Subgroups — C₂×#548×C₂₃₉ #44482235

Order 30592.

1620 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×Q₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#548×C₂₃₉	C₂	1	16	yes	maximal, direct product factor
4	C₂×#124×C₂₃₉	C₂	1	3	yes	maximal, complemented
5	C₂×C₂²⋊Q₈×C₂₃₉	C₂	1	3	yes	maximal, complemented
6	C₂×#121×C₂₃₉	C₂	1	3	yes	maximal, complemented
7	C₂×C₄×Q₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄×C₂₃₉	C₂	1	3	yes	maximal, complemented
9	C₂×#41×C₂₃₉	—	2	8	no	—
10	C₂×#41×C₂₃₉	C₂²	1	8	yes	complemented
11	C₂²×Q₈×C₂₃₉	—	2	8	no	—
12	C₂²×Q₈×C₂₃₉	C₂²	1	3	yes	—
13	C₂²×Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₂₃₉	C₂²	1	8	yes	complemented
15	C₂²×D₄×C₂₃₉	C₂²	1	3	yes	—
16	C₂²×D₄×C₂₃₉	C₂²	1	3	yes	—
17	C₂³×C₄×C₂₃₉	C₂²	1	3	yes	—
18	C₂³×C₄×C₂₃₉	C₂²	1	3	yes	—
19	#124×C₂₃₉	C₂²	1	24	yes	complemented
20	C₂²⋊Q₈×C₂₃₉	C₂²	1	24	yes	complemented
21	#121×C₂₃₉	C₂²	1	24	yes	complemented
22	C₄×Q₈×C₂₃₉	C₂²	1	8	yes	complemented
23	C₄×D₄×C₂₃₉	C₂²	1	24	yes	complemented
24	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	3	yes	—
25	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	3	yes	—
26	C₂×C₂²⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₂₃₉	C₂²	1	3	yes	—
28	C₂×C₂²⋊C₄×C₂₃₉	C₂²	1	6	yes	—
29	C₂×C₄²×C₂₃₉	C₂²	1	3	yes	—
30	C₂⁴×C₂₃₉	C₂³	1	3	yes	—
31	C₂⁴×C₂₃₉	C₂³	1	1	yes	characteristic
32	#41×C₂₃₉	—	2	32	no	—
33	C₂×Q₈×C₂₃₉	—	2	32	no	—
34	C₂×Q₈×C₂₃₉	—	2	12	no	—
35	C₂×Q₈×C₂₃₉	D₄	1	8	yes	complemented
36	C₂×Q₈×C₂₃₉	C₂³	1	12	yes	—
37	C₂×Q₈×C₂₃₉	C₂³	1	4	yes	—
38	C₂×D₄×C₂₃₉	—	2	12	no	—
39	C₂×D₄×C₂₃₉	—	2	12	no	—
40	C₂×D₄×C₂₃₉	C₂³	1	12	yes	—
41	C₂×D₄×C₂₃₉	C₂³	1	12	yes	—
42	C₂²×C₄×C₂₃₉	—	2	12	no	—
43	C₂²×C₄×C₂₃₉	—	2	12	no	—
44	C₂²×C₄×C₂₃₉	C₂³	1	12	yes	—
45	C₂²×C₄×C₂₃₉	C₂³	1	12	yes	—
46	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₂₃₉	C₂³	1	3	yes	—
48	C₂²×C₄×C₂₃₉	C₂³	1	3	yes	—
49	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₃₉	C₂³	1	3	yes	—
51	C₄⋊C₄×C₂₃₉	C₂³	1	12	yes	—
52	C₄⋊C₄×C₂₃₉	C₂³	1	12	yes	—
53	C₂²⋊C₄×C₂₃₉	C₂³	1	4	yes	—
54	C₂²⋊C₄×C₂₃₉	C₂³	1	12	yes	—
55	C₂²⋊C₄×C₂₃₉	C₂³	1	24	yes	—
56	C₄²×C₂₃₉	C₂³	1	12	yes	—
57	C₂³×C₂₃₉	—	2	2	no	—
58	C₂³×C₂₃₉	—	2	2	no	—
