Subgroups — #3298×C₂₃₉ #44482068

Order 30592.

420 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×Q₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#547×C₂₃₉	C₂	1	1	yes	maximal, characteristic
3	#487×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#483×C₂₃₉	C₂	1	2	yes	maximal, complemented
5	#475×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#465×C₂₃₉	C₂	1	2	yes	maximal, complemented
7	#461×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#448×C₂₃₉	C₂	1	2	yes	maximal, complemented
9	C₄×SD₁₆×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×D₄×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#103×C₂₃₉	C₂	1	2	yes	maximal, complemented
12	C₂×#41×C₂₃₉	—	2	1	no	—
13	C₂²×Q₈×C₂₃₉	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₂₃₉	C₂²	1	1	yes	characteristic
15	C₂×SD₁₆×C₂₃₉	C₂²	1	2	yes	—
16	C₂²×C₈×C₂₃₉	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₂₃₉	—	2	1	no	—
18	C₄⋊Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
20	#123×C₂₃₉	—	2	2	no	—
21	C₂²⋊Q₈×C₂₃₉	—	2	1	no	—
22	C₂²⋊Q₈×C₂₃₉	—	2	2	no	—
23	C₂²⋊Q₈×C₂₃₉	C₂²	1	2	yes	—
24	#121×C₂₃₉	C₂²	1	2	yes	—
25	C₄×Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₂₃₉	—	2	1	no	—
27	C₄×D₄×C₂₃₉	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₂₃₉	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	2	yes	—
30	#106×C₂₃₉	C₂²	1	1	yes	characteristic
31	#106×C₂₃₉	C₂²	1	2	yes	—
32	C₄⋊C₈×C₂₃₉	C₂²	1	1	yes	characteristic
33	#103×C₂₃₉	C₂²	1	1	yes	characteristic
34	#103×C₂₃₉	C₂²	1	2	yes	—
35	#103×C₂₃₉	C₂²	1	2	yes	—
36	#103×C₂₃₉	C₂²	1	2	yes	—
37	#103×C₂₃₉	C₂²	1	2	yes	—
38	#102×C₂₃₉	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₂₃₉	C₂²	1	2	yes	—
40	C₄×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
41	#41×C₂₃₉	—	4	4	no	—
42	C₂×Q₈×C₂₃₉	—	2	2	no	—
43	C₂×Q₈×C₂₃₉	—	2	1	no	—
44	C₂×Q₈×C₂₃₉	—	2	4	no	—
45	C₂×Q₈×C₂₃₉	—	4	2	no	—
46	C₂×Q₈×C₂₃₉	C₂³	1	2	yes	—
47	C₂×Q₈×C₂₃₉	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₂₃₉	—	2	2	no	—
49	C₂×D₄×C₂₃₉	D₄	1	1	yes	characteristic
50	C₂×D₄×C₂₃₉	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₂₃₉	—	2	2	no	—
52	C₂²×C₄×C₂₃₉	—	2	2	no	—
53	C₂²×C₄×C₂₃₉	—	2	1	no	—
54	C₂²×C₄×C₂₃₉	C₂³	1	2	yes	—
55	SD₁₆×C₂₃₉	—	2	2	no	—
56	SD₁₆×C₂₃₉	—	4	2	no	—
57	C₂×C₈×C₂₃₉	—	4	2	no	—
