Subgroups — C₄×#17984804 #31808749

Order 17792.

924 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₅₅₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984804	C₂	1	2	yes	maximal
3	C₂×C₄²×D₁₃₉	C₂	1	2	yes	maximal, complemented
4	C₄⋊C₄×Dic₁₃₉	C₂	1	2	yes	maximal, complemented
5	C₄×(C₂×D₁₃₉)⋊C₄	C₂	1	2	yes	maximal, complemented
6	#22580800	C₂	1	4	yes	maximal, complemented
7	C₄²×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₄⋊C₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₅₅₆	C₂²	1	1	yes	characteristic
10	C₂²×C₄×D₁₃₉	C₂²	1	4	yes	—
11	C₂²×C₄×D₁₃₉	C₂²	1	2	yes	—
12	C₄×D₅₅₆	C₄	1	8	yes	complemented
13	#17984804	C₄	1	16	yes	direct product factor
14	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
15	C₂×(C₂×D₁₃₉)⋊C₄	C₂²	1	2	yes	—
16	C₂×(C₂×D₁₃₉)⋊C₄	C₂²	1	4	yes	—
17	C₂×C₄×Dic₁₃₉	C₂²	1	2	yes	—
18	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₁₃₉	C₂²	1	4	yes	—
20	C₂×C₄×Dic₁₃₉	C₂²	1	2	yes	—
21	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
22	C₄²×D₁₃₉	—	2	8	no	—
23	C₁₃₉⋊#95	C₂²	1	4	yes	—
24	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
25	C₂×C₄²×C₁₃₉	C₂²	1	2	yes	—
26	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
27	#95×C₁₃₉	C₂²	1	2	yes	—
28	C₂³×D₁₃₉	C₂³	1	2	yes	—
29	C₂×D₅₅₆	C₂×C₄	1	8	yes	—
30	C₂×D₅₅₆	C₂×C₄	1	4	yes	—
31	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
32	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
33	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
34	C₂×C₄×D₁₃₉	—	2	8	no	—
35	C₂×C₄×D₁₃₉	—	2	8	no	—
36	C₂×C₄×D₁₃₉	—	2	4	no	—
37	C₂×C₄×D₁₃₉	—	2	4	no	—
38	C₂×C₄×D₁₃₉	C₂×C₄	1	16	yes	complemented
39	C₂×C₄×D₁₃₉	C₂×C₄	1	8	yes	complemented
40	C₄⋊Dic₁₃₉	C₂×C₄	1	4	yes	complemented
41	(C₂×D₁₃₉)⋊C₄	C₂×C₄	1	16	yes	complemented
42	(C₂×D₁₃₉)⋊C₄	C₂×C₄	1	8	yes	complemented
43	C₄×Dic₁₃₉	—	2	8	no	—
44	C₄×Dic₁₃₉	—	2	4	no	—
45	C₄×Dic₁₃₉	—	2	2	no	—
46	C₄×Dic₁₃₉	C₂×C₄	1	8	yes	complemented
47	C₄×Dic₁₃₉	D₄	1	4	yes	complemented
48	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₃₉	C₂³	1	4	yes	—
50	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
52	C₄⋊C₄×C₁₃₉	C₂×C₄	1	8	yes	complemented
53	C₄²×C₁₃₉	—	2	4	no	—
54	C₄²×C₁₃₉	C₂×C₄	1	4	yes	complemented
55	C₂²×D₁₃₉	—	2	2	no	—
56	C₂²×D₁₃₉	—	2	4	no	—
57	C₂²×D₁₃₉	—	2	2	no	—
58	C₂²×D₁₃₉	C₂²×C₄	1	8	yes	—
59	C₂²×D₁₃₉	C₂²×C₄	1	4	yes	—
60	D₅₅₆	C₄²	1	16	yes	complemented
61	C₂×Dic₁₃₉	—	2	2	no	—
62	C₂×Dic₁₃₉	—	2	2	no	—
63	C₂×Dic₁₃₉	—	2	2	no	—
64	C₂×Dic₁₃₉	—	2	2	no	—
65	C₂×Dic₁₃₉	C₂²×C₄	1	4	yes	—
66	C₂×Dic₁₃₉	C₂²×C₄	1	4	yes	—
67	C₂×Dic₁₃₉	C₂²×C₄	1	4	yes	—
68	C₂×Dic₁₃₉	C₂×D₄	1	4	yes	—
69	C₂×Dic₁₃₉	#41	1	4	yes	—
70	C₄×D₁₃₉	—	2	16	no	—
71	C₄×D₁₃₉	—	2	32	no	—
72	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
73	C₂×C₄×C₁₃₉	—	2	4	no	—
74	C₂×C₄×C₁₃₉	—	2	2	no	—
75	C₂×C₄×C₁₃₉	—	2	4	no	—
