Subgroups — #2486×C₂₃₉ #44481256

Order 30592.

624 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#123×C₂₃₉	C₂	1	2	yes	maximal
2	C₂×C₄×D₄×C₂₃₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊C₄×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#398×C₂₃₉	C₂	1	2	yes	maximal, complemented
5	#397×C₂₃₉	C₂	1	2	yes	maximal, complemented
6	#389×C₂₃₉	C₂	1	2	yes	maximal, complemented
7	#381×C₂₃₉	C₂	1	2	yes	maximal, complemented
8	#380×C₂₃₉	C₂	1	2	yes	maximal, complemented
9	#380×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₂₃₉	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₃₉	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₂₃₉	C₂²	1	2	yes	—
13	C₂³×C₄×C₂₃₉	C₂²	1	1	yes	characteristic
14	#123×C₂₃₉	—	2	8	no	—
15	C₄×D₄×C₂₃₉	—	2	4	no	—
16	C₂×C₄⋊C₄×C₂₃₉	—	2	4	no	—
17	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₄×C₂₃₉	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₂₃₉	C₂²	1	4	yes	—
24	C₂×C₂²⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
27	#95×C₂₃₉	C₂²	1	4	yes	—
28	#95×C₂₃₉	C₂²	1	2	yes	—
29	#95×C₂₃₉	C₂²	1	4	yes	—
30	C₂⁴×C₂₃₉	C₂³	1	1	yes	characteristic
31	C₂⁴×C₂₃₉	C₂³	1	1	yes	characteristic
32	C₂×D₄×C₂₃₉	—	2	2	no	—
33	C₂×D₄×C₂₃₉	—	2	2	no	—
34	C₂×D₄×C₂₃₉	D₄	1	4	yes	—
35	C₂²×C₄×C₂₃₉	—	2	4	no	—
36	C₂²×C₄×C₂₃₉	—	2	4	no	—
37	C₂²×C₄×C₂₃₉	—	2	4	no	—
38	C₂²×C₄×C₂₃₉	—	2	4	no	—
39	C₂²×C₄×C₂₃₉	—	2	2	no	—
40	C₂²×C₄×C₂₃₉	—	2	4	no	—
41	C₂²×C₄×C₂₃₉	D₄	1	4	yes	—
42	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₂₃₉	C₂³	1	4	yes	—
45	C₂²×C₄×C₂₃₉	C₂³	1	4	yes	—
46	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₂₃₉	—	2	4	no	—
50	C₄⋊C₄×C₂₃₉	—	2	4	no	—
51	C₄⋊C₄×C₂₃₉	—	2	2	no	—
52	C₄⋊C₄×C₂₃₉	—	2	8	no	—
53	C₄⋊C₄×C₂₃₉	—	2	4	no	—
54	C₂²⋊C₄×C₂₃₉	—	2	2	no	—
55	C₂²⋊C₄×C₂₃₉	—	2	4	no	—
56	C₂²⋊C₄×C₂₃₉	—	2	8	no	—
57	C₂²⋊C₄×C₂₃₉	—	2	2	no	—
58	C₄²×C₂₃₉	—	2	2	no	—
59	C₂³×C₂₃₉	—	2	2	no	—
60	C₂³×C₂₃₉	—	2	2	no	—
61	C₂³×C₂₃₉	—	2	1	no	—
62	C₂³×C₂₃₉	—	2	2	no	—
63	C₂³×C₂₃₉	—	2	1	no	—
64	C₂³×C₂₃₉	—	2	2	no	—
65	C₂³×C₂₃₉	C₂×D₄	1	2	yes	—
66	C₂³×C₂₃₉	C₂×D₄	1	2	yes	—
67	C₂³×C₂₃₉	C₂×D₄	1	2	yes	—
68	C₂³×C₂₃₉	#41	1	2	yes	—
69	C₂³×C₂₃₉	C₂⁴	1	1	yes	characteristic
70	D₄×C₂₃₉	—	2	8	no	—
71	C₂×C₄×C₂₃₉	—	2	1	no	—
72	C₂×C₄×C₂₃₉	—	2	2	no	—
73	C₂×C₄×C₂₃₉	—	2	1	no	—
74	C₂×C₄×C₂₃₉	—	2	8	no	—
75	C₂×C₄×C₂₃₉	—	2	1	no	—
76	C₂×C₄×C₂₃₉	—	2	4	no	—
77	C₂×C₄×C₂₃₉	—	2	4	no	—
78	C₂×C₄×C₂₃₉	—	2	1	no	—
79	C₂×C₄×C₂₃₉	—	2	4	no	—
80	C₂×C₄×C₂₃₉	—	2	4	no	—
81	C₂×C₄×C₂₃₉	—	2	4	no	—
82	C₂×C₄×C₂₃₉	—	2	2	no	—
83	C₂×C₄×C₂₃₉	—	2	2	no	—
84	C₂×C₄×C₂₃₉	—	2	4	no	—
85	C₂×C₄×C₂₃₉	—	4	8	no	—
