Subgroups — #2319×C₂₃₉ #44481089

Order 30592.

540 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₂₃₉	C₂	1	1	yes	maximal, characteristic
2	#390×C₂₃₉	C₂	1	2	yes	maximal
3	#389×C₂₃₉	C₂	1	2	yes	maximal, complemented
4	#388×C₂₃₉	C₂	1	4	yes	maximal, complemented
5	#381×C₂₃₉	C₂	1	2	yes	maximal, complemented
6	#380×C₂₃₉	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊C₄×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#376×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₂₃₉	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₂₃₉	C₂²	1	2	yes	—
11	C₄×D₄×C₂₃₉	C₄	1	8	yes	complemented
12	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₂₃₉	C₂²	1	4	yes	—
16	C₂×C₂²⋊C₄×C₂₃₉	C₂²	1	4	yes	—
17	C₂×C₂²⋊C₄×C₂₃₉	C₂²	1	2	yes	—
18	C₂×C₄²×C₂₃₉	C₂²	1	2	yes	—
19	C₂×C₄²×C₂₃₉	C₂²	1	2	yes	—
20	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
21	#95×C₂₃₉	C₂²	1	1	yes	characteristic
22	#95×C₂₃₉	C₂²	1	1	yes	characteristic
23	#95×C₂₃₉	C₂²	1	2	yes	—
24	#95×C₂₃₉	C₂²	1	2	yes	—
25	#95×C₂₃₉	C₂²	1	4	yes	—
26	C₂⁴×C₂₃₉	C₂³	1	2	yes	—
27	C₂×D₄×C₂₃₉	—	2	4	no	—
28	C₂×D₄×C₂₃₉	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₂₃₉	—	4	4	no	—
30	C₂²×C₄×C₂₃₉	C₂×C₄	1	8	yes	—
31	C₂²×C₄×C₂₃₉	C₂³	1	2	yes	—
32	C₂²×C₄×C₂₃₉	C₂³	1	2	yes	—
33	C₂²×C₄×C₂₃₉	C₂³	1	4	yes	—
34	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₂₃₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₂₃₉	—	2	4	no	—
39	C₄⋊C₄×C₂₃₉	—	4	4	no	—
40	C₄⋊C₄×C₂₃₉	C₂×C₄	1	4	yes	—
41	C₂²⋊C₄×C₂₃₉	—	4	4	no	—
42	C₂²⋊C₄×C₂₃₉	—	4	4	no	—
43	C₂²⋊C₄×C₂₃₉	C₂×C₄	1	8	yes	—
44	C₄²×C₂₃₉	—	2	4	no	—
45	C₄²×C₂₃₉	—	4	2	no	—
46	C₄²×C₂₃₉	C₂×C₄	1	4	yes	—
47	C₂³×C₂₃₉	—	2	4	no	—
48	C₂³×C₂₃₉	—	2	2	no	—
49	C₂³×C₂₃₉	—	2	4	no	—
50	C₂³×C₂₃₉	—	2	2	no	—
51	C₂³×C₂₃₉	C₂²×C₄	1	4	yes	—
52	C₂³×C₂₃₉	C₂⁴	1	1	yes	characteristic
53	D₄×C₂₃₉	—	2	8	no	—
54	C₂×C₄×C₂₃₉	—	2	1	no	—
55	C₂×C₄×C₂₃₉	—	2	4	no	—
56	C₂×C₄×C₂₃₉	—	2	2	no	—
57	C₂×C₄×C₂₃₉	—	2	1	no	—
58	C₂×C₄×C₂₃₉	—	2	2	no	—
59	C₂×C₄×C₂₃₉	—	2	2	no	—
60	C₂×C₄×C₂₃₉	—	2	4	no	—
61	C₂×C₄×C₂₃₉	—	2	4	no	—
62	C₂×C₄×C₂₃₉	—	2	2	no	—
63	C₂×C₄×C₂₃₉	—	2	2	no	—
64	C₂×C₄×C₂₃₉	—	2	4	no	—
65	C₂×C₄×C₂₃₉	—	2	2	no	—
66	C₂×C₄×C₂₃₉	—	2	2	no	—
67	C₂×C₄×C₂₃₉	—	4	8	no	—
