Subgroups — C₂×#22581434 #31819599

Order 17792.

1640 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555785	C₂	1	1	yes	maximal, characteristic
2	#22581434	C₂	1	16	yes	maximal, direct product factor
3	C₂×#17984807	C₂	1	2	yes	maximal, complemented
4	C₂×#17984804	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₂²⋊D₁₃₉	C₂	1	4	yes	maximal, complemented
6	C₂×#17984801	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#17984800	C₂	1	2	yes	maximal, complemented
8	C₂×C₄⋊C₄×D₁₃₉	C₂	1	2	yes	maximal, complemented
9	C₂²×C₄⋊C₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14555785	C₄	1	16	yes	complemented
11	C₂×#14555785	C₂²	1	8	yes	—
12	C₂²×Dic₂₇₈	C₂²	1	1	yes	characteristic
13	C₂²×D₅₅₆	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₁₃₉	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₃₉	C₂²	1	4	yes	—
16	C₂²×C₄×D₁₃₉	C₂²	1	2	yes	—
17	#17984807	C₂²	1	16	yes	complemented
18	#17984804	C₂²	1	16	yes	complemented
19	C₄×C₂²⋊D₁₃₉	C₂²	1	32	yes	complemented
20	#17984801	C₂²	1	8	yes	complemented
21	#17984800	C₂²	1	16	yes	complemented
22	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₃₉⋊C₄	C₂²	1	4	yes	—
25	C₄⋊C₄×D₁₃₉	C₂²	1	16	yes	complemented
26	C₂×C₂²⋊Dic₁₃₉	C₂²	1	2	yes	—
27	C₂×(C₂×D₁₃₉)⋊C₄	C₂²	1	4	yes	—
28	C₂×C₄×Dic₁₃₉	C₂²	1	4	yes	—
29	C₂×C₄×Dic₁₃₉	C₂²	1	2	yes	—
30	C₂³×C₄×C₁₃₉	C₂²	1	2	yes	—
31	C₂³×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
33	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
34	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	8	yes	complemented
35	C₂³×D₁₃₉	C₂³	1	2	yes	—
36	#14555785	C₂×C₄	1	64	yes	complemented
37	C₂×Dic₂₇₈	C₂×C₄	1	8	yes	complemented
38	C₂×Dic₂₇₈	C₂³	1	4	yes	—
39	C₂×D₅₅₆	C₂×C₄	1	8	yes	complemented
40	C₂×D₅₅₆	C₂³	1	4	yes	—
41	C₂×C₂²⋊D₁₃₉	C₂×C₄	1	16	yes	complemented
42	C₂×C₂²⋊D₁₃₉	C₂³	1	8	yes	—
43	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
44	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
45	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
46	C₂×C₄×D₁₃₉	—	4	8	no	—
47	C₂×C₄×D₁₃₉	C₂×C₄	1	16	yes	complemented
48	C₂×C₄×D₁₃₉	C₂³	1	16	yes	—
49	C₂×C₄×D₁₃₉	C₂³	1	8	yes	—
50	C₄⋊Dic₁₃₉	C₂³	1	4	yes	—
51	C₄⋊Dic₁₃₉	C₂³	1	4	yes	—
52	Dic₁₃₉⋊C₄	C₂³	1	16	yes	—
53	C₂²⋊Dic₁₃₉	C₂³	1	8	yes	—
54	(C₂×D₁₃₉)⋊C₄	C₂³	1	16	yes	—
55	C₄×Dic₁₃₉	C₂³	1	16	yes	—
56	C₄×Dic₁₃₉	C₂³	1	8	yes	—
