Subgroups — C₂×#22581766 #31821167

Order 17792.

1408 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984890	C₂	1	2	yes	maximal, complemented
2	#22581766	C₂	1	16	yes	maximal, direct product factor
3	C₂×#17984874	C₂	1	4	yes	maximal, complemented
4	C₂×#17984873	C₂	1	2	yes	maximal, complemented
5	C₂×#17984865	C₂	1	2	yes	maximal, complemented
6	C₂×#17984864	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Q₁₆×D₁₃₉	C₂	1	2	yes	maximal, complemented
8	C₂×#17984848	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₁₆×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#17984890	—	2	8	no	—
11	#17984890	C₂²	1	16	yes	complemented
12	C₂×#14555786	—	2	2	no	—
13	C₂×#14555786	C₂²	1	4	yes	—
14	C₂×#14555785	—	2	4	no	—
15	C₂×#14555785	C₂²	1	2	yes	—
16	C₂×#14555785	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₁₃₉	—	2	2	no	—
18	C₂×Q₈×D₁₃₉	C₂²	1	4	yes	—
19	#17984874	C₂²	1	32	yes	complemented
20	#17984873	C₂²	1	16	yes	complemented
21	#17984865	C₂²	1	16	yes	complemented
22	#17984864	C₂²	1	8	yes	complemented
23	C₂×Dic₅₅₆	C₂²	1	1	yes	characteristic
24	C₂×#14555775	C₂²	1	4	yes	—
25	Q₁₆×D₁₃₉	C₂²	1	16	yes	complemented
26	C₂×#14555774	C₂²	1	2	yes	—
27	C₂×#14555772	C₂²	1	4	yes	—
28	C₂×D₁₁₁₂	C₂²	1	1	yes	characteristic
29	#17984848	C₂²	1	8	yes	complemented
30	C₂×#14555769	C₂²	1	1	yes	characteristic
31	C₂×#14555768	C₂²	1	2	yes	—
32	C₂×C₈×D₁₃₉	C₂²	1	2	yes	—
33	C₂²×Q₈×C₁₃₉	C₂²	1	2	yes	—
34	C₂×Q₁₆×C₁₃₉	C₂²	1	2	yes	—
35	C₂×Q₁₆×C₁₃₉	C₂²	1	2	yes	—
36	C₂×Q₁₆×C₁₃₉	C₂²	1	8	yes	complemented
37	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
38	#14555786	—	2	16	no	—
39	#14555786	—	2	8	no	—
40	#14555786	C₂³	1	16	yes	—
41	#14555785	—	2	32	no	—
42	#14555785	—	2	4	no	—
43	#14555785	D₄	1	4	yes	—
44	#14555785	C₂³	1	8	yes	—
45	#14555785	C₂³	1	4	yes	—
46	C₂×Dic₂₇₈	—	2	4	no	—
47	C₂×Dic₂₇₈	D₄	1	1	yes	characteristic
48	C₂×Dic₂₇₈	C₂³	1	2	yes	—
49	Q₈×D₁₃₉	—	2	16	no	—
50	Q₈×D₁₃₉	—	2	8	no	—
51	Q₈×D₁₃₉	C₂³	1	16	yes	—
52	C₂×D₅₅₆	—	2	4	no	—
53	C₂×D₅₅₆	D₄	1	1	yes	characteristic
54	C₂×D₅₅₆	C₂³	1	2	yes	—
55	C₂×C₂²⋊D₁₃₉	—	2	2	no	—
56	C₂×C₂²⋊D₁₃₉	D₄	1	2	yes	—
57	C₂×C₄×D₁₃₉	—	2	4	no	—
58	C₂×C₄×D₁₃₉	—	2	4	no	—
59	C₂×C₄×D₁₃₉	—	2	2	no	—
60	C₂×C₄×D₁₃₉	C₂³	1	2	yes	—
61	Dic₅₅₆	C₂³	1	4	yes	—
