Subgroups — C₂×Q₃₂×#6923 #30624283

Order 15616.

678 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×#6923	C₂	1	2	yes	maximal
2	Q₃₂×#6923	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16840688	C₂	1	1	yes	maximal, characteristic
4	C₂×#16840675	C₂	1	2	yes	maximal
5	C₂×C₁₆×#6923	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₃₂×D₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₈×#6923	—	2	2	no	—
8	C₂×D₆₁⋊C₁₆	C₂²	1	1	yes	characteristic
9	Q₁₆×#6923	—	2	4	no	—
10	Q₁₆×#6923	C₂²	1	8	yes	—
11	C₂×#14519846	C₂²	1	2	yes	—
12	C₂×#14519840	—	2	2	no	—
13	C₂×C₈×#6923	C₂²	1	1	yes	characteristic
14	#16840688	C₂²	1	4	yes	—
15	#16840675	C₂²	1	8	yes	—
16	C₁₆×#6923	C₂²	1	4	yes	—
17	C₂×Q₁₆×D₆₁	C₂²	1	2	yes	—
18	C₂×Dic₄₈₈	C₄	1	1	yes	characteristic, complemented
19	C₂×#14519723	C₄	1	2	yes	complemented
20	Q₃₂×D₆₁	C₄	1	4	yes	complemented
21	Q₃₂×D₆₁	C₂²	1	4	yes	—
22	C₂×C₁₆×D₆₁	C₂²	1	1	yes	characteristic
23	C₂×Q₃₂×C₆₁	C₄	1	1	yes	characteristic, complemented
24	C₂×D₆₁⋊C₈	D₄	1	1	yes	characteristic
25	Q₈×#6923	—	2	8	no	—
26	Q₈×#6923	—	4	4	no	—
27	C₂×C₄⋊#6923	—	2	2	no	—
28	C₂×C₄⋊#6923	—	4	2	no	—
29	C₂×C₄×#6923	—	4	2	no	—
30	C₂×C₄×#6923	D₄	1	1	yes	characteristic
31	D₆₁⋊C₁₆	—	2	1	no	—
32	D₆₁⋊C₁₆	C₂³	1	2	yes	—
33	C₂×#11758187	—	2	1	no	—
34	#14519846	—	2	2	no	—
35	#14519846	C₂³	1	4	yes	—
36	#14519840	—	2	8	no	—
37	C₈×#6923	D₄	1	2	yes	—
38	C₈×#6923	C₂³	1	2	yes	—
39	C₂×Q₈×D₆₁	—	2	2	no	—
40	C₂×Dic₂₄₄	C₂×C₄	1	2	yes	—
41	C₂×#11758202	—	2	2	no	—
42	Q₁₆×D₆₁	—	2	2	no	—
43	Q₁₆×D₆₁	—	2	2	no	—
44	Q₁₆×D₆₁	C₂×C₄	1	4	yes	—
45	Q₁₆×D₆₁	C₂³	1	4	yes	—
46	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
47	Dic₄₈₈	C₂×C₄	1	4	yes	complemented
48	#14519723	C₂×C₄	1	8	yes	complemented
49	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
50	C₁₆×D₆₁	C₂×C₄	1	2	yes	—
51	C₁₆×D₆₁	C₂³	1	2	yes	—
52	C₂×Q₁₆×C₆₁	C₂×C₄	1	2	yes	—
53	Q₃₂×C₆₁	C₂×C₄	1	4	yes	complemented
54	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
55	C₂²×#6923	—	4	2	no	—
56	C₂²×#6923	D₈	1	1	yes	characteristic
57	C₂²×#6923	D₈	1	1	yes	characteristic
58	C₂×Q₈×C₆₁	—	2	2	no	—
59	Q₈×D₆₁	—	2	4	no	—
60	Q₈×D₆₁	—	2	4	no	—
61	Q₈×D₆₁	—	4	2	no	—
62	Q₈×D₆₁	—	4	2	no	—
63	C₂×C₄×D₆₁	—	4	2	no	—
64	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
65	C₂×Dic₁₂₂	—	2	2	no	—
66	C₂×Dic₁₂₂	—	4	2	no	—
