Subgroups — #2538×C₂₄₁ #44504205

Order 30848.

620 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	C₂×C₂²⋊Q₈×C₂₄₁	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×Q₈×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₄×C₂₄₁	C₂	1	1	yes	maximal, characteristic
4	#396×C₂₄₁	C₂	1	4	yes	maximal, complemented
5	#390×C₂₄₁	C₂	1	2	yes	maximal
6	#389×C₂₄₁	C₂	1	2	yes	maximal, complemented
7	#384×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#380×C₂₄₁	C₂	1	2	yes	maximal, complemented
9	C₂²×Q₈×C₂₄₁	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₂₄₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₄₁	C₂²	1	2	yes	—
12	C₂²⋊Q₈×C₂₄₁	—	2	8	no	—
13	C₄×Q₈×C₂₄₁	—	2	4	no	—
14	C₄×D₄×C₂₄₁	—	2	4	no	—
15	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	4	yes	—
20	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₂₄₁	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₂₄₁	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₂₄₁	C₂²	1	4	yes	—
24	C₂×C₂²⋊C₄×C₂₄₁	C₂²	1	4	yes	—
25	C₂×C₄²×C₂₄₁	C₂²	1	2	yes	—
26	C₂×C₄²×C₂₄₁	C₂²	1	1	yes	characteristic
27	#95×C₂₄₁	C₂²	1	4	yes	—
28	C₂⁴×C₂₄₁	C₂³	1	2	yes	—
29	C₂×Q₈×C₂₄₁	—	2	2	no	—
30	C₂×Q₈×C₂₄₁	—	2	2	no	—
31	C₂×Q₈×C₂₄₁	D₄	1	4	yes	complemented
32	C₂×D₄×C₂₄₁	—	2	2	no	—
33	C₂×D₄×C₂₄₁	—	2	2	no	—
34	C₂×D₄×C₂₄₁	Q₈	1	4	yes	complemented
35	C₂²×C₄×C₂₄₁	—	2	4	no	—
36	C₂²×C₄×C₂₄₁	—	2	4	no	—
37	C₂²×C₄×C₂₄₁	—	2	4	no	—
38	C₂²×C₄×C₂₄₁	C₂³	1	2	yes	—
39	C₂²×C₄×C₂₄₁	C₂³	1	4	yes	—
40	C₂²×C₄×C₂₄₁	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₂₄₁	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₂₄₁	—	2	2	no	—
46	C₄⋊C₄×C₂₄₁	—	2	4	no	—
47	C₄⋊C₄×C₂₄₁	—	2	2	no	—
48	C₄⋊C₄×C₂₄₁	—	2	4	no	—
49	C₄⋊C₄×C₂₄₁	—	2	8	no	—
50	C₄⋊C₄×C₂₄₁	—	2	4	no	—
51	C₄⋊C₄×C₂₄₁	—	4	2	no	—
52	C₂²⋊C₄×C₂₄₁	—	2	4	no	—
53	C₂²⋊C₄×C₂₄₁	—	2	8	no	—
54	C₂²⋊C₄×C₂₄₁	—	4	4	no	—
55	C₄²×C₂₄₁	—	2	4	no	—
56	C₄²×C₂₄₁	D₄	1	4	yes	—
57	C₂³×C₂₄₁	—	2	2	no	—
58	C₂³×C₂₄₁	—	2	2	no	—
59	C₂³×C₂₄₁	—	2	2	no	—
60	C₂³×C₂₄₁	—	2	2	no	—
61	C₂³×C₂₄₁	—	2	2	no	—
62	C₂³×C₂₄₁	—	2	2	no	—
