Subgroups — #334×#6923 #30598005

Order 15616.

470 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₈×#6923	C₂	1	2	yes	maximal
2	C₄×C₈×#6923	C₂	1	1	yes	maximal, characteristic
3	D₆₁⋊#334	C₂	1	1	yes	maximal, characteristic
4	#21460648	C₂	1	2	yes	maximal
5	#334×D₆₁	C₂	1	1	yes	maximal, characteristic
6	D₆₁⋊(C₄⋊C₈)	C₂²	1	2	yes	—
7	C₄×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×#6923	—	2	2	no	—
9	C₂×C₈×#6923	C₄	1	2	yes	—
10	C₂×C₈×#6923	C₄	1	2	yes	—
11	C₄²×#6923	C₂²	1	1	yes	characteristic
12	(C₂×#6923)⋊C₈	—	2	2	no	—
13	#16840624	C₄	1	2	yes	complemented
14	#16840623	C₄	1	2	yes	complemented
15	C₄⋊C₈×D₆₁	C₂²	1	2	yes	—
16	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
17	#16839513	C₄	1	1	yes	characteristic, complemented
18	#16839512	C₄	1	2	yes	complemented
19	C₆₁×#334	C₄	1	1	yes	characteristic, complemented
20	C₂×D₆₁⋊C₈	—	2	2	no	—
21	C₂×D₆₁⋊C₈	C₂×C₄	1	2	yes	—
22	C₂×C₄×#6923	C₂×C₄	1	2	yes	—
23	C₂×C₄×#6923	D₄	1	1	yes	characteristic
24	C₂×C₄×#6923	D₄	1	1	yes	characteristic
25	C₂×C₄×#6923	Q₈	1	2	yes	—
26	#14519844	—	2	2	no	—
27	C₄⋊#92420	C₂×C₄	1	4	yes	—
28	C₄×#92420	—	2	1	no	—
29	C₄×#92420	—	2	1	no	—
30	C₈×#6923	—	2	4	no	—
31	C₈×#6923	—	4	4	no	—
32	C₈×#6923	C₈	1	8	yes	complemented
33	C₂×C₈×D₆₁	—	2	2	no	—
34	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
36	C₄²×D₆₁	C₂³	1	1	yes	characteristic
37	C₄⋊#92418	C₂×C₄	1	2	yes	—
38	#14519706	—	2	2	no	—
39	C₄×#92418	C₂×C₄	1	1	yes	characteristic
40	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
41	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
42	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×#6923	C₄⋊C₄	1	2	yes	—
45	C₂²×#6923	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×D₆₁	C₂×Q₈	1	1	yes	characteristic
49	C₂×#92420	—	2	2	no	—
50	C₂×#92420	—	2	4	no	—
51	C₄×#6923	—	2	4	no	—
52	C₄×#6923	—	2	2	no	—
53	C₄×#6923	C₂×C₈	1	4	yes	—
54	C₄×#6923	M₁₆	1	4	yes	—
55	C₄×#6923	SD₁₆	1	4	yes	—
56	D₆₁⋊C₈	—	2	2	no	—
57	D₆₁⋊C₈	—	4	2	no	—
58	D₆₁⋊C₈	—	4	2	no	—
59	D₆₁⋊C₈	C₂×C₈	1	4	yes	—
60	C₂×C₈×C₆₁	—	2	2	no	—
61	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
62	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
63	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
64	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
65	C₄×Dic₆₁	#41	1	1	yes	characteristic
66	C₄×Dic₆₁	#41	1	1	yes	characteristic
67	C₂×#92418	—	2	2	no	—
68	C₂×#92418	C₄²	1	1	yes	characteristic
69	C₂×#92418	C₄²	1	1	yes	characteristic
70	C₈×D₆₁	—	2	2	no	—
71	C₈×D₆₁	—	4	2	no	—
72	C₈×D₆₁	—	4	2	no	—
73	C₈×D₆₁	C₂×C₈	1	4	yes	—
74	C₂²×D₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×#6923	—	2	2	no	—
