Subgroups — C₂×#29504533 #44484255

Order 30592.

488 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#21439919	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₂₃₉	C₂	1	1	yes	maximal, characteristic
3	#29504533	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₁₆×C₂₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₂₃₉	—	2	1	no	—
6	#21439919	—	2	1	no	—
7	#21439919	—	2	1	no	—
8	#21439919	C₄	1	2	yes	complemented
9	#21439919	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₂₃₉	C₄	1	2	yes	complemented
11	C₂×C₂²⋊Dic₂₃₉	—	2	1	no	—
12	C₂×C₄×Dic₂₃₉	—	2	2	no	—
13	C₂×C₄×Dic₂₃₉	—	2	1	no	—
14	C₂×C₄×Dic₂₃₉	—	2	1	no	—
15	C₂×C₄×Dic₂₃₉	C₄	1	4	yes	complemented
16	C₂×C₄×Dic₂₃₉	C₂²	1	2	yes	—
17	C₂³×C₄×C₂₃₉	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₂₃₉	—	2	2	no	—
19	C₂×M₁₆×C₂₃₉	—	2	1	no	—
20	C₂×M₁₆×C₂₃₉	C₄	1	4	yes	complemented
21	C₂×M₁₆×C₂₃₉	C₂²	1	2	yes	—
22	C₂²×C₈×C₂₃₉	—	2	1	no	—
23	C₂²×Dic₂₃₉	—	2	8	no	—
24	C₂²×Dic₂₃₉	—	2	4	no	—
25	C₂²×Dic₂₃₉	—	2	2	no	—
26	C₂²×Dic₂₃₉	—	2	2	no	—
27	C₄⋊Dic₂₃₉	—	2	2	no	—
28	C₄⋊Dic₂₃₉	C₂×C₄	1	4	yes	complemented
29	C₂²⋊Dic₂₃₉	—	2	2	no	—
30	C₂²⋊Dic₂₃₉	—	2	2	no	—
31	C₄×Dic₂₃₉	—	2	4	no	—
32	C₄×Dic₂₃₉	—	2	2	no	—
33	C₄×Dic₂₃₉	—	2	2	no	—
34	C₄×Dic₂₃₉	—	2	2	no	—
35	C₄×Dic₂₃₉	C₂×C₄	1	4	yes	complemented
36	C₂⁴×C₂₃₉	D₄	1	1	yes	characteristic
37	C₂²×C₄×C₂₃₉	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₂₃₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₃₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₃₉	C₂×C₄	1	2	yes	—
41	C₂²×C₄×C₂₃₉	D₄	1	2	yes	—
42	C₂²×C₄×C₂₃₉	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₃₉	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃₉	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃₉	Q₈	1	2	yes	—
46	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
47	M₁₆×C₂₃₉	—	2	4	no	—
48	M₁₆×C₂₃₉	—	2	2	no	—
49	M₁₆×C₂₃₉	C₂×C₄	1	4	yes	complemented
50	C₂×C₈×C₂₃₉	—	2	4	no	—
51	C₂×C₈×C₂₃₉	—	2	2	no	—
52	C₂×Dic₂₃₉	—	2	16	no	—
53	C₂×Dic₂₃₉	—	2	8	no	—
54	C₂×Dic₂₃₉	—	2	4	no	—
55	C₂×Dic₂₃₉	—	2	4	no	—
56	C₂×Dic₂₃₉	—	4	4	no	—
57	C₂³×C₂₃₉	—	2	4	no	—
58	C₂³×C₂₃₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂³×C₂₃₉	C₂²⋊C₄	1	2	yes	—
60	C₂³×C₂₃₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂³×C₂₃₉	C₄⋊C₄	1	2	yes	—
62	C₂³×C₂₃₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₂₃₉	C₄²	1	2	yes	—
64	C₂×C₄×C₂₃₉	C₄²	1	2	yes	—
65	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	2	yes	—
66	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	2	yes	—
69	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	2	yes	—
70	C₂×C₄×C₂₃₉	C₄⋊C₄	1	2	yes	—
