Subgroups — #1854×C₂₁₁ #41009474

Order 27008.

316 conjugacy classes of subgroups.

The subgroup lattice has 242 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	C₂×#107×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#106×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂²⋊C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	#380×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	#340×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#340×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
11	#117×C₂₁₁	—	2	2	no	—
12	#117×C₂₁₁	—	2	2	no	—
13	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₄×C₂₁₁	—	2	1	no	—
18	C₂×C₂²⋊C₄×C₂₁₁	—	2	1	no	—
19	C₂×C₄²×C₂₁₁	—	2	1	no	—
20	#107×C₂₁₁	—	2	2	no	—
21	#106×C₂₁₁	—	2	2	no	—
22	C₂²⋊C₈×C₂₁₁	C₄	1	4	yes	complemented
23	#95×C₂₁₁	—	2	1	no	—
24	C₂⁴×C₂₁₁	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₁₁	—	2	1	no	—
26	C₂²×C₄×C₂₁₁	—	2	1	no	—
27	C₂²×C₄×C₂₁₁	—	2	1	no	—
28	C₂²×C₄×C₂₁₁	—	2	1	no	—
29	C₂²×C₄×C₂₁₁	—	2	1	no	—
30	C₂²×C₄×C₂₁₁	—	2	1	no	—
31	C₂²×C₄×C₂₁₁	—	2	1	no	—
32	C₂²×C₄×C₂₁₁	—	2	1	no	—
33	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₂₁₁	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₂₁₁	Q₈	1	2	yes	—
36	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₂₁₁	—	2	1	no	—
38	C₂×C₈×C₂₁₁	—	2	1	no	—
39	C₂×C₈×C₂₁₁	—	2	1	no	—
40	C₂×C₈×C₂₁₁	—	2	1	no	—
41	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
42	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
43	C₄⋊C₄×C₂₁₁	—	2	1	no	—
44	C₄⋊C₄×C₂₁₁	—	2	1	no	—
45	C₄⋊C₄×C₂₁₁	—	2	1	no	—
46	C₄⋊C₄×C₂₁₁	—	2	1	no	—
47	C₄⋊C₄×C₂₁₁	—	2	1	no	—
48	C₄⋊C₄×C₂₁₁	—	2	1	no	—
49	C₄⋊C₄×C₂₁₁	D₄	1	2	yes	—
50	C₄⋊C₄×C₂₁₁	D₄	1	2	yes	—
51	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
52	C₂²⋊C₄×C₂₁₁	—	4	2	no	—
53	C₄²×C₂₁₁	—	2	4	no	—
54	C₂³×C₂₁₁	—	2	1	no	—
55	C₂³×C₂₁₁	—	2	1	no	—
56	C₂³×C₂₁₁	—	2	1	no	—
57	C₂³×C₂₁₁	—	2	1	no	—
58	C₂³×C₂₁₁	—	2	1	no	—
59	C₂³×C₂₁₁	—	2	1	no	—
60	C₂³×C₂₁₁	C₄⋊C₄	1	2	yes	—
61	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₂₁₁	—	2	1	no	—
63	C₂×C₄×C₂₁₁	—	2	1	no	—
64	C₂×C₄×C₂₁₁	—	2	2	no	—
65	C₂×C₄×C₂₁₁	—	2	2	no	—
66	C₂×C₄×C₂₁₁	—	2	2	no	—
67	C₂×C₄×C₂₁₁	—	2	2	no	—
68	C₂×C₄×C₂₁₁	—	2	2	no	—
69	C₂×C₄×C₂₁₁	—	2	2	no	—
70	C₂×C₄×C₂₁₁	—	4	1	no	—
71	C₂×C₄×C₂₁₁	—	4	1	no	—
72	C₂×C₄×C₂₁₁	—	4	1	no	—
73	C₂×C₄×C₂₁₁	—	4	1	no	—
74	C₂×C₄×C₂₁₁	—	4	1	no	—
75	C₂×C₄×C₂₁₁	—	4	1	no	—
76	C₂×C₄×C₂₁₁	—	4	1	no	—
77	C₂×C₄×C₂₁₁	—	4	1	no	—
78	C₂×C₄×C₂₁₁	—	4	1	no	—
79	C₂×C₄×C₂₁₁	—	4	1	no	—
80	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
81	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
86	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
