Subgroups — #3375×C₂₄₁ #44505042

Order 30848.

350 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#559×C₂₄₁	C₂	1	1	yes	maximal, characteristic
2	Q₈²×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#489×C₂₄₁	C₂	1	2	yes	maximal, complemented
4	#486×C₂₄₁	C₂	1	1	yes	maximal, characteristic
5	#477×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#474×C₂₄₁	C₂	1	2	yes	maximal, complemented
7	#464×C₂₄₁	C₂	1	2	yes	maximal, complemented
8	#460×C₂₄₁	C₂	1	1	yes	maximal, characteristic
9	#446×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#440×C₂₄₁	C₂	1	2	yes	maximal, complemented
11	C₄×Q₁₆×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×C₂₄₁	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×C₂₄₁	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₂₄₁	—	2	1	no	—
15	C₄⋊Q₈×C₂₄₁	—	2	2	no	—
16	C₄⋊Q₈×C₂₄₁	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₂₄₁	C₂²	1	2	yes	—
18	#127×C₂₄₁	C₂²	1	1	yes	characteristic
19	#124×C₂₄₁	—	2	1	no	—
20	#124×C₂₄₁	C₂²	1	2	yes	—
21	#123×C₂₄₁	—	2	2	no	—
22	C₄×Q₈×C₂₄₁	—	2	1	no	—
23	C₄×Q₈×C₂₄₁	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₂₄₁	C₂²	1	2	yes	—
25	C₄×Q₈×C₂₄₁	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₂₄₁	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₂₄₁	—	2	1	no	—
28	C₄×D₄×C₂₄₁	C₂²	1	2	yes	—
29	#107×C₂₄₁	C₂²	1	1	yes	characteristic
30	#107×C₂₄₁	C₂²	1	2	yes	—
31	C₄⋊C₈×C₂₄₁	C₂²	1	1	yes	characteristic
32	C₄⋊C₈×C₂₄₁	C₂²	1	2	yes	—
33	#103×C₂₄₁	C₂²	1	1	yes	characteristic
34	#103×C₂₄₁	C₂²	1	2	yes	—
35	#103×C₂₄₁	C₂²	1	1	yes	characteristic
36	#102×C₂₄₁	C₂²	1	2	yes	—
37	#102×C₂₄₁	C₂²	1	2	yes	—
38	#102×C₂₄₁	C₂²	1	2	yes	—
39	#97×C₂₄₁	C₂²	1	2	yes	—
40	C₄×C₈×C₂₄₁	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₂₄₁	—	2	2	no	—
42	C₂×Q₈×C₂₄₁	—	2	1	no	—
43	C₂×Q₈×C₂₄₁	D₄	1	1	yes	characteristic
44	C₂×Q₈×C₂₄₁	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₂₄₁	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₂₄₁	—	2	1	no	—
47	C₂×D₄×C₂₄₁	—	2	1	no	—
48	C₂×D₄×C₂₄₁	C₂³	1	2	yes	—
49	C₂²×C₄×C₂₄₁	—	2	2	no	—
50	Q₁₆×C₂₄₁	—	2	2	no	—
51	SD₁₆×C₂₄₁	—	2	4	no	—
52	C₂×C₈×C₂₄₁	C₂³	1	2	yes	—
53	C₂×C₈×C₂₄₁	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₂₄₁	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×C₂₄₁	—	2	2	no	—
56	C₄⋊C₄×C₂₄₁	—	2	1	no	—
57	C₄⋊C₄×C₂₄₁	—	2	1	no	—
58	C₄⋊C₄×C₂₄₁	—	2	2	no	—
59	C₄⋊C₄×C₂₄₁	—	2	1	no	—
60	C₄⋊C₄×C₂₄₁	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₂₄₁	D₄	1	2	yes	—
62	C₄⋊C₄×C₂₄₁	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₂₄₁	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₂₄₁	C₂³	1	2	yes	—
65	C₂²⋊C₄×C₂₄₁	—	2	2	no	—
66	C₂²⋊C₄×C₂₄₁	—	2	2	no	—
67	C₂²⋊C₄×C₂₄₁	—	2	2	no	—
68	C₄²×C₂₄₁	—	2	1	no	—
69	C₄²×C₂₄₁	—	2	2	no	—
70	C₄²×C₂₄₁	—	2	1	no	—
71	C₄²×C₂₄₁	C₂³	1	2	yes	—
72	C₄²×C₂₄₁	C₂³	1	1	yes	characteristic
