Subgroups — C₄×#17984800 #31808744

Order 17792.

752 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984800	C₂	1	2	yes	maximal
2	C₂×C₄²×D₁₃₉	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₁₃₉	C₂	1	2	yes	maximal, complemented
4	C₄×C₄⋊Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×(C₂×D₁₃₉)⋊C₄	C₂	1	2	yes	maximal
6	#22580800	C₂	1	2	yes	maximal
7	#22580798	C₂	1	2	yes	maximal, complemented
8	C₄²×Dic₁₃₉	C₂	1	2	yes	maximal, complemented
9	C₄×C₄⋊C₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
12	C₂²×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
13	#17984800	C₄	1	16	yes	direct product factor
14	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
15	C₂×(C₂×D₁₃₉)⋊C₄	C₂²	1	4	yes	—
16	C₂×C₄×Dic₁₃₉	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	1	yes	characteristic
20	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃₉	C₂²	1	4	yes	—
22	C₄²×D₁₃₉	C₄	1	8	yes	complemented
23	C₁₃₉⋊#95	C₂²	1	2	yes	—
24	C₁₃₉⋊#95	C₂²	1	4	yes	—
25	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
26	C₂×C₄²×C₁₃₉	C₂²	1	2	yes	—
27	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
28	#95×C₁₃₉	C₂²	1	2	yes	—
29	C₂³×D₁₃₉	C₂³	1	1	yes	characteristic
30	C₂²×Dic₁₃₉	C₂³	1	4	yes	—
31	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
32	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
33	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
34	C₂×C₄×D₁₃₉	—	2	4	no	—
35	C₂×C₄×D₁₃₉	—	2	2	no	—
36	C₂×C₄×D₁₃₉	—	2	2	no	—
37	C₂×C₄×D₁₃₉	C₂×C₄	1	4	yes	—
38	C₂×C₄×D₁₃₉	C₂×C₄	1	4	yes	—
39	C₂×C₄×D₁₃₉	C₂×C₄	1	4	yes	—
40	C₂×C₄×D₁₃₉	C₂×C₄	1	8	yes	—
41	C₄⋊Dic₁₃₉	C₂×C₄	1	8	yes	complemented
42	(C₂×D₁₃₉)⋊C₄	C₂×C₄	1	16	yes	—
43	C₄×Dic₁₃₉	—	2	8	no	—
44	C₄×Dic₁₃₉	—	2	4	no	—
45	C₄×Dic₁₃₉	C₂×C₄	1	4	yes	—
46	C₄×Dic₁₃₉	C₂×C₄	1	4	yes	—
47	C₄×Dic₁₃₉	C₂×C₄	1	4	yes	—
48	C₄×Dic₁₃₉	C₂×C₄	1	16	yes	complemented
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	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
53	C₄⋊C₄×C₁₃₉	C₂×C₄	1	8	yes	complemented
54	C₄²×C₁₃₉	—	2	4	no	—
55	C₄²×C₁₃₉	C₂×C₄	1	4	yes	—
56	C₂²×D₁₃₉	—	2	2	no	—
57	C₂²×D₁₃₉	—	2	1	no	—
58	C₂²×D₁₃₉	—	2	1	no	—
59	C₂²×D₁₃₉	C₂²×C₄	1	4	yes	—
60	C₂²×D₁₃₉	C₂²×C₄	1	2	yes	—
61	C₂×Dic₁₃₉	—	2	4	no	—
62	C₂×Dic₁₃₉	—	2	4	no	—
63	C₂×Dic₁₃₉	—	2	1	no	—
64	C₂×Dic₁₃₉	—	2	1	no	—
65	C₂×Dic₁₃₉	C₂²×C₄	1	2	yes	—
66	C₂×Dic₁₃₉	C₂²×C₄	1	8	yes	—
67	C₂×Dic₁₃₉	C₂²×C₄	1	2	yes	—
68	C₂×Dic₁₃₉	C₂²×C₄	1	4	yes	—
69	C₂×Dic₁₃₉	C₂²×C₄	1	2	yes	—
70	C₂×Dic₁₃₉	#41	1	4	yes	—
71	C₄×D₁₃₉	—	2	8	no	—
72	C₄×D₁₃₉	—	2	8	no	—
73	C₄×D₁₃₉	C₄²	1	16	yes	complemented
74	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