87	C₈×C₂₁₁	—	2	2	no	—
88	C₈×C₂₁₁	—	2	2	no	—
89	C₂²×C₂₁₁	—	2	2	no	—
90	C₂²×C₂₁₁	—	2	2	no	—
91	C₂²×C₂₁₁	—	2	2	no	—
92	C₂²×C₂₁₁	—	4	1	no	—
93	C₂²×C₂₁₁	—	4	1	no	—
94	C₂²×C₂₁₁	—	4	1	no	—
95	C₂²×C₂₁₁	—	4	1	no	—
96	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₂₁₁	#120	1	1	yes	characteristic
98	C₂²×C₂₁₁	#123	1	1	yes	characteristic
99	C₂²×C₂₁₁	#135	1	1	yes	characteristic
100	C₂²×C₂₁₁	#135	1	1	yes	characteristic
101	C₂²×C₂₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂²×C₂₁₁	#137	1	1	yes	characteristic
103	C₄×C₂₁₁	—	2	2	no	—
104	C₄×C₂₁₁	—	4	2	no	—
105	C₄×C₂₁₁	—	4	2	no	—
106	C₄×C₂₁₁	—	8	1	no	—
107	C₄×C₂₁₁	—	8	1	no	—
108	C₄×C₂₁₁	C₄×D₄	1	1	yes	characteristic
109	C₄×C₂₁₁	C₄×D₄	1	1	yes	characteristic
110	C₄×C₂₁₁	C₂²⋊Q₈	1	1	yes	characteristic
111	C₄×C₂₁₁	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₂₁₁	—	4	2	no	—
113	C₂×C₂₁₁	#385	1	1	yes	characteristic
114	C₂×C₂₁₁	#427	1	1	yes	characteristic
115	C₂×C₂₁₁	#428	1	1	yes	characteristic
116	C₂×C₂₁₁	#449	1	1	yes	characteristic
117	C₂×C₂₁₁	#452	1	1	yes	characteristic
118	C₂×C₂₁₁	#482	1	1	yes	characteristic
119	C₂×C₂₁₁	#485	1	1	yes	characteristic
120	C₂₁₁	#1854	1	1	yes	characteristic, direct product factor, Sylow
121	#1854	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂×#117	C₂×C₂₁₁	1	1	yes	characteristic
123	C₂×#107	C₂×C₂₁₁	1	1	yes	characteristic, complemented
124	C₂×#106	C₂×C₂₁₁	1	1	yes	characteristic, complemented
125	C₂×C₂²⋊C₈	C₂×C₂₁₁	1	1	yes	characteristic
126	#380	C₂×C₂₁₁	1	1	yes	characteristic
127	#340	C₂×C₂₁₁	1	1	yes	characteristic, complemented
128	#340	C₂×C₂₁₁	1	1	yes	characteristic, complemented
129	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
130	C₂²×C₈	C₂²×C₂₁₁	1	1	yes	characteristic
131	C₂²×C₈	C₂²×C₂₁₁	1	1	yes	characteristic
132	#117	—	2	2	no	—
133	#117	—	2	2	no	—
134	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
135	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
136	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
137	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	1	yes	characteristic
138	C₂×C₂²⋊C₄	—	2	1	no	—
139	C₂×C₂²⋊C₄	—	2	1	no	—
140	C₂×C₄²	—	2	1	no	—
141	#107	—	2	2	no	—
142	#106	—	2	2	no	—
143	C₂²⋊C₈	C₄×C₂₁₁	1	4	yes	complemented
144	#95	—	2	1	no	—
145	C₂⁴	D₄×C₂₁₁	1	1	yes	characteristic
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	—	2	1	no	—
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	—	2	1	no	—
154	C₂²×C₄	C₂×C₄×C₂₁₁	1	2	yes	—
155	C₂²×C₄	D₄×C₂₁₁	1	1	yes	characteristic
156	C₂²×C₄	Q₈×C₂₁₁	1	2	yes	—
157	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	C₂×C₄×C₂₁₁	1	2	yes	—
