Subgroups — C₂²×#555×C₈₉ #36440555

Order 22784.

8710 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#555×C₈₉	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#124×C₈₉	C₂	1	4	yes	maximal, complemented
4	C₂²×#121×C₈₉	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄×D₄×C₈₉	C₂	1	2	yes	maximal, complemented
6	C₂²×#117×C₈₉	C₂	1	4	yes	maximal, complemented
7	C₂³×Q₈×C₈₉	C₂²	1	2	yes	—
8	C₂³×D₄×C₈₉	C₂²	1	2	yes	—
9	C₂³×D₄×C₈₉	C₂²	1	4	yes	—
10	C₂⁴×C₄×C₈₉	C₂²	1	4	yes	—
11	#555×C₈₉	C₂²	1	256	yes	direct product factor
12	C₂×C₄⋊Q₈×C₈₉	C₂²	1	24	yes	complemented
13	C₂×#124×C₈₉	C₂²	1	96	yes	complemented
14	C₂×#121×C₈₉	C₂²	1	96	yes	complemented
15	C₂×C₄×D₄×C₈₉	C₂²	1	48	yes	complemented
16	C₂×#117×C₈₉	C₂²	1	96	yes	complemented
17	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
19	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	8	yes	—
20	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
21	C₂²×C₄²×C₈₉	C₂²	1	4	yes	—
22	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
23	C₂⁵×C₈₉	C₂³	1	4	yes	—
24	C₂²×Q₈×C₈₉	—	4	8	no	—
25	C₂²×Q₈×C₈₉	C₂³	1	24	yes	—
26	C₂²×D₄×C₈₉	—	2	16	no	—
27	C₂²×D₄×C₈₉	—	4	16	no	—
28	C₂²×D₄×C₈₉	C₂³	1	24	yes	—
29	C₂²×D₄×C₈₉	C₂³	1	48	yes	—
30	C₂³×C₄×C₈₉	—	2	32	no	—
31	C₂³×C₄×C₈₉	C₂³	1	48	yes	—
32	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
33	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
35	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
36	C₄⋊Q₈×C₈₉	C₂³	1	64	yes	complemented
37	#124×C₈₉	C₂³	1	256	yes	complemented
38	#121×C₈₉	C₂³	1	256	yes	complemented
39	C₄×D₄×C₈₉	C₂³	1	128	yes	complemented
40	#117×C₈₉	C₂³	1	256	yes	complemented
41	C₂×C₄⋊C₄×C₈₉	C₂³	1	24	yes	—
42	C₂×C₄⋊C₄×C₈₉	C₂³	1	48	yes	—
43	C₂×C₂²⋊C₄×C₈₉	C₂³	1	96	yes	—
44	C₂×C₂²⋊C₄×C₈₉	C₂³	1	48	yes	—
45	C₂×C₄²×C₈₉	C₂³	1	48	yes	—
46	C₂×C₄²×C₈₉	C₂³	1	12	yes	—
47	C₂⁴×C₈₉	—	2	16	no	—
48	C₂⁴×C₈₉	—	2	16	no	—
49	C₂⁴×C₈₉	—	2	16	no	—
50	C₂⁴×C₈₉	C₂⁴	1	24	yes	—
51	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
52	C₂×Q₈×C₈₉	—	4	48	no	—
53	C₂×Q₈×C₈₉	C₂⁴	1	32	yes	—
54	C₂×D₄×C₈₉	—	2	96	no	—
55	C₂×D₄×C₈₉	—	4	96	no	—
56	C₂×D₄×C₈₉	C₂⁴	1	32	yes	—
57	C₂×D₄×C₈₉	C₂⁴	1	64	yes	—
58	C₂²×C₄×C₈₉	—	2	16	no	—
59	C₂²×C₄×C₈₉	—	2	16	no	—
60	C₂²×C₄×C₈₉	—	2	4	no	—
61	C₂²×C₄×C₈₉	—	2	192	no	—
62	C₂²×C₄×C₈₉	#41	1	16	yes	—