75	C₂×C₄×C₁₃₉	—	2	4	no	—
76	C₂×C₄×C₁₃₉	—	2	4	no	—
77	C₂×C₄×C₁₃₉	—	2	2	no	—
78	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₁₃₉	C₂²×C₄	1	8	yes	—
80	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
81	C₂×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₁₃₉	#41	1	2	yes	—
84	C₂×C₄×C₁₃₉	#41	1	2	yes	—
85	C₂²×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
86	C₂²×C₁₃₉	C₂³×C₄	1	2	yes	—
87	C₂²×C₁₃₉	C₂×#41	1	2	yes	—
88	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
89	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
90	C₂×D₁₃₉	—	2	2	no	—
91	C₂×D₁₃₉	—	2	2	no	—
92	C₂×D₁₃₉	—	2	4	no	—
93	C₂×D₁₃₉	—	2	4	no	—
94	C₂×D₁₃₉	C₂×C₄²	1	4	yes	—
95	C₄×C₁₃₉	—	2	2	no	—
96	C₄×C₁₃₉	—	2	8	no	—
97	C₄×C₁₃₉	C₂×C₄²	1	4	yes	—
98	C₄×C₁₃₉	#117	1	4	yes	complemented
99	Dic₁₃₉	—	2	2	no	—
100	Dic₁₃₉	—	2	8	no	—
101	Dic₁₃₉	C₂×C₄²	1	4	yes	—
102	Dic₁₃₉	#117	1	4	yes	complemented
103	C₂×C₁₃₉	C₂²×C₄²	1	1	yes	characteristic
104	C₂×C₁₃₉	C₂×#117	1	1	yes	characteristic
105	C₂×C₁₃₉	C₂×#117	1	1	yes	characteristic
106	C₂×C₁₃₉	C₄×#41	1	2	yes	—
107	C₂×C₁₃₉	C₄×#41	1	2	yes	—
108	D₁₃₉	—	2	4	no	—
109	C₁₃₉	C₄×#117	1	1	yes	characteristic, complemented, Sylow
110	C₄×#117	—	139	1	no	maximal, Sylow
111	C₂×#117	—	139	2	no	—
112	C₂²×C₄²	—	139	1	no	—
113	C₄×C₄⋊C₄	—	139	2	no	—
114	C₄×C₄⋊C₄	—	139	1	no	—
115	C₄×C₄⋊C₄	D₁₃₉	1	1	yes	characteristic, complemented
116	C₄×C₂²⋊C₄	—	139	2	no	—
117	C₄×C₂²⋊C₄	—	139	2	no	—
118	#376	—	139	2	no	—
119	C₄³	—	139	2	no	—
120	C₂³×C₄	—	139	1	no	—
121	C₂³×C₄	—	139	1	no	—
122	C₂³×C₄	—	139	1	no	—
123	#117	—	139	16	no	—
124	C₂×C₄⋊C₄	—	139	2	no	—
125	C₂×C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
126	C₂×C₂²⋊C₄	—	139	4	no	—
127	C₂×C₄²	—	139	4	no	—
128	C₂×C₄²	—	139	2	no	—
129	C₂×C₄²	—	139	4	no	—
130	C₂×C₄²	—	139	1	no	—
131	C₂×C₄²	—	139	1	no	—
132	C₂×C₄²	—	139	1	no	—
133	C₂×C₄²	—	139	8	no	—
134	C₂×C₄²	C₂×D₁₃₉	1	2	yes	—
135	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
136	#95	—	139	4	no	—
137	#95	—	139	2	no	—
138	#95	C₂×D₁₃₉	1	2	yes	—
139	C₂⁴	—	139	1	no	—
140	C₂²×C₄	—	278	4	no	—
141	C₂²×C₄	—	278	2	no	—
142	C₂²×C₄	—	278	2	no	—
143	C₂²×C₄	—	139	1	no	—
144	C₂²×C₄	—	139	4	no	—
145	C₂²×C₄	—	139	4	no	—
146	C₂²×C₄	—	139	4	no	—
147	C₂²×C₄	—	139	1	no	—
148	C₂²×C₄	—	139	1	no	—
149	C₂²×C₄	—	139	4	no	—
150	C₂²×C₄	—	139	8	no	—
151	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
152	C₂²×C₄	C₂²×D₁₃₉	1	4	yes	—
153	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
155	C₄⋊C₄	—	139	8	no	—
156	C₄⋊C₄	C₄×D₁₃₉	1	8	yes	complemented
157	C₂²⋊C₄	—	139	16	no	—
158	C₄²	—	2	4	no	—
159	C₄²	—	278	8	no	—
