Subgroups — #3126×C₂₃₉ #44481896

Order 30592.

382 conjugacy classes of subgroups.

The subgroup lattice has 204 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	1	yes	maximal, characteristic
2	#519×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#482×C₂₃₉	C₂	1	2	yes	maximal, complemented
4	#479×C₂₃₉	C₂	1	2	yes	maximal, complemented
5	#460×C₂₃₉	C₂	1	2	yes	maximal, complemented
6	#448×C₂₃₉	C₂	1	2	yes	maximal, complemented
7	#440×C₂₃₉	C₂	1	2	yes	maximal, complemented
8	C₄×SD₁₆×C₂₃₉	C₂	1	2	yes	maximal, complemented
9	#433×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
11	C₂×SD₁₆×C₂₃₉	C₂²	1	2	yes	—
12	C₂×SD₁₆×C₂₃₉	C₂²	1	2	yes	—
13	#127×C₂₃₉	C₂²	1	1	yes	characteristic
14	#125×C₂₃₉	C₂²	1	1	yes	characteristic
15	#123×C₂₃₉	—	2	2	no	—
16	#121×C₂₃₉	—	2	2	no	—
17	#121×C₂₃₉	C₂²	1	2	yes	—
18	C₄×Q₈×C₂₃₉	—	2	2	no	—
19	C₄×Q₈×C₂₃₉	C₂²	1	2	yes	—
20	C₄×Q₈×C₂₃₉	C₂²	1	2	yes	—
21	C₄×D₄×C₂₃₉	C₂²	1	2	yes	—
22	#117×C₂₃₉	—	2	2	no	—
23	#117×C₂₃₉	C₂²	1	1	yes	characteristic
24	#117×C₂₃₉	C₂²	1	1	yes	characteristic
25	#107×C₂₃₉	C₂²	1	2	yes	—
26	#106×C₂₃₉	C₂²	1	2	yes	—
27	C₄⋊C₈×C₂₃₉	C₂²	1	2	yes	—
28	#103×C₂₃₉	C₂²	1	2	yes	—
29	#103×C₂₃₉	C₂²	1	2	yes	—
30	#102×C₂₃₉	C₂²	1	2	yes	—
31	#102×C₂₃₉	C₂²	1	2	yes	—
32	C₂²⋊C₈×C₂₃₉	C₂²	1	2	yes	—
33	#97×C₂₃₉	C₂²	1	1	yes	characteristic
34	C₄×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₂₃₉	—	2	4	no	—
36	C₂×Q₈×C₂₃₉	—	2	1	no	—
37	C₂×Q₈×C₂₃₉	C₂³	1	2	yes	—
38	C₂×D₄×C₂₃₉	—	2	2	no	—
39	C₂×D₄×C₂₃₉	—	2	2	no	—
40	C₂×D₄×C₂₃₉	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃₉	—	2	1	no	—
42	C₂²×C₄×C₂₃₉	—	2	2	no	—
43	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
44	SD₁₆×C₂₃₉	—	2	4	no	—
45	SD₁₆×C₂₃₉	—	2	4	no	—
46	C₂×C₈×C₂₃₉	C₂³	1	2	yes	—
47	C₂×C₈×C₂₃₉	C₂³	1	2	yes	—
48	C₄⋊C₄×C₂₃₉	—	2	2	no	—
49	C₄⋊C₄×C₂₃₉	—	2	2	no	—
50	C₄⋊C₄×C₂₃₉	—	2	2	no	—
51	C₄⋊C₄×C₂₃₉	D₄	1	2	yes	—
52	C₄⋊C₄×C₂₃₉	C₂³	1	2	yes	—
53	C₄⋊C₄×C₂₃₉	C₂³	1	2	yes	—
54	C₂²⋊C₄×C₂₃₉	—	2	1	no	—
55	C₂²⋊C₄×C₂₃₉	—	2	2	no	—
56	C₂²⋊C₄×C₂₃₉	—	2	2	no	—
57	C₂²⋊C₄×C₂₃₉	D₄	1	2	yes	—
58	C₄²×C₂₃₉	—	2	2	no	—
59	C₄²×C₂₃₉	—	2	2	no	—
60	C₄²×C₂₃₉	—	2	1	no	—
61	C₄²×C₂₃₉	C₂³	1	1	yes	characteristic
62	C₄²×C₂₃₉	C₂³	1	1	yes	characteristic
63	C₂³×C₂₃₉	—	2	2	no	—
64	C₂³×C₂₃₉	C₂×D₄	1	1	yes	characteristic
65	Q₈×C₂₃₉	—	2	4	no	—
66	Q₈×C₂₃₉	—	2	2	no	—
67	Q₈×C₂₃₉	#41	1	4	yes	—
68	D₄×C₂₃₉	—	2	2	no	—
