Subgroups — #2314×C₂₁₁ #41009934

Order 27008.

484 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#117×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	#388×C₂₁₁	C₂	1	4	yes	maximal
3	#387×C₂₁₁	C₂	1	4	yes	maximal, complemented
4	#384×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#382×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#380×C₂₁₁	C₂	1	2	yes	maximal
7	C₄×C₄⋊C₄×C₂₁₁	C₂	1	2	yes	maximal, complemented
8	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
9	#117×C₂₁₁	C₄	1	8	yes	complemented
10	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	4	yes	—
13	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
14	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	4	yes	—
15	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
16	C₂×C₄²×C₂₁₁	C₂²	1	4	yes	—
17	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
19	#95×C₂₁₁	C₂²	1	4	yes	—
20	#95×C₂₁₁	C₂²	1	4	yes	—
21	#95×C₂₁₁	C₂²	1	4	yes	—
22	C₂⁴×C₂₁₁	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₂₁₁	—	2	2	no	—
24	C₂²×C₄×C₂₁₁	—	2	2	no	—
25	C₂²×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₂₁₁	C₂³	1	4	yes	—
27	C₂²×C₄×C₂₁₁	C₂³	1	4	yes	—
28	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
30	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
31	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₂₁₁	—	2	4	no	—
33	C₄⋊C₄×C₂₁₁	—	2	4	no	—
34	C₄⋊C₄×C₂₁₁	—	4	4	no	—
35	C₄⋊C₄×C₂₁₁	C₂×C₄	1	8	yes	—
36	C₂²⋊C₄×C₂₁₁	—	4	4	no	—
37	C₂²⋊C₄×C₂₁₁	C₂×C₄	1	8	yes	—
38	C₄²×C₂₁₁	—	4	4	no	—
39	C₄²×C₂₁₁	C₂×C₄	1	4	yes	—
40	C₄²×C₂₁₁	C₂×C₄	1	4	yes	—
41	C₂³×C₂₁₁	—	2	1	no	—
42	C₂³×C₂₁₁	—	2	1	no	—
43	C₂³×C₂₁₁	—	2	2	no	—
44	C₂³×C₂₁₁	—	2	2	no	—
45	C₂³×C₂₁₁	C₂²×C₄	1	2	yes	—
46	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
47	C₂×C₄×C₂₁₁	—	2	1	no	—
48	C₂×C₄×C₂₁₁	—	2	1	no	—
49	C₂×C₄×C₂₁₁	—	2	2	no	—
50	C₂×C₄×C₂₁₁	—	2	2	no	—
51	C₂×C₄×C₂₁₁	—	2	2	no	—
52	C₂×C₄×C₂₁₁	—	2	2	no	—
53	C₂×C₄×C₂₁₁	—	2	8	no	—
54	C₂×C₄×C₂₁₁	—	2	4	no	—
55	C₂×C₄×C₂₁₁	—	2	4	no	—
56	C₂×C₄×C₂₁₁	—	2	4	no	—
57	C₂×C₄×C₂₁₁	—	2	4	no	—
58	C₂×C₄×C₂₁₁	—	2	4	no	—
59	C₂×C₄×C₂₁₁	—	2	4	no	—
60	C₂×C₄×C₂₁₁	C₂²×C₄	1	4	yes	—
61	C₂×C₄×C₂₁₁	C₂²×C₄	1	4	yes	—
62	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₂₁₁	#41	1	2	yes	—
65	C₂×C₄×C₂₁₁	#41	1	2	yes	—
66	C₂²×C₂₁₁	—	2	2	no	—
67	C₂²×C₂₁₁	—	2	4	no	—
68	C₂²×C₂₁₁	—	4	1	no	—
69	C₂²×C₂₁₁	—	4	1	no	—
70	C₂²×C₂₁₁	—	4	2	no	—
71	C₂²×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
72	C₂²×C₂₁₁	C₂×#41	1	1	yes	characteristic
73	C₂²×C₂₁₁	C₂×#41	1	1	yes	characteristic
74	C₂²×C₂₁₁	H(5,𝔽₂)	1	2	yes	—
75	C₂²×C₂₁₁	2₋¹⁺⁴	1	2	yes	—
76	C₄×C₂₁₁	—	2	2	no	—
77	C₄×C₂₁₁	—	2	4	no	—
78	C₄×C₂₁₁	—	2	4	no	—
79	C₄×C₂₁₁	—	4	4	no	—
80	C₄×C₂₁₁	—	4	4	no	—
81	C₄×C₂₁₁	#117	1	4	yes	—
82	C₂×C₂₁₁	—	4	2	no	—
83	C₂×C₂₁₁	C₂×#117	1	1	yes	characteristic
84	C₂×C₂₁₁	#520	1	2	yes	—
85	C₂×C₂₁₁	#540	1	1	yes	characteristic
86	C₂×C₂₁₁	#541	1	1	yes	characteristic
87	C₂×C₂₁₁	#542	1	2	yes	—
88	C₂₁₁	#2314	1	1	yes	characteristic, direct product factor, Sylow
89	#2314	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂×#117	C₂×C₂₁₁	1	1	yes	characteristic
91	#388	C₂×C₂₁₁	1	4	yes	—
92	#387	C₂×C₂₁₁	1	4	yes	complemented
93	#384	C₂×C₂₁₁	1	1	yes	characteristic, complemented
94	#382	C₂×C₂₁₁	1	1	yes	characteristic, complemented
