Subgroups — C₂×#22581768 #31821169

Order 17792.

1408 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984889	C₂	1	1	yes	maximal, characteristic, complemented
2	#22581768	C₂	1	16	yes	maximal, direct product factor
3	C₂×#17984873	C₂	1	3	yes	maximal, complemented
4	C₂×#17984872	C₂	1	3	yes	maximal, complemented
5	C₂×#17984862	C₂	1	3	yes	maximal, complemented
6	C₂²×#14555775	C₂	1	3	yes	maximal, complemented
7	C₂×#17984849	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×2₋¹⁺⁴×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#17984889	—	2	4	no	—
10	#17984889	C₂²	1	8	yes	complemented
11	C₂×#14555785	C₂²	1	3	yes	—
12	C₂×#14555784	—	2	3	no	—
13	C₂²×Dic₂₇₈	C₂²	1	3	yes	—
14	C₂×Q₈×D₁₃₉	—	2	1	no	—
15	#17984873	C₂²	1	24	yes	complemented
16	#17984872	C₂²	1	24	yes	complemented
17	#17984862	C₂²	1	24	yes	complemented
18	C₂×#14555775	C₂²	1	3	yes	—
19	C₂×#14555775	C₂²	1	6	yes	—
20	C₂×#14555775	C₂²	1	24	yes	complemented
21	C₂×#14555773	C₂²	1	3	yes	—
22	C₂×#14555772	C₂²	1	3	yes	—
23	C₂×#14555770	C₂²	1	1	yes	characteristic
24	#17984849	C₂²	1	8	yes	complemented
25	C₂×#14555769	C₂²	1	3	yes	—
26	C₂²×C₁₃₉⋊C₈	C₂²	1	3	yes	—
27	2₋¹⁺⁴×C₁₃₉	—	2	4	no	—
28	2₋¹⁺⁴×C₁₃₉	C₂²	1	8	yes	complemented
29	C₂×#41×C₁₃₉	—	2	3	no	—
30	C₂×#41×C₁₃₉	C₂²	1	3	yes	—
31	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic
32	C₂²×Q₈×C₁₃₉	—	2	1	no	—
33	C₂²×Q₈×C₁₃₉	C₂²	1	3	yes	—
34	#14555785	—	2	6	no	—
35	#14555785	C₂³	1	12	yes	—
36	#14555784	—	2	24	no	—
37	C₂×Dic₂₇₈	—	2	3	no	—
38	C₂×Dic₂₇₈	—	2	6	no	—
39	C₂×Dic₂₇₈	C₂³	1	3	yes	—
40	C₂×Dic₂₇₈	C₂³	1	12	yes	—
41	Q₈×D₁₃₉	—	2	8	no	—
42	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
43	C₂×C₂²⋊D₁₃₉	—	2	3	no	—
44	C₂²×Dic₁₃₉	—	2	3	no	—
45	C₂×C₄×D₁₃₉	—	2	3	no	—
46	#14555775	C₂³	1	12	yes	—
47	#14555775	C₂³	1	24	yes	—
48	#14555773	C₂³	1	12	yes	—
49	#14555772	C₂³	1	12	yes	—
50	#14555770	C₂³	1	4	yes	—
51	#14555769	C₂³	1	12	yes	—
52	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
53	C₂×C₁₃₉⋊C₈	C₂³	1	3	yes	—
54	C₂×C₁₃₉⋊C₈	C₂³	1	12	yes	—
55	#41×C₁₃₉	—	2	24	no	—
