Subgroups — #31815248

Order 17792.

470 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581436	C₂	1	1	yes	maximal, characteristic
2	#22581431	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581060	C₂	1	2	yes	maximal, complemented
4	#22581054	C₂	1	2	yes	maximal, complemented
5	Q₁₆×Dic₁₃₉	C₂	1	2	yes	maximal, complemented
6	SD₁₆×Dic₁₃₉	C₂	1	2	yes	maximal, complemented
7	#22580995	C₂	1	1	yes	maximal, characteristic, complemented
8	#22580949	C₂	1	1	yes	maximal, characteristic
9	C₂×#17984766	C₂	1	1	yes	maximal, characteristic, complemented
10	#22580911	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#137×C₁₃₉	C₂	1	1	yes	maximal, characteristic
12	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
13	Q₈×Dic₁₃₉	—	2	1	no	—
14	Q₈×Dic₁₃₉	—	2	1	no	—
15	Q₈×Dic₁₃₉	C₂²	1	2	yes	—
16	Q₈×Dic₁₃₉	C₂²	1	2	yes	—
17	Q₈×Dic₁₃₉	C₂²	1	2	yes	—
18	D₄×Dic₁₃₉	—	2	2	no	—
19	D₄×Dic₁₃₉	C₂²	1	2	yes	—
20	#17984801	—	2	1	no	—
21	#17984801	—	2	2	no	—
22	#17984801	C₂²	1	1	yes	characteristic
23	#17984801	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃₉	—	2	1	no	—
25	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
26	#17984776	C₂²	1	2	yes	—
27	#17984774	C₂²	1	2	yes	—
28	#17984766	C₂²	1	2	yes	—
29	#17984766	C₂²	1	1	yes	characteristic
30	#17984766	C₂²	1	2	yes	—
31	#17984766	C₂²	1	1	yes	characteristic
32	#17984763	C₂²	1	1	yes	characteristic
33	#17984763	C₂²	1	1	yes	characteristic
34	#17984752	C₂²	1	2	yes	—
35	C₈×Dic₁₃₉	C₂²	1	2	yes	—
36	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic
37	C₂²×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
38	#137×C₁₃₉	C₄	1	8	yes	complemented
39	C₂×Q₁₆×C₁₃₉	C₂²	1	2	yes	—
40	C₂×SD₁₆×C₁₃₉	C₂²	1	2	yes	—
41	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
42	C₂²×Dic₁₃₉	—	2	2	no	—
43	C₂²×Dic₁₃₉	—	2	1	no	—
44	C₂×C₁₃₉⋊C₈	—	2	2	no	—
45	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
46	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
47	C₄⋊Dic₁₃₉	—	2	2	no	—
48	C₄⋊Dic₁₃₉	—	2	2	no	—
49	C₄⋊Dic₁₃₉	—	2	2	no	—
50	C₄⋊Dic₁₃₉	—	2	1	no	—
51	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
52	C₄⋊Dic₁₃₉	D₄	1	1	yes	characteristic
53	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
54	C₄⋊Dic₁₃₉	C₂³	1	2	yes	—
55	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
56	C₂²⋊Dic₁₃₉	—	2	2	no	—
57	C₂²⋊Dic₁₃₉	—	2	1	no	—
58	C₂²⋊Dic₁₃₉	D₄	1	2	yes	—
59	C₄×Dic₁₃₉	—	2	2	no	—
60	C₄×Dic₁₃₉	—	2	2	no	—
61	C₄×Dic₁₃₉	—	2	2	no	—
62	C₄×Dic₁₃₉	—	2	1	no	—
63	C₄×Dic₁₃₉	C₂³	1	2	yes	—
64	#41×C₁₃₉	—	2	2	no	—
65	#41×C₁₃₉	C₂×C₄	1	4	yes	—
66	C₂×Q₈×C₁₃₉	—	2	1	no	—
67	C₂×Q₈×C₁₃₉	—	2	2	no	—
68	C₂×Q₈×C₁₃₉	C₂×C₄	1	4	yes	complemented
69	C₂×Q₈×C₁₃₉	C₂³	1	2	yes	—
70	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
71	C₂×D₄×C₁₃₉	—	2	1	no	—
72	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
73	C₂²×C₄×C₁₃₉	—	2	1	no	—
74	C₂²×C₄×C₁₃₉	—	2	1	no	—
75	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
76	Q₁₆×C₁₃₉	C₂×C₄	1	8	yes	complemented
77	SD₁₆×C₁₃₉	C₂×C₄	1	8	yes	complemented
78	M₁₆×C₁₃₉	C₂×C₄	1	4	yes	complemented
79	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
80	C₂×Dic₁₃₉	—	2	1	no	—
81	C₂×Dic₁₃₉	—	2	1	no	—
82	C₂×Dic₁₃₉	—	2	2	no	—
83	C₂×Dic₁₃₉	—	8	2	no	—
84	C₂×Dic₁₃₉	—	8	1	no	—
85	C₂×Dic₁₃₉	C₂×D₄	1	2	yes	—
86	C₂×Dic₁₃₉	C₂×D₄	1	2	yes	—
87	C₁₃₉⋊C₈	—	2	1	no	—
88	C₁₃₉⋊C₈	—	2	1	no	—
89	C₂³×C₁₃₉	—	2	1	no	—
90	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
91	Q₈×C₁₃₉	—	2	4	no	—
92	Q₈×C₁₃₉	—	2	2	no	—
93	Q₈×C₁₃₉	—	2	1	no	—
94	Q₈×C₁₃₉	C₂²×C₄	1	4	yes	—
95	Q₈×C₁₃₉	C₂²×C₄	1	2	yes	—
96	D₄×C₁₃₉	—	2	4	no	—
97	D₄×C₁₃₉	—	2	1	no	—
98	D₄×C₁₃₉	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₁₃₉	—	2	4	no	—
100	C₂×C₄×C₁₃₉	—	2	1	no	—
101	C₂×C₄×C₁₃₉	—	2	2	no	—
102	C₂×C₄×C₁₃₉	—	2	4	no	—
103	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
104	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃₉	#41	1	2	yes	—
106	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
107	C₈×C₁₃₉	C₂²×C₄	1	4	yes	—
108	C₂²×C₁₃₉	—	2	1	no	—
109	C₂²×C₁₃₉	—	2	1	no	—
110	C₂²×C₁₃₉	—	2	2	no	—
111	C₂²×C₁₃₉	—	4	1	no	—
112	C₂²×C₁₃₉	—	4	1	no	—
113	C₂²×C₁₃₉	C₄×D₄	1	2	yes	—
114	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₁₃₉	—	2	2	no	—
116	C₄×C₁₃₉	—	2	4	no	—
117	C₄×C₁₃₉	C₄×D₄	1	2	yes	—