63	C₂²×C₄×C₈₉	C₂⁴	1	24	yes	—
64	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
65	C₂²×C₄×C₈₉	C₂⁴	1	24	yes	—
66	C₂²×C₄×C₈₉	C₂⁴	1	64	yes	—
67	C₂²×C₄×C₈₉	C₂⁴	1	12	yes	—
68	C₄⋊C₄×C₈₉	C₂⁴	1	32	yes	—
69	C₄⋊C₄×C₈₉	C₂⁴	1	64	yes	—
70	C₂²⋊C₄×C₈₉	C₂⁴	1	128	yes	—
71	C₂²⋊C₄×C₈₉	C₂⁴	1	64	yes	—
72	C₄²×C₈₉	C₂⁴	1	64	yes	—
73	C₄²×C₈₉	C₂⁴	1	16	yes	—
74	C₂³×C₈₉	—	2	48	no	—
75	C₂³×C₈₉	—	2	48	no	—
76	C₂³×C₈₉	—	2	48	no	—
77	C₂³×C₈₉	—	4	64	no	—
78	C₂³×C₈₉	C₂×#41	1	8	yes	—
79	C₂³×C₈₉	H(5,𝔽₂)	1	4	yes	—
80	C₂³×C₈₉	C₂⁵	1	16	yes	—
81	C₂³×C₈₉	C₂⁵	1	3	yes	—
82	Q₈×C₈₉	—	4	32	no	—
83	D₄×C₈₉	—	2	64	no	—
84	D₄×C₈₉	—	4	64	no	—
85	C₂×C₄×C₈₉	—	2	48	no	—
86	C₂×C₄×C₈₉	—	2	48	no	—
87	C₂×C₄×C₈₉	—	2	12	no	—
88	C₂×C₄×C₈₉	—	2	128	no	—
89	C₂×C₄×C₈₉	C₂×#41	1	48	yes	—
90	C₂×C₄×C₈₉	C₂⁵	1	16	yes	—
91	C₂×C₄×C₈₉	C₂⁵	1	4	yes	—
92	C₂×C₄×C₈₉	C₂⁵	1	16	yes	—
93	C₂×C₄×C₈₉	C₂⁵	1	8	yes	—
94	C₂²×C₈₉	—	2	16	no	—
95	C₂²×C₈₉	—	2	16	no	—
96	C₂²×C₈₉	—	2	16	no	—
97	C₂²×C₈₉	—	4	96	no	—
98	C₂²×C₈₉	#555	1	16	yes	direct product factor
99	C₂²×C₈₉	C₂²×#41	1	12	yes	—
100	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	6	yes	—
101	C₂²×C₈₉	C₂⁶	1	1	yes	characteristic
102	C₄×C₈₉	—	2	16	no	—
103	C₄×C₈₉	—	2	16	no	—
104	C₄×C₈₉	—	2	4	no	—
105	C₄×C₈₉	C₂²×#41	1	16	yes	—
106	C₂²×#555	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂×C₈₉	—	4	16	no	—
108	C₂×C₈₉	C₂×#555	1	12	yes	direct product factor
109	C₂×C₈₉	C₂³×#41	1	2	yes	—
110	C₂×C₈₉	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
111	C₂×#555	C₂×C₈₉	1	48	yes	direct product factor
112	C₂²×C₄⋊Q₈	C₂×C₈₉	1	1	yes	characteristic, complemented
113	C₂²×#124	C₂×C₈₉	1	4	yes	complemented
114	C₂²×#121	C₂×C₈₉	1	4	yes	complemented
115	C₂²×C₄×D₄	C₂×C₈₉	1	2	yes	complemented
116	C₂²×#117	C₂×C₈₉	1	4	yes	complemented
117	C₈₉	C₂²×#555	1	1	yes	characteristic, direct product factor, Sylow
118	C₂³×Q₈	C₂²×C₈₉	1	2	yes	—
119	C₂³×D₄	C₂²×C₈₉	1	2	yes	—
120	C₂³×D₄	C₂²×C₈₉	1	4	yes	—
121	C₂⁴×C₄	C₂²×C₈₉	1	4	yes	—
122	#555	C₂²×C₈₉	1	256	yes	direct product factor
123	C₂×C₄⋊Q₈	C₂²×C₈₉	1	24	yes	complemented
124	C₂×#124	C₂²×C₈₉	1	96	yes	complemented
125	C₂×#121	C₂²×C₈₉	1	96	yes	complemented
126	C₂×C₄×D₄	C₂²×C₈₉	1	48	yes	complemented
127	C₂×#117	C₂²×C₈₉	1	96	yes	complemented
128	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
