Subgroups — #2349×C₂₁₁ #41009969

Order 27008.

704 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₂₁₁	C₂	1	2	yes	maximal
2	C₂×C₄×D₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	#389×C₂₁₁	C₂	1	2	yes	maximal, complemented
4	#385×C₂₁₁	C₂	1	4	yes	maximal, complemented
5	#382×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#380×C₂₁₁	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊C₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#95×C₂₁₁	C₂	1	2	yes	maximal, complemented
9	C₂²×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₂₁₁	C₂²	1	4	yes	—
11	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
12	C₄×D₄×C₂₁₁	—	2	8	no	—
13	C₄×D₄×C₂₁₁	C₄	1	8	yes	complemented
14	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	4	yes	—
19	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
20	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
22	#95×C₂₁₁	C₂²	1	1	yes	characteristic
23	#95×C₂₁₁	C₂²	1	2	yes	—
24	#95×C₂₁₁	C₂²	1	4	yes	—
25	#95×C₂₁₁	C₂²	1	1	yes	characteristic
26	#95×C₂₁₁	C₂²	1	2	yes	—
27	C₂⁴×C₂₁₁	C₂³	1	2	yes	—
28	C₂×D₄×C₂₁₁	C₂×C₄	1	4	yes	—
29	C₂×D₄×C₂₁₁	D₄	1	4	yes	—
30	C₂×D₄×C₂₁₁	Q₈	1	4	yes	—
31	C₂²×C₄×C₂₁₁	—	2	8	no	—
32	C₂²×C₄×C₂₁₁	—	2	8	no	—
33	C₂²×C₄×C₂₁₁	—	4	4	no	—
34	C₂²×C₄×C₂₁₁	—	4	4	no	—
35	C₂²×C₄×C₂₁₁	C₂×C₄	1	8	yes	complemented
36	C₂²×C₄×C₂₁₁	C₂³	1	4	yes	—
37	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
38	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₂₁₁	—	2	4	no	—
44	C₄⋊C₄×C₂₁₁	—	2	4	no	—
45	C₄⋊C₄×C₂₁₁	—	4	4	no	—
46	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	complemented
47	C₂²⋊C₄×C₂₁₁	—	2	8	no	—
48	C₂²⋊C₄×C₂₁₁	C₂×C₄	1	8	yes	complemented
49	C₄²×C₂₁₁	—	2	4	no	—
50	C₄²×C₂₁₁	C₂×C₄	1	4	yes	complemented
51	C₂³×C₂₁₁	—	2	4	no	—
52	C₂³×C₂₁₁	—	2	4	no	—
53	C₂³×C₂₁₁	C₂²×C₄	1	4	yes	—
54	C₂³×C₂₁₁	C₂×D₄	1	4	yes	—
55	C₂³×C₂₁₁	C₂×Q₈	1	4	yes	—
56	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
57	D₄×C₂₁₁	C₄⋊C₄	1	16	yes	complemented
58	C₂×C₄×C₂₁₁	—	2	2	no	—
59	C₂×C₄×C₂₁₁	—	2	2	no	—
60	C₂×C₄×C₂₁₁	—	2	4	no	—
61	C₂×C₄×C₂₁₁	—	2	2	no	—
62	C₂×C₄×C₂₁₁	—	2	2	no	—
63	C₂×C₄×C₂₁₁	—	2	2	no	—
64	C₂×C₄×C₂₁₁	—	2	4	no	—
65	C₂×C₄×C₂₁₁	—	2	2	no	—
66	C₂×C₄×C₂₁₁	—	2	4	no	—
67	C₂×C₄×C₂₁₁	—	2	2	no	—
68	C₂×C₄×C₂₁₁	—	2	4	no	—
69	C₂×C₄×C₂₁₁	—	4	8	no	—
70	C₂×C₄×C₂₁₁	—	4	16	no	—
71	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₂₁₁	C₂²×C₄	1	4	yes	—
74	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
77	C₂×C₄×C₂₁₁	#41	1	4	yes	—
78	C₂²×C₂₁₁	—	2	4	no	—
79	C₂²×C₂₁₁	—	2	4	no	—
80	C₂²×C₂₁₁	—	2	8	no	—
81	C₂²×C₂₁₁	—	2	8	no	—
82	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	8	yes	—
83	C₂²×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
