Subgroups — C₂×#540×C₂₃₉ #44482222

Order 30592.

1260 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#540×C₂₃₉	C₂	1	16	yes	maximal, direct product factor
2	C₂×#127×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#125×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#123×C₂₃₉	C₂	1	4	yes	maximal, complemented
5	C₂×#121×C₂₃₉	C₂	1	4	yes	maximal, complemented
6	C₂×#121×C₂₃₉	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×D₄×C₂₃₉	C₂	1	2	yes	maximal, complemented
8	C₂×#117×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₂₃₉	C₂²	1	2	yes	—
10	C₂²×D₄×C₂₃₉	C₂²	1	4	yes	—
11	C₂²×D₄×C₂₃₉	C₂²	1	4	yes	—
12	C₂³×C₄×C₂₃₉	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₂₃₉	C₂²	1	4	yes	—
14	#127×C₂₃₉	C₂²	1	8	yes	complemented
15	#125×C₂₃₉	C₂²	1	8	yes	complemented
16	#123×C₂₃₉	C₂²	1	32	yes	complemented
17	#121×C₂₃₉	C₂²	1	32	yes	complemented
18	#121×C₂₃₉	C₂²	1	16	yes	complemented
19	C₄×D₄×C₂₃₉	C₂²	1	16	yes	complemented
20	#117×C₂₃₉	C₂²	1	8	yes	complemented
21	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	4	yes	—
24	C₂×C₂²⋊C₄×C₂₃₉	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₂₃₉	C₂²	1	4	yes	—
26	C₂×C₂²⋊C₄×C₂₃₉	C₂²	1	4	yes	—
27	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
29	C₂⁴×C₂₃₉	C₂³	1	1	yes	characteristic
30	C₂⁴×C₂₃₉	C₂³	1	4	yes	—
31	C₂×D₄×C₂₃₉	—	2	8	no	—
32	C₂×D₄×C₂₃₉	—	4	8	no	—
33	C₂×D₄×C₂₃₉	—	4	8	no	—
34	C₂×D₄×C₂₃₉	C₂³	1	8	yes	—
35	C₂×D₄×C₂₃₉	C₂³	1	16	yes	—
36	C₂×D₄×C₂₃₉	C₂³	1	16	yes	—
37	C₂²×C₄×C₂₃₉	—	2	4	no	—
38	C₂²×C₄×C₂₃₉	—	4	8	no	—
39	C₂²×C₄×C₂₃₉	C₂³	1	4	yes	—
40	C₂²×C₄×C₂₃₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃₉	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₂₃₉	C₂³	1	4	yes	—
44	C₂²×C₄×C₂₃₉	C₂³	1	16	yes	—
45	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₂₃₉	C₂³	1	8	yes	—
47	C₄⋊C₄×C₂₃₉	C₂³	1	8	yes	—
48	C₄⋊C₄×C₂₃₉	C₂³	1	16	yes	—
49	C₂²⋊C₄×C₂₃₉	C₂³	1	8	yes	—
50	C₂²⋊C₄×C₂₃₉	C₂³	1	16	yes	—
51	C₂²⋊C₄×C₂₃₉	C₂³	1	16	yes	—
52	C₄²×C₂₃₉	C₂³	1	4	yes	—
53	C₄²×C₂₃₉	C₂³	1	4	yes	—
54	C₂³×C₂₃₉	—	2	2	no	—
55	C₂³×C₂₃₉	—	2	8	no	—
56	C₂³×C₂₃₉	—	2	8	no	—
57	C₂³×C₂₃₉	—	2	4	no	—
58	C₂³×C₂₃₉	—	2	8	no	—
59	C₂³×C₂₃₉	C₂⁴	1	2	yes	—
60	C₂³×C₂₃₉	C₂⁴	1	8	yes	—
61	C₂³×C₂₃₉	C₂⁴	1	1	yes	characteristic
62	D₄×C₂₃₉	—	2	16	no	—
63	D₄×C₂₃₉	—	4	16	no	—
64	D₄×C₂₃₉	—	4	16	no	—
65	C₂×C₄×C₂₃₉	—	2	2	no	—
66	C₂×C₄×C₂₃₉	—	2	4	no	—
67	C₂×C₄×C₂₃₉	—	2	8	no	—
68	C₂×C₄×C₂₃₉	—	2	4	no	—
69	C₂×C₄×C₂₃₉	—	2	8	no	—
70	C₂×C₄×C₂₃₉	—	4	16	no	—
71	C₂×C₄×C₂₃₉	#41	1	4	yes	—
72	C₂×C₄×C₂₃₉	C₂⁴	1	4	yes	—
73	C₂×C₄×C₂₃₉	C₂⁴	1	4	yes	—
74	C₂×C₄×C₂₃₉	C₂⁴	1	2	yes	—
