Subgroups — #2353×C₂₃₉ #44481123

Order 30592.

588 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₂₃₉	C₂	1	1	yes	maximal, characteristic
2	#388×C₂₃₉	C₂	1	2	yes	maximal, complemented
3	#387×C₂₃₉	C₂	1	2	yes	maximal, complemented
4	#386×C₂₃₉	C₂	1	2	yes	maximal
5	#383×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#381×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#380×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₂²⋊C₄×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₂²⋊C₄×C₂₃₉	C₂	1	2	yes	maximal, complemented
10	#376×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#95×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₂₃₉	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₂₃₉	C₂²	1	2	yes	—
14	C₂³×C₄×C₂₃₉	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₂₃₉	C₂²	1	1	yes	characteristic
16	C₄×D₄×C₂₃₉	C₄	1	8	yes	complemented
17	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₂₃₉	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₂₃₉	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₂₃₉	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₂₃₉	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₂₃₉	C₂²	1	2	yes	—
25	C₂×C₄²×C₂₃₉	C₂²	1	2	yes	—
26	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
27	#95×C₂₃₉	C₂²	1	1	yes	characteristic
28	#95×C₂₃₉	C₂²	1	1	yes	characteristic
29	#95×C₂₃₉	C₂²	1	2	yes	—
30	#95×C₂₃₉	C₂²	1	1	yes	characteristic
31	#95×C₂₃₉	C₂²	1	2	yes	—
32	#95×C₂₃₉	C₂²	1	1	yes	characteristic
33	#95×C₂₃₉	C₂²	1	2	yes	—
34	#95×C₂₃₉	C₂²	1	2	yes	—
35	#95×C₂₃₉	C₂²	1	2	yes	—
36	C₂⁴×C₂₃₉	C₂³	1	1	yes	characteristic
37	C₂⁴×C₂₃₉	C₂³	1	1	yes	characteristic
38	C₂×D₄×C₂₃₉	—	2	2	no	—
39	C₂×D₄×C₂₃₉	—	2	2	no	—
40	C₂×D₄×C₂₃₉	C₂×C₄	1	4	yes	—
41	C₂²×C₄×C₂₃₉	—	2	4	no	—
42	C₂²×C₄×C₂₃₉	—	2	4	no	—
43	C₂²×C₄×C₂₃₉	—	4	2	no	—
44	C₂²×C₄×C₂₃₉	—	4	2	no	—
45	C₂²×C₄×C₂₃₉	—	4	2	no	—
46	C₂²×C₄×C₂₃₉	C₂×C₄	1	4	yes	—
47	C₂²×C₄×C₂₃₉	C₂×C₄	1	4	yes	complemented
48	C₂²×C₄×C₂₃₉	C₂³	1	2	yes	—
49	C₂²×C₄×C₂₃₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₂₃₉	C₂³	1	2	yes	—
51	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₂₃₉	C₂³	1	2	yes	—
57	C₄⋊C₄×C₂₃₉	—	4	2	no	—
58	C₄⋊C₄×C₂₃₉	C₂×C₄	1	4	yes	complemented
59	C₂²⋊C₄×C₂₃₉	—	2	4	no	—
60	C₂²⋊C₄×C₂₃₉	—	2	4	no	—
61	C₂²⋊C₄×C₂₃₉	—	4	2	no	—
62	C₂²⋊C₄×C₂₃₉	C₂×C₄	1	4	yes	—
63	C₂²⋊C₄×C₂₃₉	C₂×C₄	1	4	yes	complemented
64	C₄²×C₂₃₉	—	2	4	no	—