86	C₂×C₄×C₂₃₉	—	4	4	no	—
87	C₂×C₄×C₂₃₉	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₂₃₉	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₂₃₉	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₂₃₉	#41	1	4	yes	—
91	C₂²×C₂₃₉	—	2	2	no	—
92	C₂²×C₂₃₉	—	2	2	no	—
93	C₂²×C₂₃₉	—	2	4	no	—
94	C₂²×C₂₃₉	—	2	4	no	—
95	C₂²×C₂₃₉	—	2	2	no	—
96	C₂²×C₂₃₉	—	2	4	no	—
97	C₂²×C₂₃₉	—	4	1	no	—
98	C₂²×C₂₃₉	—	4	1	no	—
99	C₂²×C₂₃₉	—	4	2	no	—
100	C₂²×C₂₃₉	#121	1	4	yes	—
101	C₂²×C₂₃₉	C₂²×D₄	1	1	yes	characteristic
102	C₂²×C₂₃₉	C₂²×D₄	1	1	yes	characteristic
103	C₂²×C₂₃₉	C₂×#41	1	1	yes	characteristic
104	C₂²×C₂₃₉	C₂×#41	1	2	yes	—
105	C₂²×C₂₃₉	2₋¹⁺⁴	1	2	yes	—
106	C₄×C₂₃₉	—	2	2	no	—
107	C₄×C₂₃₉	—	2	2	no	—
108	C₄×C₂₃₉	—	4	4	no	—
109	C₄×C₂₃₉	—	4	1	no	—
110	C₄×C₂₃₉	—	4	1	no	—
111	C₄×C₂₃₉	—	4	4	no	—
112	C₄×C₂₃₉	#123	1	4	yes	complemented
113	C₂×C₂₃₉	—	2	4	no	—
114	C₂×C₂₃₉	—	4	2	no	—
115	C₂×C₂₃₉	C₂×#121	1	1	yes	characteristic
116	C₂×C₂₃₉	C₂×#123	1	1	yes	characteristic
117	C₂×C₂₃₉	#536	1	1	yes	characteristic
118	C₂×C₂₃₉	#547	1	2	yes	—
119	C₂×C₂₃₉	#552	1	2	yes	—
120	C₂₃₉	#2486	1	1	yes	characteristic, direct product factor, Sylow
121	#2486	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂×#123	C₂×C₂₃₉	1	2	yes	—
123	C₂×C₄×D₄	C₂×C₂₃₉	1	1	yes	characteristic
124	C₂²×C₄⋊C₄	C₂×C₂₃₉	1	1	yes	characteristic, complemented
125	#398	C₂×C₂₃₉	1	2	yes	complemented
126	#397	C₂×C₂₃₉	1	2	yes	complemented
127	#389	C₂×C₂₃₉	1	2	yes	complemented
128	#381	C₂×C₂₃₉	1	2	yes	complemented
129	#380	C₂×C₂₃₉	1	2	yes	complemented
130	#380	C₂×C₂₃₉	1	1	yes	characteristic, complemented
131	C₂²×D₄	C₂²×C₂₃₉	1	1	yes	characteristic
132	C₂³×C₄	C₂²×C₂₃₉	1	2	yes	—
133	C₂³×C₄	C₂²×C₂₃₉	1	1	yes	characteristic
134	C₂³×C₄	C₂²×C₂₃₉	1	1	yes	characteristic
135	#123	—	2	8	no	—
136	C₄×D₄	—	2	4	no	—
137	C₂×C₄⋊C₄	—	2	4	no	—
138	C₂×C₄⋊C₄	C₂²×C₂₃₉	1	2	yes	—
139	C₂×C₄⋊C₄	C₂²×C₂₃₉	1	2	yes	—
140	C₂×C₄⋊C₄	C₂²×C₂₃₉	1	1	yes	characteristic
141	C₂×C₄⋊C₄	C₂²×C₂₃₉	1	4	yes	—
142	C₂×C₄⋊C₄	C₂²×C₂₃₉	1	2	yes	—
143	C₂×C₂²⋊C₄	C₂²×C₂₃₉	1	1	yes	characteristic
144	C₂×C₂²⋊C₄	C₂²×C₂₃₉	1	1	yes	characteristic
145	C₂×C₂²⋊C₄	C₂²×C₂₃₉	1	2	yes	—
146	C₂×C₂²⋊C₄	C₂²×C₂₃₉	1	4	yes	—
147	C₂×C₄²	C₂²×C₂₃₉	1	1	yes	characteristic
148	#95	C₂²×C₂₃₉	1	4	yes	—
149	#95	C₂²×C₂₃₉	1	2	yes	—
150	#95	C₂²×C₂₃₉	1	4	yes	—
151	C₂⁴	C₂³×C₂₃₉	1	1	yes	characteristic
152	C₂⁴	C₂³×C₂₃₉	1	1	yes	characteristic
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	D₄×C₂₃₉	1	4	yes	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	D₄×C₂₃₉	1	4	yes	—
163	C₂²×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