163	C₂×C₈	C₂×C₄×C₂₁₁	1	2	yes	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	D₄×C₂₁₁	1	2	yes	—
171	C₄⋊C₄	D₄×C₂₁₁	1	2	yes	—
172	C₂²⋊C₄	—	2	4	no	—
173	C₂²⋊C₄	—	4	2	no	—
174	C₄²	—	2	4	no	—
175	C₂³	—	2	1	no	—
176	C₂³	—	2	1	no	—
177	C₂³	—	2	1	no	—
178	C₂³	—	2	1	no	—
179	C₂³	—	2	1	no	—
180	C₂³	—	2	1	no	—
181	C₂³	C₄⋊C₄×C₂₁₁	1	2	yes	—
182	C₂³	C₂×D₄×C₂₁₁	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	1	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	C₄⋊C₄×C₂₁₁	1	2	yes	—
202	C₂×C₄	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
203	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
204	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
205	C₂×C₄	C₂×Q₈×C₂₁₁	1	1	yes	characteristic
206	C₂×C₄	#41×C₂₁₁	1	1	yes	characteristic
207	C₂×C₄	#41×C₂₁₁	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	1	no	—
214	C₂²	—	4	1	no	—
215	C₂²	—	4	1	no	—
216	C₂²	—	4	1	no	—
217	C₂²	C₂×C₄⋊C₄×C₂₁₁	1	1	yes	characteristic
218	C₂²	#120×C₂₁₁	1	1	yes	characteristic
219	C₂²	#123×C₂₁₁	1	1	yes	characteristic
220	C₂²	#135×C₂₁₁	1	1	yes	characteristic
221	C₂²	#135×C₂₁₁	1	1	yes	characteristic
222	C₂²	AGL(1,ℤ/8)×C₂₁₁	1	1	yes	characteristic
223	C₂²	#137×C₂₁₁	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	4	2	no	—
226	C₄	—	4	2	no	—
227	C₄	—	8	1	no	—
228	C₄	—	8	1	no	—
229	C₄	C₄×D₄×C₂₁₁	1	1	yes	characteristic
230	C₄	C₄×D₄×C₂₁₁	1	1	yes	characteristic
231	C₄	C₂²⋊Q₈×C₂₁₁	1	1	yes	characteristic
232	C₄	C₂²⋊Q₈×C₂₁₁	1	1	yes	characteristic
233	C₂	—	4	2	no	—
234	C₂	#385×C₂₁₁	1	1	yes	characteristic
235	C₂	#427×C₂₁₁	1	1	yes	characteristic
236	C₂	#428×C₂₁₁	1	1	yes	characteristic
237	C₂	#449×C₂₁₁	1	1	yes	characteristic
238	C₂	#452×C₂₁₁	1	1	yes	characteristic
239	C₂	#482×C₂₁₁	1	1	yes	characteristic
240	C₂	#485×C₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₁₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₁₁
Frattini, cosocle	C₂²×C₄	C₂³×C₂₁₁
prime radical	C₂⁴×C₂₁₁	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₁₁ ◁ C₂³×C₄×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#107×C₂₁₁	C₂	semidirect
3. C₂×#106×C₂₁₁	C₂	semidirect
6. #340×C₂₁₁	C₂	semidirect
7. #340×C₂₁₁	C₂	semidirect
22. C₂²⋊C₈×C₂₁₁	C₄	semidirect
120. C₂₁₁	#1854	direct (N × H)
121. #1854	C₂₁₁	direct (N × H)
123. C₂×#107	C₂×C₂₁₁	semidirect
124. C₂×#106	C₂×C₂₁₁	semidirect
127. #340	C₂×C₂₁₁	semidirect
128. #340	C₂×C₂₁₁	semidirect
143. C₂²⋊C₈	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
C₂×#107×C₂₁₁	2
C₂×#106×C₂₁₁	2
C₂×C₂²⋊C₈×C₂₁₁	2
#380×C₂₁₁	2
#340×C₂₁₁	2
#1854	211

Subgroups — #1854×C211 #41009474

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1854×C₂₁₁ #41009474