84	C₂²×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
85	C₂²×C₂₁₁	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×C₂₁₁	C₂×#41	1	2	yes	—
87	C₂²×C₂₁₁	H(5,𝔽₂)	1	2	yes	—
88	C₄×C₂₁₁	—	2	2	no	—
89	C₄×C₂₁₁	—	2	2	no	—
90	C₄×C₂₁₁	—	2	4	no	—
91	C₄×C₂₁₁	—	2	4	no	—
92	C₄×C₂₁₁	—	4	2	no	—
93	C₄×C₂₁₁	—	4	4	no	—
94	C₄×C₂₁₁	C₂×C₄⋊C₄	1	4	yes	—
95	C₂×C₂₁₁	—	2	8	no	—
96	C₂×C₂₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₂₁₁	C₄×#41	1	1	yes	characteristic
98	C₂×C₂₁₁	#518	1	1	yes	characteristic
99	C₂×C₂₁₁	#546	1	2	yes	—
100	C₂×C₂₁₁	#554	1	2	yes	—
101	C₂₁₁	#2349	1	1	yes	characteristic, direct product factor, Sylow
102	#2349	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂×C₄×D₄	C₂×C₂₁₁	1	2	yes	—
104	C₂×C₄×D₄	C₂×C₂₁₁	1	1	yes	characteristic
105	#389	C₂×C₂₁₁	1	2	yes	complemented
106	#385	C₂×C₂₁₁	1	4	yes	complemented
107	#382	C₂×C₂₁₁	1	1	yes	characteristic, complemented
108	#380	C₂×C₂₁₁	1	2	yes	complemented
109	C₄×C₄⋊C₄	C₂×C₂₁₁	1	1	yes	characteristic, complemented
110	C₂×#95	C₂×C₂₁₁	1	2	yes	complemented
111	C₂²×D₄	C₂²×C₂₁₁	1	1	yes	characteristic
112	C₂³×C₄	C₂²×C₂₁₁	1	4	yes	—
113	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
114	C₄×D₄	—	2	8	no	—
115	C₄×D₄	C₄×C₂₁₁	1	8	yes	complemented
116	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
117	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
118	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	4	yes	—
119	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
120	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	4	yes	—
121	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
122	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
123	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
124	#95	C₂²×C₂₁₁	1	1	yes	characteristic
125	#95	C₂²×C₂₁₁	1	2	yes	—
126	#95	C₂²×C₂₁₁	1	4	yes	—
127	#95	C₂²×C₂₁₁	1	1	yes	characteristic
128	#95	C₂²×C₂₁₁	1	2	yes	—
129	C₂⁴	C₂³×C₂₁₁	1	2	yes	—
130	C₂×D₄	C₂×C₄×C₂₁₁	1	4	yes	—
131	C₂×D₄	D₄×C₂₁₁	1	4	yes	—
132	C₂×D₄	Q₈×C₂₁₁	1	4	yes	—
133	C₂²×C₄	—	2	8	no	—
134	C₂²×C₄	—	2	8	no	—
135	C₂²×C₄	—	4	4	no	—
136	C₂²×C₄	—	4	4	no	—
137	C₂²×C₄	C₂×C₄×C₂₁₁	1	8	yes	complemented
138	C₂²×C₄	C₂³×C₂₁₁	1	4	yes	—
139	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
140	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
141	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
142	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
143	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
144	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	4	4	no	—
148	C₄⋊C₄	C₂×C₄×C₂₁₁	1	4	yes	complemented
149	C₂²⋊C₄	—	2	8	no	—
150	C₂²⋊C₄	C₂×C₄×C₂₁₁	1	8	yes	complemented
