Subgroups — #2357×C₂₁₁ #41009977

Order 27008.

560 conjugacy classes of subgroups.

The subgroup lattice has 196 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	1	yes	maximal, characteristic
2	#390×C₂₁₁	C₂	1	2	yes	maximal
3	#388×C₂₁₁	C₂	1	4	yes	maximal, complemented
4	#382×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#381×C₂₁₁	C₂	1	2	yes	maximal, complemented
6	C₄×C₄⋊C₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₄⋊C₄×C₂₁₁	C₂	1	2	yes	maximal, complemented
8	C₄×C₂²⋊C₄×C₂₁₁	C₂	1	2	yes	maximal, complemented
9	C₂²×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
11	C₄×D₄×C₂₁₁	C₄	1	8	yes	complemented
12	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	4	yes	—
16	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	4	yes	—
17	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
18	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
19	C₂×C₄²×C₂₁₁	C₂²	1	4	yes	—
20	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
21	#95×C₂₁₁	C₂²	1	1	yes	characteristic
22	#95×C₂₁₁	C₂²	1	2	yes	—
23	#95×C₂₁₁	C₂²	1	1	yes	characteristic
24	#95×C₂₁₁	C₂²	1	4	yes	—
25	#95×C₂₁₁	C₂²	1	2	yes	—
26	C₂⁴×C₂₁₁	C₂³	1	2	yes	—
27	C₂×D₄×C₂₁₁	—	2	2	no	—
28	C₂×D₄×C₂₁₁	—	2	2	no	—
29	C₂×D₄×C₂₁₁	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₂₁₁	—	4	4	no	—
31	C₂²×C₄×C₂₁₁	C₂×C₄	1	8	yes	—
32	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
33	C₂²×C₄×C₂₁₁	C₂³	1	4	yes	—
34	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
36	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
38	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₂₁₁	—	2	4	no	—
40	C₄⋊C₄×C₂₁₁	—	2	4	no	—
41	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	—
42	C₂²⋊C₄×C₂₁₁	—	2	8	no	—
43	C₂²⋊C₄×C₂₁₁	—	4	4	no	—
44	C₂²⋊C₄×C₂₁₁	C₂×C₄	1	8	yes	—
45	C₄²×C₂₁₁	—	2	4	no	—
46	C₄²×C₂₁₁	—	4	4	no	—
47	C₄²×C₂₁₁	C₂×C₄	1	4	yes	—
48	C₂³×C₂₁₁	—	2	4	no	—
49	C₂³×C₂₁₁	—	2	2	no	—
50	C₂³×C₂₁₁	—	2	2	no	—
51	C₂³×C₂₁₁	—	2	4	no	—
52	C₂³×C₂₁₁	C₂²×C₄	1	4	yes	—
53	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
54	D₄×C₂₁₁	—	2	8	no	—
55	C₂×C₄×C₂₁₁	—	2	2	no	—
56	C₂×C₄×C₂₁₁	—	2	2	no	—
57	C₂×C₄×C₂₁₁	—	2	2	no	—
58	C₂×C₄×C₂₁₁	—	2	4	no	—
59	C₂×C₄×C₂₁₁	—	2	2	no	—
60	C₂×C₄×C₂₁₁	—	2	2	no	—
61	C₂×C₄×C₂₁₁	—	2	4	no	—
62	C₂×C₄×C₂₁₁	—	2	2	no	—
63	C₂×C₄×C₂₁₁	—	2	4	no	—
64	C₂×C₄×C₂₁₁	—	2	4	no	—
65	C₂×C₄×C₂₁₁	—	2	2	no	—
66	C₂×C₄×C₂₁₁	—	4	8	no	—
67	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₂₁₁	C₂²×C₄	1	4	yes	—
70	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₂₁₁	#41	1	4	yes	—
72	C₂×C₄×C₂₁₁	#41	1	2	yes	—
73	C₂×C₄×C₂₁₁	#41	1	2	yes	—
74	C₂²×C₂₁₁	—	2	4	no	—
75	C₂²×C₂₁₁	—	2	4	no	—
76	C₂²×C₂₁₁	—	2	4	no	—
77	C₂²×C₂₁₁	—	4	4	no	—
78	C₂²×C₂₁₁	—	4	4	no	—
79	C₂²×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
80	C₂²×C₂₁₁	C₂×#41	1	1	yes	characteristic
81	C₂²×C₂₁₁	C₂×#41	1	1	yes	characteristic
82	C₂²×C₂₁₁	C₂×#41	1	2	yes	—
83	C₂²×C₂₁₁	H(5,𝔽₂)	1	2	yes	—
84	C₄×C₂₁₁	—	2	2	no	—
85	C₄×C₂₁₁	—	2	2	no	—
86	C₄×C₂₁₁	—	2	4	no	—
87	C₄×C₂₁₁	—	2	4	no	—
88	C₄×C₂₁₁	—	4	2	no	—
89	C₄×C₂₁₁	—	4	4	no	—
90	C₄×C₂₁₁	#117	1	4	yes	—
91	C₂×C₂₁₁	—	4	4	no	—
92	C₂×C₂₁₁	C₂×#117	1	1	yes	characteristic
93	C₂×C₂₁₁	C₄×#41	1	1	yes	characteristic
94	C₂×C₂₁₁	#518	1	1	yes	characteristic
95	C₂×C₂₁₁	#553	1	2	yes	—
96	C₂×C₂₁₁	#559	1	2	yes	—
97	C₂₁₁	#2357	1	1	yes	characteristic, direct product factor, Sylow
98	#2357	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂×C₄×D₄	C₂×C₂₁₁	1	1	yes	characteristic
100	#390	C₂×C₂₁₁	1	2	yes	—
101	#388	C₂×C₂₁₁	1	4	yes	complemented
102	#382	C₂×C₂₁₁	1	1	yes	characteristic, complemented
103	#381	C₂×C₂₁₁	1	2	yes	complemented
104	C₄×C₄⋊C₄	C₂×C₂₁₁	1	1	yes	characteristic, complemented
