Subgroups — #3486×C₂₃₉ #44482256

Order 30592.

1492 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×2₋¹⁺⁴×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×Q₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Q₈×C₂₃₉	C₂	1	8	yes	maximal, complemented
4	#548×C₂₃₉	C₂	1	8	yes	maximal, complemented
5	#536×C₂₃₉	C₂	1	6	yes	maximal, complemented
6	#525×C₂₃₉	C₂	1	6	yes	maximal, complemented
7	#519×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	2₋¹⁺⁴×C₂₃₉	—	2	8	no	—
9	C₂×#41×C₂₃₉	C₂²	1	6	yes	—
10	C₂×#41×C₂₃₉	C₂²	1	4	yes	complemented
11	C₂²×Q₈×C₂₃₉	—	2	8	no	—
12	C₂²×Q₈×C₂₃₉	C₂²	1	3	yes	—
13	C₂²×Q₈×C₂₃₉	C₂²	1	8	yes	—
14	C₂²×Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
15	C₂²×Q₈×C₂₃₉	C₂²	1	4	yes	complemented
16	C₂³×C₄×C₂₃₉	C₂²	1	3	yes	—
17	C₄⋊Q₈×C₂₃₉	C₂²	1	12	yes	complemented
18	#124×C₂₃₉	C₂²	1	12	yes	complemented
19	#123×C₂₃₉	C₂²	1	12	yes	—
20	C₂²⋊Q₈×C₂₃₉	C₂²	1	12	yes	—
21	C₂²⋊Q₈×C₂₃₉	C₂²	1	24	yes	complemented
22	#121×C₂₃₉	C₂²	1	12	yes	—
23	#120×C₂₃₉	C₂²	1	4	yes	—
24	C₄×Q₈×C₂₃₉	C₂²	1	8	yes	complemented
25	C₄×D₄×C₂₃₉	C₂²	1	24	yes	—
26	#117×C₂₃₉	C₂²	1	6	yes	—
27	C₂⁴×C₂₃₉	C₂³	1	1	yes	characteristic
28	#41×C₂₃₉	—	2	16	no	—
29	#41×C₂₃₉	—	2	24	no	—
30	C₂×Q₈×C₂₃₉	—	2	32	no	—
31	C₂×Q₈×C₂₃₉	—	2	16	no	—
32	C₂×Q₈×C₂₃₉	—	2	12	no	—
33	C₂×Q₈×C₂₃₉	D₄	1	8	yes	complemented
34	C₂×Q₈×C₂₃₉	C₂³	1	6	yes	—
35	C₂×Q₈×C₂₃₉	C₂³	1	12	yes	—
36	C₂×Q₈×C₂₃₉	C₂³	1	4	yes	—
37	C₂×D₄×C₂₃₉	C₂³	1	8	yes	—
38	C₂×D₄×C₂₃₉	C₂³	1	4	yes	—
39	C₂×D₄×C₂₃₉	C₂³	1	6	yes	—
40	C₂²×C₄×C₂₃₉	—	2	12	no	—
41	C₂²×C₄×C₂₃₉	C₂³	1	12	yes	—
42	C₂²×C₄×C₂₃₉	C₂³	1	12	yes	—
43	C₂²×C₄×C₂₃₉	C₂³	1	3	yes	—
44	C₂²×C₄×C₂₃₉	C₂³	1	3	yes	—
45	C₄⋊C₄×C₂₃₉	C₂³	1	12	yes	—
46	C₄⋊C₄×C₂₃₉	C₂³	1	24	yes	—
47	C₂²⋊C₄×C₂₃₉	C₂³	1	4	yes	—
48	C₂²⋊C₄×C₂₃₉	C₂³	1	8	yes	—
49	C₂²⋊C₄×C₂₃₉	C₂³	1	24	yes	—
50	C₄²×C₂₃₉	C₂³	1	12	yes	—
51	C₂³×C₂₃₉	—	2	2	no	—
52	C₂³×C₂₃₉	—	2	4	no	—
53	C₂³×C₂₃₉	C₂⁴	1	4	yes	—
54	C₂³×C₂₃₉	C₂⁴	1	1	yes	characteristic
55	C₂³×C₂₃₉	C₂⁴	1	2	yes	—
56	Q₈×C₂₃₉	—	2	24	no	—
57	Q₈×C₂₃₉	—	2	12	no	—
58	Q₈×C₂₃₉	C₂×D₄	1	16	yes	—
59	D₄×C₂₃₉	—	2	8	no	—
60	D₄×C₂₃₉	—	2	16	no	—
61	D₄×C₂₃₉	—	2	12	no	—
62	C₂×C₄×C₂₃₉	—	2	6	no	—
63	C₂×C₄×C₂₃₉	—	2	24	no	—
64	C₂×C₄×C₂₃₉	—	2	24	no	—
65	C₂×C₄×C₂₃₉	C₂×D₄	1	12	yes	—
66	C₂×C₄×C₂₃₉	C₂⁴	1	8	yes	—
67	C₂×C₄×C₂₃₉	C₂⁴	1	6	yes	—
68	C₂×C₄×C₂₃₉	C₂⁴	1	4	yes	—