65	C₄²×C₂₃₉	—	4	2	no	—
66	C₄²×C₂₃₉	C₂×C₄	1	4	yes	complemented
67	C₂³×C₂₃₉	—	2	2	no	—
68	C₂³×C₂₃₉	—	2	2	no	—
69	C₂³×C₂₃₉	—	2	2	no	—
70	C₂³×C₂₃₉	—	2	1	no	—
71	C₂³×C₂₃₉	—	2	1	no	—
72	C₂³×C₂₃₉	—	2	2	no	—
73	C₂³×C₂₃₉	C₂²×C₄	1	2	yes	—
74	C₂³×C₂₃₉	C₂²×C₄	1	2	yes	—
75	C₂³×C₂₃₉	#41	1	2	yes	—
76	C₂³×C₂₃₉	#41	1	2	yes	—
77	C₂³×C₂₃₉	C₂⁴	1	1	yes	characteristic
78	D₄×C₂₃₉	—	2	8	no	—
79	C₂×C₄×C₂₃₉	—	2	2	no	—
80	C₂×C₄×C₂₃₉	—	2	2	no	—
81	C₂×C₄×C₂₃₉	—	2	1	no	—
82	C₂×C₄×C₂₃₉	—	2	1	no	—
83	C₂×C₄×C₂₃₉	—	2	2	no	—
84	C₂×C₄×C₂₃₉	—	2	2	no	—
85	C₂×C₄×C₂₃₉	—	2	2	no	—
86	C₂×C₄×C₂₃₉	—	2	2	no	—
87	C₂×C₄×C₂₃₉	—	2	2	no	—
88	C₂×C₄×C₂₃₉	—	2	2	no	—
89	C₂×C₄×C₂₃₉	—	2	2	no	—
90	C₂×C₄×C₂₃₉	—	2	2	no	—
91	C₂×C₄×C₂₃₉	—	2	2	no	—
92	C₂×C₄×C₂₃₉	—	2	2	no	—
93	C₂×C₄×C₂₃₉	—	2	2	no	—
94	C₂×C₄×C₂₃₉	—	2	2	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	8	no	—
99	C₂×C₄×C₂₃₉	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₂₃₉	C₂²×C₄	1	2	yes	—
101	C₂×C₄×C₂₃₉	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₂₃₉	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₂₃₉	C₂²×C₄	1	2	yes	—
104	C₂×C₄×C₂₃₉	#41	1	4	yes	—
105	C₂²×C₂₃₉	—	2	2	no	—
106	C₂²×C₂₃₉	—	2	2	no	—
107	C₂²×C₂₃₉	—	2	4	no	—
108	C₂²×C₂₃₉	—	2	4	no	—
109	C₂²×C₂₃₉	—	2	2	no	—
110	C₂²×C₂₃₉	—	2	2	no	—
111	C₂²×C₂₃₉	—	2	2	no	—
112	C₂²×C₂₃₉	—	4	2	no	—
113	C₂²×C₂₃₉	—	4	2	no	—
114	C₂²×C₂₃₉	#117	1	4	yes	—
115	C₂²×C₂₃₉	C₂³×C₄	1	1	yes	characteristic
116	C₂²×C₂₃₉	C₂×#41	1	1	yes	characteristic
117	C₂²×C₂₃₉	C₂×#41	1	1	yes	characteristic
118	C₂²×C₂₃₉	C₂×#41	1	2	yes	—
119	C₂²×C₂₃₉	H(5,𝔽₂)	1	2	yes	—
120	C₄×C₂₃₉	—	2	2	no	—
121	C₄×C₂₃₉	—	2	2	no	—
122	C₄×C₂₃₉	—	2	2	no	—
123	C₄×C₂₃₉	—	2	2	no	—
124	C₄×C₂₃₉	—	2	2	no	—
125	C₄×C₂₃₉	—	2	4	no	—
126	C₄×C₂₃₉	—	4	2	no	—
127	C₄×C₂₃₉	—	4	2	no	—
128	C₄×C₂₃₉	—	4	2	no	—
129	C₂×C₂₃₉	—	2	4	no	—
130	C₂×C₂₃₉	—	4	2	no	—
131	C₂×C₂₃₉	C₂×#117	1	1	yes	characteristic
132	C₂×C₂₃₉	C₄×#41	1	1	yes	characteristic
133	C₂×C₂₃₉	#518	1	1	yes	characteristic
134	C₂×C₂₃₉	#551	1	2	yes	—
135	C₂×C₂₃₉	#553	1	2	yes	—
136	C₂₃₉	#2353	1	1	yes	characteristic, direct product factor, Sylow
137	#2353	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂×C₄×D₄	C₂×C₂₃₉	1	1	yes	characteristic
139	#388	C₂×C₂₃₉	1	2	yes	complemented
140	#387	C₂×C₂₃₉	1	2	yes	complemented
141	#386	C₂×C₂₃₉	1	2	yes	—
142	#383	C₂×C₂₃₉	1	1	yes	characteristic, complemented
143	#381	C₂×C₂₃₉	1	1	yes	characteristic, complemented
