Subgroups — C₄×#16840660 #30606403

Order 15616.

490 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840660	C₂	1	2	yes	maximal
2	C₄×#14519851	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×#11758187	C₂	1	1	yes	maximal, characteristic, complemented
4	#21460873	C₂	1	2	yes	maximal, complemented
5	C₄×#14519784	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519851	C₂²	1	2	yes	—
7	C₂²×#11758187	C₂²	1	2	yes	—
8	#16840660	C₄	1	8	yes	direct product factor
9	#16840650	—	2	1	no	—
10	C₄×#11758187	—	2	1	no	—
11	C₄×#11758187	—	2	1	no	—
12	C₄×#11758187	—	2	1	no	—
13	C₂×#14519784	C₄	1	1	yes	characteristic
14	C₂×#14519784	C₂²	1	1	yes	characteristic
15	#16839718	—	2	1	no	—
16	#16839718	—	2	1	no	—
17	#16839718	C₄	1	1	yes	characteristic
18	#16839718	C₂²	1	1	yes	characteristic
19	C₄×#11758194	—	2	1	no	—
20	C₄×#11758194	C₄	1	1	yes	characteristic
21	C₂×C₄×#92418	—	2	1	no	—
22	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
23	C₄×#41×C₆₁	C₄	1	1	yes	characteristic
24	#14519851	—	2	2	no	—
25	#14519851	C₂×C₄	1	4	yes	complemented
26	C₂×#11758187	—	2	2	no	—
27	C₂×#11758187	—	2	2	no	—
28	C₂×#11758187	—	2	2	no	—
29	C₂×#11758187	C₂×C₄	1	4	yes	complemented
30	#14519784	—	2	2	no	—
31	#14519784	—	2	2	no	—
32	#14519784	C₈	1	4	yes	—
33	#14519784	C₂×C₄	1	4	yes	—
34	C₂×#11758194	—	2	1	no	—
35	C₂×#11758194	—	2	1	no	—
36	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
37	C₂×#11758194	C₂×C₄	1	1	yes	characteristic
38	C₂²×#92418	—	2	1	no	—
39	C₂²×#92418	—	2	1	no	—
40	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
41	C₂²×#92418	C₂³	1	1	yes	characteristic
42	#14519688	—	2	1	no	—
43	#14519688	—	2	1	no	—
44	#14519688	—	2	1	no	—
45	C₄×#92418	—	2	1	no	—
46	C₄×#92418	—	2	1	no	—
47	C₄×#92418	—	2	1	no	—
48	C₄×#92418	D₄	1	1	yes	characteristic
49	C₄×#92418	D₄	1	1	yes	characteristic
50	C₄×#92418	D₄	1	1	yes	characteristic
51	C₄×#92418	D₄	1	1	yes	characteristic
52	C₂×#41×C₆₁	C₂×C₄	1	1	yes	characteristic
53	C₄×Q₈×C₆₁	C₈	1	2	yes	—
54	C₄×D₄×C₆₁	—	2	2	no	—
55	C₄×D₄×C₆₁	C₈	1	2	yes	—
56	#117×C₆₁	—	2	1	no	—
57	#117×C₆₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₁	—	2	1	no	—
59	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
60	#41×C₆₁	—	2	1	no	—
61	#41×C₆₁	—	2	1	no	—
62	#41×C₆₁	C₄²	1	2	yes	—
63	#41×C₆₁	C₂×C₈	1	2	yes	—
64	C₂×Q₈×C₆₁	C₂×C₈	1	1	yes	characteristic
65	C₂×D₄×C₆₁	—	2	1	no	—
66	C₂×D₄×C₆₁	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₆₁	—	2	1	no	—
68	C₂²×C₄×C₆₁	—	2	2	no	—
69	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
70	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
71	C₄⋊C₄×C₆₁	—	2	2	no	—
72	C₄⋊C₄×C₆₁	C₂×C₈	1	2	yes	—
73	C₂²⋊C₄×C₆₁	—	2	1	no	—
74	C₂²⋊C₄×C₆₁	—	2	2	no	—
75	C₄²×C₆₁	—	2	2	no	—
76	C₄²×C₆₁	—	2	1	no	—
77	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²×C₆₁	C₂×C₈	1	2	yes	—
80	#11758194	—	2	1	no	—
81	#11758194	—	2	1	no	—
82	#11758194	—	2	4	no	—
83	#11758194	—	2	4	no	—
84	#11758194	C₄²	1	2	yes	—
85	#11758194	C₂×C₈	1	2	yes	—
86	C₂×#92418	—	2	1	no	—
87	C₂×#92418	—	2	1	no	—
88	C₂×#92418	—	2	4	no	—
89	C₂×#92418	—	2	4	no	—
90	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×#92418	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×#92418	C₂×C₈	1	2	yes	—