58	C₂×C₈×C₂₃₉	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₂₃₉	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₂₃₉	C₂³	1	2	yes	—
61	C₄⋊C₄×C₂₃₉	—	2	2	no	—
62	C₄⋊C₄×C₂₃₉	—	2	1	no	—
63	C₄⋊C₄×C₂₃₉	—	2	1	no	—
64	C₄⋊C₄×C₂₃₉	—	2	1	no	—
65	C₄⋊C₄×C₂₃₉	—	2	2	no	—
66	C₄⋊C₄×C₂₃₉	—	2	1	no	—
67	C₄⋊C₄×C₂₃₉	D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₂₃₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₂₃₉	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₂₃₉	C₂³	1	2	yes	—
71	C₂²⋊C₄×C₂₃₉	—	2	2	no	—
72	C₂²⋊C₄×C₂₃₉	—	2	1	no	—
73	C₂²⋊C₄×C₂₃₉	—	2	2	no	—
74	C₂²⋊C₄×C₂₃₉	D₄	1	2	yes	—
75	C₄²×C₂₃₉	—	2	1	no	—
76	C₄²×C₂₃₉	C₂³	1	1	yes	characteristic
77	C₂³×C₂₃₉	—	2	1	no	—
78	C₂³×C₂₃₉	C₂×D₄	1	2	yes	—
79	Q₈×C₂₃₉	—	2	1	no	—
80	Q₈×C₂₃₉	—	2	2	no	—
81	Q₈×C₂₃₉	—	2	2	no	—
82	Q₈×C₂₃₉	—	4	4	no	—
83	Q₈×C₂₃₉	—	4	2	no	—
84	Q₈×C₂₃₉	#41	1	2	yes	complemented
85	D₄×C₂₃₉	—	2	1	no	—
86	D₄×C₂₃₉	—	2	1	no	—
87	D₄×C₂₃₉	—	4	4	no	—
88	D₄×C₂₃₉	SD₁₆	1	4	yes	complemented
89	C₂×C₄×C₂₃₉	—	2	1	no	—
90	C₂×C₄×C₂₃₉	—	2	2	no	—
91	C₂×C₄×C₂₃₉	—	2	2	no	—
92	C₂×C₄×C₂₃₉	—	2	1	no	—
93	C₂×C₄×C₂₃₉	—	2	2	no	—
94	C₂×C₄×C₂₃₉	—	2	1	no	—
95	C₂×C₄×C₂₃₉	—	2	2	no	—
96	C₂×C₄×C₂₃₉	—	4	4	no	—
97	C₂×C₄×C₂₃₉	—	4	4	no	—
98	C₂×C₄×C₂₃₉	—	4	2	no	—
99	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₂₃₉	C₂×D₄	1	2	yes	—
101	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₂₃₉	C₂⁴	1	1	yes	characteristic
103	C₈×C₂₃₉	—	2	1	no	—
104	C₈×C₂₃₉	—	2	1	no	—
105	C₈×C₂₃₉	—	2	2	no	—
106	C₂²×C₂₃₉	—	2	2	no	—
107	C₂²×C₂₃₉	—	2	2	no	—
108	C₂²×C₂₃₉	—	4	1	no	—
109	C₂²×C₂₃₉	—	4	1	no	—
110	C₂²×C₂₃₉	—	4	1	no	—
111	C₂²×C₂₃₉	C₂×SD₁₆	1	4	yes	—
112	C₂²×C₂₃₉	C₂²×D₄	1	1	yes	characteristic
113	C₄×C₂₃₉	—	2	2	no	—
114	C₄×C₂₃₉	—	2	2	no	—
115	C₄×C₂₃₉	—	2	1	no	—
116	C₄×C₂₃₉	—	4	2	no	—
117	C₄×C₂₃₉	—	4	1	no	—
118	C₄×C₂₃₉	—	4	2	no	—
119	C₄×C₂₃₉	—	4	1	no	—
120	C₄×C₂₃₉	C₂×SD₁₆	1	2	yes	—
121	C₄×C₂₃₉	C₂×#41	1	1	yes	characteristic
122	C₄×C₂₃₉	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×C₂₃₉	—	2	4	no	—
124	C₂×C₂₃₉	—	8	1	no	—
125	C₂×C₂₃₉	#546	1	1	yes	characteristic
126	C₂×C₂₃₉	C₂²×SD₁₆	1	1	yes	characteristic
127	C₂×C₂₃₉	#578	1	1	yes	characteristic
128	C₂₃₉	#3298	1	1	yes	characteristic, direct product factor, Sylow
129	#3298	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	D₄×Q₈	C₂×C₂₃₉	1	1	yes	characteristic, complemented
131	#547	C₂×C₂₃₉	1	1	yes	characteristic
