Subgroups — #3043×C₂₃₉ #44481813

Order 30592.

490 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#137×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×SD₁₆×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#535×C₂₃₉	C₂	1	1	yes	maximal, characteristic
4	#519×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#464×C₂₃₉	C₂	1	4	yes	maximal, complemented
6	#460×C₂₃₉	C₂	1	4	yes	maximal, complemented
7	#424×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#418×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#416×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₂₃₉	C₂²	1	1	yes	characteristic
11	C₂²×Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₂₃₉	C₂²	1	1	yes	characteristic
13	#137×C₂₃₉	—	2	4	no	—
14	C₂×Q₁₆×C₂₃₉	C₂²	1	2	yes	complemented
15	C₂×SD₁₆×C₂₃₉	—	2	4	no	—
16	C₂×SD₁₆×C₂₃₉	C₂²	1	4	yes	—
17	C₂×SD₁₆×C₂₃₉	C₂²	1	2	yes	complemented
18	C₂×M₁₆×C₂₃₉	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
20	#127×C₂₃₉	C₂²	1	2	yes	—
21	#124×C₂₃₉	C₂²	1	2	yes	—
22	#121×C₂₃₉	—	2	2	no	—
23	#120×C₂₃₉	—	2	2	no	—
24	C₄×Q₈×C₂₃₉	—	2	2	no	—
25	C₄×Q₈×C₂₃₉	C₂²	1	4	yes	complemented
26	#117×C₂₃₉	—	2	2	no	—
27	#117×C₂₃₉	C₂²	1	1	yes	characteristic
28	#117×C₂₃₉	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₂₃₉	C₂²	1	4	yes	complemented
30	#103×C₂₃₉	C₂²	1	2	yes	complemented
31	#102×C₂₃₉	C₂²	1	4	yes	—
32	#102×C₂₃₉	C₂²	1	2	yes	complemented
33	C₂⁴×C₂₃₉	—	2	1	no	—
34	#41×C₂₃₉	—	2	2	no	—
35	#41×C₂₃₉	—	2	2	no	—
36	C₂×Q₈×C₂₃₉	—	2	1	no	—
37	C₂×Q₈×C₂₃₉	—	2	2	no	—
38	C₂×Q₈×C₂₃₉	D₄	1	4	yes	complemented
39	C₂×Q₈×C₂₃₉	C₂³	1	1	yes	characteristic
40	C₂×Q₈×C₂₃₉	C₂³	1	2	yes	—
41	C₂×D₄×C₂₃₉	—	2	1	no	—
42	C₂×D₄×C₂₃₉	—	2	2	no	—
43	C₂×D₄×C₂₃₉	—	2	1	no	—
44	C₂×D₄×C₂₃₉	—	2	4	no	—
45	C₂×D₄×C₂₃₉	—	2	2	no	—
46	C₂×D₄×C₂₃₉	C₂³	1	2	yes	—
47	C₂×D₄×C₂₃₉	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₂₃₉	—	2	1	no	—
49	C₂²×C₄×C₂₃₉	—	2	1	no	—
50	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
51	Q₁₆×C₂₃₉	—	2	4	no	—
52	SD₁₆×C₂₃₉	—	2	8	no	—
53	SD₁₆×C₂₃₉	—	2	4	no	—
54	M₁₆×C₂₃₉	—	2	2	no	—
55	C₂×C₈×C₂₃₉	—	2	2	no	—
56	C₂×C₈×C₂₃₉	C₂³	1	2	yes	—
57	C₂×C₈×C₂₃₉	C₂³	1	2	yes	—
