Subgroups — C₂²×#16840628 #30613837

Order 15616.

3086 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#92420	C₂	1	2	yes	maximal
2	C₂×#16840628	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₄×Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×#92420	—	2	8	no	—
5	C₂³×#92420	—	2	8	no	—
6	C₂³×#92420	C₄	1	16	yes	complemented
7	C₂³×#92420	C₂²	1	12	yes	—
8	#16840628	C₂²	1	16	yes	direct product factor
9	C₂⁴×Dic₆₁	C₄	1	1	yes	characteristic
10	C₂⁴×Dic₆₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄×Dic₆₁	—	2	8	no	—
12	C₂²×C₄×Dic₆₁	C₄	1	6	yes	—
13	C₂²×C₄×Dic₆₁	C₂²	1	6	yes	—
14	C₂⁴×C₄×C₆₁	C₄	1	1	yes	characteristic
15	C₂²×#92420	—	2	64	no	—
16	C₂²×#92420	—	2	24	no	—
17	C₂²×#92420	—	2	24	no	—
18	C₂²×#92420	C₂×C₄	1	48	yes	complemented
19	C₂²×#92420	C₂³	1	8	yes	—
20	C₂³×Dic₆₁	—	2	4	no	—
21	C₂³×Dic₆₁	—	2	4	no	—
22	C₂³×Dic₆₁	C₈	1	8	yes	complemented
23	C₂³×Dic₆₁	C₂×C₄	1	6	yes	—
24	C₂³×Dic₆₁	C₂×C₄	1	3	yes	—
25	C₂³×Dic₆₁	C₂×C₄	1	8	yes	—
26	C₂³×Dic₆₁	D₄	1	4	yes	—
27	C₂³×Dic₆₁	D₄	1	8	yes	—
28	C₂³×Dic₆₁	Q₈	1	4	yes	—
29	C₂³×Dic₆₁	C₂³	1	3	yes	—
30	C₂×C₄×Dic₆₁	—	2	24	no	—
31	C₂×C₄×Dic₆₁	—	2	24	no	—
32	C₂×C₄×Dic₆₁	C₂×C₄	1	12	yes	—
33	C₂×C₄×Dic₆₁	C₂³	1	4	yes	—
34	C₂⁵×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₆₁	—	2	4	no	—
36	C₂³×C₄×C₆₁	—	2	4	no	—
37	C₂³×C₄×C₆₁	C₈	1	8	yes	complemented
38	C₂³×C₄×C₆₁	C₂×C₄	1	6	yes	—
39	C₂⁴×C₆₁	C₄²	1	4	yes	—
40	C₂⁴×C₆₁	C₂²⋊C₄	1	4	yes	—
41	C₂⁴×C₆₁	C₄⋊C₄	1	4	yes	—
42	C₂⁴×C₆₁	C₂×C₈	1	4	yes	—
43	C₂⁴×C₆₁	C₂×C₈	1	4	yes	—
44	C₂⁴×C₆₁	M₁₆	1	4	yes	—
45	C₂⁴×C₆₁	M₁₆	1	4	yes	—
46	C₂⁴×C₆₁	C₂²×C₄	1	3	yes	—
47	C₂²×C₄×C₆₁	—	2	32	no	—
48	C₂²×C₄×C₆₁	—	2	12	no	—
49	C₂²×C₄×C₆₁	—	2	12	no	—
50	C₂²×C₄×C₆₁	C₂×C₈	1	24	yes	complemented
51	C₂²×C₄×C₆₁	C₂²×C₄	1	4	yes	—
52	C₂²×Dic₆₁	—	2	32	no	—
53	C₂²×Dic₆₁	—	2	12	no	—
54	C₂²×Dic₆₁	—	2	12	no	—
55	C₂²×Dic₆₁	C₄²	1	12	yes	—
56	C₂²×Dic₆₁	C₂²⋊C₄	1	12	yes	—
57	C₂²×Dic₆₁	C₂²⋊C₄	1	32	yes	—
58	C₂²×Dic₆₁	C₄⋊C₄	1	12	yes	—
59	C₂²×Dic₆₁	C₄⋊C₄	1	32	yes	—
60	C₂²×Dic₆₁	C₂×C₈	1	24	yes	complemented
61	C₂²×Dic₆₁	C₂²×C₄	1	4	yes	—
62	C₂²×Dic₆₁	C₂²×C₄	1	3	yes	—
63	C₂²×Dic₆₁	C₂²×C₄	1	12	yes	—
64	C₂²×Dic₆₁	C₂×D₄	1	6	yes	—
65	C₂²×Dic₆₁	C₂×D₄	1	12	yes	—
66	C₂²×Dic₆₁	C₂×Q₈	1	6	yes	—
67	C₂²×Dic₆₁	C₂⁴	1	1	yes	characteristic
68	C₂×#92420	—	2	96	no	—
69	C₂×#92420	—	2	8	no	—
70	C₂×#92420	—	2	8	no	—
71	C₂×#92420	C₂²×C₄	1	16	yes	complemented
72	C₄×Dic₆₁	—	2	24	no	—
73	C₄×Dic₆₁	—	2	8	no	—
74	C₂³×C₆₁	#95	1	16	yes	—