144	#380	C₂×C₂₃₉	1	1	yes	characteristic, complemented
145	C₄×C₂²⋊C₄	C₂×C₂₃₉	1	1	yes	characteristic, complemented
146	C₄×C₂²⋊C₄	C₂×C₂₃₉	1	2	yes	complemented
147	#376	C₂×C₂₃₉	1	1	yes	characteristic, complemented
148	C₂×#95	C₂×C₂₃₉	1	1	yes	characteristic, complemented
149	C₂²×D₄	C₂²×C₂₃₉	1	1	yes	characteristic
150	C₂³×C₄	C₂²×C₂₃₉	1	2	yes	—
151	C₂³×C₄	C₂²×C₂₃₉	1	1	yes	characteristic
152	C₂³×C₄	C₂²×C₂₃₉	1	1	yes	characteristic
153	C₄×D₄	C₄×C₂₃₉	1	8	yes	complemented
154	C₂×C₄⋊C₄	C₂²×C₂₃₉	1	2	yes	—
155	C₂×C₄⋊C₄	C₂²×C₂₃₉	1	1	yes	characteristic
156	C₂×C₂²⋊C₄	C₂²×C₂₃₉	1	2	yes	—
157	C₂×C₂²⋊C₄	C₂²×C₂₃₉	1	2	yes	—
158	C₂×C₂²⋊C₄	C₂²×C₂₃₉	1	2	yes	—
159	C₂×C₂²⋊C₄	C₂²×C₂₃₉	1	1	yes	characteristic
160	C₂×C₂²⋊C₄	C₂²×C₂₃₉	1	1	yes	characteristic
161	C₂×C₄²	C₂²×C₂₃₉	1	2	yes	—
162	C₂×C₄²	C₂²×C₂₃₉	1	2	yes	—
163	C₂×C₄²	C₂²×C₂₃₉	1	1	yes	characteristic
164	#95	C₂²×C₂₃₉	1	1	yes	characteristic
165	#95	C₂²×C₂₃₉	1	1	yes	characteristic
166	#95	C₂²×C₂₃₉	1	2	yes	—
167	#95	C₂²×C₂₃₉	1	1	yes	characteristic
168	#95	C₂²×C₂₃₉	1	2	yes	—
169	#95	C₂²×C₂₃₉	1	1	yes	characteristic
170	#95	C₂²×C₂₃₉	1	2	yes	—
171	#95	C₂²×C₂₃₉	1	2	yes	—
172	#95	C₂²×C₂₃₉	1	2	yes	—
173	C₂⁴	C₂³×C₂₃₉	1	1	yes	characteristic
174	C₂⁴	C₂³×C₂₃₉	1	1	yes	characteristic
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	C₂×C₄×C₂₃₉	1	4	yes	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	C₂×C₄×C₂₃₉	1	4	yes	—
184	C₂²×C₄	C₂×C₄×C₂₃₉	1	4	yes	complemented
185	C₂²×C₄	C₂³×C₂₃₉	1	2	yes	—
186	C₂²×C₄	C₂³×C₂₃₉	1	2	yes	—
187	C₂²×C₄	C₂³×C₂₃₉	1	2	yes	—
188	C₂²×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
189	C₂²×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
190	C₂²×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
191	C₂²×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
192	C₂²×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
193	C₂²×C₄	C₂³×C₂₃₉	1	2	yes	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	C₂×C₄×C₂₃₉	1	4	yes	complemented
196	C₂²⋊C₄	—	2	4	no	—
197	C₂²⋊C₄	—	2	4	no	—
198	C₂²⋊C₄	—	4	2	no	—
199	C₂²⋊C₄	C₂×C₄×C₂₃₉	1	4	yes	—
200	C₂²⋊C₄	C₂×C₄×C₂₃₉	1	4	yes	complemented
201	C₄²	—	2	4	no	—
202	C₄²	—	4	2	no	—
203	C₄²	C₂×C₄×C₂₃₉	1	4	yes	complemented
204	C₂³	—	2	2	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	2	1	no	—
209	C₂³	—	2	2	no	—
210	C₂³	C₂²×C₄×C₂₃₉	1	2	yes	—
211	C₂³	C₂²×C₄×C₂₃₉	1	2	yes	—
212	C₂³	#41×C₂₃₉	1	2	yes	—