58	C₄⋊C₄×C₂₃₉	—	2	4	no	—
59	C₄⋊C₄×C₂₃₉	D₄	1	2	yes	—
60	C₄⋊C₄×C₂₃₉	C₂³	1	2	yes	—
61	C₂²⋊C₄×C₂₃₉	—	2	1	no	—
62	C₂²⋊C₄×C₂₃₉	—	2	4	no	—
63	C₂²⋊C₄×C₂₃₉	D₄	1	2	yes	—
64	C₄²×C₂₃₉	—	2	4	no	—
65	C₄²×C₂₃₉	—	2	1	no	—
66	C₄²×C₂₃₉	C₂³	1	2	yes	—
67	C₂³×C₂₃₉	—	2	2	no	—
68	C₂³×C₂₃₉	—	2	1	no	—
69	C₂³×C₂₃₉	—	4	2	no	—
70	C₂³×C₂₃₉	—	8	2	no	—
71	C₂³×C₂₃₉	C₂×D₄	1	1	yes	characteristic
72	Q₈×C₂₃₉	—	2	4	no	—
73	Q₈×C₂₃₉	—	2	2	no	—
74	Q₈×C₂₃₉	—	2	1	no	—
75	Q₈×C₂₃₉	—	2	1	no	—
76	Q₈×C₂₃₉	C₂×D₄	1	4	yes	—
77	D₄×C₂₃₉	—	2	2	no	—
78	D₄×C₂₃₉	—	2	1	no	—
79	D₄×C₂₃₉	—	2	2	no	—
80	D₄×C₂₃₉	—	2	1	no	—
81	D₄×C₂₃₉	—	4	2	no	—
82	D₄×C₂₃₉	—	4	2	no	—
83	D₄×C₂₃₉	—	8	4	no	—
84	C₂×C₄×C₂₃₉	—	2	4	no	—
85	C₂×C₄×C₂₃₉	—	2	2	no	—
86	C₂×C₄×C₂₃₉	—	2	2	no	—
87	C₂×C₄×C₂₃₉	—	2	1	no	—
88	C₂×C₄×C₂₃₉	—	4	2	no	—
89	C₂×C₄×C₂₃₉	C₂×D₄	1	4	yes	—
90	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃₉	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₂₃₉	#41	1	2	yes	—
93	C₂×C₄×C₂₃₉	C₂⁴	1	1	yes	characteristic
94	C₈×C₂₃₉	—	2	2	no	—
95	C₈×C₂₃₉	—	2	2	no	—
96	C₂²×C₂₃₉	—	2	2	no	—
97	C₂²×C₂₃₉	—	4	1	no	—
98	C₂²×C₂₃₉	—	4	2	no	—
99	C₂²×C₂₃₉	—	4	2	no	—
100	C₂²×C₂₃₉	—	4	4	no	—
101	C₂²×C₂₃₉	—	8	4	no	—
102	C₂²×C₂₃₉	#121	1	2	yes	—
103	C₂²×C₂₃₉	C₂²×D₄	1	1	yes	characteristic
104	C₄×C₂₃₉	—	2	4	no	—
105	C₄×C₂₃₉	—	2	4	no	—
106	C₄×C₂₃₉	—	4	1	no	—
107	C₄×C₂₃₉	—	4	2	no	—
108	C₄×C₂₃₉	#121	1	2	yes	—
109	C₄×C₂₃₉	C₂²×D₄	1	1	yes	characteristic
110	C₄×C₂₃₉	C₂×#41	1	1	yes	characteristic
111	C₂×C₂₃₉	—	2	2	no	—
112	C₂×C₂₃₉	—	8	1	no	—
113	C₂×C₂₃₉	—	8	2	no	—
114	C₂×C₂₃₉	C₂×#121	1	1	yes	characteristic
115	C₂×C₂₃₉	#577	1	2	yes	—
116	C₂₃₉	#3043	1	1	yes	characteristic, direct product factor, Sylow
117	#3043	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂×#137	C₂×C₂₃₉	1	1	yes	characteristic, complemented
119	C₂²×SD₁₆	C₂×C₂₃₉	1	1	yes	characteristic, complemented
120	#535	C₂×C₂₃₉	1	1	yes	characteristic
121	#519	C₂×C₂₃₉	1	1	yes	characteristic, complemented
122	#464	C₂×C₂₃₉	1	4	yes	complemented
123	#460	C₂×C₂₃₉	1	4	yes	complemented
124	#424	C₂×C₂₃₉	1	1	yes	characteristic, complemented
125	#418	C₂×C₂₃₉	1	1	yes	characteristic, complemented