69	C₂×C₄×C₂₃₉	C₂⁴	1	6	yes	—
70	C₂²×C₂₃₉	—	2	8	no	—
71	C₂²×C₂₃₉	—	2	2	no	—
72	C₂²×C₂₃₉	—	2	2	no	—
73	C₂²×C₂₃₉	—	2	2	no	—
74	C₂²×C₂₃₉	—	2	4	no	—
75	C₂²×C₂₃₉	—	2	8	no	—
76	C₂²×C₂₃₉	C₂²×D₄	1	2	yes	—
77	C₂²×C₂₃₉	2₋¹⁺⁴	1	4	yes	complemented
78	C₂²×C₂₃₉	C₂⁵	1	1	yes	characteristic
79	C₄×C₂₃₉	—	2	4	no	—
80	C₄×C₂₃₉	—	2	6	no	—
81	C₄×C₂₃₉	—	2	8	no	—
82	C₄×C₂₃₉	C₂²×D₄	1	12	yes	—
83	C₂×C₂₃₉	—	2	2	no	—
84	C₂×C₂₃₉	—	2	4	no	—
85	C₂×C₂₃₉	—	4	4	no	—
86	C₂×C₂₃₉	C₂³×D₄	1	1	yes	characteristic
87	C₂×C₂₃₉	C₂×2₋¹⁺⁴	1	2	yes	—
88	C₂₃₉	#3486	1	1	yes	characteristic, direct product factor, Sylow
89	#3486	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂×2₋¹⁺⁴	C₂×C₂₃₉	1	1	yes	characteristic, complemented
91	C₂³×Q₈	C₂×C₂₃₉	1	1	yes	characteristic, complemented
92	D₄×Q₈	C₂×C₂₃₉	1	8	yes	complemented
93	#548	C₂×C₂₃₉	1	8	yes	complemented
94	#536	C₂×C₂₃₉	1	6	yes	complemented
95	#525	C₂×C₂₃₉	1	6	yes	complemented
96	#519	C₂×C₂₃₉	1	1	yes	characteristic, complemented
97	2₋¹⁺⁴	—	2	8	no	—
98	C₂×#41	C₂²×C₂₃₉	1	6	yes	—
99	C₂×#41	C₂²×C₂₃₉	1	4	yes	complemented
100	C₂²×Q₈	—	2	8	no	—
101	C₂²×Q₈	C₂²×C₂₃₉	1	3	yes	—
102	C₂²×Q₈	C₂²×C₂₃₉	1	8	yes	—
103	C₂²×Q₈	C₂²×C₂₃₉	1	1	yes	characteristic
104	C₂²×Q₈	C₂²×C₂₃₉	1	4	yes	complemented
105	C₂³×C₄	C₂²×C₂₃₉	1	3	yes	—
106	C₄⋊Q₈	C₂²×C₂₃₉	1	12	yes	complemented
107	#124	C₂²×C₂₃₉	1	12	yes	complemented
108	#123	C₂²×C₂₃₉	1	12	yes	—
109	C₂²⋊Q₈	C₂²×C₂₃₉	1	12	yes	—
110	C₂²⋊Q₈	C₂²×C₂₃₉	1	24	yes	complemented
111	#121	C₂²×C₂₃₉	1	12	yes	—
112	#120	C₂²×C₂₃₉	1	4	yes	—
113	C₄×Q₈	C₂²×C₂₃₉	1	8	yes	complemented
114	C₄×D₄	C₂²×C₂₃₉	1	24	yes	—
115	#117	C₂²×C₂₃₉	1	6	yes	—
116	C₂⁴	C₂³×C₂₃₉	1	1	yes	characteristic
117	#41	—	2	16	no	—
118	#41	—	2	24	no	—
119	C₂×Q₈	—	2	32	no	—
120	C₂×Q₈	—	2	16	no	—
121	C₂×Q₈	—	2	12	no	—
122	C₂×Q₈	D₄×C₂₃₉	1	8	yes	complemented
123	C₂×Q₈	C₂³×C₂₃₉	1	6	yes	—
124	C₂×Q₈	C₂³×C₂₃₉	1	12	yes	—
125	C₂×Q₈	C₂³×C₂₃₉	1	4	yes	—
126	C₂×D₄	C₂³×C₂₃₉	1	8	yes	—
127	C₂×D₄	C₂³×C₂₃₉	1	4	yes	—
128	C₂×D₄	C₂³×C₂₃₉	1	6	yes	—
129	C₂²×C₄	—	2	12	no	—
130	C₂²×C₄	C₂³×C₂₃₉	1	12	yes	—
131	C₂²×C₄	C₂³×C₂₃₉	1	12	yes	—
132	C₂²×C₄	C₂³×C₂₃₉	1	3	yes	—
133	C₂²×C₄	C₂³×C₂₃₉	1	3	yes	—
134	C₄⋊C₄	C₂³×C₂₃₉	1	12	yes	—
135	C₄⋊C₄	C₂³×C₂₃₉	1	24	yes	—
136	C₂²⋊C₄	C₂³×C₂₃₉	1	4	yes	—
137	C₂²⋊C₄	C₂³×C₂₃₉	1	8	yes	—
138	C₂²⋊C₄	C₂³×C₂₃₉	1	24	yes	—
139	C₄²	C₂³×C₂₃₉	1	12	yes	—