62	#14555775	C₂³	1	16	yes	—
63	#14555774	C₂³	1	8	yes	—
64	#14555772	C₂³	1	16	yes	—
65	D₁₁₁₂	C₂³	1	4	yes	—
66	#14555769	C₂³	1	4	yes	—
67	#14555768	C₂³	1	8	yes	—
68	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
69	C₈×D₁₃₉	C₂³	1	8	yes	—
70	C₂×Q₈×C₁₃₉	—	2	2	no	—
71	C₂×Q₈×C₁₃₉	—	2	4	no	—
72	C₂×Q₈×C₁₃₉	C₂³	1	4	yes	—
73	C₂×Q₈×C₁₃₉	C₂³	1	8	yes	—
74	C₂²×C₄×C₁₃₉	—	2	2	no	—
75	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
76	Q₁₆×C₁₃₉	C₂³	1	8	yes	—
77	Q₁₆×C₁₃₉	C₂³	1	8	yes	—
78	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₁₃₉	C₂³	1	4	yes	—
81	C₂²×D₁₃₉	—	2	2	no	—
82	C₂²×D₁₃₉	C₂×D₄	1	2	yes	—
83	Dic₂₇₈	—	2	2	no	—
84	Dic₂₇₈	—	2	16	no	—
85	Dic₂₇₈	C₂×D₄	1	4	yes	—
86	Dic₂₇₈	C₂⁴	1	4	yes	—
87	D₅₅₆	—	2	16	no	—
88	D₅₅₆	—	2	2	no	—
89	D₅₅₆	C₂×D₄	1	4	yes	—
90	D₅₅₆	C₂⁴	1	4	yes	—
91	C₂²⋊D₁₃₉	—	2	8	no	—
92	C₂²⋊D₁₃₉	C₂×D₄	1	8	yes	—
93	C₂×Dic₁₃₉	—	2	2	no	—
94	C₂×Dic₁₃₉	C₂×D₄	1	2	yes	—
95	C₄×D₁₃₉	—	2	16	no	—
96	C₄×D₁₃₉	—	2	8	no	—
97	C₄×D₁₃₉	—	2	16	no	—
98	C₄×D₁₃₉	C₂×D₄	1	4	yes	—
99	C₄×D₁₃₉	C₂⁴	1	4	yes	—
100	C₁₃₉⋊C₈	C₂⁴	1	4	yes	—
101	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
102	Q₈×C₁₃₉	—	2	8	no	—
103	Q₈×C₁₃₉	—	2	4	no	—
104	Q₈×C₁₃₉	C₂⁴	1	8	yes	—
105	C₂×C₄×C₁₃₉	—	2	4	no	—
106	C₂×C₄×C₁₃₉	—	2	8	no	—
107	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
110	C₂×C₄×C₁₃₉	C₂⁴	1	2	yes	—
111	C₈×C₁₃₉	C₂⁴	1	2	yes	—
112	C₈×C₁₃₉	C₂⁴	1	2	yes	—
113	C₂²×C₁₃₉	—	2	2	no	—
114	C₂²×C₁₃₉	C₂²×D₄	1	2	yes	—
115	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
116	C₂×D₁₃₉	—	2	4	no	—
117	C₂×D₁₃₉	—	2	4	no	—
118	C₂×D₁₃₉	—	4	4	no	—
119	C₂×D₁₃₉	C₂²×D₄	1	4	yes	—
120	C₄×C₁₃₉	—	2	8	no	—
121	C₄×C₁₃₉	C₂²×D₄	1	2	yes	—
122	C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
123	C₄×C₁₃₉	C₂⁵	1	1	yes	characteristic
124	Dic₁₃₉	—	2	4	no	—
125	Dic₁₃₉	C₂²×D₄	1	4	yes	—
126	C₂×C₁₃₉	—	2	2	no	—
127	C₂×C₁₃₉	#578	1	2	yes	complemented
128	C₂×C₁₃₉	C₂³×D₄	1	1	yes	characteristic
129	D₁₃₉	—	2	4	no	—
130	D₁₃₉	—	4	4	no	—
131	C₁₃₉	C₂×#578	1	1	yes	characteristic, complemented, Sylow
132	C₂×#578	—	139	1	no	maximal, Sylow
133	C₂×2₋¹⁺⁴	—	139	2	no	—
134	#578	—	139	16	no	—
135	C₂×#137	—	139	2	no	—
136	C₂×#137	—	139	4	no	—
137	C₂×#135	—	139	2	no	—
138	C₂×#135	—	139	1	no	—
139	C₂²×Q₁₆	—	139	2	no	—
140	C₂²×Q₁₆	D₁₃₉	1	1	yes	characteristic, complemented