56	#41×C₁₃₉	—	2	6	no	—
57	#41×C₁₃₉	D₄	1	4	yes	—
58	#41×C₁₃₉	C₂³	1	12	yes	—
59	#41×C₁₃₉	C₂³	1	4	yes	—
60	C₂×Q₈×C₁₃₉	—	2	8	no	—
61	C₂×Q₈×C₁₃₉	—	2	3	no	—
62	C₂×Q₈×C₁₃₉	—	2	6	no	—
63	C₂×Q₈×C₁₃₉	D₄	1	1	yes	characteristic
64	C₂×Q₈×C₁₃₉	C₂³	1	3	yes	—
65	C₂×Q₈×C₁₃₉	C₂³	1	12	yes	—
66	C₂×D₄×C₁₃₉	—	2	3	no	—
67	C₂×D₄×C₁₃₉	D₄	1	3	yes	—
68	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₃₉	—	2	3	no	—
70	C₂²×C₄×C₁₃₉	—	2	3	no	—
71	C₂²×C₄×C₁₃₉	C₂³	1	3	yes	—
72	C₂²×D₁₃₉	—	2	1	no	—
73	Dic₂₇₈	—	2	12	no	—
74	Dic₂₇₈	—	2	3	no	—
75	Dic₂₇₈	C₂⁴	1	6	yes	—
76	D₅₅₆	—	2	1	no	—
77	D₅₅₆	C₂⁴	1	2	yes	—
78	C₂²⋊D₁₃₉	—	2	12	no	—
79	C₂×Dic₁₃₉	—	2	3	no	—
80	C₂×Dic₁₃₉	—	2	12	no	—
81	C₄×D₁₃₉	—	2	12	no	—
82	C₁₃₉⋊C₈	C₂⁴	1	2	yes	—
83	C₁₃₉⋊C₈	C₂⁴	1	6	yes	—
84	C₂³×C₁₃₉	—	2	1	no	—
85	C₂³×C₁₃₉	C₂×D₄	1	3	yes	—
86	Q₈×C₁₃₉	—	2	12	no	—
87	Q₈×C₁₃₉	—	2	3	no	—
88	Q₈×C₁₃₉	C₂×D₄	1	4	yes	—
89	Q₈×C₁₃₉	C₂⁴	1	6	yes	—
90	D₄×C₁₃₉	—	2	12	no	—
91	D₄×C₁₃₉	—	2	1	no	—
92	D₄×C₁₃₉	C₂×D₄	1	12	yes	—
93	D₄×C₁₃₉	C₂⁴	1	2	yes	—
94	C₂×C₄×C₁₃₉	—	2	3	no	—
95	C₂×C₄×C₁₃₉	—	2	12	no	—
96	C₂×C₄×C₁₃₉	—	2	12	no	—
97	C₂×C₄×C₁₃₉	C₂×D₄	1	3	yes	—
98	C₂×C₄×C₁₃₉	C₂×D₄	1	6	yes	—
99	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
100	C₂×C₄×C₁₃₉	C₂⁴	1	6	yes	—
101	C₂²×C₁₃₉	—	2	6	no	—
102	C₂²×C₁₃₉	—	2	2	no	—
103	C₂²×C₁₃₉	—	4	2	no	—
104	C₂²×C₁₃₉	C₂²×D₄	1	6	yes	—
105	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
106	C₂×D₁₃₉	—	2	2	no	—
107	C₂×D₁₃₉	—	4	2	no	—
108	C₄×C₁₃₉	—	2	6	no	—
109	C₄×C₁₃₉	C₂²×D₄	1	6	yes	—
110	C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
111	C₄×C₁₃₉	C₂⁵	1	1	yes	characteristic
112	Dic₁₃₉	—	2	6	no	—
113	C₂×C₁₃₉	—	2	6	no	—
114	C₂×C₁₃₉	—	4	2	no	—
115	C₂×C₁₃₉	#578	1	2	yes	complemented
116	C₂×C₁₃₉	C₂³×D₄	1	1	yes	characteristic
117	D₁₃₉	—	4	2	no	—
118	C₁₃₉	C₂×#578	1	1	yes	characteristic, complemented, Sylow
119	C₂×#578	—	139	1	no	maximal, Sylow
120	C₂×2₋¹⁺⁴	—	139	1	no	—
121	C₂×2₋¹⁺⁴	D₁₃₉	1	1	yes	characteristic, complemented
122	#578	—	139	16	no	—
123	C₂×#137	—	139	3	no	—
124	C₂×#137	—	139	3	no	—
125	C₂×#135	—	139	3	no	—
126	C₂²×Q₁₆	—	139	3	no	—