164	C₂²×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₂₃₉	1	4	yes	—
166	C₂²×C₄	C₂³×C₂₃₉	1	4	yes	—
167	C₂²×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	8	no	—
174	C₄⋊C₄	—	2	4	no	—
175	C₂²⋊C₄	—	2	2	no	—
176	C₂²⋊C₄	—	2	4	no	—
177	C₂²⋊C₄	—	2	8	no	—
178	C₂²⋊C₄	—	2	2	no	—
179	C₄²	—	2	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	1	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	2	1	no	—
185	C₂³	—	2	2	no	—
186	C₂³	C₂×D₄×C₂₃₉	1	2	yes	—
187	C₂³	C₂×D₄×C₂₃₉	1	2	yes	—
188	C₂³	C₂×D₄×C₂₃₉	1	2	yes	—
189	C₂³	#41×C₂₃₉	1	2	yes	—
190	C₂³	C₂⁴×C₂₃₉	1	1	yes	characteristic
191	D₄	—	2	8	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	8	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	8	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	C₂×D₄×C₂₃₉	1	2	yes	—
209	C₂×C₄	C₂×D₄×C₂₃₉	1	2	yes	—
210	C₂×C₄	C₂×D₄×C₂₃₉	1	2	yes	—
211	C₂×C₄	#41×C₂₃₉	1	4	yes	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	4	no	—
218	C₂²	—	4	1	no	—
219	C₂²	—	4	1	no	—
220	C₂²	—	4	2	no	—
221	C₂²	#121×C₂₃₉	1	4	yes	—
222	C₂²	C₂²×D₄×C₂₃₉	1	1	yes	characteristic
223	C₂²	C₂²×D₄×C₂₃₉	1	1	yes	characteristic
224	C₂²	C₂×#41×C₂₃₉	1	1	yes	characteristic
225	C₂²	C₂×#41×C₂₃₉	1	2	yes	—
226	C₂²	2₋¹⁺⁴×C₂₃₉	1	2	yes	—
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	4	4	no	—
230	C₄	—	4	1	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	4	no	—
233	C₄	#123×C₂₃₉	1	4	yes	complemented
234	C₂	—	2	4	no	—
235	C₂	—	4	2	no	—
236	C₂	C₂×#121×C₂₃₉	1	1	yes	characteristic
237	C₂	C₂×#123×C₂₃₉	1	1	yes	characteristic
238	C₂	#536×C₂₃₉	1	1	yes	characteristic
239	C₂	#547×C₂₃₉	1	2	yes	—
240	C₂	#552×C₂₃₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃₉
prime radical	C₂²×D₄×C₂₃₉	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄⋊C₄×C₂₃₉	C₂	semidirect
4. #398×C₂₃₉	C₂	semidirect
5. #397×C₂₃₉	C₂	semidirect
6. #389×C₂₃₉	C₂	semidirect
7. #381×C₂₃₉	C₂	semidirect
8. #380×C₂₃₉	C₂	semidirect
9. #380×C₂₃₉	C₂	semidirect
112. C₄×C₂₃₉	#123	semidirect
120. C₂₃₉	#2486	direct (N × H)
121. #2486	C₂₃₉	direct (N × H)
124. C₂²×C₄⋊C₄	C₂×C₂₃₉	semidirect
125. #398	C₂×C₂₃₉	semidirect
126. #397	C₂×C₂₃₉	semidirect
127. #389	C₂×C₂₃₉	semidirect
128. #381	C₂×C₂₃₉	semidirect
129. #380	C₂×C₂₃₉	semidirect
130. #380	C₂×C₂₃₉	semidirect
233. C₄	#123×C₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#123×C₂₃₉	2
C₂×C₄×D₄×C₂₃₉	2
C₂²×C₄⋊C₄×C₂₃₉	2
#398×C₂₃₉	2
#397×C₂₃₉	2
#389×C₂₃₉	2
#381×C₂₃₉	2
#380×C₂₃₉	2
#2486	239

Subgroups — #2486×C239 #44481256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2486×C₂₃₉ #44481256