75	C₂³×C₆₁	C₄×C₈	1	16	yes	—
76	C₂³×C₆₁	#97	1	16	yes	—
77	C₂³×C₆₁	C₂²⋊C₈	1	16	yes	—
78	C₂³×C₆₁	C₂²⋊C₈	1	16	yes	—
79	C₂³×C₆₁	C₄⋊C₈	1	16	yes	—
80	C₂³×C₆₁	C₄⋊C₈	1	16	yes	—
81	C₂³×C₆₁	C₂×C₄²	1	6	yes	—
82	C₂³×C₆₁	C₂×C₂²⋊C₄	1	6	yes	—
83	C₂³×C₆₁	C₂×C₄⋊C₄	1	6	yes	—
84	C₂³×C₆₁	C₂²×C₈	1	6	yes	—
85	C₂³×C₆₁	C₂²×C₈	1	6	yes	—
86	C₂³×C₆₁	C₂×M₁₆	1	6	yes	—
87	C₂³×C₆₁	C₂×M₁₆	1	6	yes	—
88	C₂³×C₆₁	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₆₁	—	2	4	no	—
90	C₂×C₄×C₆₁	—	2	4	no	—
91	C₂×C₄×C₆₁	—	2	48	no	—
92	C₂×C₄×C₆₁	C₂²×C₈	1	8	yes	complemented
93	C₂×Dic₆₁	—	2	4	no	—
94	C₂×Dic₆₁	—	2	4	no	—
95	C₂×Dic₆₁	—	2	48	no	—
96	C₂×Dic₆₁	#95	1	48	yes	—
97	C₂×Dic₆₁	C₂×C₄²	1	6	yes	—
98	C₂×Dic₆₁	C₂×C₂²⋊C₄	1	24	yes	—
99	C₂×Dic₆₁	C₂×C₂²⋊C₄	1	6	yes	—
100	C₂×Dic₆₁	C₂×C₄⋊C₄	1	24	yes	—
101	C₂×Dic₆₁	C₂×C₄⋊C₄	1	6	yes	—
102	C₂×Dic₆₁	C₂²×C₈	1	8	yes	complemented
103	C₂×Dic₆₁	C₂³×C₄	1	2	yes	—
104	C₂×Dic₆₁	C₂²×D₄	1	1	yes	characteristic
105	C₂×Dic₆₁	C₂²×D₄	1	2	yes	—
106	C₂×Dic₆₁	C₂²×Q₈	1	1	yes	characteristic
107	#92420	—	2	16	no	—
108	C₂²×#336	—	61	1	no	maximal, Sylow
109	C₂²×C₆₁	#336	1	64	yes	complemented
110	C₂²×C₆₁	C₂×#95	1	12	yes	—
111	C₂²×C₆₁	C₂×C₄×C₈	1	12	yes	—
112	C₂²×C₆₁	C₂×#97	1	12	yes	—
113	C₂²×C₆₁	C₂×C₂²⋊C₈	1	12	yes	—
114	C₂²×C₆₁	C₂×C₂²⋊C₈	1	12	yes	—
115	C₂²×C₆₁	C₂×C₄⋊C₈	1	12	yes	—
116	C₂²×C₆₁	C₂×C₄⋊C₈	1	12	yes	—
117	C₂²×C₆₁	C₂²×C₄²	1	1	yes	characteristic
118	C₂²×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
119	C₂²×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
121	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
122	C₂²×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
123	C₂²×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
124	C₄×C₆₁	—	2	8	no	—
125	Dic₆₁	—	2	8	no	—
126	Dic₆₁	C₂×#95	1	12	yes	—
127	Dic₆₁	C₂²×C₂²⋊C₄	1	2	yes	—
128	Dic₆₁	C₂²×C₄⋊C₄	1	2	yes	—
129	C₂⁴×C₈	—	61	2	no	—
130	C₂³×C₄²	—	61	1	no	—
131	C₂×#336	—	61	12	no	—
132	C₂×C₆₁	C₂×#336	1	24	yes	complemented
133	C₂×C₆₁	C₂²×#95	1	1	yes	characteristic
134	C₂×C₆₁	C₂²×C₄×C₈	1	1	yes	characteristic
135	C₂×C₆₁	C₂²×#97	1	1	yes	characteristic
136	C₂×C₆₁	C₂²×C₂²⋊C₈	1	1	yes	characteristic
137	C₂×C₆₁	C₂²×C₂²⋊C₈	1	1	yes	characteristic
138	C₂×C₆₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
139	C₂×C₆₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
140	C₂⁴×C₄	—	61	1	no	—
141	C₂⁴×C₄	—	61	1	no	—
142	C₂⁴×C₄	#6923	1	1	yes	characteristic
143	C₂³×C₈	—	61	16	no	—