127	C₂×#131	—	139	1	no	—
128	2₋¹⁺⁴	—	2	4	no	—
129	2₋¹⁺⁴	—	278	4	no	—
130	2₋¹⁺⁴	—	139	8	no	—
131	2₋¹⁺⁴	C₂×D₁₃₉	1	8	yes	complemented
132	C₂×#41	—	2	3	no	—
133	C₂×#41	—	278	3	no	—
134	C₂×#41	—	139	3	no	—
135	C₂×#41	C₂×D₁₃₉	1	3	yes	—
136	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic
137	C₂²×Q₈	—	2	1	no	—
138	C₂²×Q₈	—	278	1	no	—
139	C₂²×Q₈	—	139	3	no	—
140	C₂²×Q₈	C₂×D₁₃₉	1	3	yes	—
141	#137	—	139	24	no	—
142	#137	—	139	24	no	—
143	#135	—	139	24	no	—
144	C₂×Q₁₆	—	139	3	no	—
145	C₂×Q₁₆	—	139	6	no	—
146	C₂×Q₁₆	—	139	24	no	—
147	C₂×SD₁₆	—	139	3	no	—
148	C₂×SD₁₆	—	139	3	no	—
149	C₂×D₈	—	139	1	no	—
150	#131	—	139	8	no	—
151	C₂×M₁₆	—	139	3	no	—
152	C₂²×C₈	—	139	3	no	—
153	#41	—	2	24	no	—
154	#41	—	2	6	no	—
155	#41	—	278	24	no	—
156	#41	—	278	6	no	—
157	#41	—	139	12	no	—
158	#41	C₂²⋊D₁₃₉	1	4	yes	—
159	#41	C₂²×D₁₃₉	1	12	yes	—
160	#41	C₂²×D₁₃₉	1	4	yes	—
161	C₂×Q₈	—	2	8	no	—
162	C₂×Q₈	—	2	3	no	—
163	C₂×Q₈	—	2	6	no	—
164	C₂×Q₈	—	278	8	no	—
165	C₂×Q₈	—	278	3	no	—
166	C₂×Q₈	—	278	6	no	—
167	C₂×Q₈	—	139	3	no	—
168	C₂×Q₈	—	139	12	no	—
169	C₂×Q₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
170	C₂×Q₈	C₂²×D₁₃₉	1	3	yes	—
171	C₂×Q₈	C₂²×D₁₃₉	1	12	yes	—
172	C₂×D₄	—	2	3	no	—
173	C₂×D₄	—	278	3	no	—
174	C₂×D₄	—	139	1	no	—
175	C₂×D₄	C₂²⋊D₁₃₉	1	3	yes	—
176	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
177	C₂²×C₄	—	2	3	no	—
178	C₂²×C₄	—	2	3	no	—
179	C₂²×C₄	—	278	3	no	—
180	C₂²×C₄	—	278	3	no	—
181	C₂²×C₄	C₂²×D₁₃₉	1	3	yes	—
182	Q₁₆	—	139	12	no	—
183	Q₁₆	—	139	24	no	—
184	SD₁₆	—	139	12	no	—
185	SD₁₆	—	139	12	no	—
186	D₈	—	139	4	no	—
187	M₁₆	—	139	12	no	—
188	C₂×C₈	—	139	1	no	—
189	C₂×C₈	—	139	3	no	—
190	C₂×C₈	—	139	12	no	—
191	C₂³	—	2	1	no	—
192	C₂³	—	278	1	no	—
193	C₂³	C₂×C₂²⋊D₁₃₉	1	3	yes	—
194	Q₈	—	2	12	no	—
195	Q₈	—	2	3	no	—
196	Q₈	—	278	12	no	—
197	Q₈	—	278	3	no	—
198	Q₈	—	139	6	no	—
199	Q₈	C₂×C₂²⋊D₁₃₉	1	4	yes	—
200	Q₈	C₂³×D₁₃₉	1	6	yes	—
201	D₄	—	2	12	no	—
202	D₄	—	2	1	no	—
203	D₄	—	278	12	no	—
204	D₄	—	278	1	no	—
205	D₄	—	139	2	no	—
206	D₄	C₂×C₂²⋊D₁₃₉	1	12	yes	—