71	C₂×C₄×C₂₃₉	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₂₃₉	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₂₃₉	C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₂₃₉	C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
81	C₈×C₂₃₉	—	2	4	no	—
82	C₂²×C₂₃₉	—	2	4	no	—
83	C₂²×C₂₃₉	—	2	4	no	—
84	C₂²×C₂₃₉	—	2	2	no	—
85	C₂²×C₂₃₉	—	2	2	no	—
86	C₂²×C₂₃₉	#95	1	2	yes	—
87	C₂²×C₂₃₉	#95	1	2	yes	—
88	C₂²×C₂₃₉	#104	1	4	yes	—
89	C₂²×C₂₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₂₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₂₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₂₃₉	#95	1	2	yes	—
93	C₄×C₂₃₉	#95	1	2	yes	—
94	C₄×C₂₃₉	C₂×C₄²	1	1	yes	characteristic
95	C₄×C₂₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₄×C₂₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₄×C₂₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
98	Dic₂₃₉	—	2	8	no	—
99	Dic₂₃₉	—	4	4	no	—
100	C₂×C₂₃₉	—	2	1	no	—
101	C₂×C₂₃₉	—	2	2	no	—
102	C₂×C₂₃₉	—	2	1	no	—
103	C₂×C₂₃₉	#339	1	4	yes	complemented
104	C₂×C₂₃₉	C₂×#95	1	1	yes	characteristic
105	C₂×C₂₃₉	C₂×#104	1	2	yes	—
106	C₂₃₉	C₂×#339	1	1	yes	characteristic, complemented, Sylow
107	C₂×#339	—	239	1	no	maximal, Sylow
108	C₂²×M₁₆	D₂₃₉	1	1	yes	characteristic
109	C₂×#117	—	239	1	no	—
110	C₂²×C₄²	—	239	1	no	—
111	#339	—	239	4	no	—
112	C₂³×C₄	—	478	1	no	—
113	C₂³×C₄	C₂×D₂₃₉	1	1	yes	characteristic
114	C₂×M₁₆	—	2	2	no	—
115	C₂×M₁₆	—	2	1	no	—
116	C₂×M₁₆	Dic₂₃₉	1	4	yes	complemented
117	C₂×M₁₆	C₂×D₂₃₉	1	2	yes	—
118	C₂²×C₈	—	2	1	no	—
119	#117	—	478	1	no	—
120	#117	—	478	1	no	—
121	#117	—	239	2	no	—
122	#117	—	239	2	no	—
123	C₂×C₄⋊C₄	—	239	2	no	—
124	C₂×C₂²⋊C₄	—	478	1	no	—
125	C₂×C₄²	—	478	2	no	—
126	C₂×C₄²	—	478	1	no	—
127	C₂×C₄²	—	478	1	no	—
128	C₂×C₄²	—	239	4	no	—
129	C₂×C₄²	—	239	2	no	—
130	C₂⁴	D₉₅₆	1	1	yes	characteristic
131	C₂²×C₄	—	478	2	no	—
132	C₂²×C₄	—	478	8	no	—
133	C₂²×C₄	—	478	4	no	—
134	C₂²×C₄	—	478	2	no	—
135	C₂²×C₄	C₄×D₂₃₉	1	2	yes	—
136	C₂²×C₄	C₄×D₂₃₉	1	1	yes	characteristic
137	C₂²×C₄	C₄×D₂₃₉	1	1	yes	characteristic
138	C₂²×C₄	C₂×Dic₂₃₉	1	2	yes	—
139	C₂²×C₄	C₂²⋊D₂₃₉	1	2	yes	—
140	C₂²×C₄	C₂²⋊D₂₃₉	1	1	yes	characteristic
141	C₂²×C₄	C₂²⋊D₂₃₉	1	1	yes	characteristic
142	C₂²×C₄	D₉₅₆	1	1	yes	characteristic
143	C₂²×C₄	Dic₄₇₈	1	2	yes	—
144	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
145	M₁₆	—	2	4	no	—
146	M₁₆	—	2	2	no	—
147	M₁₆	C₂×Dic₂₃₉	1	4	yes	complemented
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	2	2	no	—
150	C₄⋊C₄	—	478	2	no	—
151	C₄⋊C₄	—	239	4	no	—
152	C₂²⋊C₄	—	478	2	no	—
153	C₂²⋊C₄	—	478	2	no	—
154	C₄²	—	478	4	no	—
155	C₄²	—	478	2	no	—
156	C₄²	—	478	2	no	—
157	C₄²	—	478	2	no	—
158	C₄²	—	239	4	no	—
159	C₂³	—	2	4	no	—
160	C₂³	(C₂×D₂₃₉)⋊C₄	1	1	yes	characteristic