75	C₂×C₄×C₂₃₉	C₂⁴	1	8	yes	—
76	C₂×C₄×C₂₃₉	C₂⁴	1	2	yes	—
77	C₂²×C₂₃₉	—	2	2	no	—
78	C₂²×C₂₃₉	—	2	8	no	—
79	C₂²×C₂₃₉	—	2	8	no	—
80	C₂²×C₂₃₉	—	2	4	no	—
81	C₂²×C₂₃₉	—	2	8	no	—
82	C₂²×C₂₃₉	—	4	4	no	—
83	C₂²×C₂₃₉	—	4	16	no	—
84	C₂²×C₂₃₉	C₂×#41	1	2	yes	—
85	C₂²×C₂₃₉	H(5,𝔽₂)	1	4	yes	—
86	C₂²×C₂₃₉	C₂⁵	1	1	yes	characteristic
87	C₄×C₂₃₉	—	2	2	no	—
88	C₄×C₂₃₉	—	2	4	no	—
89	C₄×C₂₃₉	—	2	8	no	—
90	C₄×C₂₃₉	—	2	4	no	—
91	C₄×C₂₃₉	C₂×#41	1	4	yes	—
92	C₂×C₂₃₉	—	4	2	no	—
93	C₂×C₂₃₉	—	4	8	no	—
94	C₂×C₂₃₉	#540	1	4	yes	direct product factor
95	C₂×C₂₃₉	C₂²×#41	1	1	yes	characteristic
96	C₂×C₂₃₉	C₂×H(5,𝔽₂)	1	2	yes	—
97	C₂₃₉	C₂×#540	1	1	yes	characteristic, direct product factor, Sylow
98	C₂×#540	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	#540	C₂×C₂₃₉	1	16	yes	direct product factor
100	C₂×#127	C₂×C₂₃₉	1	1	yes	characteristic, complemented
101	C₂×#125	C₂×C₂₃₉	1	1	yes	characteristic, complemented
102	C₂×#123	C₂×C₂₃₉	1	4	yes	complemented
103	C₂×#121	C₂×C₂₃₉	1	4	yes	complemented
104	C₂×#121	C₂×C₂₃₉	1	2	yes	complemented
105	C₂×C₄×D₄	C₂×C₂₃₉	1	2	yes	complemented
106	C₂×#117	C₂×C₂₃₉	1	1	yes	characteristic, complemented
107	C₂²×D₄	C₂²×C₂₃₉	1	2	yes	—
108	C₂²×D₄	C₂²×C₂₃₉	1	4	yes	—
109	C₂²×D₄	C₂²×C₂₃₉	1	4	yes	—
110	C₂³×C₄	C₂²×C₂₃₉	1	1	yes	characteristic
111	C₂³×C₄	C₂²×C₂₃₉	1	4	yes	—
112	#127	C₂²×C₂₃₉	1	8	yes	complemented
113	#125	C₂²×C₂₃₉	1	8	yes	complemented
114	#123	C₂²×C₂₃₉	1	32	yes	complemented
115	#121	C₂²×C₂₃₉	1	32	yes	complemented
116	#121	C₂²×C₂₃₉	1	16	yes	complemented
117	C₄×D₄	C₂²×C₂₃₉	1	16	yes	complemented
118	#117	C₂²×C₂₃₉	1	8	yes	complemented
119	C₂×C₄⋊C₄	C₂²×C₂₃₉	1	2	yes	—
120	C₂×C₄⋊C₄	C₂²×C₂₃₉	1	2	yes	—
121	C₂×C₄⋊C₄	C₂²×C₂₃₉	1	4	yes	—
122	C₂×C₂²⋊C₄	C₂²×C₂₃₉	1	2	yes	—
123	C₂×C₂²⋊C₄	C₂²×C₂₃₉	1	4	yes	—
124	C₂×C₂²⋊C₄	C₂²×C₂₃₉	1	4	yes	—
125	C₂×C₄²	C₂²×C₂₃₉	1	1	yes	characteristic
126	C₂×C₄²	C₂²×C₂₃₉	1	1	yes	characteristic
127	C₂⁴	C₂³×C₂₃₉	1	1	yes	characteristic
128	C₂⁴	C₂³×C₂₃₉	1	4	yes	—
129	C₂×D₄	—	2	8	no	—
130	C₂×D₄	—	4	8	no	—
131	C₂×D₄	—	4	8	no	—
132	C₂×D₄	C₂³×C₂₃₉	1	8	yes	—
133	C₂×D₄	C₂³×C₂₃₉	1	16	yes	—
134	C₂×D₄	C₂³×C₂₃₉	1	16	yes	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	—	4	8	no	—
137	C₂²×C₄	C₂³×C₂₃₉	1	2	yes	—
138	C₂²×C₄	C₂³×C₂₃₉	1	2	yes	—
139	C₂²×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
140	C₂²×C₄	C₂³×C₂₃₉	1	4	yes	—
141	C₂²×C₄	C₂³×C₂₃₉	1	4	yes	—
142	C₂²×C₄	C₂³×C₂₃₉	1	16	yes	—
143	C₂²×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
144	C₄⋊C₄	C₂³×C₂₃₉	1	8	yes	—
145	C₄⋊C₄	C₂³×C₂₃₉	1	8	yes	—
146	C₄⋊C₄	C₂³×C₂₃₉	1	16	yes	—
147	C₂²⋊C₄	C₂³×C₂₃₉	1	8	yes	—
148	C₂²⋊C₄	C₂³×C₂₃₉	1	16	yes	—
