Subgroups — C₄×#5101 #10792526

Order 768.

1526 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#5101	C₂	1	4	yes	maximal
3	C₄×#926	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#208	C₂	1	1	yes	maximal, characteristic
5	#69749	C₂	1	4	yes	maximal, complemented
6	C₄×#815	C₂	1	2	yes	maximal, complemented
7	C₄×C₈×Dic₃	C₂	1	2	yes	maximal, complemented
8	C₄×#405	—	3	1	no	maximal, Sylow
9	C₂³×C₃×C₈	C₂²	1	2	yes	—
10	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
11	C₂×#926	C₂²	1	2	yes	—
12	C₂²×#208	C₂²	1	2	yes	—
13	C₂×C₃×C₄×C₈	C₄	1	8	yes	complemented
14	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
16	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₄×#217	C₄	1	4	yes	—
18	#5101	C₄	1	32	yes	direct product factor
19	C₂×#815	C₂²	1	8	yes	—
20	C₂×C₈×Dic₃	C₂²	1	8	yes	—
21	C₄×C₄⋊Dic₃	C₄	1	4	yes	—
22	#4932	C₂²	1	2	yes	—
23	C₄²×Dic₃	C₂²	1	2	yes	—
24	C₄×#208	C₄	1	8	yes	complemented
25	C₂×#803	C₂²	1	2	yes	—
26	C₂×C₄×C₃⋊C₈	C₂²	1	2	yes	—
27	C₂×#405	—	3	4	no	—
28	C₂×C₄×M₁₆	—	3	1	no	—
29	C₂²×C₄×C₈	D₃	1	1	yes	characteristic
30	C₄×#117	—	3	1	no	—
31	#1726	—	3	4	no	—
32	C₄×#97	—	3	2	no	—
33	C₄²×C₈	—	3	2	no	—
34	C₂³×C₃×C₄	C₂×C₄	1	2	yes	—
35	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
36	C₂²×C₃×C₈	C₂×C₄	1	16	yes	—
37	C₂²×C₃×C₈	C₂×C₄	1	8	yes	—
38	C₂²×C₃×C₈	C₂³	1	2	yes	—
39	C₂²×C₃×C₈	C₂³	1	2	yes	—
40	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
41	C₂×C₃×C₄²	C₂×C₄	1	4	yes	—
42	C₂×C₃×C₄²	C₂×C₄	1	4	yes	—
43	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
44	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
45	C₂×#217	C₂×C₄	1	4	yes	—
46	#926	C₂×C₄	1	16	yes	—
47	C₂×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
48	C₂×C₄×Dic₃	C₂×C₄	1	8	yes	—
49	C₂×C₄×Dic₃	C₂³	1	4	yes	—
50	C₂×#208	C₂×C₄	1	8	yes	—
51	C₂×#208	C₂×C₄	1	16	yes	—
52	C₂²×C₃⋊C₈	C₂³	1	4	yes	—
53	C₃×C₄×C₈	C₂×C₄	1	4	yes	—
54	C₃×C₄×C₈	C₂×C₄	1	8	yes	—
55	C₃×C₄×C₈	C₂×C₄	1	4	yes	—
56	C₃⋊#95	C₂×C₄	1	8	yes	—
57	#815	C₂×C₄	1	32	yes	complemented
58	C₈×Dic₃	C₂×C₄	1	32	yes	complemented
59	#803	C₂×C₄	1	8	yes	—
60	C₄×C₃⋊C₈	C₂×C₄	1	8	yes	—
61	C₂²×M₁₆	—	3	2	no	—
62	C₂³×C₈	C₂×D₃	1	2	yes	—
63	C₂×#117	—	3	2	no	—
64	C₂²×C₄²	C₂×D₃	1	1	yes	characteristic
65	#405	—	3	32	no	—
66	C₄×M₁₆	—	3	8	no	—
67	C₂×#97	—	3	8	no	—
68	C₂×#97	—	3	2	no	—
69	C₂×C₄×C₈	—	3	8	no	—
70	C₂×C₄×C₈	—	3	2	no	—
71	C₂×C₄×C₈	Dic₃	1	8	yes	complemented
72	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
74	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
75	C₄×C₄⋊C₄	—	3	4	no	—
76	C₄×C₂²⋊C₄	—	3	4	no	—
77	#376	—	3	2	no	—
78	C₄³	—	3	2	no	—
79	C₂⁴×C₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₃×C₄	—	2	8	no	—
81	C₂²×C₃×C₄	C₄²	1	8	yes	—
82	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
83	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
85	C₂²×C₃×C₄	C₂²×C₄	1	8	yes	—