57	C₂⁴×C₁₃₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₉	—	4	4	no	—
59	C₂²×C₄×C₁₃₉	C₂×C₄	1	8	yes	complemented
60	C₂²×C₄×C₁₃₉	C₂³	1	4	yes	—
61	C₂²×C₄×C₁₃₉	C₂³	1	8	yes	—
62	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₉	C₂³	1	4	yes	—
65	C₄⋊C₄×C₁₃₉	C₂³	1	8	yes	—
66	C₄⋊C₄×C₁₃₉	C₂³	1	8	yes	—
67	C₂²×D₁₃₉	—	2	4	no	—
68	C₂²×D₁₃₉	—	2	4	no	—
69	C₂²×D₁₃₉	C₂²×C₄	1	8	yes	—
70	C₂²×D₁₃₉	C₂⁴	1	4	yes	—
71	Dic₂₇₈	C₂²×C₄	1	16	yes	complemented
72	D₅₅₆	C₂²×C₄	1	16	yes	complemented
73	C₂²⋊D₁₃₉	C₂²×C₄	1	32	yes	complemented
74	C₂×Dic₁₃₉	—	2	4	no	—
75	C₂×Dic₁₃₉	—	2	4	no	—
76	C₂×Dic₁₃₉	C₂²×C₄	1	8	yes	—
77	C₂×Dic₁₃₉	C₂⁴	1	4	yes	—
78	C₂×Dic₁₃₉	C₂⁴	1	4	yes	—
79	C₂×Dic₁₃₉	C₂⁴	1	4	yes	—
80	C₄×D₁₃₉	—	4	16	no	—
81	C₄×D₁₃₉	C₂²×C₄	1	32	yes	complemented
82	C₂³×C₁₃₉	—	2	2	no	—
83	C₂³×C₁₃₉	—	2	2	no	—
84	C₂³×C₁₃₉	C₂²×C₄	1	4	yes	—
85	C₂³×C₁₃₉	C₂⁴	1	2	yes	—
86	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
87	C₂×C₄×C₁₃₉	—	2	8	no	—
88	C₂×C₄×C₁₃₉	—	4	8	no	—
89	C₂×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
90	C₂×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
91	C₂×C₄×C₁₃₉	C₂²×C₄	1	16	yes	complemented
92	C₂×C₄×C₁₃₉	C₂⁴	1	8	yes	—
93	C₂×C₄×C₁₃₉	C₂⁴	1	2	yes	—
94	C₂×C₄×C₁₃₉	C₂⁴	1	2	yes	—
95	C₂²×C₁₃₉	—	2	8	no	—
96	C₂²×C₁₃₉	—	2	2	no	—
97	C₂²×C₁₃₉	—	2	2	no	—
98	C₂²×C₁₃₉	C₂³×C₄	1	4	yes	—
99	C₂²×C₁₃₉	C₂³×C₄	1	2	yes	—
100	C₂²×C₁₃₉	H(5,𝔽₂)	1	2	yes	—
101	C₂²×C₁₃₉	2₋¹⁺⁴	1	2	yes	—
102	C₂²×C₁₃₉	C₂⁵	1	1	yes	characteristic
103	C₂×D₁₃₉	—	2	16	no	—
104	C₂×D₁₃₉	—	2	4	no	—
105	C₂×D₁₃₉	—	2	4	no	—
106	C₂×D₁₃₉	C₂³×C₄	1	8	yes	—
107	C₄×C₁₃₉	—	2	8	no	—
108	C₄×C₁₃₉	C₂³×C₄	1	4	yes	—
109	C₄×C₁₃₉	C₂³×C₄	1	4	yes	—
110	Dic₁₃₉	—	2	4	no	—
111	Dic₁₃₉	—	2	4	no	—
112	Dic₁₃₉	C₂³×C₄	1	8	yes	—
113	C₂×C₁₃₉	—	2	4	no	—
114	C₂×C₁₃₉	#520	1	4	yes	complemented
115	C₂×C₁₃₉	C₂⁴×C₄	1	1	yes	characteristic
116	C₂×C₁₃₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
117	C₂×C₁₃₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
118	D₁₃₉	—	2	8	no	—
119	C₁₃₉	C₂×#520	1	1	yes	characteristic, complemented, Sylow
120	C₂×#520	—	139	1	no	maximal, Sylow
121	C₂²×#41	—	139	1	no	—
122	#520	—	139	16	no	—
123	C₂×C₄×Q₈	—	139	2	no	—
124	C₂×C₄×D₄	—	139	4	no	—
125	C₂×C₄×D₄	—	139	2	no	—
126	C₂×#117	—	139	1	no	—
127	C₂×#117	—	139	2	no	—
128	C₂²×C₄⋊C₄	—	139	2	no	—
129	C₂²×C₄⋊C₄	D₁₃₉	1	1	yes	characteristic, complemented
130	C₂×#41	—	139	16	no	—
131	C₂×#41	—	139	8	no	—