76	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₁₃₉	C₂²×C₄	1	8	yes	—
78	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
80	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
82	C₂×C₄×C₁₃₉	#41	1	2	yes	—
83	C₂²×C₁₃₉	C₂³×C₄	1	2	yes	—
84	C₂²×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
85	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
86	C₂²×C₁₃₉	C₂×#41	1	2	yes	—
87	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
88	C₂×D₁₃₉	—	2	8	no	—
89	C₂×D₁₃₉	—	2	4	no	—
90	C₂×D₁₃₉	—	2	4	no	—
91	C₂×D₁₃₉	—	2	8	no	—
92	C₂×D₁₃₉	C₂×C₄²	1	8	yes	complemented
93	C₄×C₁₃₉	—	2	8	no	—
94	C₄×C₁₃₉	—	2	2	no	—
95	C₄×C₁₃₉	C₂×C₄²	1	4	yes	complemented
96	C₄×C₁₃₉	C₄×D₄	1	4	yes	complemented
97	Dic₁₃₉	—	2	4	no	—
98	Dic₁₃₉	—	2	4	no	—
99	Dic₁₃₉	C₄×D₄	1	8	yes	complemented
100	C₂×C₁₃₉	C₂²×C₄²	1	1	yes	characteristic
101	C₂×C₁₃₉	C₂×C₄×D₄	1	2	yes	—
102	C₂×C₁₃₉	C₂×C₄×D₄	1	1	yes	characteristic
103	C₂×C₁₃₉	C₄×#41	1	1	yes	characteristic
104	C₂×C₁₃₉	C₄×#41	1	2	yes	—
105	D₁₃₉	—	2	8	no	—
106	C₁₃₉	C₄²×D₄	1	1	yes	characteristic, complemented, Sylow
107	C₄²×D₄	—	139	1	no	maximal, Sylow
108	C₂×C₄×D₄	—	139	2	no	—
109	C₂×C₄×D₄	—	139	1	no	—
110	C₂²×C₄²	—	139	2	no	—
111	C₄×C₄⋊C₄	—	139	2	no	—
112	C₄×C₄⋊C₄	D₁₃₉	1	1	yes	characteristic, complemented
113	C₄×C₂²⋊C₄	—	139	2	no	—
114	C₄×C₂²⋊C₄	—	139	4	no	—
115	C₄³	—	139	1	no	—
116	C₂²×D₄	—	139	1	no	—
117	C₂³×C₄	—	139	4	no	—
118	C₂³×C₄	—	139	2	no	—
119	C₄×D₄	—	139	16	no	—
120	C₄×D₄	—	139	8	no	—
121	C₂×C₄⋊C₄	—	139	1	no	—
122	C₂×C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
123	C₂×C₂²⋊C₄	—	139	4	no	—
124	C₂×C₂²⋊C₄	—	139	2	no	—
125	C₂×C₄²	—	278	8	no	—
126	C₂×C₄²	—	139	2	no	—
127	C₂×C₄²	—	139	4	no	—
128	C₂×C₄²	—	139	1	no	—
129	C₂×C₄²	—	139	2	no	—
130	C₂×C₄²	—	139	1	no	—
131	C₂×C₄²	C₂×D₁₃₉	1	2	yes	—
132	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
133	#95	—	139	4	no	—
134	#95	C₂×D₁₃₉	1	2	yes	—
135	C₂⁴	—	139	2	no	—
136	C₂×D₄	—	139	8	no	—
137	C₂×D₄	—	139	4	no	—
138	C₂²×C₄	—	278	8	no	—
139	C₂²×C₄	—	278	8	no	—
140	C₂²×C₄	—	278	4	no	—
141	C₂²×C₄	—	278	4	no	—
142	C₂²×C₄	—	139	2	no	—
143	C₂²×C₄	—	139	2	no	—
144	C₂²×C₄	—	139	16	no	—
145	C₂²×C₄	—	139	8	no	—
146	C₂²×C₄	—	139	2	no	—
147	C₂²×C₄	C₂²×D₁₃₉	1	4	yes	—
148	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
149	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
151	C₄⋊C₄	—	139	4	no	—
152	C₄⋊C₄	C₄×D₁₃₉	1	8	yes	complemented
153	C₂²⋊C₄	—	139	16	no	—
154	C₂²⋊C₄	—	139	8	no	—
155	C₄²	—	2	4	no	—
156	C₄²	—	278	8	no	—
157	C₄²	—	278	4	no	—
158	C₄²	—	278	2	no	—
159	C₄²	—	139	8	no	—
160	C₄²	—	139	4	no	—
161	C₄²	C₄×D₁₃₉	1	4	yes	complemented
162	C₂³	—	278	2	no	—
163	C₂³	—	278	4	no	—
164	C₂³	—	278	2	no	—
165	C₂³	—	139	8	no	—
166	C₂³	—	139	4	no	—
167	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