63	C₂³×C₂₄₁	C₂×Q₈	1	4	yes	—
64	C₂³×C₂₄₁	C₂⁴	1	1	yes	characteristic
65	Q₈×C₂₄₁	—	2	8	no	—
66	D₄×C₂₄₁	—	2	8	no	—
67	C₂×C₄×C₂₄₁	—	2	2	no	—
68	C₂×C₄×C₂₄₁	—	2	2	no	—
69	C₂×C₄×C₂₄₁	—	2	1	no	—
70	C₂×C₄×C₂₄₁	—	2	1	no	—
71	C₂×C₄×C₂₄₁	—	2	2	no	—
72	C₂×C₄×C₂₄₁	—	2	4	no	—
73	C₂×C₄×C₂₄₁	—	2	2	no	—
74	C₂×C₄×C₂₄₁	—	2	2	no	—
75	C₂×C₄×C₂₄₁	—	2	4	no	—
76	C₂×C₄×C₂₄₁	—	2	2	no	—
77	C₂×C₄×C₂₄₁	—	2	2	no	—
78	C₂×C₄×C₂₄₁	—	2	4	no	—
79	C₂×C₄×C₂₄₁	—	2	2	no	—
80	C₂×C₄×C₂₄₁	—	4	8	no	—
81	C₂×C₄×C₂₄₁	C₂×D₄	1	2	yes	—
82	C₂×C₄×C₂₄₁	C₂×D₄	1	4	yes	—
83	C₂×C₄×C₂₄₁	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₂₄₁	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₂₄₁	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₂₄₁	C₂×Q₈	1	2	yes	—
87	C₂×C₄×C₂₄₁	#41	1	2	yes	—
88	C₂×C₄×C₂₄₁	#41	1	2	yes	—
89	C₂²×C₂₄₁	—	2	4	no	—
90	C₂²×C₂₄₁	—	2	4	no	—
91	C₂²×C₂₄₁	—	2	4	no	—
92	C₂²×C₂₄₁	—	4	2	no	—
93	C₂²×C₂₄₁	—	4	2	no	—
94	C₂²×C₂₄₁	—	4	2	no	—
95	C₂²×C₂₄₁	—	4	2	no	—
96	C₂²×C₂₄₁	C₂²×D₄	1	1	yes	characteristic
97	C₂²×C₂₄₁	C₂²×D₄	1	1	yes	characteristic
98	C₂²×C₂₄₁	C₂²×Q₈	1	1	yes	characteristic
99	C₂²×C₂₄₁	C₂×#41	1	1	yes	characteristic
100	C₂²×C₂₄₁	C₂×#41	1	1	yes	characteristic
101	C₂²×C₂₄₁	H(5,𝔽₂)	1	1	yes	characteristic
102	C₂²×C₂₄₁	2₋¹⁺⁴	1	1	yes	characteristic
103	C₄×C₂₄₁	—	2	4	no	—
104	C₄×C₂₄₁	—	2	2	no	—
105	C₄×C₂₄₁	—	4	2	no	—
106	C₄×C₂₄₁	—	4	2	no	—
107	C₄×C₂₄₁	—	4	4	no	—
108	C₄×C₂₄₁	#121	1	4	yes	—
109	C₄×C₂₄₁	C₂²⋊Q₈	1	4	yes	complemented
110	C₂×C₂₄₁	—	4	4	no	—
111	C₂×C₂₄₁	C₂×#121	1	1	yes	characteristic
112	C₂×C₂₄₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₂₄₁	#537	1	1	yes	characteristic
114	C₂×C₂₄₁	D₄×Q₈	1	1	yes	characteristic
115	C₂×C₂₄₁	#550	1	1	yes	characteristic
116	C₂×C₂₄₁	#554	1	1	yes	characteristic
117	C₂×C₂₄₁	#556	1	1	yes	characteristic
118	C₂₄₁	#2538	1	1	yes	characteristic, direct product factor, Sylow
119	#2538	C₂₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂×C₂²⋊Q₈	C₂×C₂₄₁	1	2	yes	complemented
121	C₂×C₄×Q₈	C₂×C₂₄₁	1	1	yes	characteristic, complemented
122	C₂×C₄×D₄	C₂×C₂₄₁	1	1	yes	characteristic