59	C₂³×C₂₃₉	—	2	6	no	—
60	C₂³×C₂₃₉	—	2	6	no	—
61	C₂³×C₂₃₉	—	2	6	no	—
62	C₂³×C₂₃₉	#41	1	4	yes	complemented
63	C₂³×C₂₃₉	C₂⁴	1	6	yes	—
64	C₂³×C₂₃₉	C₂⁴	1	2	yes	—
65	C₂³×C₂₃₉	C₂⁴	1	1	yes	characteristic
66	Q₈×C₂₃₉	—	2	24	no	—
67	Q₈×C₂₃₉	C₂×D₄	1	16	yes	complemented
68	D₄×C₂₃₉	—	2	24	no	—
69	D₄×C₂₃₉	—	2	24	no	—
70	C₂×C₄×C₂₃₉	—	2	2	no	—
71	C₂×C₄×C₂₃₉	—	2	6	no	—
72	C₂×C₄×C₂₃₉	—	2	6	no	—
73	C₂×C₄×C₂₃₉	—	2	2	no	—
74	C₂×C₄×C₂₃₉	—	2	24	no	—
75	C₂×C₄×C₂₃₉	—	2	24	no	—
76	C₂×C₄×C₂₃₉	C₂×D₄	1	12	yes	—
77	C₂×C₄×C₂₃₉	C₂⁴	1	2	yes	—
78	C₂×C₄×C₂₃₉	C₂⁴	1	6	yes	—
79	C₂×C₄×C₂₃₉	C₂⁴	1	6	yes	—
80	C₂×C₄×C₂₃₉	C₂⁴	1	2	yes	—
81	C₂×C₄×C₂₃₉	C₂⁴	1	6	yes	—
82	C₂²×C₂₃₉	—	2	8	no	—
83	C₂²×C₂₃₉	—	2	2	no	—
84	C₂²×C₂₃₉	—	2	2	no	—
85	C₂²×C₂₃₉	—	2	6	no	—
86	C₂²×C₂₃₉	—	2	6	no	—
87	C₂²×C₂₃₉	—	2	6	no	—
88	C₂²×C₂₃₉	—	4	12	no	—
89	C₂²×C₂₃₉	C₂²×D₄	1	2	yes	—
90	C₂²×C₂₃₉	C₂×#41	1	2	yes	—
91	C₂²×C₂₃₉	C₂×#41	1	4	yes	complemented
92	C₂²×C₂₃₉	2₋¹⁺⁴	1	2	yes	—
93	C₂²×C₂₃₉	C₂⁵	1	1	yes	characteristic
94	C₄×C₂₃₉	—	2	2	no	—
95	C₄×C₂₃₉	—	2	6	no	—
96	C₄×C₂₃₉	—	2	6	no	—
97	C₄×C₂₃₉	—	2	2	no	—
98	C₄×C₂₃₉	C₂²×D₄	1	12	yes	—
99	C₂×C₂₃₉	—	2	4	no	—
100	C₂×C₂₃₉	—	4	6	no	—
101	C₂×C₂₃₉	#548	1	4	yes	direct product factor
102	C₂×C₂₃₉	C₂³×D₄	1	1	yes	characteristic
103	C₂×C₂₃₉	C₂²×#41	1	1	yes	characteristic
104	C₂×C₂₃₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
105	C₂₃₉	C₂×#548	1	1	yes	characteristic, direct product factor, Sylow
106	C₂×#548	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂²×#41	C₂×C₂₃₉	1	1	yes	characteristic, complemented
108	C₂³×Q₈	C₂×C₂₃₉	1	1	yes	characteristic, complemented
109	#548	C₂×C₂₃₉	1	16	yes	direct product factor
110	C₂×#124	C₂×C₂₃₉	1	3	yes	complemented
111	C₂×C₂²⋊Q₈	C₂×C₂₃₉	1	3	yes	complemented
112	C₂×#121	C₂×C₂₃₉	1	3	yes	complemented
113	C₂×C₄×Q₈	C₂×C₂₃₉	1	1	yes	characteristic, complemented
114	C₂×C₄×D₄	C₂×C₂₃₉	1	3	yes	complemented
115	C₂×#41	—	2	8	no	—
116	C₂×#41	C₂²×C₂₃₉	1	8	yes	complemented
117	C₂²×Q₈	—	2	8	no	—
118	C₂²×Q₈	C₂²×C₂₃₉	1	3	yes	—
119	C₂²×Q₈	C₂²×C₂₃₉	1	1	yes	characteristic
120	C₂²×Q₈	C₂²×C₂₃₉	1	8	yes	complemented
121	C₂²×D₄	C₂²×C₂₃₉	1	3	yes	—
122	C₂²×D₄	C₂²×C₂₃₉	1	3	yes	—
