Subgroups — C₂×#26018322 #36465996

Order 22784.

286 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#13083549	C₂	1	2	yes	maximal
2	#26018322	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₈²×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13083549	—	2	2	no	—
5	C₄⋊#13083549	—	2	2	no	—
6	C₄⋊#13083549	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₈₉	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₈₉	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
10	C₈²×C₈₉	C₄	1	2	yes	—
11	C₈²×C₈₉	C₂²	1	2	yes	—
12	C₂×#13083549	—	2	4	no	—
13	C₂×#13083549	—	4	4	no	—
14	C₂²×C₈×C₈₉	—	2	1	no	—
15	C₂²×C₈×C₈₉	C₈	1	1	yes	characteristic
16	C₂²×C₈×C₈₉	C₈	1	1	yes	characteristic
17	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
19	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₈₉	—	2	1	no	—
21	C₄×C₈×C₈₉	C₈	1	2	yes	—
22	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
26	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
28	#13083549	—	4	4	no	—
29	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
31	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
32	C₂×C₈×C₈₉	—	2	4	no	—
33	C₂×C₈×C₈₉	—	2	2	no	—
34	C₂×C₈×C₈₉	—	2	2	no	—
35	C₂×C₈×C₈₉	—	2	2	no	—
36	C₂×C₈×C₈₉	C₁₆	1	4	yes	complemented
37	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₈₉	D₈	1	2	yes	—
42	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
43	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
45	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
48	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	C₂³×C₈₉	C₄⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₈₉	—	2	2	no	—
51	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₈₉	#107	1	2	yes	—
55	C₂×C₄×C₈₉	#108	1	2	yes	—
56	C₂×C₄×C₈₉	C₂×C₁₆	1	2	yes	—
57	C₂×C₄×C₈₉	M₃₂	1	2	yes	—
58	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
60	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
61	C₈×C₈₉	—	2	4	no	—
62	C₈×C₈₉	—	2	1	no	—
63	C₈×C₈₉	—	2	1	no	—
64	C₈×C₈₉	—	2	2	no	—
65	C₈×C₈₉	#107	1	2	yes	—
66	C₈×C₈₉	C₂×C₁₆	1	4	yes	complemented
67	C₈×C₈₉	C₂×D₈	1	1	yes	characteristic
68	C₈×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
69	C₂²×C₈₉	#335	1	2	yes	—
70	C₂²×C₈₉	C₄⋊C₁₆	1	2	yes	—
71	C₂²×C₈₉	#364	1	2	yes	—
72	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
73	C₄×C₈₉	—	2	1	no	—
74	C₄×C₈₉	—	2	1	no	—
75	C₄×C₈₉	#335	1	2	yes	—
76	C₄×C₈₉	C₄⋊C₁₆	1	2	yes	—
77	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
78	C₄×C₈₉	C₂×#108	1	1	yes	characteristic
79	C₄×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
80	C₄×C₈₉	C₂×M₃₂	1	1	yes	characteristic
81	C₂×#1370	—	89	1	no	maximal, Sylow
82	C₂×C₈₉	#1370	1	4	yes	complemented
83	C₂×C₈₉	C₂×#335	1	1	yes	characteristic
84	C₂×C₈₉	C₂×C₄⋊C₁₆	1	1	yes	characteristic
85	C₂×C₈₉	C₂×#364	1	1	yes	characteristic
86	C₂×C₄⋊C₁₆	—	89	2	no	—
87	C₂×C₈²	D₈₉	1	1	yes	characteristic
88	#1370	—	89	4	no	—
89	C₈₉	C₂×#1370	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₁₆	—	178	2	no	—
91	C₂×C₄×C₈	Dic₈₉	1	1	yes	characteristic
92	C₂×C₄×C₈	Dic₈₉	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
94	C₄⋊C₁₆	—	89	4	no	—
95	C₄⋊C₁₆	—	178	2	no	—
96	C₈²	Dic₈₉	1	2	yes	—
97	C₈²	C₂×D₈₉	1	2	yes	—
98	C₂²×C₈	—	2	1	no	—
99	C₂²×C₈	#10623494	1	1	yes	characteristic
100	C₂²×C₈	#10623494	1	1	yes	characteristic
101	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
102	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