69	D₄×C₂₃₉	—	2	2	no	—
70	D₄×C₂₃₉	—	8	2	no	—
71	D₄×C₂₃₉	—	8	2	no	—
72	C₂×C₄×C₂₃₉	—	2	4	no	—
73	C₂×C₄×C₂₃₉	—	2	2	no	—
74	C₂×C₄×C₂₃₉	—	2	2	no	—
75	C₂×C₄×C₂₃₉	—	2	2	no	—
76	C₂×C₄×C₂₃₉	—	2	2	no	—
77	C₂×C₄×C₂₃₉	—	8	2	no	—
78	C₂×C₄×C₂₃₉	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₂₃₉	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₂₃₉	C₂⁴	1	1	yes	characteristic
82	C₈×C₂₃₉	—	2	2	no	—
83	C₈×C₂₃₉	—	2	2	no	—
84	C₂²×C₂₃₉	—	2	1	no	—
85	C₂²×C₂₃₉	—	2	2	no	—
86	C₂²×C₂₃₉	—	4	2	no	—
87	C₂²×C₂₃₉	—	4	2	no	—
88	C₂²×C₂₃₉	—	4	2	no	—
89	C₂²×C₂₃₉	C₂²×D₄	1	1	yes	characteristic
90	C₄×C₂₃₉	—	2	4	no	—
91	C₄×C₂₃₉	—	2	1	no	—
92	C₄×C₂₃₉	—	2	2	no	—
93	C₄×C₂₃₉	—	2	2	no	—
94	C₄×C₂₃₉	—	4	2	no	—
95	C₄×C₂₃₉	—	4	2	no	—
96	C₄×C₂₃₉	C₂×#41	1	2	yes	—
97	C₂×C₂₃₉	—	4	1	no	—
98	C₂×C₂₃₉	—	8	2	no	—
99	C₂×C₂₃₉	#525	1	1	yes	characteristic
100	C₂×C₂₃₉	#577	1	2	yes	—
101	C₂₃₉	#3126	1	1	yes	characteristic, direct product factor, Sylow
102	#3126	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	#540	C₂×C₂₃₉	1	1	yes	characteristic
104	#519	C₂×C₂₃₉	1	1	yes	characteristic, complemented
105	#482	C₂×C₂₃₉	1	2	yes	complemented
106	#479	C₂×C₂₃₉	1	2	yes	complemented
107	#460	C₂×C₂₃₉	1	2	yes	complemented
108	#448	C₂×C₂₃₉	1	2	yes	complemented
109	#440	C₂×C₂₃₉	1	2	yes	complemented
110	C₄×SD₁₆	C₂×C₂₃₉	1	2	yes	complemented
111	#433	C₂×C₂₃₉	1	1	yes	characteristic, complemented
112	C₂²×Q₈	C₂²×C₂₃₉	1	1	yes	characteristic
113	C₂×SD₁₆	C₂²×C₂₃₉	1	2	yes	—
114	C₂×SD₁₆	C₂²×C₂₃₉	1	2	yes	—
115	#127	C₂²×C₂₃₉	1	1	yes	characteristic
116	#125	C₂²×C₂₃₉	1	1	yes	characteristic
117	#123	—	2	2	no	—
118	#121	—	2	2	no	—
119	#121	C₂²×C₂₃₉	1	2	yes	—
120	C₄×Q₈	—	2	2	no	—
121	C₄×Q₈	C₂²×C₂₃₉	1	2	yes	—
122	C₄×Q₈	C₂²×C₂₃₉	1	2	yes	—
123	C₄×D₄	C₂²×C₂₃₉	1	2	yes	—
124	#117	—	2	2	no	—
125	#117	C₂²×C₂₃₉	1	1	yes	characteristic
126	#117	C₂²×C₂₃₉	1	1	yes	characteristic
127	#107	C₂²×C₂₃₉	1	2	yes	—
128	#106	C₂²×C₂₃₉	1	2	yes	—
129	C₄⋊C₈	C₂²×C₂₃₉	1	2	yes	—
130	#103	C₂²×C₂₃₉	1	2	yes	—
131	#103	C₂²×C₂₃₉	1	2	yes	—
132	#102	C₂²×C₂₃₉	1	2	yes	—
133	#102	C₂²×C₂₃₉	1	2	yes	—
134	C₂²⋊C₈	C₂²×C₂₃₉	1	2	yes	—
135	#97	C₂²×C₂₃₉	1	1	yes	characteristic
136	C₄×C₈	C₂²×C₂₃₉	1	1	yes	characteristic
137	C₂×Q₈	—	2	4	no	—
138	C₂×Q₈	—	2	1	no	—
139	C₂×Q₈	C₂³×C₂₃₉	1	2	yes	—
140	C₂×D₄	—	2	2	no	—
141	C₂×D₄	—	2	2	no	—
142	C₂×D₄	C₂³×C₂₃₉	1	2	yes	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
