Subgroups — C₂×#5124 #83754

Order 384.

460 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×#208	C₂	1	1	yes	maximal, characteristic
3	C₂²×#208	C₂	1	1	yes	maximal, characteristic
4	#5124	C₂	1	8	yes	maximal, direct product factor
5	C₂×#822	C₂	1	4	yes	maximal, complemented
6	C₂×#430	—	3	1	no	maximal, Sylow
7	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₂×C₃×M₁₆	—	2	1	no	—
9	C₂×C₃×M₁₆	—	2	2	no	—
10	C₂×C₃×M₁₆	C₂²	1	2	yes	—
11	C₂×C₃×M₁₆	C₂²	1	4	yes	—
12	C₂²×C₃×C₈	—	2	1	no	—
13	C₂×#208	—	2	1	no	—
14	C₂×#208	—	2	2	no	—
15	C₂×#208	C₄	1	2	yes	—
16	C₂×#208	C₄	1	2	yes	—
17	C₂×#208	C₄	1	4	yes	—
18	C₂×#208	C₂²	1	2	yes	—
19	C₂×#208	C₂²	1	4	yes	—
20	C₂×#208	C₂²	1	4	yes	—
21	C₂²×C₃⋊C₈	—	2	1	no	—
22	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
23	#822	C₂²	1	16	yes	complemented
24	C₂²×M₁₆	—	3	1	no	—
25	C₂²×M₁₆	—	3	1	no	—
26	C₂²×M₁₆	D₃	1	1	yes	characteristic
27	#430	—	3	8	no	—
28	C₂×#108	—	3	4	no	—
29	C₂⁴×C₃	Q₈	1	1	yes	characteristic
30	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
33	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
34	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
35	C₂²×C₃×C₄	D₄	1	2	yes	—
36	C₂²×C₃×C₄	Q₈	1	1	yes	characteristic
37	C₂²×C₃×C₄	Q₈	1	2	yes	—
38	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
39	C₂²×C₃×C₄	C₂³	1	2	yes	—
40	C₃×M₁₆	—	2	4	no	—
41	C₃×M₁₆	—	2	2	no	—
42	C₃×M₁₆	C₂³	1	4	yes	—
43	C₂×C₃×C₈	—	2	4	no	—
44	C₂×C₃×C₈	—	2	2	no	—
45	#208	—	2	4	no	—
46	#208	—	2	2	no	—
47	#208	C₂×C₄	1	8	yes	—
48	#208	C₂×C₄	1	8	yes	—
49	#208	C₂³	1	4	yes	—
50	C₂×C₃⋊C₈	—	2	4	no	—
51	C₂×C₃⋊C₈	—	2	2	no	—
52	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
53	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
54	C₂×C₃⋊C₈	C₂³	1	4	yes	—
55	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
56	C₂×M₁₆	—	2	2	no	—
57	C₂×M₁₆	—	2	1	no	—
58	C₂×M₁₆	—	3	2	no	—
59	C₂×M₁₆	—	3	2	no	—
60	C₂×M₁₆	—	3	4	no	—
61	C₂×M₁₆	—	3	4	no	—
62	C₂×M₁₆	—	3	2	no	—
63	C₂×M₁₆	—	3	4	no	—
64	C₂×M₁₆	—	6	2	no	—
65	C₂×M₁₆	—	6	1	no	—
66	C₂×M₁₆	C₂×D₃	1	2	yes	—
67	C₂×M₁₆	C₂×D₃	1	4	yes	—
68	C₂²×C₈	—	2	1	no	—
69	C₂²×C₈	—	3	2	no	—
70	C₂²×C₈	—	6	1	no	—
71	#108	—	3	16	no	—
72	C₂³×C₃	—	4	2	no	—
73	C₂³×C₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂³×C₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂³×C₃	C₄⋊C₄	1	2	yes	—
76	C₂³×C₃	C₂×Q₈	1	1	yes	characteristic
77	C₂³×C₃	C₂×Q₈	1	2	yes	—
78	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
79	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
80	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
81	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
82	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
83	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
88	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
89	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
90	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
91	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
94	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
95	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
96	C₃×C₈	—	2	4	no	—
97	C₃⋊C₈	—	2	4	no	—
98	C₃⋊C₈	C₂²×C₄	1	8	yes	—
99	C₂⁴	Dic₆	1	1	yes	characteristic
100	C₂²×C₄	Dic₆	1	1	yes	characteristic
101	C₂²×C₄	Dic₆	1	2	yes	—
102	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
103	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
104	C₂²×C₄	C₄×D₃	1	2	yes	—
105	C₂²×C₄	#67	1	2	yes	—
106	C₂²×C₄	#67	1	1	yes	characteristic
107	C₂²×C₄	#67	1	1	yes	characteristic
108	C₂²×C₄	C₂²×D₃	1	2	yes	—
109	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