144	C₂³×C₈	—	61	12	no	—
145	C₂³×C₈	—	122	8	no	—
146	C₂³×C₈	—	122	8	no	—
147	C₂²×C₄²	—	61	6	no	—
148	C₂²×C₄²	—	61	6	no	—
149	C₂²×C₄²	—	122	8	no	—
150	#336	—	61	16	no	—
151	C₆₁	C₂²×#336	1	1	yes	characteristic, complemented, Sylow
152	C₂⁵	C₂×#6923	1	1	yes	characteristic
153	C₂³×C₄	—	2	4	no	—
154	C₂³×C₄	—	2	4	no	—
155	C₂³×C₄	—	61	8	no	—
156	C₂³×C₄	—	61	4	no	—
157	C₂³×C₄	—	61	4	no	—
158	C₂³×C₄	—	61	8	no	—
159	C₂³×C₄	—	61	3	no	—
160	C₂³×C₄	—	61	3	no	—
161	C₂³×C₄	—	61	8	no	—
162	C₂³×C₄	—	61	6	no	—
163	C₂³×C₄	—	122	4	no	—
164	C₂³×C₄	—	122	4	no	—
165	C₂³×C₄	#92420	1	8	yes	complemented
166	C₂³×C₄	C₂×#6923	1	6	yes	—
167	C₂²×C₈	—	61	48	no	—
168	C₂²×C₈	—	61	8	no	—
169	C₂²×C₈	—	122	64	no	—
170	C₂²×C₈	—	122	24	no	—
171	C₂²×C₈	—	122	24	no	—
172	C₂×C₄²	—	61	4	no	—
173	C₂×C₄²	—	61	12	no	—
174	C₂×C₄²	—	122	24	no	—
175	C₂×C₄²	—	122	24	no	—
176	C₂⁴	D₆₁⋊C₈	1	4	yes	—
177	C₂⁴	#11758213	1	4	yes	—
178	C₂⁴	C₄×#6923	1	4	yes	—
179	C₂⁴	C₄⋊#6923	1	4	yes	—
180	C₂⁴	C₂×#92420	1	4	yes	—
181	C₂⁴	#11758217	1	4	yes	—
182	C₂⁴	C₂²⋊#6923	1	4	yes	—
183	C₂⁴	C₂²×#6923	1	3	yes	—
184	C₂²×C₄	—	2	32	no	—
185	C₂²×C₄	—	2	12	no	—
186	C₂²×C₄	—	2	12	no	—
187	C₂²×C₄	—	61	32	no	—
188	C₂²×C₄	—	61	32	no	—
189	C₂²×C₄	—	61	12	no	—
190	C₂²×C₄	—	61	12	no	—
191	C₂²×C₄	—	61	12	no	—
192	C₂²×C₄	—	61	12	no	—
193	C₂²×C₄	—	61	6	no	—
194	C₂²×C₄	—	61	12	no	—
195	C₂²×C₄	—	61	6	no	—
196	C₂²×C₄	—	61	1	no	—
197	C₂²×C₄	—	61	3	no	—
198	C₂²×C₄	—	61	24	no	—
199	C₂²×C₄	—	61	4	no	—
200	C₂²×C₄	—	122	32	no	—
201	C₂²×C₄	—	122	12	no	—
202	C₂²×C₄	—	122	12	no	—
203	C₂²×C₄	C₂×#92420	1	24	yes	complemented
204	C₂²×C₄	C₂²×#6923	1	4	yes	—
205	C₂×C₈	—	61	16	no	—
206	C₂×C₈	—	122	8	no	—
207	C₂×C₈	—	122	8	no	—
208	C₂×C₈	—	122	96	no	—
209	C₄²	—	122	24	no	—
210	C₄²	—	122	8	no	—
211	C₂³	(C₂×#6923)⋊C₄	1	16	yes	—
212	C₂³	C₄×#92420	1	16	yes	—
213	C₂³	#14519832	1	16	yes	—
214	C₂³	C₂²⋊#92420	1	16	yes	—
215	C₂³	#14519834	1	16	yes	—
216	C₂³	C₄⋊#92420	1	16	yes	—
217	C₂³	#14519844	1	16	yes	—
218	C₂³	C₂×C₄×#6923	1	6	yes	—
219	C₂³	C₂×C₂²⋊#6923	1	6	yes	—
220	C₂³	C₂×C₄⋊#6923	1	6	yes	—
221	C₂³	C₂²×#92420	1	6	yes	—
222	C₂³	C₂×D₆₁⋊C₈	1	6	yes	—
223	C₂³	C₂×#11758217	1	6	yes	—
224	C₂³	C₂×#11758213	1	6	yes	—
225	C₂³	C₂³×#6923	1	1	yes	characteristic
226	C₂×C₄	—	2	48	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	61	6	no	—
230	C₂×C₄	—	61	2	no	—
231	C₂×C₄	—	61	6	no	—
232	C₂×C₄	—	61	2	no	—
233	C₂×C₄	—	61	6	no	—
234	C₂×C₄	—	61	1	no	—