73	C₂³×C₂₄₁	—	2	2	no	—
74	Q₈×C₂₄₁	—	2	2	no	—
75	Q₈×C₂₄₁	—	2	1	no	—
76	Q₈×C₂₄₁	—	2	1	no	—
77	Q₈×C₂₄₁	—	2	1	no	—
78	Q₈×C₂₄₁	—	8	2	no	—
79	Q₈×C₂₄₁	—	8	1	no	—
80	Q₈×C₂₄₁	#41	1	2	yes	—
81	D₄×C₂₄₁	—	2	2	no	—
82	D₄×C₂₄₁	—	2	2	no	—
83	D₄×C₂₄₁	—	4	2	no	—
84	D₄×C₂₄₁	—	8	1	no	—
85	C₂×C₄×C₂₄₁	—	2	1	no	—
86	C₂×C₄×C₂₄₁	—	2	1	no	—
87	C₂×C₄×C₂₄₁	—	2	1	no	—
88	C₂×C₄×C₂₄₁	—	2	2	no	—
89	C₂×C₄×C₂₄₁	—	2	1	no	—
90	C₂×C₄×C₂₄₁	—	8	2	no	—
91	C₂×C₄×C₂₄₁	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₂₄₁	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₂₄₁	C₂⁴	1	1	yes	characteristic
96	C₈×C₂₄₁	—	2	2	no	—
97	C₈×C₂₄₁	—	2	1	no	—
98	C₈×C₂₄₁	—	2	1	no	—
99	C₂²×C₂₄₁	—	4	2	no	—
100	C₂²×C₂₄₁	—	4	2	no	—
101	C₂²×C₂₄₁	—	4	2	no	—
102	C₂²×C₂₄₁	C₂²×D₄	1	1	yes	characteristic
103	C₄×C₂₄₁	—	2	2	no	—
104	C₄×C₂₄₁	—	2	2	no	—
105	C₄×C₂₄₁	—	2	2	no	—
106	C₄×C₂₄₁	—	2	1	no	—
107	C₄×C₂₄₁	—	4	1	no	—
108	C₄×C₂₄₁	—	4	1	no	—
109	C₄×C₂₄₁	—	4	1	no	—
110	C₄×C₂₄₁	—	4	2	no	—
111	C₄×C₂₄₁	#137	1	2	yes	—
112	C₄×C₂₄₁	C₂×#41	1	1	yes	characteristic
113	C₄×C₂₄₁	2₋¹⁺⁴	1	1	yes	characteristic
114	C₂×C₂₄₁	—	8	2	no	—
115	C₂×C₂₄₁	#548	1	1	yes	characteristic
116	C₂×C₂₄₁	C₂×#137	1	1	yes	characteristic
117	C₂×C₂₄₁	#576	1	1	yes	characteristic
118	C₂₄₁	#3375	1	1	yes	characteristic, direct product factor, Sylow
119	#3375	C₂₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	#559	C₂×C₂₄₁	1	1	yes	characteristic
121	Q₈²	C₂×C₂₄₁	1	1	yes	characteristic, complemented
122	#489	C₂×C₂₄₁	1	2	yes	complemented
123	#486	C₂×C₂₄₁	1	1	yes	characteristic
124	#477	C₂×C₂₄₁	1	1	yes	characteristic, complemented
125	#474	C₂×C₂₄₁	1	2	yes	complemented
126	#464	C₂×C₂₄₁	1	2	yes	complemented
127	#460	C₂×C₂₄₁	1	1	yes	characteristic
128	#446	C₂×C₂₄₁	1	1	yes	characteristic, complemented
129	#440	C₂×C₂₄₁	1	2	yes	complemented
130	C₄×Q₁₆	C₂×C₂₄₁	1	1	yes	characteristic, complemented
131	C₂×Q₁₆	C₂²×C₂₄₁	1	1	yes	characteristic
132	C₂×SD₁₆	C₂²×C₂₄₁	1	2	yes	—
133	C₄⋊Q₈	—	2	1	no	—
134	C₄⋊Q₈	—	2	2	no	—
135	C₄⋊Q₈	C₂²×C₂₄₁	1	2	yes	—
136	C₄⋊Q₈	C₂²×C₂₄₁	1	1	yes	characteristic
137	#127	C₂²×C₂₄₁	1	1	yes	characteristic
138	#124	—	2	1	no	—
139	#124	C₂²×C₂₄₁	1	2	yes	—
140	#123	—	2	2	no	—
141	C₄×Q₈	—	2	1	no	—
142	C₄×Q₈	C₂²×C₂₄₁	1	1	yes	characteristic
143	C₄×Q₈	C₂²×C₂₄₁	1	2	yes	—
144	C₄×Q₈	C₂²×C₂₄₁	1	1	yes	characteristic
145	C₄×Q₈	C₂²×C₂₄₁	1	1	yes	characteristic
146	C₄×D₄	—	2	1	no	—
147	C₄×D₄	C₂²×C₂₄₁	1	2	yes	—
148	#107	C₂²×C₂₄₁	1	1	yes	characteristic
149	#107	C₂²×C₂₄₁	1	2	yes	—
150	C₄⋊C₈	C₂²×C₂₄₁	1	1	yes	characteristic
151	C₄⋊C₈	C₂²×C₂₄₁	1	2	yes	—
152	#103	C₂²×C₂₄₁	1	1	yes	characteristic
153	#103	C₂²×C₂₄₁	1	1	yes	characteristic
154	#103	C₂²×C₂₄₁	1	2	yes	—
155	#102	C₂²×C₂₄₁	1	2	yes	—
156	#102	C₂²×C₂₄₁	1	2	yes	—
157	#102	C₂²×C₂₄₁	1	2	yes	—
158	#97	C₂²×C₂₄₁	1	2	yes	—
159	C₄×C₈	C₂²×C₂₄₁	1	1	yes	characteristic
160	C₂×Q₈	—	2	2	no	—
