8274	—	61	1	no	maximal, Sylow
102	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₆₁	#423	1	1	yes	characteristic
104	C₂²×C₆₁	#423	1	1	yes	characteristic
105	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
106	C₂²×C₆₁	#427	1	1	yes	characteristic
107	C₂²×C₆₁	C₂×#108	1	1	yes	characteristic
108	C₂²×C₆₁	C₂×#108	1	1	yes	characteristic
109	C₄×C₆₁	—	2	2	no	—
110	C₄×C₆₁	—	2	2	no	—
111	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
112	C₄×C₆₁	C₄×M₁₆	1	1	yes	characteristic
113	C₄×C₆₁	#432	1	1	yes	characteristic
114	C₄×C₆₁	#432	1	1	yes	characteristic
115	Dic₆₁	—	2	2	no	—
116	Dic₆₁	—	2	2	no	—
117	Dic₆₁	—	2	2	no	—
118	Dic₆₁	C₄×D₈	1	2	yes	—
119	Dic₆₁	C₄×Q₁₆	1	2	yes	—
120	#1767	—	61	2	no	—
121	C₄²×C₈	—	61	1	no	—
122	C₂×#335	—	61	2	no	—
123	#1277	—	61	2	no	—
124	C₂×C₆₁	#1567	1	1	yes	characteristic
125	C₂×C₆₁	#1567	1	1	yes	characteristic
126	C₂×C₆₁	#1568	1	1	yes	characteristic
127	C₂×C₆₁	#1568	1	1	yes	characteristic
128	C₂×C₆₁	#1767	1	1	yes	characteristic
129	C₂×C₆₁	C₄×#107	1	1	yes	characteristic
130	C₂×C₆₁	C₄×#108	1	1	yes	characteristic
131	C₂×C₄⋊C₈	—	61	4	no	—
132	C₂×#97	—	122	2	no	—
133	C₂×C₄×C₈	—	61	1	no	—
134	C₂×C₄×C₈	—	61	1	no	—
135	C₂×C₄×C₈	—	61	1	no	—
136	C₂×C₄×C₈	—	61	1	no	—
137	C₂×C₄×C₈	—	61	1	no	—
138	C₂×C₄×C₈	#6923	1	1	yes	characteristic
139	C₄³	—	61	1	no	—
140	#336	—	122	2	no	—
141	#335	—	61	8	no	—
142	C₆₁	#8274	1	1	yes	characteristic, complemented, Sylow
143	C₂²×C₈	—	61	1	no	—
144	C₂²×C₈	—	61	1	no	—
145	C₂²×C₈	—	122	4	no	—
146	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
147	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
148	C₂×C₄²	—	61	1	no	—
149	C₂×C₄²	—	61	1	no	—
150	C₂×C₄²	—	61	1	no	—
151	C₂×C₄²	—	61	1	no	—
152	C₂×C₄²	—	61	1	no	—
153	C₂×C₄²	—	61	1	no	—
154	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
155	C₄⋊C₈	—	61	8	no	—
156	C₄⋊C₈	—	122	4	no	—
157	#97	—	244	4	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	61	4	no	—
160	C₄×C₈	—	122	2	no	—
161	C₄×C₈	—	122	2	no	—
162	C₄×C₈	—	122	2	no	—
163	C₄×C₈	—	122	2	no	—
164	C₂²×C₄	—	61	1	no	—
165	C₂²×C₄	—	61	1	no	—
166	C₂²×C₄	—	61	1	no	—
167	C₂²×C₄	—	61	1	no	—
168	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
169	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
170	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	61	2	no	—
176	C₂×C₈	—	61	2	no	—
177	C₂×C₈	—	61	2	no	—
178	C₂×C₈	—	61	2	no	—
179	C₂×C₈	—	122	1	no	—
180	C₂×C₈	—	122	1	no	—
181	C₂×C₈	—	122	8	no	—
182	C₂×C₈	—	244	4	no	—
183	C₂×C₈	—	244	4	no	—
184	C₂×C₈	C₄×#6923	1	2	yes	—
185	C₂×C₈	#11758217	1	2	yes	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	61	2	no	—
189	C₄²	—	61	2	no	—
190	C₄²	—	61	2	no	—
191	C₄²	—	61	2	no	—
192	C₄²	—	122	2	no	—
193	C₄²	—	122	2	no	—
194	C₄²	—	122	2	no	—
195	C₄²	—	122	1	no	—
196	C₄²	—	122	1	no	—
197	C₂³	C₂×C₄⋊#6923	1	1	yes	characteristic
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	61	1	no	—
203	C₂×C₄	—	61	1	no	—
204	C₂×C₄	—	61	2	no	—
205	C₂×C₄	—	61	2	no	—
206	C₂×C₄	—	61	2	no	—
207	C₂×C₄	—	61	2	no	—
208	C₂×C₄	—	61	2	no	—
209	C₂×C₄	—	122	1	no	—
210	C₂×C₄	—	122	1	no	—
211	C₂×C₄	—	122	1	no	—
212	C₂×C₄	—	122	1	no	—
213	C₂×C₄	—	122	1	no	—
214	C₂×C₄	—	122	1	no	—
215	C₂×C₄	#14519846	1	2	yes	—
216	C₂×C₄	#14519847	1	2	yes	—
217	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
218	C₂×C₄	#14519856	1	1	yes	characteristic
219	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
220	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
221	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
222	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	2	2	no	—
225	C₈	—	61	4	no	—
226	C₈	—	122	2	no	—
227	C₈	—	244	8	no	—
228	C₂²	C₄×C₄⋊#6923	1	1	yes	characteristic
229	C₂²	#16840765	1	1	yes	characteristic
230	C₂²	#16840767	1	1	yes	characteristic
231	C₂²	C₂×#14519846	1	1	yes	characteristic
232	C₂²	#16840772	1	1	yes	characteristic
233	C₂²	C₂×#14519847	1	1	yes	characteristic
234	C₂²	D₆₁⋊#108	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	2	no	—
237	C₄	—	61	2	no	—
238	C₄	—	61	2	no	—
239	C₄	—	122	2	no	—
240	C₄	—	122	2	no	—
241	C₄	—	122	2	no	—
242	C₄	#16840721	1	1	yes	characteristic
243	C₄	C₄×#11758217	1	1	yes	characteristic
244	C₄	#16840782	1	1	yes	characteristic
245	C₄	#16840785	1	1	yes	characteristic
246	C₂	#21461376	1	1	yes	characteristic
247	C₂	#21461378	1	1	yes	characteristic
248	C₂	#21461379	1	1	yes	characteristic
249	C₂	#21461382	1	1	yes	characteristic
250	C₂	#21461604	1	1	yes	characteristic
251	C₂	C₄×#14519846	1	1	yes	characteristic
252	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	#8274
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. #16840626	C₄	semidirect
100. #92418	#97	semidirect
142. C₆₁	#8274	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×#16840626	2
#21460649	2
C₄×C₈×Dic₆₁	2
#8274	61