68	C₂×C₄×C₂₃₉	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₂₃₉	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₂₃₉	C₂²×C₄	1	4	yes	—
71	C₂×C₄×C₂₃₉	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₂₃₉	#41	1	4	yes	—
73	C₂²×C₂₃₉	—	2	4	no	—
74	C₂²×C₂₃₉	—	2	4	no	—
75	C₂²×C₂₃₉	—	2	4	no	—
76	C₂²×C₂₃₉	—	4	4	no	—
77	C₂²×C₂₃₉	—	4	4	no	—
78	C₂²×C₂₃₉	C₂³×C₄	1	1	yes	characteristic
79	C₂²×C₂₃₉	C₂×#41	1	2	yes	—
80	C₂²×C₂₃₉	H(5,𝔽₂)	1	2	yes	—
81	C₂²×C₂₃₉	H(5,𝔽₂)	1	1	yes	characteristic
82	C₂²×C₂₃₉	2₋¹⁺⁴	1	1	yes	characteristic
83	C₄×C₂₃₉	—	2	2	no	—
84	C₄×C₂₃₉	—	2	4	no	—
85	C₄×C₂₃₉	—	2	2	no	—
86	C₄×C₂₃₉	—	4	2	no	—
87	C₄×C₂₃₉	—	4	4	no	—
88	C₄×C₂₃₉	—	4	2	no	—
89	C₄×C₂₃₉	#117	1	4	yes	—
90	C₂×C₂₃₉	—	4	4	no	—
91	C₂×C₂₃₉	C₂×#117	1	1	yes	characteristic
92	C₂×C₂₃₉	#518	1	1	yes	characteristic
93	C₂×C₂₃₉	#520	1	1	yes	characteristic
94	C₂×C₂₃₉	#540	1	2	yes	—
95	C₂×C₂₃₉	#542	1	2	yes	—
96	C₂₃₉	#2319	1	1	yes	characteristic, direct product factor, Sylow
97	#2319	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂×C₄×D₄	C₂×C₂₃₉	1	1	yes	characteristic
99	#390	C₂×C₂₃₉	1	2	yes	—
100	#389	C₂×C₂₃₉	1	2	yes	complemented
101	#388	C₂×C₂₃₉	1	4	yes	complemented
102	#381	C₂×C₂₃₉	1	2	yes	complemented
103	#380	C₂×C₂₃₉	1	2	yes	complemented
104	C₄×C₄⋊C₄	C₂×C₂₃₉	1	1	yes	characteristic, complemented
105	#376	C₂×C₂₃₉	1	1	yes	characteristic, complemented
106	C₂²×D₄	C₂²×C₂₃₉	1	1	yes	characteristic
107	C₂³×C₄	C₂²×C₂₃₉	1	2	yes	—
108	C₄×D₄	C₄×C₂₃₉	1	8	yes	complemented
109	C₂×C₄⋊C₄	C₂²×C₂₃₉	1	2	yes	—
110	C₂×C₄⋊C₄	C₂²×C₂₃₉	1	4	yes	—
111	C₂×C₄⋊C₄	C₂²×C₂₃₉	1	1	yes	characteristic
112	C₂×C₂²⋊C₄	C₂²×C₂₃₉	1	4	yes	—
113	C₂×C₂²⋊C₄	C₂²×C₂₃₉	1	4	yes	—
114	C₂×C₂²⋊C₄	C₂²×C₂₃₉	1	2	yes	—
115	C₂×C₄²	C₂²×C₂₃₉	1	2	yes	—
116	C₂×C₄²	C₂²×C₂₃₉	1	2	yes	—
117	C₂×C₄²	C₂²×C₂₃₉	1	1	yes	characteristic
118	#95	C₂²×C₂₃₉	1	1	yes	characteristic
119	#95	C₂²×C₂₃₉	1	1	yes	characteristic
120	#95	C₂²×C₂₃₉	1	2	yes	—
121	#95	C₂²×C₂₃₉	1	2	yes	—
122	#95	C₂²×C₂₃₉	1	4	yes	—
123	C₂⁴	C₂³×C₂₃₉	1	2	yes	—
124	C₂×D₄	—	2	4	no	—
125	C₂×D₄	C₂×C₄×C₂₃₉	1	4	yes	—
126	C₂²×C₄	—	4	4	no	—
127	C₂²×C₄	C₂×C₄×C₂₃₉	1	8	yes	—
128	C₂²×C₄	C₂³×C₂₃₉	1	2	yes	—
129	C₂²×C₄	C₂³×C₂₃₉	1	2	yes	—
130	C₂²×C₄	C₂³×C₂₃₉	1	4	yes	—
131	C₂²×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
132	C₂²×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