123	#396	C₂×C₂₄₁	1	4	yes	complemented
124	#390	C₂×C₂₄₁	1	2	yes	—
125	#389	C₂×C₂₄₁	1	2	yes	complemented
126	#384	C₂×C₂₄₁	1	1	yes	characteristic, complemented
127	#380	C₂×C₂₄₁	1	2	yes	complemented
128	C₂²×Q₈	C₂²×C₂₄₁	1	1	yes	characteristic
129	C₂²×D₄	C₂²×C₂₄₁	1	1	yes	characteristic
130	C₂³×C₄	C₂²×C₂₄₁	1	2	yes	—
131	C₂²⋊Q₈	—	2	8	no	—
132	C₄×Q₈	—	2	4	no	—
133	C₄×D₄	—	2	4	no	—
134	C₂×C₄⋊C₄	C₂²×C₂₄₁	1	1	yes	characteristic
135	C₂×C₄⋊C₄	C₂²×C₂₄₁	1	1	yes	characteristic
136	C₂×C₄⋊C₄	C₂²×C₂₄₁	1	2	yes	—
137	C₂×C₄⋊C₄	C₂²×C₂₄₁	1	2	yes	—
138	C₂×C₄⋊C₄	C₂²×C₂₄₁	1	4	yes	—
139	C₂×C₄⋊C₄	C₂²×C₂₄₁	1	2	yes	—
140	C₂×C₄⋊C₄	C₂²×C₂₄₁	1	2	yes	—
141	C₂×C₂²⋊C₄	C₂²×C₂₄₁	1	2	yes	—
142	C₂×C₂²⋊C₄	C₂²×C₂₄₁	1	4	yes	—
143	C₂×C₂²⋊C₄	C₂²×C₂₄₁	1	4	yes	—
144	C₂×C₄²	C₂²×C₂₄₁	1	2	yes	—
145	C₂×C₄²	C₂²×C₂₄₁	1	1	yes	characteristic
146	#95	C₂²×C₂₄₁	1	4	yes	—
147	C₂⁴	C₂³×C₂₄₁	1	2	yes	—
148	C₂×Q₈	—	2	2	no	—
149	C₂×Q₈	—	2	2	no	—
150	C₂×Q₈	D₄×C₂₄₁	1	4	yes	complemented
151	C₂×D₄	—	2	2	no	—
152	C₂×D₄	—	2	2	no	—
153	C₂×D₄	Q₈×C₂₄₁	1	4	yes	complemented
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	C₂³×C₂₄₁	1	2	yes	—
158	C₂²×C₄	C₂³×C₂₄₁	1	2	yes	—
159	C₂²×C₄	C₂³×C₂₄₁	1	4	yes	—
160	C₂²×C₄	C₂³×C₂₄₁	1	1	yes	characteristic
161	C₂²×C₄	C₂³×C₂₄₁	1	2	yes	—
162	C₂²×C₄	C₂³×C₂₄₁	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₂₄₁	1	1	yes	characteristic
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	2	8	no	—
169	C₄⋊C₄	—	2	4	no	—
170	C₄⋊C₄	—	4	2	no	—
171	C₂²⋊C₄	—	2	4	no	—
172	C₂²⋊C₄	—	2	8	no	—
173	C₂²⋊C₄	—	4	4	no	—
174	C₄²	—	2	4	no	—
175	C₄²	D₄×C₂₄₁	1	4	yes	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	2	no	—
178	C₂³	—	2	2	no	—
179	C₂³	—	2	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	C₂×Q₈×C₂₄₁	1	4	yes	—
183	C₂³	C₂⁴×C₂₄₁	1	1	yes	characteristic
184	Q₈	—	2	8	no	—
185	D₄	—	2	8	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	4	8	no	—
200	C₂×C₄	C₂×D₄×C₂₄₁	1	2	yes	—
201	C₂×C₄	C₂×D₄×C₂₄₁	1	4	yes	—
202	C₂×C₄	C₂×D₄×C₂₄₁	1	2	yes	—
203	C₂×C₄	C₂×D₄×C₂₄₁	1	2	yes	—
204	C₂×C₄	C₂×D₄×C₂₄₁	1	2	yes	—