67	C₂×#92420	—	2	1	no	—
68	C₄⋊#6923	—	2	4	no	—
69	C₄⋊#6923	—	4	8	no	—
70	C₄⋊#6923	—	4	2	no	—
71	C₄×#6923	—	4	8	no	—
72	C₄×#6923	D₈	1	2	yes	—
73	C₄×#6923	C₂×D₄	1	2	yes	—
74	D₆₁⋊C₈	—	2	1	no	—
75	D₆₁⋊C₈	C₂×D₄	1	2	yes	—
76	Q₁₆×C₆₁	—	2	2	no	—
77	Q₁₆×C₆₁	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
79	#11758202	—	2	4	no	—
80	#11758202	—	2	4	no	—
81	C₂×#92418	#41	1	1	yes	characteristic
82	Dic₂₄₄	—	2	2	no	—
83	Dic₂₄₄	C₂²×C₄	1	4	yes	—
84	C₈×D₆₁	C₂²×C₄	1	1	yes	characteristic
85	C₈×D₆₁	C₂×D₄	1	1	yes	characteristic
86	C₈×D₆₁	#41	1	1	yes	characteristic
87	C₈×D₆₁	C₂⁴	1	1	yes	characteristic
88	#11758187	—	2	2	no	—
89	C₁₆×C₆₁	C₂²×C₄	1	2	yes	—
90	#11758185	C₂²×C₄	1	2	yes	—
91	C₂²×D₆₁	C₂×D₈	1	1	yes	characteristic
92	C₂×#6923	—	2	2	no	—
93	C₂×#6923	—	4	4	no	—
94	C₂×#6923	—	8	4	no	—
95	C₂×#6923	Q₃₂	1	4	yes	direct product factor
96	C₂×#6923	C₂×D₈	1	2	yes	—
97	C₂×#6923	C₂×D₈	1	2	yes	—
98	Q₈×C₆₁	—	2	4	no	—
99	Q₈×C₆₁	—	4	2	no	—
100	C₂×C₄×C₆₁	—	4	2	no	—
101	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
102	C₂×Dic₆₁	—	4	2	no	—
103	C₂×Dic₆₁	#135	1	1	yes	characteristic
104	C₄×D₆₁	—	4	4	no	—
105	C₄×D₆₁	—	4	4	no	—
106	C₄×D₆₁	C₄×D₄	1	1	yes	characteristic
107	C₄×D₆₁	C₂×D₈	1	1	yes	characteristic
108	C₄×D₆₁	#135	1	1	yes	characteristic
109	C₄×D₆₁	C₂²×D₄	1	1	yes	characteristic
110	Dic₁₂₂	—	2	4	no	—
111	Dic₁₂₂	—	4	2	no	—
112	Dic₁₂₂	—	4	4	no	—
113	Dic₁₂₂	—	4	4	no	—
114	#92420	—	2	2	no	—
115	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
116	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
117	#92418	C₄×D₄	1	1	yes	characteristic
118	#92418	C₂×#41	1	1	yes	characteristic
119	C₂×C₄×Q₃₂	—	61	1	no	maximal, Sylow
120	C₂²×C₆₁	C₄×D₈	1	1	yes	characteristic
121	C₂×D₆₁	C₄×D₈	1	1	yes	characteristic
122	C₂×D₆₁	C₂×Q₃₂	1	2	yes	—
123	C₂×D₆₁	#508	1	2	yes	—
124	C₂×D₆₁	C₂²×D₈	1	1	yes	characteristic
125	#6923	—	2	2	no	—
126	#6923	—	8	4	no	—
127	#6923	C₂×Q₃₂	1	4	yes	direct product factor
128	C₄×C₆₁	—	4	4	no	—
129	C₄×C₆₁	C₄×D₈	1	1	yes	characteristic
130	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
131	Dic₆₁	—	4	4	no	—
132	Dic₆₁	C₄×D₈	1	1	yes	characteristic
133	Dic₆₁	C₂×#135	1	1	yes	characteristic
134	C₂²×Q₃₂	—	61	1	no	—
135	C₂×C₄×Q₁₆	—	61	2	no	—
136	C₄×Q₃₂	—	61	8	no	—
137	C₂×#366	—	61	1	no	—
138	C₂×#358	—	61	2	no	—