140	C₂³	—	2	2	no	—
141	C₂³	—	2	4	no	—
142	C₂³	C₂⁴×C₂₃₉	1	4	yes	—
143	C₂³	C₂⁴×C₂₃₉	1	1	yes	characteristic
144	C₂³	C₂⁴×C₂₃₉	1	2	yes	—
145	Q₈	—	2	24	no	—
146	Q₈	—	2	12	no	—
147	Q₈	C₂×D₄×C₂₃₉	1	16	yes	—
148	D₄	—	2	8	no	—
149	D₄	—	2	16	no	—
150	D₄	—	2	12	no	—
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	—	2	24	no	—
153	C₂×C₄	—	2	24	no	—
154	C₂×C₄	C₂×D₄×C₂₃₉	1	12	yes	—
155	C₂×C₄	C₂⁴×C₂₃₉	1	8	yes	—
156	C₂×C₄	C₂⁴×C₂₃₉	1	6	yes	—
157	C₂×C₄	C₂⁴×C₂₃₉	1	4	yes	—
158	C₂×C₄	C₂⁴×C₂₃₉	1	6	yes	—
159	C₂²	—	2	8	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	4	no	—
164	C₂²	—	2	8	no	—
165	C₂²	C₂²×D₄×C₂₃₉	1	2	yes	—
166	C₂²	2₋¹⁺⁴×C₂₃₉	1	4	yes	complemented
167	C₂²	C₂⁵×C₂₃₉	1	1	yes	characteristic
168	C₄	—	2	4	no	—
169	C₄	—	2	6	no	—
170	C₄	—	2	8	no	—
171	C₄	C₂²×D₄×C₂₃₉	1	12	yes	—
172	C₂	—	2	2	no	—
173	C₂	—	2	4	no	—
174	C₂	—	4	4	no	—
175	C₂	C₂³×D₄×C₂₃₉	1	1	yes	characteristic
176	C₂	C₂×2₋¹⁺⁴×C₂₃₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃₉	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₃₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×2₋¹⁺⁴×C₂₃₉	C₂	semidirect
2. C₂³×Q₈×C₂₃₉	C₂	semidirect
3. D₄×Q₈×C₂₃₉	C₂	semidirect
4. #548×C₂₃₉	C₂	semidirect
5. #536×C₂₃₉	C₂	semidirect
6. #525×C₂₃₉	C₂	semidirect
7. #519×C₂₃₉	C₂	semidirect
10. C₂×#41×C₂₃₉	C₂²	semidirect
15. C₂²×Q₈×C₂₃₉	C₂²	semidirect
17. C₄⋊Q₈×C₂₃₉	C₂²	semidirect
18. #124×C₂₃₉	C₂²	semidirect
21. C₂²⋊Q₈×C₂₃₉	C₂²	semidirect
24. C₄×Q₈×C₂₃₉	C₂²	semidirect
33. C₂×Q₈×C₂₃₉	D₄	semidirect
77. C₂²×C₂₃₉	2₋¹⁺⁴	semidirect
88. C₂₃₉	#3486	direct (N × H)
89. #3486	C₂₃₉	direct (N × H)
90. C₂×2₋¹⁺⁴	C₂×C₂₃₉	semidirect
91. C₂³×Q₈	C₂×C₂₃₉	semidirect
92. D₄×Q₈	C₂×C₂₃₉	semidirect
93. #548	C₂×C₂₃₉	semidirect
94. #536	C₂×C₂₃₉	semidirect
95. #525	C₂×C₂₃₉	semidirect
96. #519	C₂×C₂₃₉	semidirect
99. C₂×#41	C₂²×C₂₃₉	semidirect
104. C₂²×Q₈	C₂²×C₂₃₉	semidirect
106. C₄⋊Q₈	C₂²×C₂₃₉	semidirect
107. #124	C₂²×C₂₃₉	semidirect
110. C₂²⋊Q₈	C₂²×C₂₃₉	semidirect
113. C₄×Q₈	C₂²×C₂₃₉	semidirect
122. C₂×Q₈	D₄×C₂₃₉	semidirect
166. C₂²	2₋¹⁺⁴×C₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×2₋¹⁺⁴×C₂₃₉	2
C₂³×Q₈×C₂₃₉	2
D₄×Q₈×C₂₃₉	2
#548×C₂₃₉	2
#536×C₂₃₉	2
#525×C₂₃₉	2
#519×C₂₃₉	2
#3486	239

Subgroups — #3486×C239 #44482256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3486×C₂₃₉ #44482256