132	C₂²×Q₈	—	139	1	no	—
133	C₂²×D₄	—	139	2	no	—
134	C₂²×D₄	—	139	1	no	—
135	C₂³×C₄	—	139	4	no	—
136	C₂³×C₄	—	139	2	no	—
137	C₂³×C₄	C₂×D₁₃₉	1	2	yes	—
138	C₂³×C₄	C₂×D₁₃₉	1	1	yes	characteristic
139	C₄×Q₈	—	139	16	no	—
140	C₄×D₄	—	139	32	no	—
141	C₄×D₄	—	139	16	no	—
142	#117	—	139	8	no	—
143	#117	—	139	16	no	—
144	C₂×C₄⋊C₄	—	139	1	no	—
145	C₂×C₄⋊C₄	—	139	4	no	—
146	C₂×C₄⋊C₄	—	139	1	no	—
147	C₂×C₄⋊C₄	—	139	16	no	—
148	C₂×C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
149	C₂×C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
150	C₂×C₄⋊C₄	C₂×D₁₃₉	1	8	yes	complemented
151	C₂×C₂²⋊C₄	—	139	2	no	—
152	C₂×C₂²⋊C₄	—	139	4	no	—
153	C₂×C₄²	—	139	2	no	—
154	C₂×C₄²	—	139	4	no	—
155	C₂⁴	—	139	2	no	—
156	C₂⁴	C₂²×D₁₃₉	1	1	yes	characteristic
157	#41	—	139	64	no	—
158	C₂×Q₈	—	139	8	no	—
159	C₂×Q₈	—	139	4	no	—
160	C₂×D₄	—	139	16	no	—
161	C₂×D₄	—	139	8	no	—
162	C₂×D₄	—	139	8	no	—
163	C₂×D₄	—	139	4	no	—
164	C₂²×C₄	—	4	4	no	—
165	C₂²×C₄	—	556	8	no	—
166	C₂²×C₄	—	139	2	no	—
167	C₂²×C₄	—	139	2	no	—
168	C₂²×C₄	—	139	16	no	—
169	C₂²×C₄	—	139	2	no	—
170	C₂²×C₄	—	139	16	no	—
171	C₂²×C₄	—	139	8	no	—
172	C₂²×C₄	C₄×D₁₃₉	1	8	yes	complemented
173	C₂²×C₄	C₂²×D₁₃₉	1	4	yes	—
174	C₂²×C₄	C₂²×D₁₃₉	1	8	yes	—
175	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
176	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
177	C₂²×C₄	C₂²×D₁₃₉	1	4	yes	—
178	C₄⋊C₄	—	139	4	no	—
179	C₄⋊C₄	—	139	16	no	—
180	C₄⋊C₄	—	139	4	no	—
181	C₄⋊C₄	C₂²×D₁₃₉	1	8	yes	—
182	C₄⋊C₄	C₂²×D₁₃₉	1	8	yes	—
183	C₂²⋊C₄	—	139	8	no	—
184	C₂²⋊C₄	—	139	16	no	—
185	C₄²	—	139	8	no	—
186	C₄²	—	139	16	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	2	no	—
189	C₂³	—	278	4	no	—
190	C₂³	—	278	4	no	—
191	C₂³	—	139	8	no	—
192	C₂³	—	139	4	no	—
193	C₂³	C₂×C₄×D₁₃₉	1	4	yes	—
194	C₂³	C₂³×D₁₃₉	1	2	yes	—
195	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
196	Q₈	—	139	16	no	—
197	D₄	—	139	32	no	—
198	D₄	—	139	16	no	—
199	C₂×C₄	—	2	8	no	—
200	C₂×C₄	—	4	8	no	—
201	C₂×C₄	—	278	4	no	—
202	C₂×C₄	—	278	4	no	—
203	C₂×C₄	—	556	16	no	—
204	C₂×C₄	—	139	4	no	—
205	C₂×C₄	—	139	4	no	—
206	C₂×C₄	—	139	8	no	—
207	C₂×C₄	—	139	4	no	—
208	C₂×C₄	—	139	32	no	—
209	C₂×C₄	C₂×C₄×D₁₃₉	1	4	yes	—
210	C₂×C₄	C₂×C₄×D₁₃₉	1	4	yes	—
211	C₂×C₄	C₂×C₄×D₁₃₉	1	16	yes	complemented
212	C₂×C₄	C₂³×D₁₃₉	1	8	yes	—
213	C₂×C₄	C₂³×D₁₃₉	1	2	yes	—
