Subgroups — #29569418

Order 15616.

662 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454452	C₂	1	2	yes	maximal
2	#21443648	C₂	1	4	yes	maximal, complemented
3	C₄×#117×C₆₁	C₂	1	1	yes	maximal, characteristic
4	#16839819	—	2	2	no	—
5	#16839819	—	2	2	no	—
6	#16839815	C₄	1	4	yes	complemented
7	#16839815	C₂²	1	4	yes	—
8	#16839786	—	2	2	no	—
9	#16839726	—	2	2	no	—
10	C₄×#11758194	C₄	1	4	yes	complemented
11	C₂×#117×C₆₁	C₄	1	2	yes	—
12	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₆₁	—	2	2	no	—
14	C₄×C₂²⋊C₄×C₆₁	—	2	2	no	—
15	C₆₁×#376	C₂²	1	2	yes	—
16	C₄³×C₆₁	C₄	1	2	yes	—
17	C₂×#11758194	—	2	4	no	—
18	C₄⋊#92418	—	2	4	no	—
19	C₄⋊#92418	—	2	4	no	—
20	C₂²⋊#92418	—	2	4	no	—
21	C₂²⋊#92418	—	2	4	no	—
22	#14519688	—	2	4	no	—
23	C₄×#92418	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₆₁	D₄	1	2	yes	—
26	#117×C₆₁	—	2	8	no	—
27	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
28	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
29	C₂×C₄²×C₆₁	—	2	4	no	—
30	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₆₁	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₆₁	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₁	D₄	1	2	yes	—
37	C₂×C₄²×C₆₁	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₆₁	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
40	#95×C₆₁	—	2	4	no	—
41	C₂⁴×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₆₁	—	2	4	no	—
43	C₂²×C₄×C₆₁	—	2	4	no	—
44	C₂²×C₄×C₆₁	—	4	4	no	—
45	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
57	C₄⋊C₄×C₆₁	—	2	8	no	—
58	C₂²⋊C₄×C₆₁	—	2	8	no	—
59	C₄²×C₆₁	—	2	4	no	—
60	C₄²×C₆₁	—	4	8	no	—
61	C₄²×C₆₁	C₄²	1	2	yes	—
62	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
65	C₄²×C₆₁	C₄⋊C₄	1	2	yes	—
66	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
67	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
68	C₄²×C₆₁	M₁₆	1	8	yes	complemented
69	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
70	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
72	#11758194	—	4	8	no	—
73	C₂×#92418	—	2	8	no	—
74	C₂³×C₆₁	—	2	2	no	—
75	C₂³×C₆₁	—	4	2	no	—
76	C₂³×C₆₁	#95	1	1	yes	characteristic
77	C₂³×C₆₁	#95	1	1	yes	characteristic
78	C₂³×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₆₁	—	2	2	no	—
80	C₂×C₄×C₆₁	—	2	4	no	—
81	C₂×C₄×C₆₁	—	2	4	no	—
82	C₂×C₄×C₆₁	—	2	4	no	—
83	C₂×C₄×C₆₁	—	2	2	no	—
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	8	no	—
88	C₂×C₄×C₆₁	—	4	2	no	—
89	C₂×C₄×C₆₁	—	4	2	no	—
90	C₂×C₄×C₆₁	—	4	2	no	—
91	C₂×C₄×C₆₁	—	4	8	no	—
92	C₂×C₄×C₆₁	#95	1	2	yes	—
93	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
94	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
95	C₂×C₄×C₆₁	#95	1	2	yes	—
96	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂×M₁₆	1	4	yes	—
101	#92418	—	4	8	no	—
102	#7856	—	61	1	no	maximal, Sylow
103	C₂²×C₆₁	—	2	2	no	—
104	C₂²×C₆₁	—	2	1	no	—
105	C₂²×C₆₁	—	2	1	no	—
106	C₂²×C₆₁	—	2	2	no	—
107	C₂²×C₆₁	—	2	2	no	—
108	C₂²×C₆₁	—	4	4	no	—
109	C₂²×C₆₁	C₂×#95	1	1	yes	characteristic
110	C₂²×C₆₁	#407	1	2	yes	—
111	C₄×C₆₁	—	2	1	no	—
112	C₄×C₆₁	—	2	1	no	—
113	C₄×C₆₁	—	2	2	no	—
114	C₄×C₆₁	—	2	2	no	—
115	C₄×C₆₁	—	2	2	no	—
116	C₄×C₆₁	—	4	8	no	—
117	C₄×C₆₁	H(3,ℤ/4)	1	4	yes	—
118	C₄×C₆₁	C₄×M₁₆	1	2	yes	—
119	C₄×C₆₁	#423	1	2	yes	—
120	#2920	—	61	2	no	—
121	C₄×#117	D₆₁	1	1	yes	characteristic
122	#1274	—	61	4	no	—
123	C₂×C₆₁	—	2	2	no	—
124	C₂×C₆₁	—	4	1	no	—
125	C₂×C₆₁	—	4	1	no	—
126	C₂×C₆₁	#1728	1	1	yes	characteristic
127	C₂×C₆₁	C₂×H(3,ℤ/4)	1	2	yes	—
128	C₂×#117	Dic₆₁	1	2	yes	—
129	C₂²×C₄²	C₂×D₆₁	1	1	yes	characteristic
130	#432	—	122	2	no	—
131	#432	—	122	2	no	—
132	#431	—	61	4	no	—
133	#431	—	61	4	no	—
134	#423	—	122	2	no	—
135	#407	—	122	2	no	—
136	C₄×M₁₆	—	61	4	no	—