139	C₂×C₄×C₁₆	—	61	1	no	—
140	C₂×C₆₁	C₄×Q₃₂	1	2	yes	complemented
141	C₂×C₆₁	C₂×C₄×D₈	1	1	yes	characteristic
142	D₆₁	C₄×Q₃₂	1	2	yes	complemented
143	D₆₁	C₂²×Q₃₂	1	1	yes	characteristic
144	D₆₁	C₂×#508	1	1	yes	characteristic
145	C₂²×Q₁₆	—	61	2	no	—
146	C₂×C₄×Q₈	—	122	2	no	—
147	C₂×Q₃₂	—	61	2	no	—
148	C₂×Q₃₂	—	61	4	no	—
149	C₂×Q₃₂	—	61	4	no	—
150	C₂×Q₃₂	—	61	1	no	—
151	C₂×Q₃₂	#6923	1	1	yes	characteristic, direct product factor
152	C₂²×C₁₆	—	61	1	no	—
153	C₂²×C₁₆	—	61	1	no	—
154	C₄×Q₁₆	—	61	8	no	—
155	C₄×Q₁₆	—	122	4	no	—
156	C₂×#107	—	61	2	no	—
157	C₂×#103	—	122	2	no	—
158	C₂×C₄×C₈	—	61	1	no	—
159	#366	—	61	4	no	—
160	#358	—	61	8	no	—
161	C₄×C₁₆	—	61	4	no	—
162	C₆₁	C₂×C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
163	C₂²×Q₈	—	122	2	no	—
164	C₂×Q₁₆	—	61	4	no	—
165	C₂×Q₁₆	—	61	4	no	—
166	C₂×Q₁₆	—	61	2	no	—
167	C₂×Q₁₆	—	122	2	no	—
168	C₂×Q₁₆	—	122	2	no	—
169	C₂×Q₁₆	—	122	2	no	—
170	C₂×Q₁₆	C₂×#6923	1	2	yes	—
171	C₂²×C₈	—	61	1	no	—
172	C₂²×C₈	—	61	1	no	—
173	C₄×Q₈	—	122	8	no	—
174	C₄×Q₈	—	244	4	no	—
175	C₂×C₄⋊C₄	—	122	2	no	—
176	C₂×C₄⋊C₄	—	244	2	no	—
177	C₂×C₄²	—	61	1	no	—
178	C₂×C₄²	—	244	2	no	—
179	Q₃₂	—	61	8	no	—
180	Q₃₂	—	61	4	no	—
181	Q₃₂	C₂×#6923	1	4	yes	direct product factor
182	C₂×C₁₆	—	61	1	no	—
183	C₂×C₁₆	—	61	2	no	—
184	C₂×C₁₆	—	61	2	no	—
185	C₂×C₁₆	—	61	2	no	—
186	C₂×C₁₆	—	122	1	no	—
187	C₂×C₁₆	—	122	1	no	—
188	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
189	#107	—	61	4	no	—
190	#107	—	122	2	no	—
191	#103	—	122	8	no	—
192	C₄×C₈	—	61	2	no	—
193	C₄×C₈	—	61	2	no	—
194	C₂×Q₈	—	2	2	no	—
195	C₂×Q₈	—	122	4	no	—
196	C₂×Q₈	—	122	4	no	—
197	C₂×Q₈	—	122	2	no	—
198	C₂×Q₈	—	244	2	no	—
199	C₂×Q₈	—	244	2	no	—
200	C₂×Q₈	—	244	2	no	—
201	C₂²×C₄	—	61	1	no	—
202	C₂²×C₄	—	61	1	no	—
203	C₂²×C₄	—	61	1	no	—
204	C₂²×C₄	—	244	2	no	—
205	C₂²×C₄	—	244	2	no	—
206	Q₁₆	—	2	2	no	—
207	Q₁₆	—	61	4	no	—
208	Q₁₆	—	122	4	no	—
209	Q₁₆	—	122	4	no	—
210	Q₁₆	—	122	2	no	—
211	Q₁₆	C₂²×#6923	1	4	yes	—
212	C₂×C₈	—	61	1	no	—
213	C₂×C₈	—	61	1	no	—
214	C₂×C₈	—	61	1	no	—
215	C₂×C₈	—	61	1	no	—
216	C₂×C₈	—	61	1	no	—
217	C₂×C₈	—	61	2	no	—
218	C₂×C₈	—	122	1	no	—
219	C₂×C₈	—	122	1	no	—
220	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
221	C₄⋊C₄	—	122	4	no	—
222	C₄⋊C₄	—	244	2	no	—
223	C₄⋊C₄	—	244	8	no	—
224	C₄²	—	61	2	no	—
