Subgroups — C₂×#5207 #84022

Order 384.

1008 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×#208	C₂	1	2	yes	maximal, complemented
3	#5207	C₂	1	8	yes	maximal, direct product factor
4	C₂×#830	C₂	1	4	yes	maximal, complemented
5	C₂×#407	—	3	1	no	maximal, Sylow
6	C₂⁵×C₃	C₄	1	1	yes	characteristic
7	C₂³×C₃×C₄	—	2	8	no	—
8	C₂³×C₃×C₄	C₄	1	1	yes	characteristic
9	C₂³×C₃×C₄	C₄	1	4	yes	—
10	C₂³×C₃×C₄	C₄	1	2	yes	—
11	C₂³×C₃×C₄	C₂²	1	2	yes	—
12	C₂³×C₃×C₄	C₂²	1	4	yes	—
13	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
14	C₂×#208	—	2	8	no	—
15	C₂×#208	C₂²	1	8	yes	complemented
16	C₂²×C₃⋊C₈	C₂²	1	4	yes	complemented
17	#830	C₂²	1	16	yes	complemented
18	C₂⁴×C₄	D₃	1	1	yes	characteristic
19	C₂²×M₁₆	—	3	2	no	—
20	#407	—	3	8	no	—
21	C₂×C₂²⋊C₈	—	3	4	no	—
22	C₂⁴×C₃	—	2	4	no	—
23	C₂⁴×C₃	—	2	8	no	—
24	C₂⁴×C₃	C₂×C₄	1	4	yes	—
25	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
26	C₂⁴×C₃	C₂×C₄	1	2	yes	—
27	C₂²×C₃×C₄	—	2	4	no	—
28	C₂²×C₃×C₄	—	2	16	no	—
29	C₂²×C₃×C₄	—	2	8	no	—
30	C₂²×C₃×C₄	—	2	8	no	—
31	C₂²×C₃×C₄	—	2	16	no	—
32	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
33	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
34	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
35	C₂²×C₃×C₄	C₂×C₄	1	8	yes	—
36	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
37	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₃×C₄	D₄	1	8	yes	—
39	C₂²×C₃×C₄	C₂³	1	4	yes	—
40	C₂²×C₃×C₄	C₂³	1	2	yes	—
41	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
42	#208	—	2	16	no	—
43	C₂×C₃⋊C₈	—	2	8	no	—
44	C₂×C₃⋊C₈	C₂³	1	8	yes	complemented
45	C₂⁵	Dic₃	1	1	yes	characteristic
46	C₂³×C₄	—	2	8	no	—
47	C₂³×C₄	Dic₃	1	1	yes	characteristic
48	C₂³×C₄	Dic₃	1	4	yes	—
49	C₂³×C₄	Dic₃	1	2	yes	—
50	C₂³×C₄	C₂×D₃	1	4	yes	—
51	C₂³×C₄	C₂×D₃	1	2	yes	—
52	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
53	C₂×M₁₆	—	3	8	no	—
54	C₂×M₁₆	—	6	8	no	—
55	C₂²×C₈	—	3	4	no	—
56	C₂²⋊C₈	—	3	16	no	—
57	C₂³×C₃	—	2	32	no	—
58	C₂³×C₃	—	2	8	no	—
59	C₂³×C₃	—	2	4	no	—
60	C₂³×C₃	—	2	4	no	—
61	C₂³×C₃	—	2	4	no	—
62	C₂³×C₃	—	2	16	no	—
63	C₂³×C₃	C₂²⋊C₄	1	4	yes	—
64	C₂³×C₃	M₁₆	1	8	yes	complemented
65	C₂³×C₃	C₂²×C₄	1	2	yes	—
66	C₂³×C₃	C₂²×C₄	1	4	yes	—
67	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₄	—	2	8	no	—
69	C₂×C₃×C₄	—	2	4	no	—
70	C₂×C₃×C₄	—	2	8	no	—
71	C₂×C₃×C₄	—	2	16	no	—
72	C₂×C₃×C₄	—	2	8	no	—
73	C₂×C₃×C₄	C₂²⋊C₄	1	8	yes	—
74	C₂×C₃×C₄	C₂²⋊C₄	1	4	yes	—
75	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
76	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
77	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
79	C₂×C₃×C₄	C₂×D₄	1	8	yes	—
80	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
81	C₃⋊C₈	—	2	8	no	—
82	C₂⁴	—	2	8	no	—
83	C₂⁴	—	2	4	no	—
84	C₂⁴	C₂×Dic₃	1	4	yes	—
85	C₂⁴	C₂×Dic₃	1	2	yes	—
86	C₂⁴	C₂×Dic₃	1	1	yes	characteristic
87	C₂²×C₄	—	2	4	no	—
88	C₂²×C₄	—	2	16	no	—
89	C₂²×C₄	—	2	8	no	—
90	C₂²×C₄	—	2	8	no	—
91	C₂²×C₄	—	2	16	no	—
92	C₂²×C₄	C₂×Dic₃	1	4	yes	—
93	C₂²×C₄	C₂×Dic₃	1	2	yes	—
94	C₂²×C₄	C₂×Dic₃	1	8	yes	—