133	C₂²×C₄	C₂³×C₂₃₉	1	2	yes	—
134	C₂²×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
135	C₄⋊C₄	—	2	4	no	—
136	C₄⋊C₄	—	4	4	no	—
137	C₄⋊C₄	C₂×C₄×C₂₃₉	1	4	yes	—
138	C₂²⋊C₄	—	4	4	no	—
139	C₂²⋊C₄	—	4	4	no	—
140	C₂²⋊C₄	C₂×C₄×C₂₃₉	1	8	yes	—
141	C₄²	—	2	4	no	—
142	C₄²	—	4	2	no	—
143	C₄²	C₂×C₄×C₂₃₉	1	4	yes	—
144	C₂³	—	2	4	no	—
145	C₂³	—	2	2	no	—
146	C₂³	—	2	4	no	—
147	C₂³	—	2	2	no	—
148	C₂³	C₂²×C₄×C₂₃₉	1	4	yes	—
149	C₂³	C₂⁴×C₂₃₉	1	1	yes	characteristic
150	D₄	—	2	8	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	4	8	no	—
165	C₂×C₄	C₂²×C₄×C₂₃₉	1	2	yes	—
166	C₂×C₄	C₂²×C₄×C₂₃₉	1	2	yes	—
167	C₂×C₄	C₂²×C₄×C₂₃₉	1	4	yes	—
168	C₂×C₄	C₂²×C₄×C₂₃₉	1	2	yes	—
169	C₂×C₄	#41×C₂₃₉	1	4	yes	—
170	C₂²	—	2	4	no	—
171	C₂²	—	2	4	no	—
172	C₂²	—	2	4	no	—
173	C₂²	—	4	4	no	—
174	C₂²	—	4	4	no	—
175	C₂²	C₂³×C₄×C₂₃₉	1	1	yes	characteristic
176	C₂²	C₂×#41×C₂₃₉	1	2	yes	—
177	C₂²	H(5,𝔽₂)×C₂₃₉	1	2	yes	—
178	C₂²	H(5,𝔽₂)×C₂₃₉	1	1	yes	characteristic
179	C₂²	2₋¹⁺⁴×C₂₃₉	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	4	no	—
182	C₄	—	2	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	4	4	no	—
185	C₄	—	4	2	no	—
186	C₄	#117×C₂₃₉	1	4	yes	—
187	C₂	—	4	4	no	—
188	C₂	C₂×#117×C₂₃₉	1	1	yes	characteristic
189	C₂	#518×C₂₃₉	1	1	yes	characteristic
190	C₂	#520×C₂₃₉	1	1	yes	characteristic
191	C₂	#540×C₂₃₉	1	2	yes	—
192	C₂	#542×C₂₃₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₃₉
Frattini, cosocle	C₂³	C₂⁴×C₂₃₉
prime radical	C₂²×D₄×C₂₃₉	C₂²
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
3. #389×C₂₃₉	C₂	semidirect
4. #388×C₂₃₉	C₂	semidirect
5. #381×C₂₃₉	C₂	semidirect
6. #380×C₂₃₉	C₂	semidirect
7. C₄×C₄⋊C₄×C₂₃₉	C₂	semidirect
8. #376×C₂₃₉	C₂	semidirect
11. C₄×D₄×C₂₃₉	C₄	semidirect
96. C₂₃₉	#2319	direct (N × H)
97. #2319	C₂₃₉	direct (N × H)
100. #389	C₂×C₂₃₉	semidirect
101. #388	C₂×C₂₃₉	semidirect
102. #381	C₂×C₂₃₉	semidirect
103. #380	C₂×C₂₃₉	semidirect
104. C₄×C₄⋊C₄	C₂×C₂₃₉	semidirect
105. #376	C₂×C₂₃₉	semidirect
108. C₄×D₄	C₄×C₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₂₃₉	2
#390×C₂₃₉	2
#389×C₂₃₉	2
#388×C₂₃₉	2
#381×C₂₃₉	2
#380×C₂₃₉	2
C₄×C₄⋊C₄×C₂₃₉	2
#376×C₂₃₉	2
#2319	239

Subgroups — #2319×C239 #44481089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2319×C₂₃₉ #44481089