118	C₄×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
119	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
120	Dic₁₃₉	—	2	2	no	—
121	Dic₁₃₉	—	2	2	no	—
122	Dic₁₃₉	—	4	1	no	—
123	Dic₁₃₉	—	4	2	no	—
124	Dic₁₃₉	—	4	1	no	—
125	C₂×C₁₃₉	—	2	2	no	—
126	C₂×C₁₃₉	—	4	2	no	—
127	C₂×C₁₃₉	C₂×C₄×D₄	1	1	yes	characteristic
128	C₂×C₁₃₉	#577	1	1	yes	characteristic
129	C₂×C₁₃₉	#578	1	1	yes	characteristic
130	C₁₃₉	#2947	1	1	yes	characteristic, complemented, Sylow
131	#2947	—	139	1	no	maximal, Sylow
132	C₂×#137	D₁₃₉	1	1	yes	characteristic
133	#520	—	139	1	no	—
134	#519	—	139	1	no	—
135	#441	—	139	2	no	—
136	#440	—	139	2	no	—
137	C₄×Q₁₆	—	139	2	no	—
138	C₄×SD₁₆	—	139	2	no	—
139	#428	—	139	1	no	—
140	#416	—	139	1	no	—
141	C₂×#103	—	139	1	no	—
142	#405	—	139	1	no	—
143	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic
144	C₂²×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
145	#137	Dic₁₃₉	1	8	yes	complemented
146	C₂×Q₁₆	C₂×D₁₃₉	1	2	yes	—
147	C₂×SD₁₆	C₂×D₁₃₉	1	2	yes	—
148	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
149	C₂²×C₈	—	139	1	no	—
150	C₄×Q₈	—	278	1	no	—
151	C₄×Q₈	—	278	1	no	—
152	C₄×Q₈	—	139	2	no	—
153	C₄×Q₈	—	139	2	no	—
154	C₄×Q₈	—	139	2	no	—
155	C₄×D₄	—	278	2	no	—
156	C₄×D₄	—	139	2	no	—
157	#117	—	278	1	no	—
158	#117	—	278	2	no	—
159	#117	—	139	1	no	—
160	#117	—	139	1	no	—
161	C₂×C₄⋊C₄	—	278	1	no	—
162	C₂×C₄⋊C₄	—	139	1	no	—
163	#107	—	139	2	no	—
164	#106	—	139	2	no	—
165	#103	—	139	2	no	—
166	#103	—	139	2	no	—
167	#103	—	139	1	no	—
168	#103	—	139	1	no	—
169	#102	—	139	1	no	—
170	#102	—	139	1	no	—
171	#97	—	139	2	no	—
172	C₄×C₈	—	139	2	no	—
173	#41	—	2	2	no	—
174	#41	C₂×Dic₁₃₉	1	4	yes	—
175	C₂×Q₈	—	2	1	no	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	C₂×Dic₁₃₉	1	4	yes	complemented
178	C₂×Q₈	C₂²×D₁₃₉	1	2	yes	—
179	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
180	C₂×D₄	—	2	1	no	—
181	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	278	1	no	—
185	C₂²×C₄	—	278	2	no	—
186	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
187	Q₁₆	C₂×Dic₁₃₉	1	8	yes	complemented
188	SD₁₆	C₂×Dic₁₃₉	1	8	yes	complemented
189	M₁₆	C₂×Dic₁₃₉	1	4	yes	complemented
190	C₂×C₈	—	278	2	no	—
191	C₂×C₈	—	139	1	no	—
192	C₂×C₈	—	139	1	no	—
193	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
194	C₄⋊C₄	—	278	2	no	—
195	C₄⋊C₄	—	278	2	no	—
196	C₄⋊C₄	—	278	2	no	—
197	C₄⋊C₄	—	278	1	no	—
198	C₄⋊C₄	—	139	1	no	—
199	C₄⋊C₄	—	139	1	no	—
200	C₄⋊C₄	—	139	2	no	—
201	C₄⋊C₄	—	139	1	no	—
202	C₄⋊C₄	—	139	1	no	—
203	C₂²⋊C₄	—	278	1	no	—
204	C₂²⋊C₄	—	278	2	no	—
205	C₂²⋊C₄	—	139	2	no	—
206	C₄²	—	278	2	no	—
207	C₄²	—	278	2	no	—
208	C₄²	—	278	2	no	—
209	C₄²	—	278	1	no	—
210	C₄²	—	139	2	no	—
211	C₂³	—	2	1	no	—
212	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
213	Q₈	—	2	4	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	2	1	no	—
216	Q₈	C₂²×Dic₁₃₉	1	4	yes	—
217	Q₈	C₂²×Dic₁₃₉	1	2	yes	—
218	D₄	—	2	4	no	—
219	D₄	—	2	1	no	—
220	D₄	C₂²×Dic₁₃₉	1	2	yes	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	278	1	no	—
226	C₂×C₄	—	278	1	no	—
227	C₂×C₄	—	278	2	no	—
228	C₂×C₄	—	1112	2	no	—
229	C₂×C₄	—	1112	1	no	—
230	C₂×C₄	—	139	2	no	—
231	C₂×C₄	—	139	2	no	—
232	C₂×C₄	C₂²×Dic₁₃₉	1	2	yes	—
233	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
234	C₂×C₄	#14555784	1	2	yes	—
235	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
236	C₈	—	278	1	no	—
237	C₈	—	278	1	no	—
238	C₈	C₂²×Dic₁₃₉	1	4	yes	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	D₄×Dic₁₃₉	1	2	yes	—
245	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	4	no	—
248	C₄	—	278	2	no	—
249	C₄	—	278	2	no	—
250	C₄	—	556	2	no	—
251	C₄	—	556	1	no	—
252	C₄	—	556	1	no	—
253	C₄	D₄×Dic₁₃₉	1	2	yes	—
254	C₄	C₂³×Dic₁₃₉	1	1	yes	characteristic
255	C₄	C₂×#14555784	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	4	2	no	—
258	C₂	C₂×D₄×Dic₁₃₉	1	1	yes	characteristic
259	C₂	#22581762	1	1	yes	characteristic
260	C₂	#22581769	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator	C₄×C₁₃₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	C₂×#137×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×#41×C₁₃₉	C₂²
cosocle	C₂×C₄×C₁₃₉	C₂⁴
hypercenter	C₂×#137	D₁₃₉
hypocenter	C₁₃₉	#2947
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#137