149	C₂²⋊C₄	C₂³×C₂₃₉	1	16	yes	—
150	C₄²	C₂³×C₂₃₉	1	4	yes	—
151	C₄²	C₂³×C₂₃₉	1	4	yes	—
152	C₂³	—	2	2	no	—
153	C₂³	—	2	8	no	—
154	C₂³	—	2	8	no	—
155	C₂³	—	2	4	no	—
156	C₂³	—	2	8	no	—
157	C₂³	C₂⁴×C₂₃₉	1	2	yes	—
158	C₂³	C₂⁴×C₂₃₉	1	8	yes	—
159	C₂³	C₂⁴×C₂₃₉	1	1	yes	characteristic
160	D₄	—	2	16	no	—
161	D₄	—	4	16	no	—
162	D₄	—	4	16	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	—	4	16	no	—
169	C₂×C₄	#41×C₂₃₉	1	4	yes	—
170	C₂×C₄	C₂⁴×C₂₃₉	1	4	yes	—
171	C₂×C₄	C₂⁴×C₂₃₉	1	4	yes	—
172	C₂×C₄	C₂⁴×C₂₃₉	1	2	yes	—
173	C₂×C₄	C₂⁴×C₂₃₉	1	8	yes	—
174	C₂×C₄	C₂⁴×C₂₃₉	1	2	yes	—
175	C₂²	—	2	2	no	—
176	C₂²	—	2	8	no	—
177	C₂²	—	2	8	no	—
178	C₂²	—	2	4	no	—
179	C₂²	—	2	8	no	—
180	C₂²	—	4	4	no	—
181	C₂²	—	4	16	no	—
182	C₂²	C₂×#41×C₂₃₉	1	2	yes	—
183	C₂²	H(5,𝔽₂)×C₂₃₉	1	4	yes	—
184	C₂²	C₂⁵×C₂₃₉	1	1	yes	characteristic
185	C₄	—	2	2	no	—
186	C₄	—	2	4	no	—
187	C₄	—	2	8	no	—
188	C₄	—	2	4	no	—
189	C₄	C₂×#41×C₂₃₉	1	4	yes	—
190	C₂	—	4	2	no	—
191	C₂	—	4	8	no	—
192	C₂	#540×C₂₃₉	1	4	yes	direct product factor
193	C₂	C₂²×#41×C₂₃₉	1	1	yes	characteristic
194	C₂	C₂×H(5,𝔽₂)×C₂₃₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₃₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #540×C₂₃₉	C₂	direct (N × H)
2. C₂×#127×C₂₃₉	C₂	semidirect
3. C₂×#125×C₂₃₉	C₂	semidirect
4. C₂×#123×C₂₃₉	C₂	semidirect
5. C₂×#121×C₂₃₉	C₂	semidirect
6. C₂×#121×C₂₃₉	C₂	semidirect
7. C₂×C₄×D₄×C₂₃₉	C₂	semidirect
8. C₂×#117×C₂₃₉	C₂	semidirect
14. #127×C₂₃₉	C₂²	semidirect
15. #125×C₂₃₉	C₂²	semidirect
16. #123×C₂₃₉	C₂²	semidirect
17. #121×C₂₃₉	C₂²	semidirect
18. #121×C₂₃₉	C₂²	semidirect
19. C₄×D₄×C₂₃₉	C₂²	semidirect
20. #117×C₂₃₉	C₂²	semidirect
94. C₂×C₂₃₉	#540	direct (N × H)
97. C₂₃₉	C₂×#540	direct (N × H)
98. C₂×#540	C₂₃₉	direct (N × H)
99. #540	C₂×C₂₃₉	direct (N × H)
100. C₂×#127	C₂×C₂₃₉	semidirect
101. C₂×#125	C₂×C₂₃₉	semidirect
102. C₂×#123	C₂×C₂₃₉	semidirect
103. C₂×#121	C₂×C₂₃₉	semidirect
104. C₂×#121	C₂×C₂₃₉	semidirect
105. C₂×C₄×D₄	C₂×C₂₃₉	semidirect
106. C₂×#117	C₂×C₂₃₉	semidirect
112. #127	C₂²×C₂₃₉	semidirect
113. #125	C₂²×C₂₃₉	semidirect
114. #123	C₂²×C₂₃₉	semidirect
115. #121	C₂²×C₂₃₉	semidirect
116. #121	C₂²×C₂₃₉	semidirect
117. C₄×D₄	C₂²×C₂₃₉	semidirect
118. #117	C₂²×C₂₃₉	semidirect
192. C₂	#540×C₂₃₉	direct (N × H)

Maximal subgroups

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

subgroup	index
#540×C₂₃₉	2
C₂×#127×C₂₃₉	2
C₂×#125×C₂₃₉	2
C₂×#123×C₂₃₉	2
C₂×#121×C₂₃₉	2
C₂×C₄×D₄×C₂₃₉	2
C₂×#117×C₂₃₉	2
C₂×#540	239

Subgroups — C2×#540×C239 #44482222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#540×C₂₃₉ #44482222