Subgroups — #31809368

Order 17792.

464 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#22580860	C₂	1	1	yes	maximal, characteristic
3	#22580838	C₂	1	2	yes	maximal
4	#22580836	C₂	1	2	yes	maximal
5	#22580820	C₂	1	2	yes	maximal
6	#22580818	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₄×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₁₃₉⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₄×C₄⋊Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
10	#22580798	C₂	1	2	yes	maximal
11	#391×C₁₃₉	C₂	1	1	yes	maximal, characteristic
12	Q₈×Dic₁₃₉	C₄	1	8	yes	complemented
13	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
16	C₂×Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₁₃₉	C₂²	1	2	yes	—
19	C₂×C₄×Dic₁₃₉	C₂²	1	2	yes	—
20	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃₉	C₂²	1	2	yes	—
22	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
23	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
24	C₁₃₉⋊#95	C₂²	1	2	yes	—
25	C₁₃₉⋊#95	C₂²	1	2	yes	—
26	C₁₃₉⋊#95	C₂²	1	2	yes	—
27	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
28	C₁₃₉⋊#95	C₂²	1	2	yes	—
29	C₁₃₉⋊#95	C₂²	1	2	yes	—
30	C₁₃₉⋊#95	C₂²	1	2	yes	—
31	C₂²×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
34	#95×C₁₃₉	C₂²	1	2	yes	—
35	#95×C₁₃₉	C₂²	1	2	yes	—
36	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
37	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
38	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
39	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
42	C₄⋊Dic₁₃₉	—	2	2	no	—
43	C₄⋊Dic₁₃₉	C₂×C₄	1	8	yes	—
44	C₄⋊Dic₁₃₉	C₂×C₄	1	4	yes	—
45	Dic₁₃₉⋊C₄	—	2	2	no	—
46	Dic₁₃₉⋊C₄	—	2	2	no	—
47	C₄×Dic₁₃₉	—	2	4	no	—
48	C₄×Dic₁₃₉	—	2	2	no	—
49	C₄×Dic₁₃₉	—	2	4	no	—
50	C₄×Dic₁₃₉	C₂×C₄	1	8	yes	—
51	C₄×Dic₁₃₉	C₂×C₄	1	4	yes	—
52	C₂×Q₈×C₁₃₉	—	2	2	no	—
53	C₂×Q₈×C₁₃₉	—	2	2	no	—
54	C₂×Q₈×C₁₃₉	C₂×C₄	1	4	yes	—
55	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
56	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₁₃₉	—	2	2	no	—
61	C₄²×C₁₃₉	—	2	2	no	—
62	C₂×Dic₁₃₉	—	2	2	no	—
63	C₂×Dic₁₃₉	—	2	1	no	—
64	C₂×Dic₁₃₉	—	2	2	no	—
65	C₂×Dic₁₃₉	—	2	1	no	—
66	C₂×Dic₁₃₉	—	2	1	no	—
67	C₂×Dic₁₃₉	—	2	1	no	—
68	C₂×Dic₁₃₉	—	2	1	no	—
69	C₂×Dic₁₃₉	—	2	1	no	—
70	C₂×Dic₁₃₉	—	2	1	no	—
71	C₂×Dic₁₃₉	—	2	2	no	—
72	C₂×Dic₁₃₉	—	2	1	no	—
73	C₂×Dic₁₃₉	—	2	2	no	—
74	C₂×Dic₁₃₉	—	2	1	no	—
75	C₂×Dic₁₃₉	—	2	2	no	—
76	C₂×Dic₁₃₉	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₃₉	C₂²×C₄	1	4	yes	—
78	C₂×Dic₁₃₉	C₂²×C₄	1	2	yes	—
79	C₂×Dic₁₃₉	#41	1	2	yes	—
80	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
81	Q₈×C₁₃₉	—	2	8	no	—
82	C₂×C₄×C₁₃₉	—	2	1	no	—