161	C₂³	(C₂×D₂₃₉)⋊C₄	1	2	yes	—
162	C₂³	(C₂×D₂₃₉)⋊C₄	1	1	yes	characteristic
163	C₂³	C₄⋊Dic₂₃₉	1	2	yes	—
164	C₂³	C₂×D₉₅₆	1	1	yes	characteristic
165	C₂×C₄	—	956	4	no	—
166	C₂×C₄	—	478	4	no	—
167	C₂×C₄	—	478	8	no	—
168	C₂×C₄	—	478	16	no	—
169	C₂×C₄	—	478	4	no	—
170	C₂×C₄	C₄×Dic₂₃₉	1	2	yes	—
171	C₂×C₄	C₄×Dic₂₃₉	1	2	yes	—
172	C₂×C₄	(C₂×D₂₃₉)⋊C₄	1	2	yes	—
173	C₂×C₄	(C₂×D₂₃₉)⋊C₄	1	1	yes	characteristic
174	C₂×C₄	(C₂×D₂₃₉)⋊C₄	1	1	yes	characteristic
175	C₂×C₄	C₂²⋊Dic₂₃₉	1	2	yes	—
176	C₂×C₄	C₂²⋊Dic₂₃₉	1	2	yes	—
177	C₂×C₄	Dic₂₃₉⋊C₄	1	2	yes	—
178	C₂×C₄	Dic₂₃₉⋊C₄	1	2	yes	—
179	C₂×C₄	Dic₂₃₉⋊C₄	1	2	yes	—
180	C₂×C₄	Dic₂₃₉⋊C₄	1	2	yes	—
181	C₂×C₄	C₄⋊Dic₂₃₉	1	2	yes	—
182	C₂×C₄	C₂×C₄×D₂₃₉	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×D₂₃₉	1	1	yes	characteristic
184	C₂×C₄	C₂²×Dic₂₃₉	1	1	yes	characteristic
185	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
186	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
187	C₂×C₄	C₂×Dic₄₇₈	1	1	yes	characteristic
188	C₈	—	2	4	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	4	no	—
193	C₂²	C₂₃₉⋊#95	1	2	yes	—
194	C₂²	C₂₃₉⋊#95	1	2	yes	—
195	C₂²	#21439886	1	4	yes	—
196	C₂²	C₂×(C₂×D₂₃₉)⋊C₄	1	1	yes	characteristic
197	C₂²	C₂×(C₂×D₂₃₉)⋊C₄	1	1	yes	characteristic
198	C₂²	C₂×C₄⋊Dic₂₃₉	1	1	yes	characteristic
199	C₄	—	956	4	no	—
200	C₄	—	478	8	no	—
201	C₄	C₂₃₉⋊#95	1	2	yes	—
202	C₄	C₂₃₉⋊#95	1	2	yes	—
203	C₄	C₂×C₄×Dic₂₃₉	1	1	yes	characteristic
204	C₄	C₂×C₂²⋊Dic₂₃₉	1	1	yes	characteristic
205	C₄	C₂×Dic₂₃₉⋊C₄	1	1	yes	characteristic
206	C₄	C₂×Dic₂₃₉⋊C₄	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	2	no	—
209	C₂	—	2	1	no	—
210	C₂	#29504533	1	4	yes	direct product factor
211	C₂	C₂×C₂₃₉⋊#95	1	1	yes	characteristic
212	C₂	C₂×#21439886	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₂₃₉
commutator	C₄×C₂₃₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₃₉
Fitting	C₂²×M₁₆×C₂₃₉	C₂
socle	C₂³×C₂₃₉	C₂²⋊C₄
prime radical	C₂⁴×C₂₃₉	D₄
cosocle	C₂²×C₄×C₂₃₉	C₂³
hypercenter	C₂²×M₁₆	D₂₃₉
hypocenter	C₂₃₉	C₂×#339
LCS	G ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #29504533	C₂	direct (N × H)
8. #21439919	C₄	semidirect
10. C₂×C₄⋊Dic₂₃₉	C₄	semidirect
15. C₂×C₄×Dic₂₃₉	C₄	semidirect
20. C₂×M₁₆×C₂₃₉	C₄	semidirect
28. C₄⋊Dic₂₃₉	C₂×C₄	semidirect
35. C₄×Dic₂₃₉	C₂×C₄	semidirect
49. M₁₆×C₂₃₉	C₂×C₄	semidirect
103. C₂×C₂₃₉	#339	semidirect
106. C₂₃₉	C₂×#339	semidirect
116. C₂×M₁₆	Dic₂₃₉	semidirect
147. M₁₆	C₂×Dic₂₃₉	semidirect
210. C₂	#29504533	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×#21439919	2
C₂²×C₄×Dic₂₃₉	2
#29504533	2
C₂²×M₁₆×C₂₃₉	2
C₂×#339	239

Subgroups — C2×#29504533 #44484255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#29504533 #44484255