Subgroups — C₂×C₈₉×#2862 #36414480

Order 22784.

2814 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₈₉	C₂	1	2	yes	maximal
2	C₈₉×#2862	C₂	1	16	yes	maximal, direct product factor
3	C₂×#391×C₈₉	C₂	1	8	yes	maximal
4	C₂×#382×C₈₉	C₂	1	4	yes	maximal
5	C₂×C₄³×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂³×Q₈×C₈₉	C₂²	1	4	yes	—
7	C₂×C₄⋊Q₈×C₈₉	—	2	16	no	—
8	C₂×C₄⋊Q₈×C₈₉	C₂²	1	16	yes	—
9	C₂²×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
10	C₂²×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
11	C₂²×C₄²×C₈₉	C₂²	1	4	yes	—
12	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₈₉	C₂²	1	2	yes	—
14	#391×C₈₉	C₂²	1	64	yes	—
15	#382×C₈₉	C₂²	1	32	yes	—
16	C₂×#95×C₈₉	C₂²	1	16	yes	—
17	C₄³×C₈₉	C₂²	1	8	yes	—
18	C₂²×Q₈×C₈₉	—	2	16	no	—
19	C₂²×Q₈×C₈₉	—	2	32	no	—
20	C₂²×Q₈×C₈₉	C₂³	1	16	yes	—
21	C₂³×C₄×C₈₉	C₂³	1	8	yes	—
22	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
23	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
24	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
25	C₄⋊Q₈×C₈₉	—	2	64	no	—
26	C₂×C₄⋊C₄×C₈₉	—	2	16	no	—
27	C₂×C₄⋊C₄×C₈₉	—	2	16	no	—
28	C₂×C₄⋊C₄×C₈₉	C₂³	1	16	yes	—
29	C₂×C₄⋊C₄×C₈₉	C₂³	1	16	yes	—
30	C₂×C₄²×C₈₉	—	2	16	no	—
31	C₂×C₄²×C₈₉	D₄	1	8	yes	—
32	C₂×C₄²×C₈₉	Q₈	1	16	yes	complemented
33	C₂×C₄²×C₈₉	C₂³	1	16	yes	—
34	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
35	C₂×C₄²×C₈₉	C₂³	1	8	yes	—
36	#95×C₈₉	C₂³	1	64	yes	—
37	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
38	C₂×Q₈×C₈₉	—	2	32	no	—
39	C₂×Q₈×C₈₉	—	2	64	no	—
40	C₂×Q₈×C₈₉	—	4	64	no	—
41	C₂²×C₄×C₈₉	—	2	16	no	—
42	C₂²×C₄×C₈₉	—	2	16	no	—
43	C₂²×C₄×C₈₉	—	2	32	no	—
44	C₂²×C₄×C₈₉	C₂×D₄	1	8	yes	—
45	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
46	C₂²×C₄×C₈₉	C₂×Q₈	1	16	yes	—
47	C₂²×C₄×C₈₉	C₂×Q₈	1	8	yes	—
48	C₂²×C₄×C₈₉	#41	1	16	yes	—
49	C₂²×C₄×C₈₉	C₂⁴	1	16	yes	—
50	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
51	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
52	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
53	C₄⋊C₄×C₈₉	—	2	32	no	—
54	C₄⋊C₄×C₈₉	—	2	32	no	—
55	C₄²×C₈₉	—	2	32	no	—
56	C₄²×C₈₉	C₂×D₄	1	16	yes	—
57	C₄²×C₈₉	C₂×Q₈	1	32	yes	complemented
58	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
59	C₂³×C₈₉	C₂²×Q₈	1	4	yes	—
60	C₂³×C₈₉	C₂×#41	1	8	yes	—
61	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