105	C₄×C₄⋊C₄	C₂×C₂₁₁	1	2	yes	complemented
106	C₄×C₂²⋊C₄	C₂×C₂₁₁	1	2	yes	complemented
107	C₂²×D₄	C₂²×C₂₁₁	1	1	yes	characteristic
108	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
109	C₄×D₄	C₄×C₂₁₁	1	8	yes	complemented
110	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
111	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
112	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
113	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	4	yes	—
114	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	4	yes	—
115	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
116	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
117	C₂×C₄²	C₂²×C₂₁₁	1	4	yes	—
118	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
119	#95	C₂²×C₂₁₁	1	1	yes	characteristic
120	#95	C₂²×C₂₁₁	1	2	yes	—
121	#95	C₂²×C₂₁₁	1	1	yes	characteristic
122	#95	C₂²×C₂₁₁	1	4	yes	—
123	#95	C₂²×C₂₁₁	1	2	yes	—
124	C₂⁴	C₂³×C₂₁₁	1	2	yes	—
125	C₂×D₄	—	2	2	no	—
126	C₂×D₄	—	2	2	no	—
127	C₂×D₄	C₂×C₄×C₂₁₁	1	4	yes	—
128	C₂²×C₄	—	4	4	no	—
129	C₂²×C₄	C₂×C₄×C₂₁₁	1	8	yes	—
130	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
131	C₂²×C₄	C₂³×C₂₁₁	1	4	yes	—
132	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
133	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
134	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
135	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
136	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
137	C₄⋊C₄	—	2	4	no	—
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	C₂×C₄×C₂₁₁	1	4	yes	—
140	C₂²⋊C₄	—	2	8	no	—
141	C₂²⋊C₄	—	4	4	no	—
142	C₂²⋊C₄	C₂×C₄×C₂₁₁	1	8	yes	—
143	C₄²	—	2	4	no	—
144	C₄²	—	4	4	no	—
145	C₄²	C₂×C₄×C₂₁₁	1	4	yes	—
146	C₂³	—	2	4	no	—
147	C₂³	—	2	2	no	—
148	C₂³	—	2	2	no	—
149	C₂³	—	2	4	no	—
150	C₂³	C₂²×C₄×C₂₁₁	1	4	yes	—
151	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
152	D₄	—	2	8	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	4	8	no	—
165	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
166	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
167	C₂×C₄	C₂²×C₄×C₂₁₁	1	4	yes	—
168	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
169	C₂×C₄	#41×C₂₁₁	1	4	yes	—
170	C₂×C₄	#41×C₂₁₁	1	2	yes	—
171	C₂×C₄	#41×C₂₁₁	1	2	yes	—
172	C₂²	—	2	4	no	—
173	C₂²	—	2	4	no	—
174	C₂²	—	2	4	no	—
175	C₂²	—	4	4	no	—
176	C₂²	—	4	4	no	—
177	C₂²	C₂³×C₄×C₂₁₁	1	1	yes	characteristic
178	C₂²	C₂×#41×C₂₁₁	1	1	yes	characteristic
179	C₂²	C₂×#41×C₂₁₁	1	2	yes	—
180	C₂²	C₂×#41×C₂₁₁	1	1	yes	characteristic
181	C₂²	H(5,𝔽₂)×C₂₁₁	1	2	yes	—
182	C₄	—	2	4	no	—
183	C₄	—	2	2	no	—
184	C₄	—	2	2	no	—
185	C₄	—	2	4	no	—
186	C₄	—	4	2	no	—
187	C₄	—	4	4	no	—
188	C₄	#117×C₂₁₁	1	4	yes	—
189	C₂	—	4	4	no	—
190	C₂	C₂×#117×C₂₁₁	1	1	yes	characteristic
191	C₂	C₄×#41×C₂₁₁	1	1	yes	characteristic
192	C₂	#518×C₂₁₁	1	1	yes	characteristic
193	C₂	#553×C₂₁₁	1	2	yes	—
194	C₂	#559×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. #388×C₂₁₁	C₂	semidirect
4. #382×C₂₁₁	C₂	semidirect
5. #381×C₂₁₁	C₂	semidirect
6. C₄×C₄⋊C₄×C₂₁₁	C₂	semidirect
7. C₄×C₄⋊C₄×C₂₁₁	C₂	semidirect
8. C₄×C₂²⋊C₄×C₂₁₁	C₂	semidirect
11. C₄×D₄×C₂₁₁	C₄	semidirect
97. C₂₁₁	#2357	direct (N × H)
98. #2357	C₂₁₁	direct (N × H)
101. #388	C₂×C₂₁₁	semidirect
102. #382	C₂×C₂₁₁	semidirect
103. #381	C₂×C₂₁₁	semidirect
104. C₄×C₄⋊C₄	C₂×C₂₁₁	semidirect
105. C₄×C₄⋊C₄	C₂×C₂₁₁	semidirect
106. C₄×C₂²⋊C₄	C₂×C₂₁₁	semidirect
109. C₄×D₄	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
#390×C₂₁₁	2
#388×C₂₁₁	2
#382×C₂₁₁	2
#381×C₂₁₁	2
C₄×C₄⋊C₄×C₂₁₁	2
C₄×C₂²⋊C₄×C₂₁₁	2
#2357	211

Subgroups — #2357×C211 #41009977

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2357×C₂₁₁ #41009977