137	C₄×C₄⋊C₄	—	2	2	no	—
138	C₄×C₂²⋊C₄	—	2	2	no	—
139	#376	C₂×D₆₁	1	2	yes	—
140	C₄³	Dic₆₁	1	2	yes	—
141	C₆₁	#7856	1	1	yes	characteristic, complemented, Sylow
142	C₂³×C₄	C₂×Dic₆₁	1	1	yes	characteristic
143	C₂³×C₄	C₂²⋊D₆₁	1	2	yes	—
144	C₂×M₁₆	—	122	4	no	—
145	#117	—	2	8	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₂²⋊C₄	—	2	2	no	—
148	C₂×C₄²	—	2	4	no	—
149	C₂×C₄²	Dic₁₂₂	1	1	yes	characteristic
150	C₂×C₄²	Dic₁₂₂	1	1	yes	characteristic
151	C₂×C₄²	C₄×D₆₁	1	2	yes	—
152	C₂×C₄²	C₄×D₆₁	1	2	yes	—
153	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
154	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
155	C₂×C₄²	C₂×Dic₆₁	1	4	yes	—
156	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
157	C₂×C₄²	C₂²⋊D₆₁	1	2	yes	—
158	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
159	C₄⋊C₈	—	122	4	no	—
160	C₄⋊C₈	—	122	4	no	—
161	C₂²⋊C₈	—	122	4	no	—
162	C₂²⋊C₈	—	122	4	no	—
163	#97	—	122	4	no	—
164	C₄×C₈	—	61	8	no	—
165	#95	—	2	4	no	—
166	C₂⁴	C₂²⋊Dic₆₁	1	1	yes	characteristic
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	4	4	no	—
170	C₂²×C₄	C₄×Dic₆₁	1	1	yes	characteristic
171	C₂²×C₄	C₄×Dic₆₁	1	1	yes	characteristic
172	C₂²×C₄	#11758196	1	2	yes	—
173	C₂²×C₄	#11758196	1	2	yes	—
174	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
175	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
176	C₂²×C₄	#11758198	1	2	yes	—
177	C₂²×C₄	#11758198	1	2	yes	—
178	C₂²×C₄	C₂²⋊Dic₆₁	1	2	yes	—
179	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
180	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
182	M₁₆	—	244	8	no	—
183	C₂×C₈	—	122	8	no	—
184	C₄⋊C₄	—	2	8	no	—
185	C₂²⋊C₄	—	2	8	no	—
186	C₄²	—	2	4	no	—
187	C₄²	—	4	8	no	—
188	C₄²	#11758194	1	8	yes	complemented
189	C₄²	C₄×Dic₆₁	1	2	yes	—
190	C₄²	#11758196	1	2	yes	—
191	C₄²	#11758196	1	2	yes	—
192	C₄²	C₄⋊Dic₆₁	1	1	yes	characteristic
193	C₄²	C₄⋊Dic₆₁	1	1	yes	characteristic
194	C₄²	#11758198	1	2	yes	—
195	C₄²	#11758198	1	2	yes	—
196	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
197	C₄²	C₂×C₄×D₆₁	1	2	yes	—
198	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
199	C₂³	—	2	2	no	—
200	C₂³	—	4	2	no	—
201	C₂³	#14519684	1	1	yes	characteristic
202	C₂³	#14519684	1	1	yes	characteristic
203	C₂³	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	8	no	—
217	C₂×C₄	#14519684	1	2	yes	—
218	C₂×C₄	#14519684	1	1	yes	characteristic
219	C₂×C₄	#14519684	1	1	yes	characteristic
220	C₂×C₄	#14519684	1	2	yes	—
221	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
222	C₂×C₄	C₂×#11758198	1	2	yes	—
223	C₂×C₄	C₂×#11758196	1	2	yes	—
224	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
225	C₂×C₄	C₂×#11758194	1	4	yes	—
226	C₈	—	244	8	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	4	4	no	—
233	C₂²	C₂×#14519684	1	1	yes	characteristic
234	C₂²	#16839726	1	2	yes	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	4	8	no	—
241	C₄	#16839518	1	4	yes	—
242	C₄	C₄×#11758194	1	2	yes	—
243	C₄	#16839786	1	2	yes	—
244	C₂	—	2	2	no	—
245	C₂	—	4	1	no	—
246	C₂	—	4	1	no	—
247	C₂	#21445513	1	1	yes	characteristic
248	C₂	C₂×#16839518	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₆₁
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₄×#117×C₆₁	C₂
socle	C₂²×C₆₁	C₂×#95
prime radical	C₂⁴×C₆₁	C₂²⋊C₄
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₄×#117	D₆₁
hypocenter	C₆₁	#7856
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
2. #21443648	C₂	semidirect
6. #16839815	C₄	semidirect
10. C₄×#11758194	C₄	semidirect
68. C₄²×C₆₁	M₁₆	semidirect
141. C₆₁	#7856	semidirect
188. C₄²	#11758194	semidirect

Maximal subgroups

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

subgroup	index
#21454452	2
#21443648	2
C₄×#117×C₆₁	2
#7856	61