Subgroups — #36466069

Order 22784.

242 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	1	yes	maximal, characteristic
2	#26019011	C₂	1	1	yes	maximal, characteristic
3	#26018323	C₂	1	4	yes	maximal, complemented
4	C₈₉×#1567	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083549	C₄	1	2	yes	—
6	#19151622	—	2	1	no	—
7	C₄×#13083549	C₄	1	1	yes	characteristic
8	C₄×#13083549	C₄	1	1	yes	characteristic
9	C₄×#13083549	C₂²	1	2	yes	—
10	#19151351	—	2	1	no	—
11	#19151351	C₂²	1	2	yes	—
12	#423×C₈₉	—	2	1	no	—
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	#335×C₈₉	—	2	1	no	—
15	#335×C₈₉	C₂²	1	2	yes	—
16	C₂×#13083558	—	4	1	no	—
17	C₂×#13083549	—	2	4	no	—
18	C₂×#13083549	C₂×C₄	1	2	yes	—
19	C₂×#13083549	C₂×C₄	1	2	yes	—
20	C₄⋊#10623494	—	2	2	no	—
21	C₄⋊#10623494	—	4	1	no	—
22	C₂×M₁₆×C₈₉	—	4	1	no	—
23	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
25	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
26	C₄⋊C₈×C₈₉	—	2	2	no	—
27	C₄⋊C₈×C₈₉	—	4	1	no	—
28	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
30	C₄×C₈×C₈₉	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
32	#13083558	—	8	2	no	—
33	C₂×#10623494	—	4	2	no	—
34	#13083549	—	2	2	no	—
35	#13083549	M₁₆	1	4	yes	complemented
36	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₈₉	Q₁₆	1	1	yes	characteristic
39	M₁₆×C₈₉	—	8	2	no	—
40	C₂×C₈×C₈₉	—	2	2	no	—
41	C₂×C₈×C₈₉	—	2	2	no	—
42	C₂×C₈×C₈₉	—	4	2	no	—
43	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₈₉	C₄⋊C₄	1	1	yes	characteristic
48	C₄²×C₈₉	D₈	1	1	yes	characteristic
49	C₄²×C₈₉	Q₁₆	1	1	yes	characteristic
50	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
51	C₂³×C₈₉	#107	1	1	yes	characteristic
52	C₂×C₄×C₈₉	—	2	2	no	—
53	C₂×C₄×C₈₉	—	2	1	no	—
54	C₂×C₄×C₈₉	—	2	1	no	—
55	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
56	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
57	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
58	C₂×C₄×C₈₉	D₁₆	1	2	yes	—
59	C₂×C₄×C₈₉	Q₃₂	1	2	yes	—
60	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
63	C₈×C₈₉	—	2	1	no	—
64	C₈×C₈₉	—	2	1	no	—
65	C₈×C₈₉	—	8	2	no	—
66	C₈×C₈₉	#108	1	2	yes	—
67	C₈×C₈₉	C₂×M₁₆	1	2	yes	—
68	#10623494	—	8	2	no	—
69	C₂²×C₈₉	—	2	1	no	—
70	C₂²×C₈₉	—	2	1	no	—
71	C₂²×C₈₉	#366	1	2	yes	—
72	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
73	C₄×C₈₉	—	2	1	no	—
74	C₄×C₈₉	—	2	1	no	—
75	C₄×C₈₉	#366	1	2	yes	—
76	C₄×C₈₉	#368	1	2	yes	—
77	C₄×C₈₉	#423	1	1	yes	characteristic
78	C₄×C₈₉	C₂×#108	1	1	yes	characteristic
79	C₄×C₈₉	C₂×D₁₆	1	1	yes	characteristic
80	C₄×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
81	#12022	—	89	1	no	maximal, Sylow
82	C₂×C₈₉	—	2	2	no	—
83	C₂×C₈₉	#1567	1	1	yes	characteristic
84	C₂×C₈₉	C₂×#366	1	1	yes	characteristic
85	C₂×C₈₉	C₂×#368	1	1	yes	characteristic
86	C₂×C₄×C₁₆	—	89	1	no	—
87	#1567	—	89	1	no	—
88	#1567	D₈₉	1	1	yes	characteristic
89	#1371	—	89	4	no	—
90	C₈₉	#12022	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₁₆	—	89	2	no	—
92	#423	—	2	1	no	—
93	#423	—	178	1	no	—
94	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