168	D₄	—	139	16	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	278	32	no	—
173	C₂×C₄	—	278	2	no	—
174	C₂×C₄	—	278	2	no	—
175	C₂×C₄	—	278	2	no	—
176	C₂×C₄	—	278	2	no	—
177	C₂×C₄	—	278	16	no	—
178	C₂×C₄	—	139	4	no	—
179	C₂×C₄	—	139	4	no	—
180	C₂×C₄	—	139	4	no	—
181	C₂×C₄	—	139	4	no	—
182	C₂×C₄	—	139	4	no	—
183	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
184	C₂×C₄	C₂×C₄×D₁₃₉	1	8	yes	—
185	C₂×C₄	C₂×C₄×D₁₃₉	1	4	yes	—
186	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
187	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
188	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
189	C₂×C₄	#14555786	1	2	yes	—
190	C₂²	—	278	8	no	—
191	C₂²	—	278	4	no	—
192	C₂²	—	278	4	no	—
193	C₂²	—	278	8	no	—
194	C₂²	—	139	8	no	—
195	C₂²	C₂²×C₄×D₁₃₉	1	2	yes	—
196	C₂²	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
197	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
198	C₂²	#41×D₁₃₉	1	2	yes	—
199	C₂²	C₂×#14555786	1	1	yes	characteristic
200	C₄	—	2	8	no	—
201	C₄	—	2	2	no	—
202	C₄	—	278	4	no	—
203	C₄	—	278	4	no	—
204	C₄	—	139	8	no	—
205	C₄	C₄²×D₁₃₉	1	4	yes	complemented
206	C₄	#17984804	1	4	yes	direct product factor
207	C₂	—	278	8	no	—
208	C₂	C₂×C₄²×D₁₃₉	1	1	yes	characteristic
209	C₂	C₄×D₄×D₁₃₉	1	2	yes	—
210	C₂	C₂×#17984804	1	1	yes	characteristic
211	C₂	#22581421	1	1	yes	characteristic
212	C₂	C₄×#14555786	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₃₉
commutator	C₂×C₁₃₉	C₂²×C₄²
Frattini	C₂³	C₂³×D₁₃₉
Fitting	C₄×C₄⋊C₄×C₁₃₉	C₂
socle, cosocle	C₂³×C₁₃₉	C₂⁴
prime radical	C₂²×D₅₅₆	C₂²
hypercenter	C₄×C₄⋊C₄	D₁₃₉
hypocenter	C₁₃₉	C₄²×D₄
LCS	G ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. C₂×C₄²×D₁₃₉	C₂	semidirect
4. C₄⋊C₄×Dic₁₃₉	C₂	semidirect
5. C₄×(C₂×D₁₃₉)⋊C₄	C₂	semidirect
6. #22580800	C₂	semidirect
7. C₄²×Dic₁₃₉	C₂	semidirect
8. C₄×C₄⋊C₄×C₁₃₉	C₂	semidirect
12. C₄×D₅₅₆	C₄	semidirect
13. #17984804	C₄	direct (N × H)
38. C₂×C₄×D₁₃₉	C₂×C₄	semidirect
39. C₂×C₄×D₁₃₉	C₂×C₄	semidirect
40. C₄⋊Dic₁₃₉	C₂×C₄	semidirect
41. (C₂×D₁₃₉)⋊C₄	C₂×C₄	semidirect
42. (C₂×D₁₃₉)⋊C₄	C₂×C₄	semidirect
46. C₄×Dic₁₃₉	C₂×C₄	semidirect
47. C₄×Dic₁₃₉	D₄	semidirect
52. C₄⋊C₄×C₁₃₉	C₂×C₄	semidirect
54. C₄²×C₁₃₉	C₂×C₄	semidirect
60. D₅₅₆	C₄²	semidirect
92. C₂×D₁₃₉	C₂×C₄²	semidirect
95. C₄×C₁₃₉	C₂×C₄²	semidirect
96. C₄×C₁₃₉	C₄×D₄	semidirect
99. Dic₁₃₉	C₄×D₄	semidirect
106. C₁₃₉	C₄²×D₄	semidirect
112. C₄×C₄⋊C₄	D₁₃₉	semidirect
152. C₄⋊C₄	C₄×D₁₃₉	semidirect
161. C₄²	C₄×D₁₃₉	semidirect
205. C₄	C₄²×D₁₃₉	semidirect
206. C₄	#17984804	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×D₅₅₆	2
C₂×#17984804	2
C₂×C₄²×D₁₃₉	2
C₄⋊C₄×Dic₁₃₉	2
C₄×(C₂×D₁₃₉)⋊C₄	2
#22580800	2
C₄²×Dic₁₃₉	2
C₄×C₄⋊C₄×C₁₃₉	2
C₄²×D₄	139

Subgroups — C4×#17984804 #31808749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17984804 #31808749