Subgroups — #31816215

Order 17792.

416 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984841	C₂	1	1	yes	maximal, characteristic, complemented
2	#41×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#22581246	C₂	1	2	yes	maximal, complemented
4	#22581241	C₂	1	2	yes	maximal, complemented
5	#22581236	C₂	1	2	yes	maximal, complemented
6	#22581231	C₂	1	2	yes	maximal, complemented
7	#22580995	C₂	1	1	yes	maximal, characteristic, complemented
8	#22580993	C₂	1	1	yes	maximal, characteristic, complemented
9	#22580976	C₂	1	1	yes	maximal, characteristic, complemented
10	#22580954	C₂	1	1	yes	maximal, characteristic
11	#417×C₁₃₉	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₂₇₈	—	2	1	no	—
13	C₂×#14555769	C₂²	1	1	yes	characteristic
14	#17984841	—	2	1	no	—
15	#17984841	—	2	1	no	—
16	#17984841	C₂²	1	2	yes	—
17	#17984841	C₂²	1	2	yes	—
18	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
19	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic
20	D₄×Dic₁₃₉	—	2	1	no	—
21	D₄×Dic₁₃₉	—	2	1	no	—
22	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
23	D₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
24	#17984801	—	2	1	no	—
25	#17984801	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
27	C₂×Dic₁₃₉⋊C₄	—	2	1	no	—
28	C₂×C₄×Dic₁₃₉	—	2	1	no	—
29	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
30	#17984776	C₂²	1	2	yes	—
31	#17984775	C₂²	1	2	yes	—
32	#17984774	C₂²	1	2	yes	—
33	#17984773	C₂²	1	2	yes	—
34	Dic₁₃₉⋊C₈	C₂²	1	2	yes	—
35	Dic₁₃₉⋊C₈	C₂²	1	2	yes	—
36	#17984766	C₂²	1	2	yes	—
37	#17984763	C₂²	1	2	yes	—
38	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic
39	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
41	#103×C₁₃₉	C₂²	1	2	yes	—
42	#102×C₁₃₉	C₂²	1	2	yes	—
43	C₂×Dic₂₇₈	—	2	2	no	—
44	C₂×Dic₂₇₈	—	2	2	no	—
45	C₂×Dic₂₇₈	—	4	4	no	—
46	C₂²×Dic₁₃₉	—	2	1	no	—
47	C₂²×Dic₁₃₉	—	2	1	no	—
48	C₂²×Dic₁₃₉	—	2	1	no	—
49	C₂²×Dic₁₃₉	D₄	1	1	yes	characteristic
50	C₂²×Dic₁₃₉	D₄	1	1	yes	characteristic
51	#14555769	—	2	2	no	—
52	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
53	C₄⋊Dic₁₃₉	—	2	1	no	—
54	C₄⋊Dic₁₃₉	—	2	1	no	—
55	C₄⋊Dic₁₃₉	—	2	1	no	—
56	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
57	C₄⋊Dic₁₃₉	C₂³	1	2	yes	—
58	C₄⋊Dic₁₃₉	C₂³	1	1	yes	characteristic
59	Dic₁₃₉⋊C₄	—	2	2	no	—
60	Dic₁₃₉⋊C₄	—	2	2	no	—
61	C₂²⋊Dic₁₃₉	—	2	1	no	—
62	C₂²⋊Dic₁₃₉	—	2	1	no	—
63	C₂²⋊Dic₁₃₉	—	2	1	no	—
64	C₄×Dic₁₃₉	—	2	1	no	—
65	C₄×Dic₁₃₉	—	2	1	no	—
66	C₄×Dic₁₃₉	—	2	1	no	—
67	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
68	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
69	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
70	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
71	#41×C₁₃₉	—	2	2	no	—
72	#41×C₁₃₉	Q₈	1	4	yes	complemented
73	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
74	C₂×D₄×C₁₃₉	—	2	1	no	—
75	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₁₃₉	—	2	1	no	—
77	C₂²×C₄×C₁₃₉	—	2	1	no	—
78	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
79	M₁₆×C₁₃₉	—	2	2	no	—
80	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
81	C₄⋊C₄×C₁₃₉	—	2	1	no	—
82	C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