110	M₁₆	—	2	4	no	—
111	M₁₆	—	2	2	no	—
112	M₁₆	—	3	8	no	—
113	M₁₆	—	3	8	no	—
114	M₁₆	—	3	4	no	—
115	M₁₆	—	6	4	no	—
116	M₁₆	—	6	2	no	—
117	M₁₆	C₂²×D₃	1	4	yes	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	3	4	no	—
121	C₂×C₈	—	3	4	no	—
122	C₂×C₈	—	3	4	no	—
123	C₂×C₈	—	6	4	no	—
124	C₂×C₈	—	6	2	no	—
125	C₂²×C₃	—	2	2	no	—
126	C₂²×C₃	—	2	2	no	—
127	C₂²×C₃	—	2	2	no	—
128	C₂²×C₃	—	4	2	no	—
129	C₂²×C₃	—	4	2	no	—
130	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×C₃	C₂×C₄⋊C₄	1	2	yes	—
132	C₂²×C₃	C₂×C₄⋊C₄	1	2	yes	—
133	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
135	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
137	C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
138	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
140	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
141	C₂³	—	4	2	no	—
142	C₂³	Dic₃⋊C₄	1	1	yes	characteristic
143	C₂³	Dic₃⋊C₄	1	1	yes	characteristic
144	C₂³	Dic₃⋊C₄	1	2	yes	—
145	C₂³	C₂×Dic₆	1	1	yes	characteristic
146	C₂³	C₂×Dic₆	1	2	yes	—
147	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
148	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
149	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
150	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
151	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
152	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
153	C₂×C₄	Dic₃⋊C₄	1	2	yes	—
154	C₂×C₄	C₂×Dic₆	1	2	yes	—
155	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
157	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
159	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
160	C₂×C₄	C₂×#67	1	2	yes	—
161	C₂×C₄	C₂×#67	1	2	yes	—
162	C₂×C₄	C₂×#67	1	1	yes	characteristic
163	C₂×C₄	C₂×#67	1	1	yes	characteristic
164	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
165	C₈	—	2	4	no	—
166	C₈	—	3	8	no	—
167	C₈	—	6	4	no	—
168	C₂×C₃	—	2	2	no	—
169	C₂×C₃	—	2	2	no	—
170	C₂×C₃	—	2	1	no	—
171	C₂×C₃	—	2	1	no	—
172	C₂×C₃	#430	1	2	yes	complemented
173	C₂×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
174	C₂²	—	2	2	no	—
175	C₂²	—	2	2	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	4	2	no	—
178	C₂²	—	4	2	no	—
179	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
180	C₂²	C₂×Dic₃⋊C₄	1	2	yes	—
181	C₂²	C₂×Dic₃⋊C₄	1	2	yes	—
182	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
183	C₂²	C₂²×Dic₆	1	1	yes	characteristic
184	C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
185	C₄	C₂×Dic₃⋊C₄	1	2	yes	—
186	C₄	C₂×Dic₃⋊C₄	1	2	yes	—
187	C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
188	C₄	C₂²×C₄×D₃	1	1	yes	characteristic
189	C₄	C₂²×#67	1	1	yes	characteristic
190	C₃	C₂×#430	1	1	yes	characteristic, complemented, Sylow
191	C₂	—	2	2	no	—
192	C₂	—	2	2	no	—
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#5124	1	2	yes	direct product factor
196	C₂	C₂²×Dic₃⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#67
commutator	C₃×C₄	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂²×C₃×M₁₆	C₂
socle	C₂²×C₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₃	Q₈
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂²×M₁₆	D₃
hypocenter	C₃	C₂×#430
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
4. #5124	C₂	direct (N × H)
5. C₂×#822	C₂	semidirect
23. #822	C₂²	semidirect
172. C₂×C₃	#430	semidirect
190. C₃	C₂×#430	semidirect
195. C₂	#5124	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×M₁₆	2
C₂²×#208	2
#5124	2
C₂×#822	2
C₂×#430	3

Subgroups — C2×#5124 #83754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#5124 #83754