Subgroups — #2851×C₂₁₁ #41010471

Order 27008.

556 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#125×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄²×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	#389×C₂₁₁	C₂	1	2	yes	maximal, complemented
4	#387×C₂₁₁	C₂	1	4	yes	maximal, complemented
5	#385×C₂₁₁	C₂	1	4	yes	maximal
6	#382×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#380×C₂₁₁	C₂	1	2	yes	maximal
8	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
9	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
10	#125×C₂₁₁	—	2	4	no	—
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×C₄⋊C₄×C₂₁₁	C₂²	1	4	yes	—
15	C₂×C₂²⋊C₄×C₂₁₁	C₂²	1	4	yes	—
16	C₂×C₄²×C₂₁₁	—	2	4	no	—
17	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
18	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
20	#95×C₂₁₁	C₂²	1	4	yes	—
21	#95×C₂₁₁	C₂²	1	4	yes	—
22	#95×C₂₁₁	C₂²	1	4	yes	—
23	C₂⁴×C₂₁₁	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₂₁₁	—	2	4	no	—
25	C₂²×C₄×C₂₁₁	—	2	4	no	—
26	C₂²×C₄×C₂₁₁	—	2	2	no	—
27	C₂²×C₄×C₂₁₁	—	2	4	no	—
28	C₂²×C₄×C₂₁₁	—	2	2	no	—
29	C₂²×C₄×C₂₁₁	Q₈	1	4	yes	—
30	C₂²×C₄×C₂₁₁	C₂³	1	4	yes	—
31	C₂²×C₄×C₂₁₁	C₂³	1	4	yes	—
32	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
33	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₂₁₁	C₂³	1	2	yes	—
35	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₂₁₁	—	2	8	no	—
37	C₄⋊C₄×C₂₁₁	—	2	4	no	—
38	C₄⋊C₄×C₂₁₁	—	2	8	no	—
39	C₄⋊C₄×C₂₁₁	—	2	4	no	—
40	C₂²⋊C₄×C₂₁₁	—	2	8	no	—
41	C₄²×C₂₁₁	—	2	4	no	—
42	C₄²×C₂₁₁	—	2	2	no	—
43	C₄²×C₂₁₁	D₄	1	4	yes	—
44	C₂³×C₂₁₁	—	2	1	no	—
45	C₂³×C₂₁₁	—	2	1	no	—
46	C₂³×C₂₁₁	—	2	2	no	—
47	C₂³×C₂₁₁	C₂×Q₈	1	2	yes	—
48	C₂³×C₂₁₁	#41	1	4	yes	—
49	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
50	C₂×C₄×C₂₁₁	—	2	16	no	—
51	C₂×C₄×C₂₁₁	—	2	1	no	—
52	C₂×C₄×C₂₁₁	—	2	2	no	—
53	C₂×C₄×C₂₁₁	—	2	1	no	—
54	C₂×C₄×C₂₁₁	—	2	2	no	—
55	C₂×C₄×C₂₁₁	—	2	2	no	—
56	C₂×C₄×C₂₁₁	—	2	2	no	—
57	C₂×C₄×C₂₁₁	—	2	8	no	—
58	C₂×C₄×C₂₁₁	—	2	4	no	—
59	C₂×C₄×C₂₁₁	—	2	4	no	—
60	C₂×C₄×C₂₁₁	—	2	4	no	—
61	C₂×C₄×C₂₁₁	—	2	4	no	—
62	C₂×C₄×C₂₁₁	—	2	4	no	—
63	C₂×C₄×C₂₁₁	—	2	4	no	—
64	C₂×C₄×C₂₁₁	C₂×D₄	1	4	yes	—
65	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
67	C₂×C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
68	C₂×C₄×C₂₁₁	#41	1	4	yes	—
69	C₂×C₄×C₂₁₁	#41	1	2	yes	—
70	C₂²×C₂₁₁	—	2	2	no	—
71	C₂²×C₂₁₁	—	2	4	no	—
72	C₂²×C₂₁₁	—	2	4	no	—
73	C₂²×C₂₁₁	—	2	2	no	—
74	C₂²×C₂₁₁	#125	1	4	yes	complemented
75	C₂²×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
76	C₂²×C₂₁₁	C₂²×Q₈	1	1	yes	characteristic
77	C₂²×C₂₁₁	C₂×#41	1	1	yes	characteristic
78	C₂²×C₂₁₁	C₂×#41	1	2	yes	—
79	C₂²×C₂₁₁	C₂×#41	1	2	yes	—
80	C₄×C₂₁₁	—	2	2	no	—
81	C₄×C₂₁₁	—	2	4	no	—
82	C₄×C₂₁₁	—	2	4	no	—
83	C₄×C₂₁₁	—	4	4	no	—
84	C₄×C₂₁₁	—	4	4	no	—
85	C₄×C₂₁₁	C₂²⋊Q₈	1	4	yes	—
86	C₂×C₂₁₁	—	2	4	no	—
87	C₂×C₂₁₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
88	C₂×C₂₁₁	#525	1	2	yes	—
89	C₂×C₂₁₁	C₂×#125	1	1	yes	characteristic
90	C₂×C₂₁₁	#529	1	2	yes	—
91	C₂×C₂₁₁	#533	1	1	yes	characteristic
92	C₂₁₁	#2851	1	1	yes	characteristic, direct product factor, Sylow
93	#2851	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂×#125	C₂×C₂₁₁	1	1	yes	characteristic, complemented
95	C₂²×C₄²	C₂×C₂₁₁	1	1	yes	characteristic
96	#389	C₂×C₂₁₁	1	2	yes	complemented
97	#387	C₂×C₂₁₁	1	4	yes	complemented