129	C₂²×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
130	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	8	yes	—
131	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	—
132	C₂²×C₄²	C₂²×C₈₉	1	4	yes	—
133	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
134	C₂⁵	C₂³×C₈₉	1	4	yes	—
135	C₂²×Q₈	—	4	8	no	—
136	C₂²×Q₈	C₂³×C₈₉	1	24	yes	—
137	C₂²×D₄	—	2	16	no	—
138	C₂²×D₄	—	4	16	no	—
139	C₂²×D₄	C₂³×C₈₉	1	24	yes	—
140	C₂²×D₄	C₂³×C₈₉	1	48	yes	—
141	C₂³×C₄	—	2	32	no	—
142	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
143	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
144	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
145	C₂³×C₄	C₂³×C₈₉	1	48	yes	—
146	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
147	C₄⋊Q₈	C₂³×C₈₉	1	64	yes	complemented
148	#124	C₂³×C₈₉	1	256	yes	complemented
149	#121	C₂³×C₈₉	1	256	yes	complemented
150	C₄×D₄	C₂³×C₈₉	1	128	yes	complemented
151	#117	C₂³×C₈₉	1	256	yes	complemented
152	C₂×C₄⋊C₄	C₂³×C₈₉	1	24	yes	—
153	C₂×C₄⋊C₄	C₂³×C₈₉	1	48	yes	—
154	C₂×C₂²⋊C₄	C₂³×C₈₉	1	96	yes	—
155	C₂×C₂²⋊C₄	C₂³×C₈₉	1	48	yes	—
156	C₂×C₄²	C₂³×C₈₉	1	48	yes	—
157	C₂×C₄²	C₂³×C₈₉	1	12	yes	—
158	C₂⁴	—	2	16	no	—
159	C₂⁴	—	2	16	no	—
160	C₂⁴	—	2	16	no	—
161	C₂⁴	C₂⁴×C₈₉	1	24	yes	—
162	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
163	C₂×Q₈	—	4	48	no	—
164	C₂×Q₈	C₂⁴×C₈₉	1	32	yes	—
165	C₂×D₄	—	2	96	no	—
166	C₂×D₄	—	4	96	no	—
167	C₂×D₄	C₂⁴×C₈₉	1	32	yes	—
168	C₂×D₄	C₂⁴×C₈₉	1	64	yes	—
169	C₂²×C₄	—	2	16	no	—
170	C₂²×C₄	—	2	16	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	2	192	no	—
173	C₂²×C₄	#41×C₈₉	1	16	yes	—
174	C₂²×C₄	C₂⁴×C₈₉	1	24	yes	—
175	C₂²×C₄	C₂⁴×C₈₉	1	6	yes	—
176	C₂²×C₄	C₂⁴×C₈₉	1	24	yes	—
177	C₂²×C₄	C₂⁴×C₈₉	1	64	yes	—
178	C₂²×C₄	C₂⁴×C₈₉	1	12	yes	—
179	C₄⋊C₄	C₂⁴×C₈₉	1	32	yes	—
180	C₄⋊C₄	C₂⁴×C₈₉	1	64	yes	—
181	C₂²⋊C₄	C₂⁴×C₈₉	1	128	yes	—
182	C₂²⋊C₄	C₂⁴×C₈₉	1	64	yes	—
183	C₄²	C₂⁴×C₈₉	1	64	yes	—
184	C₄²	C₂⁴×C₈₉	1	16	yes	—
185	C₂³	—	2	48	no	—
186	C₂³	—	2	48	no	—
187	C₂³	—	2	48	no	—
188	C₂³	—	4	64	no	—
189	C₂³	C₂×#41×C₈₉	1	8	yes	—
190	C₂³	H(5,𝔽₂)×C₈₉	1	4	yes	—
191	C₂³	C₂⁵×C₈₉	1	16	yes	—
192	C₂³	C₂⁵×C₈₉	1	3	yes	—
193	Q₈	—	4	32	no	—
194	D₄	—	2	64	no	—
195	D₄	—	4	64	no	—
196	C₂×C₄	—	2	48	no	—
197	C₂×C₄	—	2	48	no	—
198	C₂×C₄	—	2	12	no	—
199	C₂×C₄	—	2	128	no	—
200	C₂×C₄	C₂×#41×C₈₉	1	48	yes	—