123	C₂³×C₄	C₂²×C₂₃₉	1	3	yes	—
124	C₂³×C₄	C₂²×C₂₃₉	1	3	yes	—
125	#124	C₂²×C₂₃₉	1	24	yes	complemented
126	C₂²⋊Q₈	C₂²×C₂₃₉	1	24	yes	complemented
127	#121	C₂²×C₂₃₉	1	24	yes	complemented
128	C₄×Q₈	C₂²×C₂₃₉	1	8	yes	complemented
129	C₄×D₄	C₂²×C₂₃₉	1	24	yes	complemented
130	C₂×C₄⋊C₄	C₂²×C₂₃₉	1	3	yes	—
131	C₂×C₄⋊C₄	C₂²×C₂₃₉	1	3	yes	—
132	C₂×C₂²⋊C₄	C₂²×C₂₃₉	1	1	yes	characteristic
133	C₂×C₂²⋊C₄	C₂²×C₂₃₉	1	3	yes	—
134	C₂×C₂²⋊C₄	C₂²×C₂₃₉	1	6	yes	—
135	C₂×C₄²	C₂²×C₂₃₉	1	3	yes	—
136	C₂⁴	C₂³×C₂₃₉	1	3	yes	—
137	C₂⁴	C₂³×C₂₃₉	1	1	yes	characteristic
138	#41	—	2	32	no	—
139	C₂×Q₈	—	2	32	no	—
140	C₂×Q₈	—	2	12	no	—
141	C₂×Q₈	D₄×C₂₃₉	1	8	yes	complemented
142	C₂×Q₈	C₂³×C₂₃₉	1	12	yes	—
143	C₂×Q₈	C₂³×C₂₃₉	1	4	yes	—
144	C₂×D₄	—	2	12	no	—
145	C₂×D₄	—	2	12	no	—
146	C₂×D₄	C₂³×C₂₃₉	1	12	yes	—
147	C₂×D₄	C₂³×C₂₃₉	1	12	yes	—
148	C₂²×C₄	—	2	12	no	—
149	C₂²×C₄	—	2	12	no	—
150	C₂²×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
151	C₂²×C₄	C₂³×C₂₃₉	1	3	yes	—
152	C₂²×C₄	C₂³×C₂₃₉	1	3	yes	—
153	C₂²×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
154	C₂²×C₄	C₂³×C₂₃₉	1	12	yes	—
155	C₂²×C₄	C₂³×C₂₃₉	1	12	yes	—
156	C₂²×C₄	C₂³×C₂₃₉	1	3	yes	—
157	C₄⋊C₄	C₂³×C₂₃₉	1	12	yes	—
158	C₄⋊C₄	C₂³×C₂₃₉	1	12	yes	—
159	C₂²⋊C₄	C₂³×C₂₃₉	1	4	yes	—
160	C₂²⋊C₄	C₂³×C₂₃₉	1	12	yes	—
161	C₂²⋊C₄	C₂³×C₂₃₉	1	24	yes	—
162	C₄²	C₂³×C₂₃₉	1	12	yes	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	—	2	6	no	—
166	C₂³	—	2	6	no	—
167	C₂³	—	2	6	no	—
168	C₂³	#41×C₂₃₉	1	4	yes	complemented
169	C₂³	C₂⁴×C₂₃₉	1	6	yes	—
170	C₂³	C₂⁴×C₂₃₉	1	2	yes	—
171	C₂³	C₂⁴×C₂₃₉	1	1	yes	characteristic
172	Q₈	—	2	24	no	—
173	Q₈	C₂×D₄×C₂₃₉	1	16	yes	complemented
174	D₄	—	2	24	no	—
175	D₄	—	2	24	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	6	no	—
178	C₂×C₄	—	2	6	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	24	no	—
181	C₂×C₄	—	2	24	no	—
182	C₂×C₄	C₂×D₄×C₂₃₉	1	12	yes	—
183	C₂×C₄	C₂⁴×C₂₃₉	1	2	yes	—
184	C₂×C₄	C₂⁴×C₂₃₉	1	6	yes	—
185	C₂×C₄	C₂⁴×C₂₃₉	1	6	yes	—
186	C₂×C₄	C₂⁴×C₂₃₉	1	2	yes	—
187	C₂×C₄	C₂⁴×C₂₃₉	1	6	yes	—
188	C₂²	—	2	8	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	6	no	—
192	C₂²	—	2	6	no	—
193	C₂²	—	2	6	no	—
194	C₂²	—	4	12	no	—