132	#487	C₂×C₂₃₉	1	1	yes	characteristic, complemented
133	#483	C₂×C₂₃₉	1	2	yes	complemented
134	#475	C₂×C₂₃₉	1	1	yes	characteristic, complemented
135	#465	C₂×C₂₃₉	1	2	yes	complemented
136	#461	C₂×C₂₃₉	1	1	yes	characteristic, complemented
137	#448	C₂×C₂₃₉	1	2	yes	complemented
138	C₄×SD₁₆	C₂×C₂₃₉	1	1	yes	characteristic, complemented
139	C₈×D₄	C₂×C₂₃₉	1	1	yes	characteristic, complemented
140	C₂×#103	C₂×C₂₃₉	1	2	yes	complemented
141	C₂×#41	—	2	1	no	—
142	C₂²×Q₈	C₂²×C₂₃₉	1	2	yes	—
143	C₂×SD₁₆	C₂²×C₂₃₉	1	1	yes	characteristic
144	C₂×SD₁₆	C₂²×C₂₃₉	1	2	yes	—
145	C₂²×C₈	C₂²×C₂₃₉	1	2	yes	—
146	C₄⋊Q₈	—	2	1	no	—
147	C₄⋊Q₈	C₂²×C₂₃₉	1	1	yes	characteristic
148	C₄⋊Q₈	C₂²×C₂₃₉	1	1	yes	characteristic
149	#123	—	2	2	no	—
150	C₂²⋊Q₈	—	2	1	no	—
151	C₂²⋊Q₈	—	2	2	no	—
152	C₂²⋊Q₈	C₂²×C₂₃₉	1	2	yes	—
153	#121	C₂²×C₂₃₉	1	2	yes	—
154	C₄×Q₈	C₂²×C₂₃₉	1	1	yes	characteristic
155	C₄×D₄	—	2	1	no	—
156	C₄×D₄	C₂²×C₂₃₉	1	1	yes	characteristic
157	C₄×D₄	C₂²×C₂₃₉	1	1	yes	characteristic
158	C₂×C₄⋊C₄	C₂²×C₂₃₉	1	2	yes	—
159	#106	C₂²×C₂₃₉	1	1	yes	characteristic
160	#106	C₂²×C₂₃₉	1	2	yes	—
161	C₄⋊C₈	C₂²×C₂₃₉	1	1	yes	characteristic
162	#103	C₂²×C₂₃₉	1	1	yes	characteristic
163	#103	C₂²×C₂₃₉	1	2	yes	—
164	#103	C₂²×C₂₃₉	1	2	yes	—
165	#103	C₂²×C₂₃₉	1	2	yes	—
166	#103	C₂²×C₂₃₉	1	2	yes	—
167	#102	C₂²×C₂₃₉	1	1	yes	characteristic
168	C₂²⋊C₈	C₂²×C₂₃₉	1	2	yes	—
169	C₄×C₈	C₂²×C₂₃₉	1	1	yes	characteristic
170	#41	—	4	4	no	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×Q₈	—	2	1	no	—
173	C₂×Q₈	—	2	4	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂×Q₈	C₂³×C₂₃₉	1	2	yes	—
176	C₂×Q₈	C₂³×C₂₃₉	1	1	yes	characteristic
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	D₄×C₂₃₉	1	1	yes	characteristic
179	C₂×D₄	C₂³×C₂₃₉	1	1	yes	characteristic
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	C₂³×C₂₃₉	1	2	yes	—
184	SD₁₆	—	2	2	no	—
185	SD₁₆	—	4	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	C₂³×C₂₃₉	1	1	yes	characteristic
188	C₂×C₈	C₂³×C₂₃₉	1	1	yes	characteristic
189	C₂×C₈	C₂³×C₂₃₉	1	2	yes	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	D₄×C₂₃₉	1	1	yes	characteristic
197	C₄⋊C₄	C₂³×C₂₃₉	1	1	yes	characteristic
198	C₄⋊C₄	C₂³×C₂₃₉	1	1	yes	characteristic
199	C₄⋊C₄	C₂³×C₂₃₉	1	2	yes	—
200	C₂²⋊C₄	—	2	2	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	2	no	—
203	C₂²⋊C₄	D₄×C₂₃₉	1	2	yes	—
204	C₄²	—	2	1	no	—