103	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
104	C₂×C₁₆	—	356	4	no	—
105	C₂×C₁₆	—	178	4	no	—
106	C₄×C₈	—	2	1	no	—
107	C₄×C₈	#10623494	1	2	yes	—
108	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
109	C₄×C₈	D₃₅₆	1	1	yes	characteristic
110	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
111	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
112	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
113	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
114	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
115	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
116	C₂²×C₄	#13083558	1	1	yes	characteristic
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	#13083549	1	4	yes	complemented
122	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
123	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
124	C₂×C₈	C₂×#10623494	1	2	yes	—
125	C₂×C₈	C₂×#10623494	1	2	yes	—
126	C₂×C₈	D₇₁₂	1	2	yes	—
127	C₂×C₈	Dic₃₅₆	1	2	yes	—
128	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
129	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
130	C₄²	C₄⋊Dic₈₉	1	1	yes	characteristic
131	C₄²	C₂×#10623494	1	1	yes	characteristic
132	C₄²	#13083558	1	1	yes	characteristic
133	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
134	C₁₆	—	356	4	no	—
135	C₂³	C₄⋊#10623494	1	1	yes	characteristic
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	C₄⋊#10623494	1	1	yes	characteristic
138	C₂×C₄	C₄⋊#10623494	1	1	yes	characteristic
139	C₂×C₄	C₄⋊#10623494	1	1	yes	characteristic
140	C₂×C₄	#15667946	1	2	yes	—
141	C₂×C₄	#15667948	1	2	yes	—
142	C₂×C₄	C₂×#13083549	1	2	yes	—
143	C₂×C₄	#15667952	1	2	yes	—
144	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
145	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
146	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
147	C₈	—	2	4	no	—
148	C₈	—	2	1	no	—
149	C₈	—	2	1	no	—
150	C₈	—	2	2	no	—
151	C₈	#15667946	1	2	yes	—
152	C₈	C₂×#13083549	1	4	yes	complemented
153	C₈	C₂×D₇₁₂	1	1	yes	characteristic
154	C₈	C₂×Dic₃₅₆	1	1	yes	characteristic
155	C₂²	#19151351	1	2	yes	—
156	C₂²	C₄⋊#13083549	1	2	yes	—
157	C₂²	#19151419	1	2	yes	—
158	C₂²	C₂×C₄⋊#10623494	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#19151351	1	2	yes	—
162	C₄	C₄⋊#13083549	1	2	yes	—
163	C₄	C₂×#15667946	1	1	yes	characteristic
164	C₄	C₂×#15667948	1	1	yes	characteristic
165	C₄	C₂²×#13083549	1	1	yes	characteristic
166	C₄	C₂×#15667952	1	1	yes	characteristic
167	C₂	#26018322	1	4	yes	direct product factor
168	C₂	C₂×#19151351	1	1	yes	characteristic
169	C₂	C₂×C₄⋊#13083549	1	1	yes	characteristic
170	C₂	C₂×#19151419	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	D₃₅₆
commutator	C₄×C₈₉	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₂×C₈²×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₄⋊C₈
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	C₂×C₈²	D₈₉
hypocenter	C₈₉	C₂×#1370
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
2. #26018322	C₂	direct (N × H)
36. C₂×C₈×C₈₉	C₁₆	semidirect
66. C₈×C₈₉	C₂×C₁₆	semidirect
82. C₂×C₈₉	#1370	semidirect
89. C₈₉	C₂×#1370	semidirect
121. C₂×C₈	#13083549	semidirect
152. C₈	C₂×#13083549	semidirect
167. C₂	#26018322	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×C₄⋊#13083549	2
#26018322	2
C₂×C₈²×C₈₉	2
C₂×#1370	89

Subgroups — C2×#26018322 #36465996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26018322 #36465996