141	C₂×#131	—	139	1	no	—
142	2₋¹⁺⁴	—	278	8	no	—
143	2₋¹⁺⁴	—	139	16	no	—
144	C₂×#41	—	278	4	no	—
145	C₂×#41	—	278	2	no	—
146	C₂×#41	—	139	4	no	—
147	C₂×#41	—	139	2	no	—
148	C₂×#41	—	139	1	no	—
149	C₂²×Q₈	—	278	2	no	—
150	C₂²×Q₈	—	139	4	no	—
151	C₂²×Q₈	C₂×D₁₃₉	1	2	yes	—
152	#137	—	139	16	no	—
153	#137	—	139	32	no	—
154	#135	—	139	16	no	—
155	#135	—	139	8	no	—
156	C₂×Q₁₆	—	139	4	no	—
157	C₂×Q₁₆	—	139	16	no	—
158	C₂×Q₁₆	—	139	1	no	—
159	C₂×Q₁₆	C₂×D₁₃₉	1	2	yes	—
160	C₂×Q₁₆	C₂×D₁₃₉	1	2	yes	—
161	C₂×Q₁₆	C₂×D₁₃₉	1	8	yes	complemented
162	C₂×SD₁₆	—	139	4	no	—
163	C₂×SD₁₆	—	139	2	no	—
164	C₂×D₈	—	139	1	no	—
165	#131	—	139	8	no	—
166	C₂×M₁₆	—	139	1	no	—
167	C₂×M₁₆	—	139	2	no	—
168	C₂²×C₈	—	139	2	no	—
169	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
170	#41	—	278	32	no	—
171	#41	—	278	16	no	—
172	#41	—	278	8	no	—
173	#41	—	278	4	no	—
174	#41	—	139	16	no	—
175	#41	—	139	8	no	—
176	#41	—	139	4	no	—
177	#41	—	139	4	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	278	4	no	—
181	C₂×Q₈	—	278	16	no	—
182	C₂×Q₈	—	278	8	no	—
183	C₂×Q₈	—	139	16	no	—
184	C₂×Q₈	—	139	2	no	—
185	C₂×Q₈	—	139	1	no	—
186	C₂×Q₈	C₂²×D₁₃₉	1	4	yes	—
187	C₂×Q₈	C₂²×D₁₃₉	1	8	yes	—
188	C₂×D₄	—	278	4	no	—
189	C₂×D₄	—	278	2	no	—
190	C₂×D₄	—	139	2	no	—
191	C₂×D₄	—	139	1	no	—
192	C₂×D₄	—	139	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	278	4	no	—
195	C₂²×C₄	—	278	4	no	—
196	C₂²×C₄	—	278	2	no	—
197	C₂²×C₄	—	139	2	no	—
198	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
199	Q₁₆	—	139	16	no	—
200	Q₁₆	—	139	4	no	—
201	Q₁₆	C₂²×D₁₃₉	1	8	yes	—
202	Q₁₆	C₂²×D₁₃₉	1	8	yes	—
203	SD₁₆	—	139	16	no	—
204	SD₁₆	—	139	8	no	—
205	D₈	—	139	4	no	—
206	M₁₆	—	139	4	no	—
207	M₁₆	—	139	8	no	—
208	C₂×C₈	—	139	2	no	—
209	C₂×C₈	—	139	8	no	—
210	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
211	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
212	C₂×C₈	C₂²×D₁₃₉	1	4	yes	—
213	C₂³	—	278	2	no	—
214	C₂³	—	139	2	no	—
215	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
216	Q₈	—	2	8	no	—
217	Q₈	—	2	4	no	—
218	Q₈	—	278	16	no	—
219	Q₈	—	278	2	no	—
220	Q₈	—	139	4	no	—
221	Q₈	—	139	4	no	—
222	Q₈	C₂³×D₁₃₉	1	8	yes	—
223	D₄	—	278	16	no	—
224	D₄	—	278	8	no	—
225	D₄	—	278	2	no	—