95	#380	C₂×C₂₁₁	1	2	yes	—
96	C₄×C₄⋊C₄	C₂×C₂₁₁	1	2	yes	complemented
97	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
98	#117	C₄×C₂₁₁	1	8	yes	complemented
99	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
100	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
101	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	4	yes	—
102	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
103	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	4	yes	—
104	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	2	yes	—
105	C₂×C₄²	C₂²×C₂₁₁	1	4	yes	—
106	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
107	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
108	#95	C₂²×C₂₁₁	1	4	yes	—
109	#95	C₂²×C₂₁₁	1	4	yes	—
110	#95	C₂²×C₂₁₁	1	4	yes	—
111	C₂⁴	C₂³×C₂₁₁	1	1	yes	characteristic
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	C₂×C₄×C₂₁₁	1	4	yes	—
115	C₂²×C₄	C₂³×C₂₁₁	1	4	yes	—
116	C₂²×C₄	C₂³×C₂₁₁	1	4	yes	—
117	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
118	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
119	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
120	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	—	2	4	no	—
123	C₄⋊C₄	—	4	4	no	—
124	C₄⋊C₄	C₂×C₄×C₂₁₁	1	8	yes	—
125	C₂²⋊C₄	—	4	4	no	—
126	C₂²⋊C₄	C₂×C₄×C₂₁₁	1	8	yes	—
127	C₄²	—	4	4	no	—
128	C₄²	C₂×C₄×C₂₁₁	1	4	yes	—
129	C₄²	C₂×C₄×C₂₁₁	1	4	yes	—
130	C₂³	—	2	1	no	—
131	C₂³	—	2	1	no	—
132	C₂³	—	2	2	no	—
133	C₂³	—	2	2	no	—
134	C₂³	C₂²×C₄×C₂₁₁	1	2	yes	—
135	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	8	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	C₂²×C₄×C₂₁₁	1	4	yes	—
150	C₂×C₄	C₂²×C₄×C₂₁₁	1	4	yes	—
151	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
152	C₂×C₄	C₂²×C₄×C₂₁₁	1	2	yes	—
153	C₂×C₄	#41×C₂₁₁	1	2	yes	—
154	C₂×C₄	#41×C₂₁₁	1	2	yes	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	4	no	—
157	C₂²	—	4	1	no	—
158	C₂²	—	4	1	no	—
159	C₂²	—	4	2	no	—
160	C₂²	C₂³×C₄×C₂₁₁	1	1	yes	characteristic
161	C₂²	C₂×#41×C₂₁₁	1	1	yes	characteristic
162	C₂²	C₂×#41×C₂₁₁	1	1	yes	characteristic
163	C₂²	H(5,𝔽₂)×C₂₁₁	1	2	yes	—
164	C₂²	2₋¹⁺⁴×C₂₁₁	1	2	yes	—
165	C₄	—	2	2	no	—
166	C₄	—	2	4	no	—
167	C₄	—	2	4	no	—
168	C₄	—	4	4	no	—
169	C₄	—	4	4	no	—
170	C₄	#117×C₂₁₁	1	4	yes	—
171	C₂	—	4	2	no	—
172	C₂	C₂×#117×C₂₁₁	1	1	yes	characteristic
173	C₂	#520×C₂₁₁	1	2	yes	—
174	C₂	#540×C₂₁₁	1	1	yes	characteristic
175	C₂	#541×C₂₁₁	1	1	yes	characteristic
176	C₂	#542×C₂₁₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₁₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₁₁
Frattini, cosocle	C₂³	C₂⁴×C₂₁₁
prime radical	C₂⁴×C₂₁₁	C₂³
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. #387×C₂₁₁	C₂	semidirect
4. #384×C₂₁₁	C₂	semidirect
5. #382×C₂₁₁	C₂	semidirect
7. C₄×C₄⋊C₄×C₂₁₁	C₂	semidirect
9. #117×C₂₁₁	C₄	semidirect
88. C₂₁₁	#2314	direct (N × H)
89. #2314	C₂₁₁	direct (N × H)
92. #387	C₂×C₂₁₁	semidirect
93. #384	C₂×C₂₁₁	semidirect
94. #382	C₂×C₂₁₁	semidirect
96. C₄×C₄⋊C₄	C₂×C₂₁₁	semidirect
98. #117	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×#117×C₂₁₁	2
#388×C₂₁₁	2
#387×C₂₁₁	2
#384×C₂₁₁	2
#382×C₂₁₁	2
#380×C₂₁₁	2
C₄×C₄⋊C₄×C₂₁₁	2
#2314	211

Subgroups — #2314×C211 #41009934

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2314×C₂₁₁ #41009934