76	C₂×#6923	—	2	2	no	—
77	C₂×#6923	C₄⋊C₈	1	4	yes	—
78	C₂×#6923	C₄⋊C₈	1	4	yes	—
79	C₂×#6923	#106	1	4	yes	—
80	C₂×#6923	#108	1	4	yes	—
81	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
84	C₂×Dic₆₁	#117	1	1	yes	characteristic
85	C₂×Dic₆₁	C₄×D₄	1	1	yes	characteristic
86	C₂×Dic₆₁	C₄×Q₈	1	1	yes	characteristic
87	C₄×D₆₁	—	2	2	no	—
88	C₄×D₆₁	C₂²×C₈	1	2	yes	—
89	C₄×D₆₁	C₂×M₁₆	1	2	yes	—
90	C₄×D₆₁	C₂×SD₁₆	1	2	yes	—
91	C₄×D₆₁	#135	1	2	yes	—
92	#92420	—	2	4	no	—
93	#92420	—	4	4	no	—
94	C₈×C₆₁	—	2	1	no	—
95	C₈×C₆₁	—	4	2	no	—
96	C₈×C₆₁	C₄×C₈	1	2	yes	complemented
97	#92418	—	2	1	no	—
98	#92418	—	4	2	no	—
99	#92418	C₄×C₈	1	2	yes	complemented
100	C₄×#334	—	61	1	no	maximal, Sylow
101	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
102	C₂×D₆₁	C₂×C₄⋊C₈	1	2	yes	—
103	C₂×D₆₁	C₂×#106	1	1	yes	characteristic
104	C₂×D₆₁	#427	1	1	yes	characteristic
105	C₂×D₆₁	C₂×#108	1	2	yes	—
106	#6923	—	2	4	no	—
107	#6923	#334	1	8	yes	direct product factor
108	C₄×C₆₁	—	2	1	no	—
109	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
110	C₄×C₆₁	C₄×M₁₆	1	1	yes	characteristic
111	C₄×C₆₁	C₄×SD₁₆	1	2	yes	—
112	Dic₆₁	—	2	1	no	—
113	Dic₆₁	#431	1	2	yes	—
114	Dic₆₁	C₄×SD₁₆	1	2	yes	—
115	C₄×C₄⋊C₈	—	61	2	no	—
116	C₄²×C₈	—	61	1	no	—
117	C₂×#334	—	61	1	no	—
118	C₂×#334	—	61	1	no	—
119	#1277	—	61	2	no	—
120	C₂×C₆₁	C₄×C₄⋊C₈	1	1	yes	characteristic
121	C₂×C₆₁	C₄×#106	1	1	yes	characteristic
122	C₂×C₆₁	C₄×#108	1	1	yes	characteristic
123	D₆₁	C₂×#334	1	2	yes	—
124	D₆₁	#1564	1	2	yes	—
125	C₂×C₄⋊C₈	—	61	2	no	—
126	C₂×C₄⋊C₈	—	61	2	no	—
127	C₂×C₄×C₈	—	61	2	no	—
128	C₂×C₄×C₈	—	61	1	no	—
129	C₂×C₄×C₈	—	61	2	no	—
130	C₂×C₄×C₈	—	61	1	no	—
131	C₂×C₄×C₈	—	122	2	no	—
132	C₄³	—	61	1	no	—
133	#336	—	122	2	no	—
134	#334	—	61	2	no	—
135	#334	—	61	2	no	—
136	#334	—	61	2	no	—
137	#334	—	61	1	no	—
138	#334	#6923	1	1	yes	characteristic, direct product factor
139	C₆₁	C₄×#334	1	1	yes	characteristic, complemented, Sylow
140	C₂²×C₈	—	61	1	no	—
141	C₂²×C₈	—	61	2	no	—
142	C₂²×C₈	—	61	1	no	—
143	C₂²×C₈	—	122	2	no	—
144	C₂²×C₈	—	122	2	no	—
145	C₂×C₄²	—	61	2	no	—
146	C₂×C₄²	—	61	1	no	—
147	C₂×C₄²	—	61	1	no	—
148	C₂×C₄²	—	61	1	no	—
149	C₂×C₄²	—	61	2	no	—
150	C₄⋊C₈	—	61	4	no	—
151	C₄⋊C₈	—	61	2	no	—
152	C₄⋊C₈	—	122	2	no	—
153	C₄⋊C₈	—	122	2	no	—
154	C₄⋊C₈	C₂×#6923	1	2	yes	—
155	C₄×C₈	—	61	8	no	—
156	C₄×C₈	—	61	1	no	—
157	C₄×C₈	—	61	1	no	—
158	C₄×C₈	—	61	1	no	—
159	C₄×C₈	—	122	4	no	—
160	C₄×C₈	—	122	1	no	—
161	C₄×C₈	—	122	1	no	—
162	C₄×C₈	—	244	4	no	—
163	C₄×C₈	C₂×#6923	1	1	yes	characteristic
164	C₂²×C₄	—	61	2	no	—
165	C₂²×C₄	—	61	1	no	—
166	C₂²×C₄	—	61	1	no	—
167	C₂²×C₄	—	61	2	no	—