205	C₄²	C₂³×C₂₃₉	1	1	yes	characteristic
206	C₂³	—	2	1	no	—
207	C₂³	C₂×D₄×C₂₃₉	1	2	yes	—
208	Q₈	—	2	1	no	—
209	Q₈	—	2	2	no	—
210	Q₈	—	2	2	no	—
211	Q₈	—	4	4	no	—
212	Q₈	—	4	2	no	—
213	Q₈	#41×C₂₃₉	1	2	yes	complemented
214	D₄	—	2	1	no	—
215	D₄	—	2	1	no	—
216	D₄	—	4	4	no	—
217	D₄	SD₁₆×C₂₃₉	1	4	yes	complemented
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	C₂×D₄×C₂₃₉	1	1	yes	characteristic
229	C₂×C₄	C₂×D₄×C₂₃₉	1	2	yes	—
230	C₂×C₄	C₂×D₄×C₂₃₉	1	1	yes	characteristic
231	C₂×C₄	C₂⁴×C₂₃₉	1	1	yes	characteristic
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	1	no	—
240	C₂²	C₂×SD₁₆×C₂₃₉	1	4	yes	—
241	C₂²	C₂²×D₄×C₂₃₉	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	1	no	—
245	C₄	—	4	2	no	—
246	C₄	—	4	1	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	1	no	—
249	C₄	C₂×SD₁₆×C₂₃₉	1	2	yes	—
250	C₄	C₂×#41×C₂₃₉	1	1	yes	characteristic
251	C₄	H(5,𝔽₂)×C₂₃₉	1	1	yes	characteristic
252	C₂	—	2	4	no	—
253	C₂	—	8	1	no	—
254	C₂	#546×C₂₃₉	1	1	yes	characteristic
255	C₂	C₂²×SD₁₆×C₂₃₉	1	1	yes	characteristic
256	C₂	#578×C₂₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃₉
prime radical	#547×C₂₃₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃₉ ◁ C₄×D₄×C₂₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. D₄×Q₈×C₂₃₉	C₂	semidirect
3. #487×C₂₃₉	C₂	semidirect
4. #483×C₂₃₉	C₂	semidirect
5. #475×C₂₃₉	C₂	semidirect
6. #465×C₂₃₉	C₂	semidirect
7. #461×C₂₃₉	C₂	semidirect
8. #448×C₂₃₉	C₂	semidirect
9. C₄×SD₁₆×C₂₃₉	C₂	semidirect
10. C₈×D₄×C₂₃₉	C₂	semidirect
11. C₂×#103×C₂₃₉	C₂	semidirect
84. Q₈×C₂₃₉	#41	semidirect
88. D₄×C₂₃₉	SD₁₆	semidirect
128. C₂₃₉	#3298	direct (N × H)
129. #3298	C₂₃₉	direct (N × H)
130. D₄×Q₈	C₂×C₂₃₉	semidirect
132. #487	C₂×C₂₃₉	semidirect
133. #483	C₂×C₂₃₉	semidirect
134. #475	C₂×C₂₃₉	semidirect
135. #465	C₂×C₂₃₉	semidirect
136. #461	C₂×C₂₃₉	semidirect
137. #448	C₂×C₂₃₉	semidirect
138. C₄×SD₁₆	C₂×C₂₃₉	semidirect
139. C₈×D₄	C₂×C₂₃₉	semidirect
140. C₂×#103	C₂×C₂₃₉	semidirect
213. Q₈	#41×C₂₃₉	semidirect
217. D₄	SD₁₆×C₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
D₄×Q₈×C₂₃₉	2
#547×C₂₃₉	2
#487×C₂₃₉	2
#483×C₂₃₉	2
#475×C₂₃₉	2
#465×C₂₃₉	2
#461×C₂₃₉	2
#448×C₂₃₉	2
C₄×SD₁₆×C₂₃₉	2
C₈×D₄×C₂₃₉	2
C₂×#103×C₂₃₉	2
#3298	239

Subgroups — #3298×C239 #44482068

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3298×C₂₃₉ #44482068