151	C₄²	—	2	4	no	—
152	C₄²	C₂×C₄×C₂₁₁	1	4	yes	complemented
153	C₂³	—	2	4	no	—
154	C₂³	—	2	4	no	—
155	C₂³	C₂²×C₄×C₂₁₁	1	4	yes	—
156	C₂³	C₂×D₄×C₂₁₁	1	4	yes	—
157	C₂³	C₂×Q₈×C₂₁₁	1	4	yes	—
158	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
159	D₄	C₄⋊C₄×C₂₁₁	1	16	yes	complemented
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	4	8	no	—
172	C₂×C₄	—	4	16	no	—
173	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
174	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
175	C₂×C₄	C₂²×C₄×C₂₁₁	1	4	yes	—
176	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
177	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
178	C₂×C₄	C₂×Q₈×C₂₁₁	1	2	yes	—
179	C₂×C₄	#41×C₂₁₁	1	4	yes	—
180	C₂²	—	2	8	no	—
181	C₂²	—	2	8	no	—
182	C₂²	—	2	4	no	—
183	C₂²	—	2	4	no	—
184	C₂²	C₂×C₄⋊C₄×C₂₁₁	1	8	yes	—
185	C₂²	C₂³×C₄×C₂₁₁	1	1	yes	characteristic
186	C₂²	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
187	C₂²	C₂²×Q₈×C₂₁₁	1	1	yes	characteristic
188	C₂²	C₂×#41×C₂₁₁	1	2	yes	—
189	C₂²	H(5,𝔽₂)×C₂₁₁	1	2	yes	—
190	C₄	—	2	4	no	—
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	2	4	no	—
194	C₄	—	4	4	no	—
195	C₄	—	4	2	no	—
196	C₄	C₂×C₄⋊C₄×C₂₁₁	1	4	yes	—
197	C₂	—	2	8	no	—
198	C₂	C₂²×C₄⋊C₄×C₂₁₁	1	1	yes	characteristic
199	C₂	C₄×#41×C₂₁₁	1	1	yes	characteristic
200	C₂	#518×C₂₁₁	1	1	yes	characteristic
201	C₂	#546×C₂₁₁	1	2	yes	—
202	C₂	#554×C₂₁₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₁₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₁₁
Frattini, cosocle	C₂³	C₂⁴×C₂₁₁
prime radical	C₂²×D₄×C₂₁₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. #389×C₂₁₁	C₂	semidirect
4. #385×C₂₁₁	C₂	semidirect
5. #382×C₂₁₁	C₂	semidirect
6. #380×C₂₁₁	C₂	semidirect
7. C₄×C₄⋊C₄×C₂₁₁	C₂	semidirect
8. C₂×#95×C₂₁₁	C₂	semidirect
13. C₄×D₄×C₂₁₁	C₄	semidirect
35. C₂²×C₄×C₂₁₁	C₂×C₄	semidirect
46. C₄⋊C₄×C₂₁₁	C₂×C₄	semidirect
48. C₂²⋊C₄×C₂₁₁	C₂×C₄	semidirect
50. C₄²×C₂₁₁	C₂×C₄	semidirect
57. D₄×C₂₁₁	C₄⋊C₄	semidirect
101. C₂₁₁	#2349	direct (N × H)
102. #2349	C₂₁₁	direct (N × H)
105. #389	C₂×C₂₁₁	semidirect
106. #385	C₂×C₂₁₁	semidirect
107. #382	C₂×C₂₁₁	semidirect
108. #380	C₂×C₂₁₁	semidirect
109. C₄×C₄⋊C₄	C₂×C₂₁₁	semidirect
110. C₂×#95	C₂×C₂₁₁	semidirect
115. C₄×D₄	C₄×C₂₁₁	semidirect
137. C₂²×C₄	C₂×C₄×C₂₁₁	semidirect
148. C₄⋊C₄	C₂×C₄×C₂₁₁	semidirect
150. C₂²⋊C₄	C₂×C₄×C₂₁₁	semidirect
152. C₄²	C₂×C₄×C₂₁₁	semidirect
159. D₄	C₄⋊C₄×C₂₁₁	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×C₂₁₁	2
#389×C₂₁₁	2
#385×C₂₁₁	2
#382×C₂₁₁	2
#380×C₂₁₁	2
C₄×C₄⋊C₄×C₂₁₁	2
C₂×#95×C₂₁₁	2
#2349	211

Subgroups — #2349×C211 #41009969

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2349×C₂₁₁ #41009969