98	#385	C₂×C₂₁₁	1	4	yes	—
99	#382	C₂×C₂₁₁	1	1	yes	characteristic, complemented
100	#380	C₂×C₂₁₁	1	2	yes	—
101	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
102	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
103	#125	—	2	4	no	—
104	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
105	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	4	yes	—
106	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
107	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	4	yes	—
108	C₂×C₂²⋊C₄	C₂²×C₂₁₁	1	4	yes	—
109	C₂×C₄²	—	2	4	no	—
110	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
111	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
112	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
113	#95	C₂²×C₂₁₁	1	4	yes	—
114	#95	C₂²×C₂₁₁	1	4	yes	—
115	#95	C₂²×C₂₁₁	1	4	yes	—
116	C₂⁴	C₂³×C₂₁₁	1	1	yes	characteristic
117	C₂²×C₄	—	2	4	no	—
118	C₂²×C₄	—	2	4	no	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	—	2	4	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	Q₈×C₂₁₁	1	4	yes	—
123	C₂²×C₄	C₂³×C₂₁₁	1	4	yes	—
124	C₂²×C₄	C₂³×C₂₁₁	1	4	yes	—
125	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
126	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
127	C₂²×C₄	C₂³×C₂₁₁	1	2	yes	—
128	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
129	C₄⋊C₄	—	2	8	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	2	8	no	—
132	C₄⋊C₄	—	2	4	no	—
133	C₂²⋊C₄	—	2	8	no	—
134	C₄²	—	2	4	no	—
135	C₄²	—	2	2	no	—
136	C₄²	D₄×C₂₁₁	1	4	yes	—
137	C₂³	—	2	1	no	—
138	C₂³	—	2	1	no	—
139	C₂³	—	2	2	no	—
140	C₂³	C₂×Q₈×C₂₁₁	1	2	yes	—
141	C₂³	#41×C₂₁₁	1	4	yes	—
142	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
143	C₂×C₄	—	2	16	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	8	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	C₂×D₄×C₂₁₁	1	4	yes	—
158	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
159	C₂×C₄	C₂×Q₈×C₂₁₁	1	2	yes	—
160	C₂×C₄	C₂×Q₈×C₂₁₁	1	2	yes	—
161	C₂×C₄	#41×C₂₁₁	1	4	yes	—
162	C₂×C₄	#41×C₂₁₁	1	2	yes	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	4	no	—
165	C₂²	—	2	4	no	—
166	C₂²	—	2	2	no	—
167	C₂²	#125×C₂₁₁	1	4	yes	complemented
168	C₂²	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
169	C₂²	C₂²×Q₈×C₂₁₁	1	1	yes	characteristic
170	C₂²	C₂×#41×C₂₁₁	1	1	yes	characteristic
171	C₂²	C₂×#41×C₂₁₁	1	2	yes	—
172	C₂²	C₂×#41×C₂₁₁	1	2	yes	—
173	C₄	—	2	2	no	—
174	C₄	—	2	4	no	—
175	C₄	—	2	4	no	—
176	C₄	—	4	4	no	—
177	C₄	—	4	4	no	—
178	C₄	C₂²⋊Q₈×C₂₁₁	1	4	yes	—
179	C₂	—	2	4	no	—
180	C₂	C₂×C₂²⋊Q₈×C₂₁₁	1	1	yes	characteristic
181	C₂	#525×C₂₁₁	1	2	yes	—
182	C₂	C₂×#125×C₂₁₁	1	1	yes	characteristic
183	C₂	#529×C₂₁₁	1	2	yes	—
184	C₂	#533×C₂₁₁	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×#125×C₂₁₁	C₂	semidirect
3. #389×C₂₁₁	C₂	semidirect
4. #387×C₂₁₁	C₂	semidirect
6. #382×C₂₁₁	C₂	semidirect
74. C₂²×C₂₁₁	#125	semidirect
92. C₂₁₁	#2851	direct (N × H)
93. #2851	C₂₁₁	direct (N × H)
94. C₂×#125	C₂×C₂₁₁	semidirect
96. #389	C₂×C₂₁₁	semidirect
97. #387	C₂×C₂₁₁	semidirect
99. #382	C₂×C₂₁₁	semidirect
167. C₂²	#125×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#125×C₂₁₁	2
C₂²×C₄²×C₂₁₁	2
#389×C₂₁₁	2
#387×C₂₁₁	2
#385×C₂₁₁	2
#382×C₂₁₁	2
#380×C₂₁₁	2
#2851	211

Subgroups — #2851×C211 #41010471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2851×C₂₁₁ #41010471