Complements

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

N	H ≅ G/N	type
2. #22581431	C₂	semidirect
3. #22581060	C₂	semidirect
4. #22581054	C₂	semidirect
5. Q₁₆×Dic₁₃₉	C₂	semidirect
6. SD₁₆×Dic₁₃₉	C₂	semidirect
7. #22580995	C₂	semidirect
9. C₂×#17984766	C₂	semidirect
10. #22580911	C₂	semidirect
38. #137×C₁₃₉	C₄	semidirect
68. C₂×Q₈×C₁₃₉	C₂×C₄	semidirect
76. Q₁₆×C₁₃₉	C₂×C₄	semidirect
77. SD₁₆×C₁₃₉	C₂×C₄	semidirect
78. M₁₆×C₁₃₉	C₂×C₄	semidirect
130. C₁₃₉	#2947	semidirect
145. #137	Dic₁₃₉	semidirect
177. C₂×Q₈	C₂×Dic₁₃₉	semidirect
187. Q₁₆	C₂×Dic₁₃₉	semidirect
188. SD₁₆	C₂×Dic₁₃₉	semidirect
189. M₁₆	C₂×Dic₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#22581436	2
#22581431	2
#22581060	2
#22581054	2
Q₁₆×Dic₁₃₉	2
SD₁₆×Dic₁₃₉	2
#22580995	2
#22580949	2
C₂×#17984766	2
#22580911	2
C₂×#137×C₁₃₉	2
#2947	139