83	C₂×C₄×C₁₃₉	—	2	2	no	—
84	C₂×C₄×C₁₃₉	—	2	2	no	—
85	C₂×C₄×C₁₃₉	—	2	1	no	—
86	C₂×C₄×C₁₃₉	—	2	1	no	—
87	C₂×C₄×C₁₃₉	—	2	2	no	—
88	C₂×C₄×C₁₃₉	—	2	2	no	—
89	C₂×C₄×C₁₃₉	—	2	1	no	—
90	C₂×C₄×C₁₃₉	—	2	2	no	—
91	C₂×C₄×C₁₃₉	—	2	1	no	—
92	C₂×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
93	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₃₉	#41	1	2	yes	—
95	C₂×C₄×C₁₃₉	#41	1	2	yes	—
96	C₂×C₄×C₁₃₉	#41	1	2	yes	—
97	C₂²×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
98	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
99	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
100	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
101	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
102	C₂²×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
103	C₂²×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
104	C₄×C₁₃₉	—	2	4	no	—
105	C₄×C₁₃₉	—	2	2	no	—
106	C₄×C₁₃₉	—	4	1	no	—
107	C₄×C₁₃₉	—	4	2	no	—
108	C₄×C₁₃₉	#117	1	4	yes	—
109	Dic₁₃₉	—	2	4	no	—
110	Dic₁₃₉	—	2	2	no	—
111	Dic₁₃₉	—	2	2	no	—
112	Dic₁₃₉	—	2	2	no	—
113	Dic₁₃₉	—	4	1	no	—
114	Dic₁₃₉	—	4	2	no	—
115	C₂×C₁₃₉	C₂×#117	1	1	yes	characteristic
116	C₂×C₁₃₉	C₄×#41	1	1	yes	characteristic
117	C₂×C₁₃₉	#519	1	1	yes	characteristic
118	C₂×C₁₃₉	#552	1	1	yes	characteristic
119	C₂×C₁₃₉	#552	1	1	yes	characteristic
120	C₂×C₁₃₉	#556	1	1	yes	characteristic
121	C₂×C₁₃₉	#556	1	1	yes	characteristic
122	C₁₃₉	#2356	1	1	yes	characteristic, complemented, Sylow
123	#2356	—	139	1	no	maximal, Sylow
124	C₂×C₄×Q₈	—	139	1	no	—
125	#391	—	139	1	no	—
126	#391	D₁₃₉	1	1	yes	characteristic
127	#387	—	139	2	no	—
128	#387	—	139	2	no	—
129	#383	—	139	2	no	—
130	#382	—	139	1	no	—
131	C₄×C₄⋊C₄	—	139	1	no	—
132	C₄×C₄⋊C₄	—	139	1	no	—
133	C₄×C₄⋊C₄	—	139	1	no	—
134	#376	—	139	2	no	—
135	C₂²×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
136	C₄×Q₈	—	139	8	no	—
137	C₂×C₄⋊C₄	—	139	1	no	—
138	C₂×C₄⋊C₄	—	139	1	no	—
139	C₂×C₄⋊C₄	—	139	1	no	—
140	C₂×C₄⋊C₄	—	139	1	no	—
141	C₂×C₄⋊C₄	—	139	2	no	—
142	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
143	C₂×C₄²	—	139	2	no	—
144	C₂×C₄²	—	139	2	no	—
145	C₂×C₄²	—	139	1	no	—
146	C₂×C₄²	—	139	2	no	—
147	C₂×C₄²	—	139	1	no	—
148	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
149	#95	—	139	1	no	—
150	#95	—	139	2	no	—
151	#95	—	139	2	no	—
152	#95	—	139	1	no	—
153	#95	—	139	2	no	—
154	#95	—	139	2	no	—
155	#95	—	139	2	no	—
156	#95	—	139	2	no	—
157	#95	C₂×D₁₃₉	1	2	yes	—
158	#95	C₂×D₁₃₉	1	2	yes	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	C₄×D₁₃₉	1	4	yes	—
162	C₂²×C₄	—	139	1	no	—
163	C₂²×C₄	—	139	2	no	—
164	C₂²×C₄	—	139	1	no	—
165	C₂²×C₄	—	139	1	no	—