62	Q₈×C₈₉	—	4	64	no	—
63	C₂×C₄×C₈₉	—	2	32	no	—
64	C₂×C₄×C₈₉	—	2	16	no	—
65	C₂×C₄×C₈₉	—	2	16	no	—
66	C₂×C₄×C₈₉	—	2	32	no	—
67	C₂×C₄×C₈₉	—	4	32	no	—
68	C₂×C₄×C₈₉	#124	1	32	yes	—
69	C₂×C₄×C₈₉	C₄⋊Q₈	1	16	yes	—
70	C₂×C₄×C₈₉	C₂²×D₄	1	8	yes	—
71	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
72	C₂×C₄×C₈₉	C₂²×Q₈	1	16	yes	—
73	C₂×C₄×C₈₉	C₂²×Q₈	1	8	yes	—
74	C₂×C₄×C₈₉	C₂×#41	1	16	yes	—
75	C₂²×C₈₉	C₂×#124	1	8	yes	—
76	C₂²×C₈₉	C₂×C₄⋊Q₈	1	4	yes	—
77	C₂²×C₈₉	#533	1	16	yes	—
78	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
79	C₂²×C₈₉	C₂³×Q₈	1	2	yes	—
80	C₂²×C₈₉	C₂²×#41	1	4	yes	—
81	C₄×C₈₉	—	2	16	no	—
82	C₄×C₈₉	—	4	16	no	—
83	C₄×C₈₉	C₂×#124	1	16	yes	—
84	C₄×C₈₉	C₂×C₄⋊Q₈	1	8	yes	—
85	C₂×#2862	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂×C₈₉	#2862	1	8	yes	direct product factor
87	C₂×C₈₉	C₂²×#124	1	2	yes	—
88	C₂×C₈₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
89	C₂×C₈₉	C₂×#533	1	4	yes	—
90	C₂²×C₄⋊Q₈	C₂×C₈₉	1	2	yes	—
91	#2862	C₂×C₈₉	1	16	yes	direct product factor
92	C₂×#391	C₂×C₈₉	1	8	yes	—
93	C₂×#382	C₂×C₈₉	1	4	yes	—
94	C₂×C₄³	C₂×C₈₉	1	1	yes	characteristic
95	C₈₉	C₂×#2862	1	1	yes	characteristic, direct product factor, Sylow
96	C₂³×Q₈	C₂²×C₈₉	1	4	yes	—
97	C₂×C₄⋊Q₈	—	2	16	no	—
98	C₂×C₄⋊Q₈	C₂²×C₈₉	1	16	yes	—
99	C₂²×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
100	C₂²×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
101	C₂²×C₄²	C₂²×C₈₉	1	4	yes	—
102	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
103	C₂²×C₄²	C₂²×C₈₉	1	2	yes	—
104	#391	C₂²×C₈₉	1	64	yes	—
105	#382	C₂²×C₈₉	1	32	yes	—
106	C₂×#95	C₂²×C₈₉	1	16	yes	—
107	C₄³	C₂²×C₈₉	1	8	yes	—
108	C₂²×Q₈	—	2	16	no	—
109	C₂²×Q₈	—	2	32	no	—
110	C₂²×Q₈	C₂³×C₈₉	1	16	yes	—
111	C₂³×C₄	C₂³×C₈₉	1	8	yes	—
112	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
113	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
114	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
115	C₄⋊Q₈	—	2	64	no	—
116	C₂×C₄⋊C₄	—	2	16	no	—
117	C₂×C₄⋊C₄	—	2	16	no	—
118	C₂×C₄⋊C₄	C₂³×C₈₉	1	16	yes	—
119	C₂×C₄⋊C₄	C₂³×C₈₉	1	16	yes	—
120	C₂×C₄²	—	2	16	no	—
121	C₂×C₄²	D₄×C₈₉	1	8	yes	—
122	C₂×C₄²	Q₈×C₈₉	1	16	yes	complemented
123	C₂×C₄²	C₂³×C₈₉	1	16	yes	—
124	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
125	C₂×C₄²	C₂³×C₈₉	1	8	yes	—
126	#95	C₂³×C₈₉	1	64	yes	—
127	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