86	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
87	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
88	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
89	C₂²×Dic₃	C₂²×C₄	1	8	yes	—
90	C₂×C₃×C₈	C₄²	1	32	yes	complemented
91	C₂×C₃×C₈	C₂²×C₄	1	8	yes	—
92	C₂×C₃×C₈	C₂²×C₄	1	8	yes	—
93	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
94	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
95	C₃×C₄²	C₄²	1	8	yes	—
96	C₃×C₄²	C₂²×C₄	1	2	yes	—
97	C₃×C₄²	C₂²×C₄	1	2	yes	—
98	C₃×C₄²	C₂²×C₄	1	2	yes	—
99	C₃×C₄²	C₂²×C₄	1	2	yes	—
100	#217	C₄²	1	16	yes	—
101	C₄⋊Dic₃	C₄²	1	16	yes	—
102	C₄×Dic₃	—	2	16	no	—
103	C₄×Dic₃	C₂²×C₄	1	16	yes	—
104	#208	C₄²	1	32	yes	complemented
105	C₂×C₃⋊C₈	C₂²×C₄	1	8	yes	—
106	C₂×C₃⋊C₈	C₂²×C₄	1	16	yes	—
107	C₂³×C₄	C₄×D₃	1	2	yes	—
108	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
109	C₂×M₁₆	—	3	8	no	—
110	C₂×M₁₆	—	3	16	no	—
111	C₂²×C₈	—	3	4	no	—
112	C₂²×C₈	C₄×D₃	1	16	yes	—
113	C₂²×C₈	C₂×Dic₃	1	8	yes	—
114	C₂²×C₈	C₂²×D₃	1	2	yes	—
115	C₂²×C₈	C₂²×D₃	1	2	yes	—
116	#117	—	3	16	no	—
117	C₂×C₄⋊C₄	—	3	4	no	—
118	C₂×C₂²⋊C₄	—	3	4	no	—
119	C₂×C₄²	—	3	4	no	—
120	C₂×C₄²	—	3	8	no	—
121	C₂×C₄²	C₄×D₃	1	2	yes	—
122	C₂×C₄²	C₄×D₃	1	4	yes	—
123	C₂×C₄²	C₂×Dic₃	1	4	yes	—
124	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
125	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
126	#97	—	3	32	no	—
127	#97	—	3	8	no	—
128	C₄×C₈	—	3	32	no	—
129	C₄×C₈	—	3	8	no	—
130	C₄×C₈	C₂×Dic₃	1	4	yes	—
131	C₄×C₈	C₂×Dic₃	1	8	yes	—
132	C₄×C₈	C₂×Dic₃	1	4	yes	—
133	#95	—	3	8	no	—
134	C₂³×C₃	—	2	4	no	—
135	C₂³×C₃	C₂×C₄²	1	4	yes	—
136	C₂³×C₃	C₂×C₄²	1	2	yes	—
137	C₂³×C₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₃×C₄	—	2	4	no	—
139	C₂×C₃×C₄	—	2	16	no	—
140	C₂×C₃×C₄	C₂×C₄²	1	4	yes	—
141	C₂×C₃×C₄	C₂×C₄²	1	8	yes	—
142	C₂×C₃×C₄	C₂×C₄²	1	4	yes	—
143	C₂×C₃×C₄	C₂×C₄²	1	2	yes	—
144	C₂×C₃×C₄	C₂×C₄²	1	8	yes	—
145	C₂×C₃×C₄	C₂×C₄²	1	4	yes	—
146	C₂×C₃×C₄	#131	1	8	yes	—
147	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂³×C₄	1	4	yes	—
149	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
150	C₂×Dic₃	—	2	16	no	—
151	C₂×Dic₃	C₂×C₄²	1	16	yes	—
152	C₃×C₈	C₂×C₄²	1	8	yes	—
153	C₃×C₈	C₂×C₄²	1	8	yes	—
154	C₃⋊C₈	C₂×C₄²	1	16	yes	—
155	C₂⁴	C₂×C₄×D₃	1	1	yes	characteristic
156	C₂²×C₄	—	2	8	no	—
157	C₂²×C₄	—	3	8	no	—
158	C₂²×C₄	C₄×Dic₃	1	8	yes	—
159	C₂²×C₄	C₂×C₄×D₃	1	2	yes	—
160	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
161	C₂²×C₄	C₂×C₄×D₃	1	2	yes	—
162	C₂²×C₄	C₂×C₄×D₃	1	8	yes	—
163	C₂²×C₄	C₂×C₄×D₃	1	2	yes	—
164	C₂²×C₄	C₂²×Dic₃	1	2	yes	—
165	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
166	M₁₆	—	3	32	no	—
167	C₂×C₈	—	3	8	no	—
168	C₂×C₈	—	3	16	no	—
169	C₂×C₈	C₄×Dic₃	1	32	yes	complemented
170	C₂×C₈	C₂×C₄×D₃	1	8	yes	—
171	C₂×C₈	C₂×C₄×D₃	1	8	yes	—
172	C₂×C₈	C₂²×Dic₃	1	4	yes	—
173	C₂×C₈	C₂²×Dic₃	1	4	yes	—