126	#416	C₂×C₂₃₉	1	1	yes	characteristic, complemented
127	C₂×#41	C₂²×C₂₃₉	1	1	yes	characteristic
128	C₂²×Q₈	C₂²×C₂₃₉	1	1	yes	characteristic
129	C₂²×D₄	C₂²×C₂₃₉	1	1	yes	characteristic
130	#137	—	2	4	no	—
131	C₂×Q₁₆	C₂²×C₂₃₉	1	2	yes	complemented
132	C₂×SD₁₆	—	2	4	no	—
133	C₂×SD₁₆	C₂²×C₂₃₉	1	4	yes	—
134	C₂×SD₁₆	C₂²×C₂₃₉	1	2	yes	complemented
135	C₂×M₁₆	C₂²×C₂₃₉	1	1	yes	characteristic
136	C₂²×C₈	C₂²×C₂₃₉	1	1	yes	characteristic
137	#127	C₂²×C₂₃₉	1	2	yes	—
138	#124	C₂²×C₂₃₉	1	2	yes	—
139	#121	—	2	2	no	—
140	#120	—	2	2	no	—
141	C₄×Q₈	—	2	2	no	—
142	C₄×Q₈	C₂²×C₂₃₉	1	4	yes	complemented
143	#117	—	2	2	no	—
144	#117	C₂²×C₂₃₉	1	1	yes	characteristic
145	#117	C₂²×C₂₃₉	1	1	yes	characteristic
146	C₄⋊C₈	C₂²×C₂₃₉	1	4	yes	complemented
147	#103	C₂²×C₂₃₉	1	2	yes	complemented
148	#102	C₂²×C₂₃₉	1	4	yes	—
149	#102	C₂²×C₂₃₉	1	2	yes	complemented
150	C₂⁴	—	2	1	no	—
151	#41	—	2	2	no	—
152	#41	—	2	2	no	—
153	C₂×Q₈	—	2	1	no	—
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	D₄×C₂₃₉	1	4	yes	complemented
156	C₂×Q₈	C₂³×C₂₃₉	1	1	yes	characteristic
157	C₂×Q₈	C₂³×C₂₃₉	1	2	yes	—
158	C₂×D₄	—	2	1	no	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	—	2	1	no	—
161	C₂×D₄	—	2	4	no	—
162	C₂×D₄	—	2	2	no	—
163	C₂×D₄	C₂³×C₂₃₉	1	2	yes	—
164	C₂×D₄	C₂³×C₂₃₉	1	1	yes	characteristic
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
168	Q₁₆	—	2	4	no	—
169	SD₁₆	—	2	8	no	—
170	SD₁₆	—	2	4	no	—
171	M₁₆	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	C₂³×C₂₃₉	1	2	yes	—
174	C₂×C₈	C₂³×C₂₃₉	1	2	yes	—
175	C₄⋊C₄	—	2	4	no	—
176	C₄⋊C₄	D₄×C₂₃₉	1	2	yes	—
177	C₄⋊C₄	C₂³×C₂₃₉	1	2	yes	—
178	C₂²⋊C₄	—	2	1	no	—
179	C₂²⋊C₄	—	2	4	no	—
180	C₂²⋊C₄	D₄×C₂₃₉	1	2	yes	—
181	C₄²	—	2	4	no	—
182	C₄²	—	2	1	no	—
183	C₄²	C₂³×C₂₃₉	1	2	yes	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	1	no	—
186	C₂³	—	4	2	no	—
187	C₂³	—	8	2	no	—
188	C₂³	C₂×D₄×C₂₃₉	1	1	yes	characteristic
189	Q₈	—	2	4	no	—
190	Q₈	—	2	2	no	—
191	Q₈	—	2	1	no	—
192	Q₈	—	2	1	no	—
193	Q₈	C₂×D₄×C₂₃₉	1	4	yes	—
194	D₄	—	2	2	no	—
195	D₄	—	2	1	no	—
196	D₄	—	2	2	no	—
197	D₄	—	2	1	no	—
198	D₄	—	4	2	no	—
199	D₄	—	4	2	no	—