160	C₄²	—	278	4	no	—
161	C₄²	—	139	4	no	—
162	C₄²	—	139	4	no	—
163	C₄²	—	139	4	no	—
164	C₄²	—	139	16	no	—
165	C₄²	C₄×D₁₃₉	1	4	yes	—
166	C₂³	—	278	2	no	—
167	C₂³	—	278	1	no	—
168	C₂³	—	278	1	no	—
169	C₂³	—	139	4	no	—
170	C₂³	—	139	2	no	—
171	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	278	8	no	—
176	C₂×C₄	—	278	4	no	—
177	C₂×C₄	—	278	1	no	—
178	C₂×C₄	—	278	4	no	—
179	C₂×C₄	—	278	1	no	—
180	C₂×C₄	—	278	8	no	—
181	C₂×C₄	—	139	2	no	—
182	C₂×C₄	—	139	8	no	—
183	C₂×C₄	—	139	2	no	—
184	C₂×C₄	—	139	4	no	—
185	C₂×C₄	—	139	2	no	—
186	C₂×C₄	—	139	4	no	—
187	C₂×C₄	—	139	16	no	—
188	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
189	C₂×C₄	C₂×C₄×D₁₃₉	1	8	yes	—
190	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
191	C₂×C₄	C₂×C₄×D₁₃₉	1	4	yes	—
192	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
193	C₂×C₄	#14555784	1	2	yes	—
194	C₂×C₄	#14555786	1	2	yes	—
195	C₂²	—	278	2	no	—
196	C₂²	—	278	2	no	—
197	C₂²	—	278	4	no	—
198	C₂²	—	278	4	no	—
199	C₂²	—	139	4	no	—
200	C₂²	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
201	C₂²	C₂²×C₄×D₁₃₉	1	2	yes	—
202	C₂²	#41×D₁₃₉	1	2	yes	—
203	C₂²	C₂×#14555784	1	1	yes	characteristic
204	C₂²	C₂×#14555786	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	2	8	no	—
207	C₄	—	278	2	no	—
208	C₄	—	278	8	no	—
209	C₄	—	139	4	no	—
210	C₄	—	139	4	no	—
211	C₄	C₄²×D₁₃₉	1	4	yes	—
212	C₄	#17984800	1	4	yes	direct product factor
213	C₂	—	278	4	no	—
214	C₂	C₂×C₄²×D₁₃₉	1	1	yes	characteristic
215	C₂	#117×D₁₃₉	1	1	yes	characteristic
216	C₂	C₂×#17984800	1	1	yes	characteristic
217	C₂	C₄×#14555784	1	2	yes	—
218	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	C₂³×D₁₃₉	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₃₉
hypocenter	C₁₃₉	C₄×#117
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. C₄⋊C₄×Dic₁₃₉	C₂	semidirect
4. C₄×C₄⋊Dic₁₃₉	C₂	semidirect
7. #22580798	C₂	semidirect
8. C₄²×Dic₁₃₉	C₂	semidirect
9. C₄×C₄⋊C₄×C₁₃₉	C₂	semidirect
13. #17984800	C₄	direct (N × H)
22. C₄²×D₁₃₉	C₄	semidirect
41. C₄⋊Dic₁₃₉	C₂×C₄	semidirect
48. C₄×Dic₁₃₉	C₂×C₄	semidirect
53. C₄⋊C₄×C₁₃₉	C₂×C₄	semidirect
73. C₄×D₁₃₉	C₄²	semidirect
98. C₄×C₁₃₉	#117	semidirect
102. Dic₁₃₉	#117	semidirect
109. C₁₃₉	C₄×#117	semidirect
115. C₄×C₄⋊C₄	D₁₃₉	semidirect
156. C₄⋊C₄	C₄×D₁₃₉	semidirect
212. C₄	#17984800	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×#17984800	2
C₂×C₄²×D₁₃₉	2
C₄⋊C₄×Dic₁₃₉	2
C₄×C₄⋊Dic₁₃₉	2
C₄×(C₂×D₁₃₉)⋊C₄	2
#22580800	2
#22580798	2
C₄²×Dic₁₃₉	2
C₄×C₄⋊C₄×C₁₃₉	2
C₄×#117	139

Subgroups — C4×#17984800 #31808744

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17984800 #31808744