166	C₂²×C₄	—	139	2	no	—
167	C₂²×C₄	—	139	1	no	—
168	C₂²×C₄	C₂²×D₁₃₉	1	2	yes	—
169	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₁₃₉	1	2	yes	—
171	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
172	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	278	2	no	—
175	C₄⋊C₄	—	278	2	no	—
176	C₄⋊C₄	—	278	2	no	—
177	C₄⋊C₄	—	139	8	no	—
178	C₄⋊C₄	—	139	4	no	—
179	C₄²	—	2	2	no	—
180	C₄²	—	278	4	no	—
181	C₄²	—	278	2	no	—
182	C₄²	—	278	4	no	—
183	C₄²	—	139	8	no	—
184	C₄²	—	139	4	no	—
185	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
186	Q₈	—	2	8	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	278	2	no	—
198	C₂×C₄	—	278	1	no	—
199	C₂×C₄	—	278	1	no	—
200	C₂×C₄	—	278	2	no	—
201	C₂×C₄	—	278	1	no	—
202	C₂×C₄	—	278	1	no	—
203	C₂×C₄	—	278	1	no	—
204	C₂×C₄	—	278	1	no	—
205	C₂×C₄	—	278	1	no	—
206	C₂×C₄	—	278	2	no	—
207	C₂×C₄	—	278	2	no	—
208	C₂×C₄	—	278	1	no	—
209	C₂×C₄	—	278	1	no	—
210	C₂×C₄	—	278	2	no	—
211	C₂×C₄	—	139	2	no	—
212	C₂×C₄	—	139	4	no	—
213	C₂×C₄	—	139	2	no	—
214	C₂×C₄	—	139	2	no	—
215	C₂×C₄	C₂×C₄×D₁₃₉	1	4	yes	—
216	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
217	C₂×C₄	#14555784	1	2	yes	—
218	C₂×C₄	#14555784	1	2	yes	—
219	C₂×C₄	#14555786	1	2	yes	—
220	C₂²	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
221	C₂²	#41×D₁₃₉	1	1	yes	characteristic
222	C₂²	C₂×#14555784	1	1	yes	characteristic
223	C₂²	C₂×#14555784	1	1	yes	characteristic
224	C₂²	C₂×#14555786	1	1	yes	characteristic
225	C₂²	#17984889	1	1	yes	characteristic
226	C₂²	#17984890	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	4	no	—
229	C₄	—	4	1	no	—
230	C₄	—	4	2	no	—
231	C₄	—	278	2	no	—
232	C₄	—	278	4	no	—
233	C₄	—	278	2	no	—
234	C₄	—	278	2	no	—
235	C₄	—	556	1	no	—
236	C₄	—	556	2	no	—
237	C₄	#17984798	1	4	yes	—
238	C₂	C₂×#17984798	1	1	yes	characteristic
239	C₂	C₄×#14555786	1	1	yes	characteristic
240	C₂	#22581429	1	1	yes	characteristic
241	C₂	#22581637	1	1	yes	characteristic
242	C₂	#22581645	1	1	yes	characteristic
243	C₂	#22581672	1	1	yes	characteristic
244	C₂	#22581677	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃₉
commutator	C₂²×C₁₃₉	C₂³×C₄
Fitting	#391×C₁₃₉	C₂
socle, prime radical, cosocle	C₂³×C₁₃₉	C₂⁴
hypercenter	#391	D₁₃₉
hypocenter	C₁₃₉	#2356
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ #391

Complements

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

N	H ≅ G/N	type
12. Q₈×Dic₁₃₉	C₄	semidirect
122. C₁₃₉	#2356	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×Dic₁₃₉	2
#22580860	2
#22580838	2
#22580836	2
#22580820	2
#22580818	2
C₄⋊C₄×Dic₁₃₉	2
C₄×Dic₁₃₉⋊C₄	2
C₄×C₄⋊Dic₁₃₉	2
#22580798	2
#391×C₁₃₉	2
#2356	139