168	C₂²×C₄	—	61	1	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	61	1	no	—
171	C₂×C₈	—	61	4	no	—
172	C₂×C₈	—	61	4	no	—
173	C₂×C₈	—	61	1	no	—
174	C₂×C₈	—	122	2	no	—
175	C₂×C₈	—	122	2	no	—
176	C₂×C₈	—	122	2	no	—
177	C₂×C₈	—	122	4	no	—
178	C₂×C₈	—	122	2	no	—
179	C₂×C₈	—	244	2	no	—
180	C₂×C₈	—	244	2	no	—
181	C₂×C₈	—	244	2	no	—
182	C₂×C₈	—	244	2	no	—
183	C₂×C₈	C₄×#6923	1	1	yes	characteristic
184	C₂×C₈	C₄×#6923	1	1	yes	characteristic
185	C₄²	—	61	4	no	—
186	C₄²	—	61	4	no	—
187	C₄²	—	61	4	no	—
188	C₄²	—	61	1	no	—
189	C₄²	—	61	1	no	—
190	C₄²	—	61	1	no	—
191	C₄²	—	122	4	no	—
192	C₄²	—	122	2	no	—
193	C₄²	C₂²×#6923	1	1	yes	characteristic
194	C₂³	—	61	1	no	—
195	C₂×C₄	—	61	4	no	—
196	C₂×C₄	—	61	4	no	—
197	C₂×C₄	—	61	4	no	—
198	C₂×C₄	—	61	1	no	—
199	C₂×C₄	—	61	1	no	—
200	C₂×C₄	—	61	2	no	—
201	C₂×C₄	—	61	2	no	—
202	C₂×C₄	—	61	2	no	—
203	C₂×C₄	—	61	2	no	—
204	C₂×C₄	—	61	4	no	—
205	C₂×C₄	—	61	1	no	—
206	C₂×C₄	—	122	2	no	—
207	C₂×C₄	—	122	2	no	—
208	C₂×C₄	—	122	2	no	—
209	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
210	C₂×C₄	D₄×#6923	1	1	yes	characteristic
211	C₂×C₄	Q₈×#6923	1	1	yes	characteristic
212	C₈	—	2	1	no	—
213	C₈	—	4	2	no	—
214	C₈	—	61	2	no	—
215	C₈	—	122	4	no	—
216	C₈	—	122	1	no	—
217	C₈	—	244	4	no	—
218	C₈	—	244	2	no	—
219	C₈	C₈×#6923	1	2	yes	complemented
220	C₂²	—	61	2	no	—
221	C₂²	—	61	1	no	—
222	C₂²	—	61	1	no	—
223	C₂²	—	61	2	no	—
224	C₂²	C₄⋊C₄×#6923	1	1	yes	characteristic
225	C₄	—	2	1	no	—
226	C₄	—	61	8	no	—
227	C₄	—	61	2	no	—
228	C₄	—	61	2	no	—
229	C₄	—	122	4	no	—
230	C₄	—	122	1	no	—
231	C₄	C₂×C₈×#6923	1	1	yes	characteristic
232	C₄	M₁₆×#6923	1	1	yes	characteristic
233	C₄	SD₁₆×#6923	1	2	yes	—
234	C₂	—	61	2	no	—
235	C₂	—	61	2	no	—
236	C₂	C₄⋊C₈×#6923	1	1	yes	characteristic
237	C₂	#106×#6923	1	1	yes	characteristic
238	C₂	#108×#6923	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6923
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₄²	C₂²×#6923
Fitting	C₆₁×#334	C₄
socle	C₂²×C₆₁	C₄×C₄⋊C₄
prime radical	C₂²×D₆₁	C₂×C₄⋊C₄
cosocle	C₄²×D₆₁	C₂³
hypercenter	#334	#6923
hypocenter	C₆₁	C₄×#334
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

Complements

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

N	H ≅ G/N	type
13. #16840624	C₄	semidirect
14. #16840623	C₄	semidirect
17. #16839513	C₄	semidirect
18. #16839512	C₄	semidirect
19. C₆₁×#334	C₄	semidirect
32. C₈×#6923	C₈	semidirect
96. C₈×C₆₁	C₄×C₈	semidirect
99. #92418	C₄×C₈	semidirect
107. #6923	#334	direct (N × H)
138. #334	#6923	direct (N × H)
139. C₆₁	C₄×#334	semidirect
219. C₈	C₈×#6923	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₈×#6923	2
C₄×C₈×#6923	2
D₆₁⋊#334	2
#21460648	2
#334×D₆₁	2
C₄×#334	61