225	C₄²	—	61	2	no	—
226	C₄²	—	244	8	no	—
227	C₁₆	—	61	2	no	—
228	C₁₆	—	122	2	no	—
229	C₁₆	C₂²×#6923	1	2	yes	—
230	C₂³	—	61	1	no	—
231	Q₈	—	2	4	no	—
232	Q₈	—	4	2	no	—
233	Q₈	—	122	4	no	—
234	Q₈	—	244	4	no	—
235	Q₈	—	244	4	no	—
236	Q₈	—	244	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	61	4	no	—
239	C₂×C₄	—	61	2	no	—
240	C₂×C₄	—	61	1	no	—
241	C₂×C₄	—	61	1	no	—
242	C₂×C₄	—	61	1	no	—
243	C₂×C₄	—	61	1	no	—
244	C₂×C₄	—	61	1	no	—
245	C₂×C₄	—	61	2	no	—
246	C₂×C₄	—	122	2	no	—
247	C₂×C₄	—	488	4	no	—
248	C₂×C₄	—	244	4	no	—
249	C₂×C₄	—	244	4	no	—
250	C₂×C₄	—	244	4	no	—
251	C₂×C₄	—	244	2	no	—
252	C₂×C₄	D₄×#6923	1	1	yes	characteristic
253	C₈	—	61	1	no	—
254	C₈	—	61	1	no	—
255	C₈	—	122	2	no	—
256	C₈	D₄×#6923	1	1	yes	characteristic
257	C₈	C₂³×#6923	1	1	yes	characteristic
258	C₂²	—	61	1	no	—
259	C₂²	—	61	1	no	—
260	C₂²	—	61	2	no	—
261	C₂²	—	61	2	no	—
262	C₂²	D₈×#6923	1	1	yes	characteristic
263	C₄	—	4	4	no	—
264	C₄	—	61	4	no	—
265	C₄	—	61	1	no	—
266	C₄	—	61	1	no	—
267	C₄	—	122	2	no	—
268	C₄	—	488	4	no	—
269	C₄	—	244	4	no	—
270	C₄	D₈×#6923	1	1	yes	characteristic
271	C₄	C₂×D₄×#6923	1	1	yes	characteristic
272	C₂	—	61	1	no	—
273	C₂	—	61	1	no	—
274	C₂	—	61	2	no	—
275	C₂	Q₃₂×#6923	1	2	yes	direct product factor
276	C₂	C₂×D₈×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×#6923
commutator	C₈×C₆₁	C₂³×C₄
Frattini	C₈	C₂³×#6923
Fitting	C₂×Q₃₂×C₆₁	C₄
socle	C₂²×C₆₁	C₄×D₈
prime radical	C₂²×D₆₁	C₂×D₈
cosocle	C₈×D₆₁	C₂⁴
hypercenter	C₂×Q₃₂	#6923
hypocenter	C₆₁	C₂×C₄×Q₃₂
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
2. Q₃₂×#6923	C₂	direct (N × H)
18. C₂×Dic₄₈₈	C₄	semidirect
19. C₂×#14519723	C₄	semidirect
20. Q₃₂×D₆₁	C₄	semidirect
23. C₂×Q₃₂×C₆₁	C₄	semidirect
47. Dic₄₈₈	C₂×C₄	semidirect
48. #14519723	C₂×C₄	semidirect
53. Q₃₂×C₆₁	C₂×C₄	semidirect
95. C₂×#6923	Q₃₂	direct (N × H)
127. #6923	C₂×Q₃₂	direct (N × H)
140. C₂×C₆₁	C₄×Q₃₂	semidirect
142. D₆₁	C₄×Q₃₂	semidirect
151. C₂×Q₃₂	#6923	direct (N × H)
162. C₆₁	C₂×C₄×Q₃₂	semidirect
181. Q₃₂	C₂×#6923	direct (N × H)
275. C₂	Q₃₂×#6923	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×Q₁₆×#6923	2
Q₃₂×#6923	2
C₂×#16840688	2
C₂×#16840675	2
C₂×C₁₆×#6923	2
C₂×Q₃₂×D₆₁	2
C₂×C₄×Q₃₂	61

Subgroups — C2×Q32×#6923 #30624283

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₃₂×#6923 #30624283