93	C₂×#92418	M₁₆	1	2	yes	—
94	C₂×#92418	C₂²×C₄	1	2	yes	—
95	C₂×#92418	C₂×D₄	1	1	yes	characteristic
96	C₂×#92418	C₂×D₄	1	1	yes	characteristic
97	C₂×#92418	#41	1	2	yes	—
98	#11758187	—	2	4	no	—
99	#11758187	—	2	4	no	—
100	C₂³×C₆₁	—	2	1	no	—
101	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
102	Q₈×C₆₁	—	2	1	no	—
103	Q₈×C₆₁	C₄×C₈	1	2	yes	—
104	D₄×C₆₁	—	2	2	no	—
105	D₄×C₆₁	—	2	2	no	—
106	D₄×C₆₁	—	2	1	no	—
107	D₄×C₆₁	C₄×C₈	1	2	yes	—
108	C₂×C₄×C₆₁	—	2	1	no	—
109	C₂×C₄×C₆₁	—	2	4	no	—
110	C₂×C₄×C₆₁	—	2	2	no	—
111	C₂×C₄×C₆₁	—	2	2	no	—
112	C₂×C₄×C₆₁	—	2	2	no	—
113	C₂×C₄×C₆₁	—	4	4	no	—
114	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
115	C₂×C₄×C₆₁	C₂²⋊C₈	1	2	yes	—
116	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
119	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
122	#92418	—	2	2	no	—
123	#92418	—	2	2	no	—
124	#92418	C₂²⋊C₈	1	2	yes	—
125	#92418	C₂²⋊C₈	1	2	yes	—
126	#92418	C₄×D₄	1	2	yes	—
127	#92418	C₄×D₄	1	2	yes	—
128	C₄×#350	—	61	1	no	maximal, Sylow
129	C₂²×C₆₁	—	2	2	no	—
130	C₂²×C₆₁	—	2	2	no	—
131	C₂²×C₆₁	—	4	2	no	—
132	C₂²×C₆₁	C₄×M₁₆	1	1	yes	characteristic
133	C₂²×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
134	C₂²×C₆₁	#431	1	1	yes	characteristic
135	C₄×C₆₁	—	2	2	no	—
136	C₄×C₆₁	—	2	2	no	—
137	C₄×C₆₁	—	4	2	no	—
138	C₄×C₆₁	#350	1	4	yes	complemented
139	C₄×C₆₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
140	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
141	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
142	C₄×C₆₁	#431	1	1	yes	characteristic
143	C₄×#131	—	61	1	no	—
144	C₂×#350	—	61	2	no	—
145	C₄×M₃₂	—	61	1	no	—
146	C₂×C₄×C₁₆	—	61	1	no	—
147	#1379	—	61	2	no	—
148	C₂×C₆₁	—	2	1	no	—
149	C₂×C₆₁	—	2	1	no	—
150	C₂×C₆₁	—	4	2	no	—
151	C₂×C₆₁	C₄×C₂²⋊C₈	1	1	yes	characteristic
152	C₂×C₆₁	C₂×#350	1	2	yes	—
153	C₂×#131	—	61	1	no	—
154	C₂×#131	—	61	1	no	—
155	C₄×#41	#6923	1	1	yes	characteristic
156	C₂×M₃₂	—	61	2	no	—
157	C₂²×C₁₆	—	61	2	no	—
158	#405	—	61	1	no	—
159	#405	—	61	1	no	—
160	#405	—	122	1	no	—
161	#405	—	122	1	no	—
162	C₄×M₁₆	—	61	1	no	—
163	C₄×M₁₆	—	122	1	no	—
164	C₂×C₄×C₈	—	61	1	no	—
165	C₂×C₄×C₈	—	122	1	no	—
166	#350	—	61	8	no	—
167	#346	—	122	1	no	—
168	C₄×C₁₆	—	122	1	no	—
169	C₄×C₁₆	—	122	1	no	—
170	C₄×C₁₆	—	122	1	no	—
171	C₆₁	C₄×#350	1	1	yes	characteristic, complemented, Sylow
172	C₂×#41	C₂×#6923	1	1	yes	characteristic
173	#131	—	61	4	no	—
174	#131	—	61	4	no	—
175	#131	—	122	2	no	—
176	#131	—	122	2	no	—
177	C₂×M₁₆	—	61	1	no	—
178	C₂×M₁₆	—	61	1	no	—
179	C₂×M₁₆	—	122	1	no	—
180	C₂×M₁₆	—	122	1	no	—
181	C₂²×C₈	—	61	1	no	—
182	C₂²×C₈	—	61	1	no	—
183	C₂²×C₈	—	122	1	no	—
184	C₂²×C₈	—	122	1	no	—
185	C₄×Q₈	#92420	1	2	yes	—
186	C₄×D₄	—	2	2	no	—
187	C₄×D₄	#92420	1	2	yes	—
188	#117	—	2	1	no	—
189	#117	C₂×#6923	1	1	yes	characteristic
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
192	M₃₂	—	61	4	no	—
193	M₃₂	—	122	2	no	—
194	C₂×C₁₆	—	61	4	no	—
195	C₂×C₁₆	—	122	2	no	—
196	C₂×C₁₆	—	122	2	no	—
197	C₂×C₁₆	—	122	2	no	—
198	#97	—	122	1	no	—
199	#97	—	122	1	no	—
200	#97	—	122	1	no	—
201	C₄×C₈	—	61	1	no	—
202	C₄×C₈	—	61	1	no	—
203	C₄×C₈	—	61	1	no	—
204	C₄×C₈	—	61	1	no	—