213	C₂³	#41×C₂₃₉	1	2	yes	—
214	C₂³	C₂⁴×C₂₃₉	1	1	yes	characteristic
215	D₄	—	2	8	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	8	no	—
236	C₂×C₄	C₂²×C₄×C₂₃₉	1	2	yes	—
237	C₂×C₄	C₂²×C₄×C₂₃₉	1	2	yes	—
238	C₂×C₄	C₂²×C₄×C₂₃₉	1	2	yes	—
239	C₂×C₄	C₂²×C₄×C₂₃₉	1	2	yes	—
240	C₂×C₄	C₂²×C₄×C₂₃₉	1	2	yes	—
241	C₂×C₄	#41×C₂₃₉	1	4	yes	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	4	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	#117×C₂₃₉	1	4	yes	—
252	C₂²	C₂³×C₄×C₂₃₉	1	1	yes	characteristic
253	C₂²	C₂×#41×C₂₃₉	1	1	yes	characteristic
254	C₂²	C₂×#41×C₂₃₉	1	1	yes	characteristic
255	C₂²	C₂×#41×C₂₃₉	1	2	yes	—
256	C₂²	H(5,𝔽₂)×C₂₃₉	1	2	yes	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	4	no	—
263	C₄	—	4	2	no	—
264	C₄	—	4	2	no	—
265	C₄	—	4	2	no	—
266	C₂	—	2	4	no	—
267	C₂	—	4	2	no	—
268	C₂	C₂×#117×C₂₃₉	1	1	yes	characteristic
269	C₂	C₄×#41×C₂₃₉	1	1	yes	characteristic
270	C₂	#518×C₂₃₉	1	1	yes	characteristic
271	C₂	#551×C₂₃₉	1	2	yes	—
272	C₂	#553×C₂₃₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₃₉
Frattini, cosocle	C₂³	C₂⁴×C₂₃₉
prime radical	C₂²×D₄×C₂₃₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #388×C₂₃₉	C₂	semidirect
3. #387×C₂₃₉	C₂	semidirect
5. #383×C₂₃₉	C₂	semidirect
6. #381×C₂₃₉	C₂	semidirect
7. #380×C₂₃₉	C₂	semidirect
8. C₄×C₂²⋊C₄×C₂₃₉	C₂	semidirect
9. C₄×C₂²⋊C₄×C₂₃₉	C₂	semidirect
10. #376×C₂₃₉	C₂	semidirect
11. C₂×#95×C₂₃₉	C₂	semidirect
16. C₄×D₄×C₂₃₉	C₄	semidirect
47. C₂²×C₄×C₂₃₉	C₂×C₄	semidirect
58. C₄⋊C₄×C₂₃₉	C₂×C₄	semidirect
63. C₂²⋊C₄×C₂₃₉	C₂×C₄	semidirect
66. C₄²×C₂₃₉	C₂×C₄	semidirect
136. C₂₃₉	#2353	direct (N × H)
137. #2353	C₂₃₉	direct (N × H)
139. #388	C₂×C₂₃₉	semidirect
140. #387	C₂×C₂₃₉	semidirect
142. #383	C₂×C₂₃₉	semidirect
143. #381	C₂×C₂₃₉	semidirect
144. #380	C₂×C₂₃₉	semidirect
145. C₄×C₂²⋊C₄	C₂×C₂₃₉	semidirect
146. C₄×C₂²⋊C₄	C₂×C₂₃₉	semidirect
147. #376	C₂×C₂₃₉	semidirect
148. C₂×#95	C₂×C₂₃₉	semidirect
153. C₄×D₄	C₄×C₂₃₉	semidirect
184. C₂²×C₄	C₂×C₄×C₂₃₉	semidirect
195. C₄⋊C₄	C₂×C₄×C₂₃₉	semidirect
200. C₂²⋊C₄	C₂×C₄×C₂₃₉	semidirect
203. C₄²	C₂×C₄×C₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₂₃₉	2
#388×C₂₃₉	2
#387×C₂₃₉	2
#386×C₂₃₉	2
#383×C₂₃₉	2
#381×C₂₃₉	2
#380×C₂₃₉	2
C₄×C₂²⋊C₄×C₂₃₉	2
#376×C₂₃₉	2
C₂×#95×C₂₃₉	2
#2353	239

Subgroups — #2353×C239 #44481123

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2353×C₂₃₉ #44481123