201	C₂×C₄	C₂⁵×C₈₉	1	16	yes	—
202	C₂×C₄	C₂⁵×C₈₉	1	4	yes	—
203	C₂×C₄	C₂⁵×C₈₉	1	16	yes	—
204	C₂×C₄	C₂⁵×C₈₉	1	8	yes	—
205	C₂²	—	2	16	no	—
206	C₂²	—	2	16	no	—
207	C₂²	—	2	16	no	—
208	C₂²	—	4	96	no	—
209	C₂²	#555×C₈₉	1	16	yes	direct product factor
210	C₂²	C₂²×#41×C₈₉	1	12	yes	—
211	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	6	yes	—
212	C₂²	C₂⁶×C₈₉	1	1	yes	characteristic
213	C₄	—	2	16	no	—
214	C₄	—	2	16	no	—
215	C₄	—	2	4	no	—
216	C₄	C₂²×#41×C₈₉	1	16	yes	—
217	C₂	—	4	16	no	—
218	C₂	C₂×#555×C₈₉	1	12	yes	direct product factor
219	C₂	C₂³×#41×C₈₉	1	2	yes	—
220	C₂	C₂²×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₈₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₈₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#555×C₈₉	C₂	direct (N × H)
2. C₂²×C₄⋊Q₈×C₈₉	C₂	semidirect
3. C₂²×#124×C₈₉	C₂	semidirect
4. C₂²×#121×C₈₉	C₂	semidirect
5. C₂²×C₄×D₄×C₈₉	C₂	semidirect
6. C₂²×#117×C₈₉	C₂	semidirect
11. #555×C₈₉	C₂²	direct (N × H)
12. C₂×C₄⋊Q₈×C₈₉	C₂²	semidirect
13. C₂×#124×C₈₉	C₂²	semidirect
14. C₂×#121×C₈₉	C₂²	semidirect
15. C₂×C₄×D₄×C₈₉	C₂²	semidirect
16. C₂×#117×C₈₉	C₂²	semidirect
36. C₄⋊Q₈×C₈₉	C₂³	semidirect
37. #124×C₈₉	C₂³	semidirect
38. #121×C₈₉	C₂³	semidirect
39. C₄×D₄×C₈₉	C₂³	semidirect
40. #117×C₈₉	C₂³	semidirect
98. C₂²×C₈₉	#555	direct (N × H)
106. C₂²×#555	C₈₉	direct (N × H)
108. C₂×C₈₉	C₂×#555	direct (N × H)
111. C₂×#555	C₂×C₈₉	direct (N × H)
112. C₂²×C₄⋊Q₈	C₂×C₈₉	semidirect
113. C₂²×#124	C₂×C₈₉	semidirect
114. C₂²×#121	C₂×C₈₉	semidirect
115. C₂²×C₄×D₄	C₂×C₈₉	semidirect
116. C₂²×#117	C₂×C₈₉	semidirect
117. C₈₉	C₂²×#555	direct (N × H)
122. #555	C₂²×C₈₉	direct (N × H)
123. C₂×C₄⋊Q₈	C₂²×C₈₉	semidirect
124. C₂×#124	C₂²×C₈₉	semidirect
125. C₂×#121	C₂²×C₈₉	semidirect
126. C₂×C₄×D₄	C₂²×C₈₉	semidirect
127. C₂×#117	C₂²×C₈₉	semidirect
147. C₄⋊Q₈	C₂³×C₈₉	semidirect
148. #124	C₂³×C₈₉	semidirect
149. #121	C₂³×C₈₉	semidirect
150. C₄×D₄	C₂³×C₈₉	semidirect
151. #117	C₂³×C₈₉	semidirect
209. C₂²	#555×C₈₉	direct (N × H)
218. C₂	C₂×#555×C₈₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#555×C₈₉	2
C₂²×C₄⋊Q₈×C₈₉	2
C₂²×#124×C₈₉	2
C₂²×#121×C₈₉	2
C₂²×C₄×D₄×C₈₉	2
C₂²×#117×C₈₉	2
C₂²×#555	89

Subgroups — C22×#555×C89 #36440555

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#555×C₈₉ #36440555