214	C₂×C₄	C₂³×D₁₃₉	1	2	yes	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	8	no	—
218	C₂²	—	278	4	no	—
219	C₂²	—	278	4	no	—
220	C₂²	—	278	16	no	—
221	C₂²	—	139	8	no	—
222	C₂²	C₂²×C₄×D₁₃₉	1	4	yes	—
223	C₂²	C₂²×C₄×D₁₃₉	1	2	yes	—
224	C₂²	#17984887	1	2	yes	—
225	C₂²	#17984890	1	2	yes	—
226	C₂²	C₂⁴×D₁₃₉	1	1	yes	characteristic
227	C₄	—	2	8	no	—
228	C₄	—	278	4	no	—
229	C₄	—	278	4	no	—
230	C₄	—	139	8	no	—
231	C₄	C₂²×C₄×D₁₃₉	1	4	yes	—
232	C₄	C₂²×C₄×D₁₃₉	1	4	yes	—
233	C₂	—	2	4	no	—
234	C₂	—	278	8	no	—
235	C₂	#22581434	1	4	yes	direct product factor
236	C₂	C₂³×C₄×D₁₃₉	1	1	yes	characteristic
237	C₂	C₂×#17984887	1	1	yes	characteristic
238	C₂	C₂×#17984890	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₃₉
commutator	C₂×C₁₃₉	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₃₉
Fitting	C₂²×C₄⋊C₄×C₁₃₉	C₂
socle	C₂³×C₁₃₉	C₂⁴
prime radical	C₂²×#14555785	C₂
cosocle	C₂²×C₁₃₉	C₂⁵
hypercenter	C₂²×C₄⋊C₄	D₁₃₉
hypocenter	C₁₃₉	C₂×#520
LCS	G ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #22581434	C₂	direct (N × H)
3. C₂×#17984807	C₂	semidirect
4. C₂×#17984804	C₂	semidirect
5. C₂×C₄×C₂²⋊D₁₃₉	C₂	semidirect
6. C₂×#17984801	C₂	semidirect
7. C₂×#17984800	C₂	semidirect
8. C₂×C₄⋊C₄×D₁₃₉	C₂	semidirect
9. C₂²×C₄⋊C₄×C₁₃₉	C₂	semidirect
10. C₂×#14555785	C₄	semidirect
17. #17984807	C₂²	semidirect
18. #17984804	C₂²	semidirect
19. C₄×C₂²⋊D₁₃₉	C₂²	semidirect
20. #17984801	C₂²	semidirect
21. #17984800	C₂²	semidirect
25. C₄⋊C₄×D₁₃₉	C₂²	semidirect
34. C₂×C₄⋊C₄×C₁₃₉	C₂²	semidirect
36. #14555785	C₂×C₄	semidirect
37. C₂×Dic₂₇₈	C₂×C₄	semidirect
39. C₂×D₅₅₆	C₂×C₄	semidirect
41. C₂×C₂²⋊D₁₃₉	C₂×C₄	semidirect
47. C₂×C₄×D₁₃₉	C₂×C₄	semidirect
59. C₂²×C₄×C₁₃₉	C₂×C₄	semidirect
71. Dic₂₇₈	C₂²×C₄	semidirect
72. D₅₅₆	C₂²×C₄	semidirect
73. C₂²⋊D₁₃₉	C₂²×C₄	semidirect
81. C₄×D₁₃₉	C₂²×C₄	semidirect
91. C₂×C₄×C₁₃₉	C₂²×C₄	semidirect
114. C₂×C₁₃₉	#520	semidirect
119. C₁₃₉	C₂×#520	semidirect
129. C₂²×C₄⋊C₄	D₁₃₉	semidirect
150. C₂×C₄⋊C₄	C₂×D₁₃₉	semidirect
172. C₂²×C₄	C₄×D₁₃₉	semidirect
211. C₂×C₄	C₂×C₄×D₁₃₉	semidirect
235. C₂	#22581434	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×#14555785	2
#22581434	2
C₂×#17984807	2
C₂×#17984804	2
C₂×C₄×C₂²⋊D₁₃₉	2
C₂×#17984801	2
C₂×#17984800	2
C₂×C₄⋊C₄×D₁₃₉	2
C₂²×C₄⋊C₄×C₁₃₉	2
C₂×#520	139

Subgroups — C2×#22581434 #31819599

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22581434 #31819599