205	C₂×C₄	C₂×Q₈×C₂₄₁	1	2	yes	—
206	C₂×C₄	#41×C₂₄₁	1	2	yes	—
207	C₂×C₄	#41×C₂₄₁	1	2	yes	—
208	C₂²	—	2	4	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	4	2	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	C₂²×D₄×C₂₄₁	1	1	yes	characteristic
216	C₂²	C₂²×D₄×C₂₄₁	1	1	yes	characteristic
217	C₂²	C₂²×Q₈×C₂₄₁	1	1	yes	characteristic
218	C₂²	C₂×#41×C₂₄₁	1	1	yes	characteristic
219	C₂²	C₂×#41×C₂₄₁	1	1	yes	characteristic
220	C₂²	H(5,𝔽₂)×C₂₄₁	1	1	yes	characteristic
221	C₂²	2₋¹⁺⁴×C₂₄₁	1	1	yes	characteristic
222	C₄	—	2	4	no	—
223	C₄	—	2	2	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	4	no	—
227	C₄	#121×C₂₄₁	1	4	yes	—
228	C₄	C₂²⋊Q₈×C₂₄₁	1	4	yes	complemented
229	C₂	—	4	4	no	—
230	C₂	C₂×#121×C₂₄₁	1	1	yes	characteristic
231	C₂	C₂×C₂²⋊Q₈×C₂₄₁	1	1	yes	characteristic
232	C₂	#537×C₂₄₁	1	1	yes	characteristic
233	C₂	D₄×Q₈×C₂₄₁	1	1	yes	characteristic
234	C₂	#550×C₂₄₁	1	1	yes	characteristic
235	C₂	#554×C₂₄₁	1	1	yes	characteristic
236	C₂	#556×C₂₄₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₄₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₄₁
prime radical	C₂²×D₄×C₂₄₁	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊Q₈×C₂₄₁	C₂	semidirect
2. C₂×C₄×Q₈×C₂₄₁	C₂	semidirect
4. #396×C₂₄₁	C₂	semidirect
6. #389×C₂₄₁	C₂	semidirect
7. #384×C₂₄₁	C₂	semidirect
8. #380×C₂₄₁	C₂	semidirect
31. C₂×Q₈×C₂₄₁	D₄	semidirect
34. C₂×D₄×C₂₄₁	Q₈	semidirect
109. C₄×C₂₄₁	C₂²⋊Q₈	semidirect
118. C₂₄₁	#2538	direct (N × H)
119. #2538	C₂₄₁	direct (N × H)
120. C₂×C₂²⋊Q₈	C₂×C₂₄₁	semidirect
121. C₂×C₄×Q₈	C₂×C₂₄₁	semidirect
123. #396	C₂×C₂₄₁	semidirect
125. #389	C₂×C₂₄₁	semidirect
126. #384	C₂×C₂₄₁	semidirect
127. #380	C₂×C₂₄₁	semidirect
150. C₂×Q₈	D₄×C₂₄₁	semidirect
153. C₂×D₄	Q₈×C₂₄₁	semidirect
228. C₄	C₂²⋊Q₈×C₂₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊Q₈×C₂₄₁	2
C₂×C₄×Q₈×C₂₄₁	2
C₂×C₄×D₄×C₂₄₁	2
#396×C₂₄₁	2
#390×C₂₄₁	2
#389×C₂₄₁	2
#384×C₂₄₁	2
#380×C₂₄₁	2
#2538	241

Subgroups — #2538×C241 #44504205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2538×C₂₄₁ #44504205