83	Dic₂₇₈	—	4	4	no	—
84	Dic₂₇₈	—	4	4	no	—
85	C₂×Dic₁₃₉	—	2	2	no	—
86	C₂×Dic₁₃₉	—	2	2	no	—
87	C₂×Dic₁₃₉	—	2	1	no	—
88	C₂×Dic₁₃₉	—	2	1	no	—
89	C₂×Dic₁₃₉	—	2	2	no	—
90	C₂×Dic₁₃₉	—	4	2	no	—
91	C₂×Dic₁₃₉	—	4	2	no	—
92	C₂×Dic₁₃₉	—	4	2	no	—
93	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
94	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
95	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
96	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
97	C₁₃₉⋊C₈	—	2	2	no	—
98	C₂³×C₁₃₉	—	2	1	no	—
99	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
100	Q₈×C₁₃₉	—	2	1	no	—
101	Q₈×C₁₃₉	C₂×Q₈	1	2	yes	complemented
102	D₄×C₁₃₉	—	2	4	no	—
103	D₄×C₁₃₉	—	2	1	no	—
104	D₄×C₁₃₉	C₂×Q₈	1	2	yes	complemented
105	C₂×C₄×C₁₃₉	—	2	1	no	—
106	C₂×C₄×C₁₃₉	—	2	4	no	—
107	C₂×C₄×C₁₃₉	—	2	2	no	—
108	C₂×C₄×C₁₃₉	—	4	2	no	—
109	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
111	C₂×C₄×C₁₃₉	#41	1	2	yes	—
112	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
113	C₈×C₁₃₉	—	2	2	no	—
114	C₂²×C₁₃₉	—	2	1	no	—
115	C₂²×C₁₃₉	—	2	1	no	—
116	C₂²×C₁₃₉	—	2	2	no	—
117	C₂²×C₁₃₉	—	4	1	no	—
118	C₂²×C₁₃₉	—	4	1	no	—
119	C₂²×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
120	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
121	C₄×C₁₃₉	—	2	2	no	—
122	C₄×C₁₃₉	—	4	2	no	—
123	C₄×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
124	C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
125	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
126	Dic₁₃₉	—	2	1	no	—
127	Dic₁₃₉	—	2	1	no	—
128	Dic₁₃₉	—	4	1	no	—
129	Dic₁₃₉	—	4	2	no	—
130	Dic₁₃₉	—	4	1	no	—
131	Dic₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
132	Dic₁₃₉	#137	1	2	yes	—
133	C₂×C₁₃₉	—	2	2	no	—
134	C₂×C₁₃₉	—	4	2	no	—
135	C₂×C₁₃₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
136	C₂×C₁₃₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₁₃₉	C₂×#137	1	1	yes	characteristic
138	C₁₃₉	#3079	1	1	yes	characteristic, complemented, Sylow
139	#3079	—	139	1	no	maximal, Sylow
140	C₂×C₄⋊Q₈	—	139	1	no	—
141	C₄×#41	—	139	1	no	—
142	#477	—	139	2	no	—
143	#476	—	139	2	no	—
144	#475	—	139	2	no	—
145	#474	—	139	2	no	—
146	#428	—	139	1	no	—
147	#428	—	139	1	no	—
148	#423	—	139	1	no	—
149	#417	—	139	1	no	—
150	#417	D₁₃₉	1	1	yes	characteristic
151	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic
152	C₂²×Q₈	—	278	1	no	—
153	C₂×M₁₆	—	139	1	no	—
154	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
155	C₄⋊Q₈	—	278	1	no	—
156	C₄⋊Q₈	—	278	1	no	—
157	C₄⋊Q₈	—	139	2	no	—
158	C₄⋊Q₈	—	139	2	no	—
159	C₄×Q₈	—	139	1	no	—
160	C₄×Q₈	—	139	1	no	—
161	C₄×D₄	—	278	1	no	—
162	C₄×D₄	—	278	1	no	—
163	C₄×D₄	—	139	1	no	—
164	C₄×D₄	—	139	1	no	—
165	#117	—	278	1	no	—
166	#117	—	139	1	no	—
167	C₂×C₄⋊C₄	—	278	1	no	—
168	C₂×C₄⋊C₄	—	139	1	no	—
169	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
170	C₂×C₄²	—	278	1	no	—
171	C₂×C₄²	—	139	1	no	—
172	#107	—	139	2	no	—
173	#107	—	139	2	no	—
174	#106	—	139	2	no	—
175	#106	—	139	2	no	—
176	C₄⋊C₈	—	139	2	no	—
177	C₄⋊C₈	—	139	2	no	—
178	#103	—	139	2	no	—
179	#103	C₂×D₁₃₉	1	2	yes	—
180	#102	—	139	2	no	—
181	#102	C₂×D₁₃₉	1	2	yes	—
182	#41	—	2	2	no	—