207	D₄	C₂³×D₁₃₉	1	2	yes	—
208	C₂×C₄	—	2	12	no	—
209	C₂×C₄	—	2	3	no	—
210	C₂×C₄	—	2	12	no	—
211	C₂×C₄	—	278	12	no	—
212	C₂×C₄	—	278	3	no	—
213	C₂×C₄	—	278	12	no	—
214	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	6	yes	—
215	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	3	yes	—
216	C₂×C₄	C₂³×D₁₃₉	1	6	yes	—
217	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
218	C₈	—	139	2	no	—
219	C₈	—	139	6	no	—
220	C₂²	—	2	6	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	278	2	no	—
224	C₂²	—	556	2	no	—
225	C₂²	C₂²×C₂²⋊D₁₃₉	1	6	yes	—
226	C₂²	C₂²×C₂²⋊D₁₃₉	1	1	yes	characteristic
227	C₄	—	2	6	no	—
228	C₄	—	278	6	no	—
229	C₄	C₂²×C₂²⋊D₁₃₉	1	6	yes	—
230	C₄	C₂²×C₂²⋊D₁₃₉	1	1	yes	characteristic
231	C₄	C₂⁴×D₁₃₉	1	1	yes	characteristic
232	C₂	—	2	6	no	—
233	C₂	—	4	2	no	—
234	C₂	—	556	2	no	—
235	C₂	#22581768	1	2	yes	direct product factor
236	C₂	C₂³×C₂²⋊D₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃₉
commutator, cosocle	C₄×C₁₃₉	C₂⁵
Frattini	C₄	C₂⁴×D₁₃₉
Fitting	C₂×2₋¹⁺⁴×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
hypercenter	C₂×2₋¹⁺⁴	D₁₃₉
hypocenter	C₁₃₉	C₂×#578
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. C₂×#17984889	C₂	semidirect
2. #22581768	C₂	direct (N × H)
3. C₂×#17984873	C₂	semidirect
4. C₂×#17984872	C₂	semidirect
5. C₂×#17984862	C₂	semidirect
6. C₂²×#14555775	C₂	semidirect
7. C₂×#17984849	C₂	semidirect
8. C₂×2₋¹⁺⁴×C₁₃₉	C₂	semidirect
10. #17984889	C₂²	semidirect
15. #17984873	C₂²	semidirect
16. #17984872	C₂²	semidirect
17. #17984862	C₂²	semidirect
20. C₂×#14555775	C₂²	semidirect
24. #17984849	C₂²	semidirect
28. 2₋¹⁺⁴×C₁₃₉	C₂²	semidirect
115. C₂×C₁₃₉	#578	semidirect
118. C₁₃₉	C₂×#578	semidirect
121. C₂×2₋¹⁺⁴	D₁₃₉	semidirect
131. 2₋¹⁺⁴	C₂×D₁₃₉	semidirect
235. C₂	#22581768	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×#17984889	2
#22581768	2
C₂×#17984873	2
C₂×#17984872	2
C₂×#17984862	2
C₂²×#14555775	2
C₂×#17984849	2
C₂×2₋¹⁺⁴×C₁₃₉	2
C₂×#578	139

Subgroups — C2×#22581768 #31821169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22581768 #31821169