Subgroups — #1445×C₂₁₁ #41009065

Order 27008.

520 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#95×C₂₁₁	C₂	1	4	yes	maximal
3	C₂×#95×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#95×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
6	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
7	C₂³×C₄×C₂₁₁	C₂²	1	2	yes	—
8	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₂₁₁	—	2	2	no	—
10	C₂×C₄⋊C₄×C₂₁₁	—	2	2	no	—
11	C₂×C₄⋊C₄×C₂₁₁	C₄	1	4	yes	complemented
12	#95×C₂₁₁	—	2	8	no	—
13	#95×C₂₁₁	—	2	2	no	—
14	#95×C₂₁₁	—	2	2	no	—
15	C₂⁴×C₂₁₁	C₂³	1	1	yes	characteristic
16	C₂²×C₄×C₂₁₁	—	2	4	no	—
17	C₂²×C₄×C₂₁₁	—	2	4	no	—
18	C₂²×C₄×C₂₁₁	—	2	2	no	—
19	C₂²×C₄×C₂₁₁	—	2	2	no	—
20	C₂²×C₄×C₂₁₁	—	2	4	no	—
21	C₂²×C₄×C₂₁₁	—	2	2	no	—
22	C₂²×C₄×C₂₁₁	—	2	2	no	—
23	C₂²×C₄×C₂₁₁	—	2	4	no	—
24	C₂²×C₄×C₂₁₁	—	2	2	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	4	no	—
29	C₂²×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₂₁₁	D₄	1	4	yes	—
32	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
33	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
34	C₂²×C₄×C₂₁₁	Q₈	1	4	yes	—
35	C₄⋊C₄×C₂₁₁	—	2	8	no	—
36	C₂³×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
37	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
38	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
39	C₂³×C₂₁₁	C₂×D₄	1	2	yes	—
40	C₂³×C₂₁₁	C₂×Q₈	1	2	yes	—
41	C₂³×C₂₁₁	#41	1	4	yes	—
42	C₂³×C₂₁₁	#41	1	2	yes	—
43	C₂³×C₂₁₁	#41	1	2	yes	—
44	C₂×C₄×C₂₁₁	—	2	4	no	—
45	C₂×C₄×C₂₁₁	—	2	4	no	—
46	C₂×C₄×C₂₁₁	—	2	4	no	—
47	C₂×C₄×C₂₁₁	—	2	4	no	—
48	C₂×C₄×C₂₁₁	—	2	8	no	—
49	C₂×C₄×C₂₁₁	—	2	4	no	—
50	C₂×C₄×C₂₁₁	—	2	8	no	—
51	C₂×C₄×C₂₁₁	—	2	4	no	—
52	C₂×C₄×C₂₁₁	—	2	8	no	—
53	C₂×C₄×C₂₁₁	—	4	4	no	—
54	C₂×C₄×C₂₁₁	—	4	2	no	—
55	C₂×C₄×C₂₁₁	—	4	2	no	—
56	C₂×C₄×C₂₁₁	—	4	4	no	—
57	C₂×C₄×C₂₁₁	—	4	2	no	—
58	C₂×C₄×C₂₁₁	—	4	2	no	—
59	C₂×C₄×C₂₁₁	—	4	4	no	—
60	C₂×C₄×C₂₁₁	—	4	2	no	—
61	C₂×C₄×C₂₁₁	—	4	2	no	—
62	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	4	yes	—
63	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₂₁₁	#117	1	2	yes	—
65	C₂²×C₂₁₁	C₄×D₄	1	2	yes	—
66	C₂²×C₂₁₁	C₄×Q₈	1	2	yes	—
67	C₂²×C₂₁₁	#121	1	2	yes	—
68	C₂²×C₂₁₁	C₂²⋊Q₈	1	4	yes	—
69	C₂²×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
70	C₂²×C₂₁₁	#123	1	2	yes	—
71	C₂²×C₂₁₁	#123	1	2	yes	—
72	C₂²×C₂₁₁	#123	1	4	yes	—
73	C₂²×C₂₁₁	#124	1	1	yes	characteristic
74	C₂²×C₂₁₁	#124	1	1	yes	characteristic
75	C₂²×C₂₁₁	#125	1	4	yes	—
76	C₂²×C₂₁₁	#126	1	4	yes	—
77	C₂²×C₂₁₁	#127	1	1	yes	characteristic
78	C₂²×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
79	C₄×C₂₁₁	—	2	4	no	—
80	C₄×C₂₁₁	—	4	4	no	—
81	C₄×C₂₁₁	—	4	4	no	—
82	C₄×C₂₁₁	—	4	4	no	—
83	C₂×C₂₁₁	#381	1	1	yes	characteristic
84	C₂×C₂₁₁	#387	1	4	yes	—
85	C₂×C₂₁₁	#390	1	1	yes	characteristic
86	C₂×C₂₁₁	#391	1	1	yes	characteristic
87	C₂×C₂₁₁	#397	1	2	yes	—
88	C₂×C₂₁₁	#398	1	2	yes	—
89	C₂×C₂₁₁	#399	1	2	yes	—
90	C₂×C₂₁₁	#400	1	2	yes	—
91	C₂₁₁	#1445	1	1	yes	characteristic, direct product factor, Sylow