183	#41	Dic₂₇₈	1	4	yes	complemented
184	C₂×Q₈	—	278	2	no	—
185	C₂×Q₈	—	278	2	no	—
186	C₂×Q₈	—	556	4	no	—
187	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	278	1	no	—
193	C₂²×C₄	—	278	1	no	—
194	C₂²×C₄	—	278	1	no	—
195	C₂²×C₄	—	139	1	no	—
196	C₂²×C₄	—	139	1	no	—
197	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
198	M₁₆	—	2	2	no	—
199	M₁₆	—	278	2	no	—
200	C₂×C₈	—	139	2	no	—
201	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	—	278	2	no	—
204	C₄⋊C₄	—	278	2	no	—
205	C₄⋊C₄	—	278	1	no	—
206	C₄⋊C₄	—	278	1	no	—
207	C₄⋊C₄	—	278	1	no	—
208	C₄⋊C₄	—	139	2	no	—
209	C₄⋊C₄	—	139	1	no	—
210	C₄⋊C₄	—	139	1	no	—
211	C₄⋊C₄	C₂²×D₁₃₉	1	2	yes	—
212	C₂²⋊C₄	—	278	1	no	—
213	C₂²⋊C₄	—	278	1	no	—
214	C₂²⋊C₄	—	278	1	no	—
215	C₄²	—	278	1	no	—
216	C₄²	—	278	1	no	—
217	C₄²	—	278	1	no	—
218	C₄²	—	139	1	no	—
219	C₄²	—	139	1	no	—
220	C₄²	—	139	1	no	—
221	C₄²	—	139	1	no	—
222	C₂³	—	2	1	no	—
223	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
224	Q₈	—	2	1	no	—
225	Q₈	—	556	4	no	—
226	Q₈	—	556	4	no	—
227	Q₈	C₂×Dic₂₇₈	1	2	yes	complemented
228	D₄	—	2	4	no	—
229	D₄	—	2	1	no	—
230	D₄	C₂×Dic₂₇₈	1	2	yes	complemented
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	278	2	no	—
236	C₂×C₄	—	278	2	no	—
237	C₂×C₄	—	278	1	no	—
238	C₂×C₄	—	278	1	no	—
239	C₂×C₄	—	278	2	no	—
240	C₂×C₄	—	556	2	no	—
241	C₂×C₄	—	556	2	no	—
242	C₂×C₄	—	556	2	no	—
243	C₂×C₄	—	139	1	no	—
244	C₂×C₄	—	139	1	no	—
245	C₂×C₄	—	139	1	no	—
246	C₂×C₄	—	139	1	no	—
247	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
248	C₂×C₄	C₂×Dic₂₇₈	1	2	yes	—
249	C₂×C₄	#14555784	1	2	yes	—
250	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	278	2	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	1	no	—
257	C₂²	—	4	1	no	—
258	C₂²	#17984819	1	2	yes	—
259	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	278	1	no	—
263	C₄	—	278	1	no	—
264	C₄	—	556	2	no	—
265	C₄	—	556	1	no	—
266	C₄	—	556	1	no	—
267	C₄	—	139	2	no	—
268	C₄	—	139	2	no	—
269	C₄	#17984819	1	2	yes	—
270	C₄	C₂²×Dic₂₇₈	1	1	yes	characteristic
271	C₄	C₂×#14555784	1	1	yes	characteristic
272	C₂	—	2	2	no	—
273	C₂	—	4	2	no	—
274	C₂	C₂×#17984819	1	1	yes	characteristic
275	C₂	AGL(1,ℤ/8)×D₁₃₉	1	1	yes	characteristic
276	C₂	#137×D₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#417×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×#41×C₁₃₉	C₂²
hypercenter	#417	D₁₃₉
hypocenter	C₁₃₉	#3079
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #417

Complements

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

N	H ≅ G/N	type
1. C₂×#17984841	C₂	semidirect
3. #22581246	C₂	semidirect
4. #22581241	C₂	semidirect
5. #22581236	C₂	semidirect
6. #22581231	C₂	semidirect
7. #22580995	C₂	semidirect
8. #22580993	C₂	semidirect
9. #22580976	C₂	semidirect
72. #41×C₁₃₉	Q₈	semidirect
101. Q₈×C₁₃₉	C₂×Q₈	semidirect
104. D₄×C₁₃₉	C₂×Q₈	semidirect
138. C₁₃₉	#3079	semidirect
183. #41	Dic₂₇₈	semidirect
227. Q₈	C₂×Dic₂₇₈	semidirect
230. D₄	C₂×Dic₂₇₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984841	2
#41×Dic₁₃₉	2
#22581246	2
#22581241	2
#22581236	2
#22581231	2
#22580995	2
#22580993	2
#22580976	2
#22580954	2
#417×C₁₃₉	2
#3079	139