161	C₂×Q₈	—	2	1	no	—
162	C₂×Q₈	D₄×C₂₄₁	1	1	yes	characteristic
163	C₂×Q₈	C₂³×C₂₄₁	1	1	yes	characteristic
164	C₂×Q₈	C₂³×C₂₄₁	1	1	yes	characteristic
165	C₂×D₄	—	2	1	no	—
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	C₂³×C₂₄₁	1	2	yes	—
168	C₂²×C₄	—	2	2	no	—
169	Q₁₆	—	2	2	no	—
170	SD₁₆	—	2	4	no	—
171	C₂×C₈	C₂³×C₂₄₁	1	1	yes	characteristic
172	C₂×C₈	C₂³×C₂₄₁	1	1	yes	characteristic
173	C₂×C₈	C₂³×C₂₄₁	1	2	yes	—
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	D₄×C₂₄₁	1	1	yes	characteristic
180	C₄⋊C₄	D₄×C₂₄₁	1	2	yes	—
181	C₄⋊C₄	C₂³×C₂₄₁	1	2	yes	—
182	C₄⋊C₄	C₂³×C₂₄₁	1	1	yes	characteristic
183	C₄⋊C₄	C₂³×C₂₄₁	1	1	yes	characteristic
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	—	2	2	no	—
187	C₄²	—	2	1	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	1	no	—
190	C₄²	C₂³×C₂₄₁	1	1	yes	characteristic
191	C₄²	C₂³×C₂₄₁	1	2	yes	—
192	C₂³	—	2	2	no	—
193	Q₈	—	2	2	no	—
194	Q₈	—	2	1	no	—
195	Q₈	—	2	1	no	—
196	Q₈	—	2	1	no	—
197	Q₈	—	8	2	no	—
198	Q₈	—	8	1	no	—
199	Q₈	#41×C₂₄₁	1	2	yes	—
200	D₄	—	2	2	no	—
201	D₄	—	2	2	no	—
202	D₄	—	4	2	no	—
203	D₄	—	8	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	8	2	no	—
210	C₂×C₄	C₂×D₄×C₂₄₁	1	2	yes	—
211	C₂×C₄	C₂×D₄×C₂₄₁	1	2	yes	—
212	C₂×C₄	C₂×D₄×C₂₄₁	1	1	yes	characteristic
213	C₂×C₄	C₂×D₄×C₂₄₁	1	1	yes	characteristic
214	C₂×C₄	C₂⁴×C₂₄₁	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	C₂²×D₄×C₂₄₁	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	2	1	no	—
226	C₄	—	4	1	no	—
227	C₄	—	4	1	no	—
228	C₄	—	4	1	no	—
229	C₄	—	4	2	no	—
230	C₄	#137×C₂₄₁	1	2	yes	—
231	C₄	C₂×#41×C₂₄₁	1	1	yes	characteristic
232	C₄	2₋¹⁺⁴×C₂₄₁	1	1	yes	characteristic
233	C₂	—	8	2	no	—
234	C₂	#548×C₂₄₁	1	1	yes	characteristic
235	C₂	C₂×#137×C₂₄₁	1	1	yes	characteristic
236	C₂	#576×C₂₄₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₄₁
prime radical	#127×C₂₄₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₄₁ ◁ C₄×Q₈×C₂₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈²×C₂₄₁	C₂	semidirect
3. #489×C₂₄₁	C₂	semidirect
5. #477×C₂₄₁	C₂	semidirect
6. #474×C₂₄₁	C₂	semidirect
7. #464×C₂₄₁	C₂	semidirect
9. #446×C₂₄₁	C₂	semidirect
10. #440×C₂₄₁	C₂	semidirect
11. C₄×Q₁₆×C₂₄₁	C₂	semidirect
118. C₂₄₁	#3375	direct (N × H)
119. #3375	C₂₄₁	direct (N × H)
121. Q₈²	C₂×C₂₄₁	semidirect
122. #489	C₂×C₂₄₁	semidirect
124. #477	C₂×C₂₄₁	semidirect
125. #474	C₂×C₂₄₁	semidirect
126. #464	C₂×C₂₄₁	semidirect
128. #446	C₂×C₂₄₁	semidirect
129. #440	C₂×C₂₄₁	semidirect
130. C₄×Q₁₆	C₂×C₂₄₁	semidirect

Maximal subgroups

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

subgroup	index
#559×C₂₄₁	2
Q₈²×C₂₄₁	2
#489×C₂₄₁	2
#486×C₂₄₁	2
#477×C₂₄₁	2
#474×C₂₄₁	2
#464×C₂₄₁	2
#460×C₂₄₁	2
#446×C₂₄₁	2
#440×C₂₄₁	2
C₄×Q₁₆×C₂₄₁	2
#3375	241

Subgroups — #3375×C241 #44505042

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3375×C₂₄₁ #44505042