128	C₂×Q₈	—	2	32	no	—
129	C₂×Q₈	—	2	64	no	—
130	C₂×Q₈	—	4	64	no	—
131	C₂²×C₄	—	2	16	no	—
132	C₂²×C₄	—	2	16	no	—
133	C₂²×C₄	—	2	32	no	—
134	C₂²×C₄	C₂×D₄×C₈₉	1	8	yes	—
135	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
136	C₂²×C₄	C₂×Q₈×C₈₉	1	16	yes	—
137	C₂²×C₄	C₂×Q₈×C₈₉	1	8	yes	—
138	C₂²×C₄	#41×C₈₉	1	16	yes	—
139	C₂²×C₄	C₂⁴×C₈₉	1	16	yes	—
140	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
141	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
142	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
143	C₄⋊C₄	—	2	32	no	—
144	C₄⋊C₄	—	2	32	no	—
145	C₄²	—	2	32	no	—
146	C₄²	C₂×D₄×C₈₉	1	16	yes	—
147	C₄²	C₂×Q₈×C₈₉	1	32	yes	complemented
148	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
149	C₂³	C₂²×Q₈×C₈₉	1	4	yes	—
150	C₂³	C₂×#41×C₈₉	1	8	yes	—
151	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
152	Q₈	—	4	64	no	—
153	C₂×C₄	—	2	32	no	—
154	C₂×C₄	—	2	16	no	—
155	C₂×C₄	—	2	16	no	—
156	C₂×C₄	—	2	32	no	—
157	C₂×C₄	—	4	32	no	—
158	C₂×C₄	#124×C₈₉	1	32	yes	—
159	C₂×C₄	C₄⋊Q₈×C₈₉	1	16	yes	—
160	C₂×C₄	C₂²×D₄×C₈₉	1	8	yes	—
161	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
162	C₂×C₄	C₂²×Q₈×C₈₉	1	16	yes	—
163	C₂×C₄	C₂²×Q₈×C₈₉	1	8	yes	—
164	C₂×C₄	C₂×#41×C₈₉	1	16	yes	—
165	C₂²	C₂×#124×C₈₉	1	8	yes	—
166	C₂²	C₂×C₄⋊Q₈×C₈₉	1	4	yes	—
167	C₂²	#533×C₈₉	1	16	yes	—
168	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
169	C₂²	C₂³×Q₈×C₈₉	1	2	yes	—
170	C₂²	C₂²×#41×C₈₉	1	4	yes	—
171	C₄	—	2	16	no	—
172	C₄	—	4	16	no	—
173	C₄	C₂×#124×C₈₉	1	16	yes	—
174	C₄	C₂×C₄⋊Q₈×C₈₉	1	8	yes	—
175	C₂	C₈₉×#2862	1	8	yes	direct product factor
176	C₂	C₂²×#124×C₈₉	1	2	yes	—
177	C₂	C₂²×C₄⋊Q₈×C₈₉	1	1	yes	characteristic
178	C₂	C₂×#533×C₈₉	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₈₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#2862	C₂	direct (N × H)
32. C₂×C₄²×C₈₉	Q₈	semidirect
57. C₄²×C₈₉	C₂×Q₈	semidirect
85. C₂×#2862	C₈₉	direct (N × H)
86. C₂×C₈₉	#2862	direct (N × H)
91. #2862	C₂×C₈₉	direct (N × H)
95. C₈₉	C₂×#2862	direct (N × H)
122. C₂×C₄²	Q₈×C₈₉	semidirect
147. C₄²	C₂×Q₈×C₈₉	semidirect
175. C₂	C₈₉×#2862	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₈₉	2
C₈₉×#2862	2
C₂×#391×C₈₉	2
C₂×#382×C₈₉	2
C₂×C₄³×C₈₉	2
C₂×#2862	89

Subgroups — C2×C89×#2862 #36414480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#2862 #36414480