226	D₄	—	139	8	no	—
227	D₄	—	139	4	no	—
228	D₄	—	139	4	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	278	16	no	—
232	C₂×C₄	—	278	16	no	—
233	C₂×C₄	—	278	2	no	—
234	C₂×C₄	—	278	8	no	—
235	C₂×C₄	—	139	2	no	—
236	C₂×C₄	—	139	4	no	—
237	C₂×C₄	—	139	4	no	—
238	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
239	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
240	C₂×C₄	C₂³×D₁₃₉	1	2	yes	—
241	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
242	C₈	—	139	4	no	—
243	C₈	C₂³×D₁₃₉	1	2	yes	—
244	C₈	C₂³×D₁₃₉	1	2	yes	—
245	C₂²	—	2	2	no	—
246	C₂²	—	278	4	no	—
247	C₂²	—	278	4	no	—
248	C₂²	—	556	4	no	—
249	C₂²	—	139	4	no	—
250	C₂²	C₂×D₄×D₁₃₉	1	2	yes	—
251	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
252	C₄	—	2	8	no	—
253	C₄	—	278	4	no	—
254	C₄	—	139	4	no	—
255	C₄	C₂×D₄×D₁₃₉	1	2	yes	—
256	C₄	C₂×D₄×D₁₃₉	1	1	yes	characteristic
257	C₄	C₂⁴×D₁₃₉	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	278	4	no	—
260	C₂	—	556	4	no	—
261	C₂	#22581766	1	2	yes	direct product factor
262	C₂	C₂²×D₄×D₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₄×C₁₃₉	C₂⁵
Frattini	C₄	C₂⁴×D₁₃₉
Fitting	C₂²×Q₁₆×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
hypercenter	C₂²×Q₁₆	D₁₃₉
hypocenter	C₁₃₉	C₂×#578
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#17984890	C₂	semidirect
2. #22581766	C₂	direct (N × H)
3. C₂×#17984874	C₂	semidirect
4. C₂×#17984873	C₂	semidirect
5. C₂×#17984865	C₂	semidirect
6. C₂×#17984864	C₂	semidirect
7. C₂×Q₁₆×D₁₃₉	C₂	semidirect
8. C₂×#17984848	C₂	semidirect
9. C₂²×Q₁₆×C₁₃₉	C₂	semidirect
11. #17984890	C₂²	semidirect
19. #17984874	C₂²	semidirect
20. #17984873	C₂²	semidirect
21. #17984865	C₂²	semidirect
22. #17984864	C₂²	semidirect
25. Q₁₆×D₁₃₉	C₂²	semidirect
29. #17984848	C₂²	semidirect
36. C₂×Q₁₆×C₁₃₉	C₂²	semidirect
127. C₂×C₁₃₉	#578	semidirect
131. C₁₃₉	C₂×#578	semidirect
140. C₂²×Q₁₆	D₁₃₉	semidirect
161. C₂×Q₁₆	C₂×D₁₃₉	semidirect
261. C₂	#22581766	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×#17984890	2
#22581766	2
C₂×#17984874	2
C₂×#17984873	2
C₂×#17984865	2
C₂×#17984864	2
C₂×Q₁₆×D₁₃₉	2
C₂×#17984848	2
C₂²×Q₁₆×C₁₃₉	2
C₂×#578	139

Subgroups — C2×#22581766 #31821167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22581766 #31821167