195	C₂²	C₂²×D₄×C₂₃₉	1	2	yes	—
196	C₂²	C₂×#41×C₂₃₉	1	2	yes	—
197	C₂²	C₂×#41×C₂₃₉	1	4	yes	complemented
198	C₂²	2₋¹⁺⁴×C₂₃₉	1	2	yes	—
199	C₂²	C₂⁵×C₂₃₉	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	2	6	no	—
202	C₄	—	2	6	no	—
203	C₄	—	2	2	no	—
204	C₄	C₂²×D₄×C₂₃₉	1	12	yes	—
205	C₂	—	2	4	no	—
206	C₂	—	4	6	no	—
207	C₂	#548×C₂₃₉	1	4	yes	direct product factor
208	C₂	C₂³×D₄×C₂₃₉	1	1	yes	characteristic
209	C₂	C₂²×#41×C₂₃₉	1	1	yes	characteristic
210	C₂	C₂×2₋¹⁺⁴×C₂₃₉	1	1	yes	characteristic

	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. C₂²×#41×C₂₃₉	C₂	semidirect
2. C₂³×Q₈×C₂₃₉	C₂	semidirect
3. #548×C₂₃₉	C₂	direct (N × H)
4. C₂×#124×C₂₃₉	C₂	semidirect
5. C₂×C₂²⋊Q₈×C₂₃₉	C₂	semidirect
6. C₂×#121×C₂₃₉	C₂	semidirect
7. C₂×C₄×Q₈×C₂₃₉	C₂	semidirect
8. C₂×C₄×D₄×C₂₃₉	C₂	semidirect
10. C₂×#41×C₂₃₉	C₂²	semidirect
14. C₂²×Q₈×C₂₃₉	C₂²	semidirect
19. #124×C₂₃₉	C₂²	semidirect
20. C₂²⋊Q₈×C₂₃₉	C₂²	semidirect
21. #121×C₂₃₉	C₂²	semidirect
22. C₄×Q₈×C₂₃₉	C₂²	semidirect
23. C₄×D₄×C₂₃₉	C₂²	semidirect
35. C₂×Q₈×C₂₃₉	D₄	semidirect
62. C₂³×C₂₃₉	#41	semidirect
67. Q₈×C₂₃₉	C₂×D₄	semidirect
91. C₂²×C₂₃₉	C₂×#41	semidirect
101. C₂×C₂₃₉	#548	direct (N × H)
105. C₂₃₉	C₂×#548	direct (N × H)
106. C₂×#548	C₂₃₉	direct (N × H)
107. C₂²×#41	C₂×C₂₃₉	semidirect
108. C₂³×Q₈	C₂×C₂₃₉	semidirect
109. #548	C₂×C₂₃₉	direct (N × H)
110. C₂×#124	C₂×C₂₃₉	semidirect
111. C₂×C₂²⋊Q₈	C₂×C₂₃₉	semidirect
112. C₂×#121	C₂×C₂₃₉	semidirect
113. C₂×C₄×Q₈	C₂×C₂₃₉	semidirect
114. C₂×C₄×D₄	C₂×C₂₃₉	semidirect
116. C₂×#41	C₂²×C₂₃₉	semidirect
120. C₂²×Q₈	C₂²×C₂₃₉	semidirect
125. #124	C₂²×C₂₃₉	semidirect
126. C₂²⋊Q₈	C₂²×C₂₃₉	semidirect
127. #121	C₂²×C₂₃₉	semidirect
128. C₄×Q₈	C₂²×C₂₃₉	semidirect
129. C₄×D₄	C₂²×C₂₃₉	semidirect
141. C₂×Q₈	D₄×C₂₃₉	semidirect
168. C₂³	#41×C₂₃₉	semidirect
173. Q₈	C₂×D₄×C₂₃₉	semidirect
197. C₂²	C₂×#41×C₂₃₉	semidirect
207. C₂	#548×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
C₂²×#41×C₂₃₉	2
C₂³×Q₈×C₂₃₉	2
#548×C₂₃₉	2
C₂×#124×C₂₃₉	2
C₂×C₂²⋊Q₈×C₂₃₉	2
C₂×#121×C₂₃₉	2
C₂×C₄×Q₈×C₂₃₉	2
C₂×C₄×D₄×C₂₃₉	2
C₂×#548	239

Subgroups — C2×#548×C239 #44482235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#548×C₂₃₉ #44482235