174	C₄⋊C₄	—	3	16	no	—
175	C₂²⋊C₄	—	3	16	no	—
176	C₄²	—	3	16	no	—
177	C₄²	—	6	16	no	—
178	C₄²	C₄×Dic₃	1	8	yes	—
179	C₄²	C₂×C₄×D₃	1	2	yes	—
180	C₄²	C₂×C₄×D₃	1	2	yes	—
181	C₄²	C₂²×Dic₃	1	2	yes	—
182	C₄²	C₂²×Dic₃	1	2	yes	—
183	C₂²×C₃	—	2	4	no	—
184	C₂²×C₃	—	2	8	no	—
185	C₂²×C₃	C₄³	1	4	yes	—
186	C₂²×C₃	C₂²×C₄²	1	2	yes	—
187	C₂²×C₃	C₂²×C₄²	1	1	yes	characteristic
188	C₂²×C₃	C₂×#131	1	4	yes	—
189	C₃×C₄	—	2	4	no	—
190	C₃×C₄	C₄³	1	4	yes	—
191	C₃×C₄	#405	1	8	yes	complemented
192	C₃×C₄	C₂²×C₄²	1	2	yes	—
193	C₃×C₄	C₂²×C₄²	1	2	yes	—
194	Dic₃	—	2	16	no	—
195	C₂³	—	2	4	no	—
196	C₂³	C₄²×D₃	1	4	yes	—
197	C₂³	C₂×C₄×Dic₃	1	2	yes	—
198	C₂³	C₂²×C₄×D₃	1	1	yes	characteristic
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	16	no	—
201	C₂×C₄	—	3	16	no	—
202	C₂×C₄	—	6	16	no	—
203	C₂×C₄	C₄²×D₃	1	4	yes	—
204	C₂×C₄	C₄²×D₃	1	8	yes	—
205	C₂×C₄	#901	1	8	yes	—
206	C₂×C₄	C₂×C₄×Dic₃	1	4	yes	—
207	C₂×C₄	C₂×C₄×Dic₃	1	2	yes	—
208	C₂×C₄	C₂×C₄×Dic₃	1	8	yes	—
209	C₂×C₄	C₂×C₄×Dic₃	1	4	yes	—
210	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
211	C₂×C₄	C₂²×C₄×D₃	1	4	yes	—
212	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
213	C₈	—	3	16	no	—
214	C₈	C₂×C₄×Dic₃	1	8	yes	—
215	C₈	C₂×C₄×Dic₃	1	8	yes	—
216	C₂×C₃	—	2	4	no	—
217	C₂×C₃	C₂×C₄³	1	1	yes	characteristic
218	C₂×C₃	C₂×#405	1	2	yes	—
219	C₂×C₃	C₄×#131	1	4	yes	—
220	C₂²	—	2	4	no	—
221	C₂²	—	2	8	no	—
222	C₂²	C₄²×Dic₃	1	4	yes	—
223	C₂²	C₂×C₄²×D₃	1	2	yes	—
224	C₂²	C₂×#901	1	4	yes	—
225	C₂²	C₂²×C₄×Dic₃	1	1	yes	characteristic
226	C₄	—	2	4	no	—
227	C₄	—	6	16	no	—
228	C₄	C₄²×Dic₃	1	4	yes	—
229	C₄	#5101	1	8	yes	direct product factor
230	C₄	C₂×C₄²×D₃	1	2	yes	—
231	C₄	C₂²×C₄×Dic₃	1	2	yes	—
232	C₃	C₄×#405	1	1	yes	characteristic, complemented, Sylow
233	C₂	—	2	4	no	—
234	C₂	C₄×#901	1	4	yes	—
235	C₂	C₂×C₄²×Dic₃	1	1	yes	characteristic
236	C₂	C₂×#5101	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₃
commutator	C₂×C₃	C₂×C₄³
Frattini	C₂²×C₄	C₂³×D₃
Fitting	C₂²×C₃×C₄×C₈	C₂
socle	C₂³×C₃	C₂³×C₄
prime radical	C₂⁴×C₃	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂⁴
hypercenter	C₂²×C₄×C₈	D₃
hypocenter	C₃	C₄×#405
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
5. #69749	C₂	semidirect
6. C₄×#815	C₂	semidirect
7. C₄×C₈×Dic₃	C₂	semidirect
13. C₂×C₃×C₄×C₈	C₄	semidirect
18. #5101	C₄	direct (N × H)
24. C₄×#208	C₄	semidirect
57. #815	C₂×C₄	semidirect
58. C₈×Dic₃	C₂×C₄	semidirect
71. C₂×C₄×C₈	Dic₃	semidirect
90. C₂×C₃×C₈	C₄²	semidirect
104. #208	C₄²	semidirect
169. C₂×C₈	C₄×Dic₃	semidirect
191. C₃×C₄	#405	semidirect
229. C₄	#5101	direct (N × H)
232. C₃	C₄×#405	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄×C₈	2
C₂×#5101	2
C₄×#926	2
C₂×C₄×#208	2
#69749	2
C₄×#815	2
C₄×C₈×Dic₃	2
C₄×#405	3

Subgroups — C4×#5101 #10792526

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#5101 #10792526