200	D₄	—	8	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	C₂×D₄×C₂₃₉	1	4	yes	—
207	C₂×C₄	C₂×D₄×C₂₃₉	1	1	yes	characteristic
208	C₂×C₄	C₂×D₄×C₂₃₉	1	2	yes	—
209	C₂×C₄	#41×C₂₃₉	1	2	yes	—
210	C₂×C₄	C₂⁴×C₂₃₉	1	1	yes	characteristic
211	C₈	—	2	2	no	—
212	C₈	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	4	1	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	4	no	—
218	C₂²	—	8	4	no	—
219	C₂²	#121×C₂₃₉	1	2	yes	—
220	C₂²	C₂²×D₄×C₂₃₉	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	2	4	no	—
223	C₄	—	4	1	no	—
224	C₄	—	4	2	no	—
225	C₄	#121×C₂₃₉	1	2	yes	—
226	C₄	C₂²×D₄×C₂₃₉	1	1	yes	characteristic
227	C₄	C₂×#41×C₂₃₉	1	1	yes	characteristic
228	C₂	—	2	2	no	—
229	C₂	—	8	2	no	—
230	C₂	—	8	1	no	—
231	C₂	C₂×#121×C₂₃₉	1	1	yes	characteristic
232	C₂	#577×C₂₃₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃₉
prime radical	#535×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
1. C₂×#137×C₂₃₉	C₂	semidirect
2. C₂²×SD₁₆×C₂₃₉	C₂	semidirect
4. #519×C₂₃₉	C₂	semidirect
5. #464×C₂₃₉	C₂	semidirect
6. #460×C₂₃₉	C₂	semidirect
7. #424×C₂₃₉	C₂	semidirect
8. #418×C₂₃₉	C₂	semidirect
9. #416×C₂₃₉	C₂	semidirect
14. C₂×Q₁₆×C₂₃₉	C₂²	semidirect
17. C₂×SD₁₆×C₂₃₉	C₂²	semidirect
25. C₄×Q₈×C₂₃₉	C₂²	semidirect
29. C₄⋊C₈×C₂₃₉	C₂²	semidirect
30. #103×C₂₃₉	C₂²	semidirect
32. #102×C₂₃₉	C₂²	semidirect
38. C₂×Q₈×C₂₃₉	D₄	semidirect
116. C₂₃₉	#3043	direct (N × H)
117. #3043	C₂₃₉	direct (N × H)
118. C₂×#137	C₂×C₂₃₉	semidirect
119. C₂²×SD₁₆	C₂×C₂₃₉	semidirect
121. #519	C₂×C₂₃₉	semidirect
122. #464	C₂×C₂₃₉	semidirect
123. #460	C₂×C₂₃₉	semidirect
124. #424	C₂×C₂₃₉	semidirect
125. #418	C₂×C₂₃₉	semidirect
126. #416	C₂×C₂₃₉	semidirect
131. C₂×Q₁₆	C₂²×C₂₃₉	semidirect
134. C₂×SD₁₆	C₂²×C₂₃₉	semidirect
142. C₄×Q₈	C₂²×C₂₃₉	semidirect
146. C₄⋊C₈	C₂²×C₂₃₉	semidirect
147. #103	C₂²×C₂₃₉	semidirect
149. #102	C₂²×C₂₃₉	semidirect
155. C₂×Q₈	D₄×C₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#137×C₂₃₉	2
C₂²×SD₁₆×C₂₃₉	2
#535×C₂₃₉	2
#519×C₂₃₉	2
#464×C₂₃₉	2
#460×C₂₃₉	2
#424×C₂₃₉	2
#418×C₂₃₉	2
#416×C₂₃₉	2
#3043	239

Subgroups — #3043×C239 #44481813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3043×C₂₃₉ #44481813