95	C₂²×C₄	C₂×Dic₃	1	4	yes	—
96	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
97	C₂²×C₄	C₂×Dic₃	1	2	yes	—
98	C₂²×C₄	#67	1	8	yes	—
99	C₂²×C₄	C₂²×D₃	1	4	yes	—
100	C₂²×C₄	C₂²×D₃	1	2	yes	—
101	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
102	M₁₆	—	6	16	no	—
103	C₂×C₈	—	3	8	no	—
104	C₂×C₈	—	6	8	no	—
105	C₂²×C₃	—	2	32	no	—
106	C₂²×C₃	—	2	4	no	—
107	C₂²×C₃	—	2	4	no	—
108	C₂²×C₃	—	2	4	no	—
109	C₂²×C₃	—	2	8	no	—
110	C₂²×C₃	—	2	16	no	—
111	C₂²×C₃	C₂×C₂²⋊C₄	1	4	yes	—
112	C₂²×C₃	C₂×C₂²⋊C₄	1	2	yes	—
113	C₂²×C₃	C₂×M₁₆	1	4	yes	—
114	C₂²×C₃	C₂×M₁₆	1	8	yes	complemented
115	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
116	C₃×C₄	—	2	4	no	—
117	C₃×C₄	C₂×C₂²⋊C₄	1	2	yes	—
118	C₃×C₄	C₂×C₂²⋊C₄	1	4	yes	—
119	C₃×C₄	C₂²×D₄	1	2	yes	—
120	C₂³	—	2	32	no	—
121	C₂³	—	2	4	no	—
122	C₂³	—	2	4	no	—
123	C₂³	—	2	16	no	—
124	C₂³	—	2	8	no	—
125	C₂³	—	2	4	no	—
126	C₂³	#208	1	8	yes	complemented
127	C₂³	#217	1	4	yes	—
128	C₂³	C₂²×Dic₃	1	4	yes	—
129	C₂³	C₂²×Dic₃	1	2	yes	—
130	C₂³	C₂²×Dic₃	1	1	yes	characteristic
131	C₂×C₄	—	2	8	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	16	no	—
134	C₂×C₄	—	2	8	no	—
135	C₂×C₄	—	2	8	no	—
136	C₂×C₄	#217	1	8	yes	—
137	C₂×C₄	#217	1	4	yes	—
138	C₂×C₄	C₂²×Dic₃	1	4	yes	—
139	C₂×C₄	C₂²×Dic₃	1	2	yes	—
140	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
141	C₂×C₄	C₂×#67	1	4	yes	—
142	C₂×C₄	C₂×#67	1	8	yes	—
143	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
144	C₈	—	6	8	no	—
145	C₂×C₃	—	2	4	no	—
146	C₂×C₃	—	2	8	no	—
147	C₂×C₃	#407	1	4	yes	complemented
148	C₂×C₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₃	C₂²×M₁₆	1	2	yes	—
150	C₂²	—	2	32	no	—
151	C₂²	—	2	16	no	—
152	C₂²	—	2	8	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	4	no	—
155	C₂²	—	2	4	no	—
156	C₂²	C₂×#208	1	4	yes	—
157	C₂²	C₂×#208	1	8	yes	complemented
158	C₂²	C₂×#217	1	4	yes	—
159	C₂²	C₂×#217	1	2	yes	—
160	C₂²	C₂³×Dic₃	1	1	yes	characteristic
161	C₄	—	2	4	no	—
162	C₄	C₂×#217	1	4	yes	—
163	C₄	C₂×#217	1	2	yes	—
164	C₄	C₂²×#67	1	2	yes	—
165	C₃	C₂×#407	1	1	yes	characteristic, complemented, Sylow
166	C₂	—	2	8	no	—
167	C₂	—	2	4	no	—
168	C₂	#5207	1	4	yes	direct product factor
169	C₂	C₂²×#208	1	2	yes	—
170	C₂	C₂²×#217	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂⁴×C₃×C₄	C₂
socle	C₂³×C₃	C₂²×C₄
prime radical	C₂⁵×C₃	C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂⁴×C₄	D₃
hypocenter	C₃	C₂×#407
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×#208	C₂	semidirect
3. #5207	C₂	direct (N × H)
4. C₂×#830	C₂	semidirect
15. C₂×#208	C₂²	semidirect
16. C₂²×C₃⋊C₈	C₂²	semidirect
17. #830	C₂²	semidirect
44. C₂×C₃⋊C₈	C₂³	semidirect
64. C₂³×C₃	M₁₆	semidirect
114. C₂²×C₃	C₂×M₁₆	semidirect
126. C₂³	#208	semidirect
147. C₂×C₃	#407	semidirect
157. C₂²	C₂×#208	semidirect
165. C₃	C₂×#407	semidirect
168. C₂	#5207	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃×C₄	2
C₂²×#208	2
#5207	2
C₂×#830	2
C₂×#407	3

Subgroups — C2×#5207 #84022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#5207 #84022