205	C₄×C₈	—	122	1	no	—
206	C₄×C₈	—	122	1	no	—
207	C₄×C₈	—	122	1	no	—
208	#41	—	2	1	no	—
209	#41	—	2	1	no	—
210	#41	D₆₁⋊C₈	1	2	yes	—
211	#41	C₄×#6923	1	2	yes	—
212	C₂×Q₈	C₂×#92420	1	1	yes	characteristic
213	C₂×D₄	—	2	1	no	—
214	C₂×D₄	C₂×#92420	1	1	yes	characteristic
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	#11758217	1	2	yes	—
218	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
219	M₁₆	—	61	2	no	—
220	M₁₆	—	61	2	no	—
221	M₁₆	—	122	4	no	—
222	M₁₆	—	122	4	no	—
223	M₁₆	—	122	1	no	—
224	M₁₆	—	122	1	no	—
225	C₂×C₈	—	61	2	no	—
226	C₂×C₈	—	61	1	no	—
227	C₂×C₈	—	61	1	no	—
228	C₂×C₈	—	61	1	no	—
229	C₂×C₈	—	61	1	no	—
230	C₂×C₈	—	61	2	no	—
231	C₂×C₈	—	61	2	no	—
232	C₂×C₈	—	61	2	no	—
233	C₂×C₈	—	122	4	no	—
234	C₂×C₈	—	122	4	no	—
235	C₂×C₈	—	122	1	no	—
236	C₂×C₈	—	122	1	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	C₂×#92420	1	2	yes	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	1	no	—
243	C₄²	C₂×#92420	1	2	yes	—
244	C₄²	C₂²⋊#6923	1	1	yes	characteristic
245	C₄²	C₂²⋊#6923	1	1	yes	characteristic
246	C₁₆	—	122	4	no	—
247	C₁₆	—	122	4	no	—
248	C₂³	—	2	1	no	—
249	C₂³	C₂×#11758217	1	1	yes	characteristic
250	Q₈	—	2	1	no	—
251	Q₈	C₄×#92420	1	2	yes	—
252	D₄	—	2	2	no	—
253	D₄	—	2	2	no	—
254	D₄	—	2	1	no	—
255	D₄	C₄×#92420	1	2	yes	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	C₂²⋊#92420	1	2	yes	—
263	C₂×C₄	C₂²⋊#92420	1	2	yes	—
264	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊#6923	1	1	yes	characteristic
266	C₂×C₄	#14519856	1	1	yes	characteristic
267	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
268	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
269	C₂×C₄	D₆₁⋊M₁₆	1	1	yes	characteristic
270	C₈	—	61	2	no	—
271	C₈	—	61	2	no	—
272	C₈	—	61	2	no	—
273	C₈	—	61	2	no	—
274	C₈	—	122	2	no	—
275	C₈	—	122	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	C₄×#11758217	1	1	yes	characteristic
280	C₂²	C₂×C₂²⋊#92420	1	1	yes	characteristic
281	C₂²	#16840779	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	4	2	no	—
285	C₄	#16840660	1	4	yes	direct product factor
286	C₄	C₄×C₂²⋊#6923	1	1	yes	characteristic
287	C₄	C₂×C₄×#92420	1	1	yes	characteristic
288	C₄	D₆₁⋊(C₂²⋊C₈)	1	1	yes	characteristic
289	C₄	#16840778	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	4	2	no	—
293	C₂	C₄×C₂²⋊#92420	1	1	yes	characteristic
294	C₂	C₂×#16840660	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²⋊#6923
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₄×#41×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₆₁	C₂×C₄
cosocle	C₂²×#92418	C₂³
hypercenter	C₄×#41	#6923
hypocenter	C₆₁	C₄×#350
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. C₄×#14519851	C₂	semidirect
3. C₂×C₄×#11758187	C₂	semidirect
4. #21460873	C₂	semidirect
8. #16840660	C₄	direct (N × H)
25. #14519851	C₂×C₄	semidirect
29. C₂×#11758187	C₂×C₄	semidirect
138. C₄×C₆₁	#350	semidirect
171. C₆₁	C₄×#350	semidirect
285. C₄	#16840660	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16840660	2
C₄×#14519851	2
C₂×C₄×#11758187	2
#21460873	2
C₄×#14519784	2
C₄×#350	61

Subgroups — C4×#16840660 #30606403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16840660 #30606403