235	C₂×C₄	—	61	1	no	—
236	C₂×C₄	—	61	48	no	—
237	C₂×C₄	—	61	24	no	—
238	C₂×C₄	—	61	24	no	—
239	C₂×C₄	—	61	8	no	—
240	C₂×C₄	—	122	48	no	—
241	C₂×C₄	—	122	4	no	—
242	C₂×C₄	—	122	4	no	—
243	C₂×C₄	C₂²×#92420	1	8	yes	complemented
244	C₈	—	122	16	no	—
245	C₂²	#16840628	1	64	yes	direct product factor
246	C₂²	C₂×(C₂×#6923)⋊C₄	1	12	yes	—
247	C₂²	C₂×C₄×#92420	1	12	yes	—
248	C₂²	C₂×#14519832	1	12	yes	—
249	C₂²	C₂×C₂²⋊#92420	1	12	yes	—
250	C₂²	C₂×#14519834	1	12	yes	—
251	C₂²	C₂×C₄⋊#92420	1	12	yes	—
252	C₂²	C₂×#14519844	1	12	yes	—
253	C₂²	C₂²×C₄×#6923	1	1	yes	characteristic
254	C₂²	C₂²×C₂²⋊#6923	1	1	yes	characteristic
255	C₂²	C₂²×C₄⋊#6923	1	1	yes	characteristic
256	C₂²	C₂³×#92420	1	1	yes	characteristic
257	C₂²	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
258	C₂²	C₂²×#11758217	1	1	yes	characteristic
259	C₂²	C₂²×#11758213	1	1	yes	characteristic
260	C₄	—	2	8	no	—
261	C₄	—	61	2	no	—
262	C₄	—	61	2	no	—
263	C₄	—	61	12	no	—
264	C₄	—	122	8	no	—
265	C₂	C₂×#16840628	1	24	yes	direct product factor
266	C₂	C₂²×(C₂×#6923)⋊C₄	1	1	yes	characteristic
267	C₂	C₂²×C₄×#92420	1	1	yes	characteristic
268	C₂	C₂²×#14519832	1	1	yes	characteristic
269	C₂	C₂²×C₂²⋊#92420	1	1	yes	characteristic
270	C₂	C₂²×#14519834	1	1	yes	characteristic
271	C₂	C₂²×C₄⋊#92420	1	1	yes	characteristic
272	C₂	C₂²×#14519844	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂×#6923
commutator	C₂×C₆₁	C₂²×C₄×C₈
Frattini	C₂³	C₂³×#6923
Fitting	C₂⁴×C₄×C₆₁	C₄
socle, prime radical	C₂⁵×C₆₁	C₂×C₄
cosocle	C₂²×Dic₆₁	C₂⁴
hypercenter	C₂⁴×C₄	#6923
hypocenter	C₆₁	C₂²×#336
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#16840628	C₂	direct (N × H)
6. C₂³×#92420	C₄	semidirect
8. #16840628	C₂²	direct (N × H)
18. C₂²×#92420	C₂×C₄	semidirect
22. C₂³×Dic₆₁	C₈	semidirect
37. C₂³×C₄×C₆₁	C₈	semidirect
50. C₂²×C₄×C₆₁	C₂×C₈	semidirect
60. C₂²×Dic₆₁	C₂×C₈	semidirect
71. C₂×#92420	C₂²×C₄	semidirect
92. C₂×C₄×C₆₁	C₂²×C₈	semidirect
102. C₂×Dic₆₁	C₂²×C₈	semidirect
109. C₂²×C₆₁	#336	semidirect
132. C₂×C₆₁	C₂×#336	semidirect
151. C₆₁	C₂²×#336	semidirect
165. C₂³×C₄	#92420	semidirect
203. C₂²×C₄	C₂×#92420	semidirect
243. C₂×C₄	C₂²×#92420	semidirect
245. C₂²	#16840628	direct (N × H)
265. C₂	C₂×#16840628	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×#92420	2
C₂×#16840628	2
C₂³×C₄×Dic₆₁	2
C₂²×#336	61

Subgroups — C22×#16840628 #30613837

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16840628 #30613837