146	SD₁₆	—	2	4	no	—
147	SD₁₆	—	2	4	no	—
148	C₂×C₈	C₂³×C₂₃₉	1	2	yes	—
149	C₂×C₈	C₂³×C₂₃₉	1	2	yes	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	D₄×C₂₃₉	1	2	yes	—
154	C₄⋊C₄	C₂³×C₂₃₉	1	2	yes	—
155	C₄⋊C₄	C₂³×C₂₃₉	1	2	yes	—
156	C₂²⋊C₄	—	2	1	no	—
157	C₂²⋊C₄	—	2	2	no	—
158	C₂²⋊C₄	—	2	2	no	—
159	C₂²⋊C₄	D₄×C₂₃₉	1	2	yes	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	1	no	—
163	C₄²	C₂³×C₂₃₉	1	1	yes	characteristic
164	C₄²	C₂³×C₂₃₉	1	1	yes	characteristic
165	C₂³	—	2	2	no	—
166	C₂³	C₂×D₄×C₂₃₉	1	1	yes	characteristic
167	Q₈	—	2	4	no	—
168	Q₈	—	2	2	no	—
169	Q₈	#41×C₂₃₉	1	4	yes	—
170	D₄	—	2	2	no	—
171	D₄	—	2	2	no	—
172	D₄	—	8	2	no	—
173	D₄	—	8	2	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	8	2	no	—
180	C₂×C₄	C₂×D₄×C₂₃₉	1	2	yes	—
181	C₂×C₄	C₂×D₄×C₂₃₉	1	2	yes	—
182	C₂×C₄	C₂×D₄×C₂₃₉	1	1	yes	characteristic
183	C₂×C₄	C₂⁴×C₂₃₉	1	1	yes	characteristic
184	C₈	—	2	2	no	—
185	C₈	—	2	2	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	2	2	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	4	2	no	—
191	C₂²	C₂²×D₄×C₂₃₉	1	1	yes	characteristic
192	C₄	—	2	4	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	4	2	no	—
197	C₄	—	4	2	no	—
198	C₄	C₂×#41×C₂₃₉	1	2	yes	—
199	C₂	—	4	1	no	—
200	C₂	—	8	2	no	—
201	C₂	#525×C₂₃₉	1	1	yes	characteristic
202	C₂	#577×C₂₃₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃₉
prime radical	#540×C₂₃₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃₉ ◁ #117×C₂₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #519×C₂₃₉	C₂	semidirect
3. #482×C₂₃₉	C₂	semidirect
4. #479×C₂₃₉	C₂	semidirect
5. #460×C₂₃₉	C₂	semidirect
6. #448×C₂₃₉	C₂	semidirect
7. #440×C₂₃₉	C₂	semidirect
8. C₄×SD₁₆×C₂₃₉	C₂	semidirect
9. #433×C₂₃₉	C₂	semidirect
101. C₂₃₉	#3126	direct (N × H)
102. #3126	C₂₃₉	direct (N × H)
104. #519	C₂×C₂₃₉	semidirect
105. #482	C₂×C₂₃₉	semidirect
106. #479	C₂×C₂₃₉	semidirect
107. #460	C₂×C₂₃₉	semidirect
108. #448	C₂×C₂₃₉	semidirect
109. #440	C₂×C₂₃₉	semidirect
110. C₄×SD₁₆	C₂×C₂₃₉	semidirect
111. #433	C₂×C₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
#540×C₂₃₉	2
#519×C₂₃₉	2
#482×C₂₃₉	2
#479×C₂₃₉	2
#460×C₂₃₉	2
#448×C₂₃₉	2
#440×C₂₃₉	2
C₄×SD₁₆×C₂₃₉	2
#433×C₂₃₉	2
#3126	239

Subgroups — #3126×C239 #44481896

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3126×C₂₃₉ #44481896