92	#1445	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂²×C₄⋊C₄	C₂×C₂₁₁	1	1	yes	characteristic
94	C₂×#95	C₂×C₂₁₁	1	4	yes	—
95	C₂×#95	C₂×C₂₁₁	1	1	yes	characteristic
96	C₂×#95	C₂×C₂₁₁	1	1	yes	characteristic
97	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
98	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
99	C₂³×C₄	C₂²×C₂₁₁	1	2	yes	—
100	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₄⋊C₄	—	2	2	no	—
103	C₂×C₄⋊C₄	C₄×C₂₁₁	1	4	yes	complemented
104	#95	—	2	8	no	—
105	#95	—	2	2	no	—
106	#95	—	2	2	no	—
107	C₂⁴	C₂³×C₂₁₁	1	1	yes	characteristic
108	C₂²×C₄	—	2	4	no	—
109	C₂²×C₄	—	2	4	no	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	2	4	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	—	2	2	no	—
115	C₂²×C₄	—	2	4	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	2	4	no	—
118	C₂²×C₄	—	2	2	no	—
119	C₂²×C₄	—	2	4	no	—
120	C₂²×C₄	—	2	4	no	—
121	C₂²×C₄	C₂×C₄×C₂₁₁	1	4	yes	—
122	C₂²×C₄	C₂×C₄×C₂₁₁	1	2	yes	—
123	C₂²×C₄	D₄×C₂₁₁	1	4	yes	—
124	C₂²×C₄	D₄×C₂₁₁	1	2	yes	—
125	C₂²×C₄	D₄×C₂₁₁	1	2	yes	—
126	C₂²×C₄	Q₈×C₂₁₁	1	4	yes	—
127	C₄⋊C₄	—	2	8	no	—
128	C₂³	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
129	C₂³	C₂×D₄×C₂₁₁	1	1	yes	characteristic
130	C₂³	C₂×D₄×C₂₁₁	1	1	yes	characteristic
131	C₂³	C₂×D₄×C₂₁₁	1	2	yes	—
132	C₂³	C₂×Q₈×C₂₁₁	1	2	yes	—
133	C₂³	#41×C₂₁₁	1	4	yes	—
134	C₂³	#41×C₂₁₁	1	2	yes	—
135	C₂³	#41×C₂₁₁	1	2	yes	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	8	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	8	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	4	4	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	4	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	4	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	4	yes	—
155	C₂²	C₂×C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
156	C₂²	#117×C₂₁₁	1	2	yes	—
157	C₂²	C₄×D₄×C₂₁₁	1	2	yes	—
158	C₂²	C₄×Q₈×C₂₁₁	1	2	yes	—
159	C₂²	#121×C₂₁₁	1	2	yes	—
160	C₂²	C₂²⋊Q₈×C₂₁₁	1	4	yes	—
161	C₂²	C₂²⋊Q₈×C₂₁₁	1	2	yes	—
162	C₂²	#123×C₂₁₁	1	2	yes	—
163	C₂²	#123×C₂₁₁	1	2	yes	—
164	C₂²	#123×C₂₁₁	1	4	yes	—
165	C₂²	#124×C₂₁₁	1	1	yes	characteristic
166	C₂²	#124×C₂₁₁	1	1	yes	characteristic
167	C₂²	#125×C₂₁₁	1	4	yes	—
168	C₂²	#126×C₂₁₁	1	4	yes	—
169	C₂²	#127×C₂₁₁	1	1	yes	characteristic
170	C₂²	C₄⋊Q₈×C₂₁₁	1	1	yes	characteristic
171	C₄	—	2	4	no	—
172	C₄	—	4	4	no	—
173	C₄	—	4	4	no	—
174	C₄	—	4	4	no	—
175	C₂	#381×C₂₁₁	1	1	yes	characteristic
176	C₂	#387×C₂₁₁	1	4	yes	—
177	C₂	#390×C₂₁₁	1	1	yes	characteristic
178	C₂	#391×C₂₁₁	1	1	yes	characteristic
179	C₂	#397×C₂₁₁	1	2	yes	—
180	C₂	#398×C₂₁₁	1	2	yes	—
181	C₂	#399×C₂₁₁	1	2	yes	—
182	C₂	#400×C₂₁₁	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
11. C₂×C₄⋊C₄×C₂₁₁	C₄	semidirect
91. C₂₁₁	#1445	direct (N × H)
92. #1445	C₂₁₁	direct (N × H)
103. C₂×C₄⋊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
C₂²×C₄⋊C₄×C₂₁₁	2
C₂×#95×C₂₁₁	2
#1445	211

Subgroups — #1445×C211 #41009065

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1445×C₂₁₁ #41009065