95	C₄×C₁₆	—	89	1	no	—
96	C₄×C₁₆	—	89	1	no	—
97	C₄×C₁₆	—	89	2	no	—
98	#335	—	2	1	no	—
99	#335	—	89	2	no	—
100	#335	—	178	1	no	—
101	#335	C₂×D₈₉	1	2	yes	—
102	C₂×M₁₆	—	4	1	no	—
103	C₂×M₁₆	—	356	1	no	—
104	C₂²×C₈	Dic₁₇₈	1	1	yes	characteristic
105	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
106	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
107	C₂×C₁₆	—	89	2	no	—
108	C₂×C₁₆	—	89	2	no	—
109	C₂×C₁₆	—	178	4	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	4	1	no	—
112	C₄⋊C₈	—	356	1	no	—
113	C₄⋊C₈	—	178	2	no	—
114	C₄×C₈	Dic₁₇₈	1	1	yes	characteristic
115	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
116	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
117	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
118	C₂²×C₄	#13083553	1	1	yes	characteristic
119	C₂²×C₄	#13083559	1	1	yes	characteristic
120	C₂²×C₄	#13083564	1	1	yes	characteristic
121	M₁₆	—	8	2	no	—
122	M₁₆	—	712	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	356	2	no	—
127	C₂×C₈	#13083553	1	1	yes	characteristic
128	C₂×C₈	#13083553	1	1	yes	characteristic
129	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
130	C₂×C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
131	C₄²	#13083553	1	1	yes	characteristic
132	C₄²	#13083559	1	1	yes	characteristic
133	C₄²	#13083564	1	1	yes	characteristic
134	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
135	C₁₆	—	89	4	no	—
136	C₁₆	—	178	2	no	—
137	C₂³	#15667945	1	1	yes	characteristic
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	#15667945	1	1	yes	characteristic
142	C₂×C₄	#15667945	1	1	yes	characteristic
143	C₂×C₄	#15667945	1	1	yes	characteristic
144	C₂×C₄	#15667953	1	2	yes	—
145	C₂×C₄	#15667958	1	2	yes	—
146	C₂×C₄	C₂×#13083553	1	1	yes	characteristic
147	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
148	C₂×C₄	C₂×#13083564	1	1	yes	characteristic
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	8	2	no	—
152	C₈	—	712	2	no	—
153	C₈	#15667947	1	2	yes	—
154	C₈	M₁₆×D₈₉	1	2	yes	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	#19151423	1	2	yes	—
158	C₂²	C₂×#15667945	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#19151423	1	2	yes	—
162	C₄	#19151427	1	2	yes	—
163	C₄	#19151621	1	1	yes	characteristic
164	C₄	C₂×#15667947	1	1	yes	characteristic
165	C₄	C₂×#15667953	1	1	yes	characteristic
166	C₄	C₂×#15667958	1	1	yes	characteristic
167	C₂	—	2	2	no	—
168	C₂	#26019010	1	1	yes	characteristic
169	C₂	C₂×#19151423	1	1	yes	characteristic
170	C₂	C₂×#19151427	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#13083559
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1567	C₂
socle	C₂²×C₈₉	C₂×#107
prime radical	C₂³×C₈₉	#107
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	#1567	D₈₉
hypocenter	C₈₉	#12022
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1567

Complements

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

N	H ≅ G/N	type
3. #26018323	C₂	semidirect
35. #13083549	M₁₆	semidirect
90. C₈₉	#12022	semidirect

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
#26019011	2
#26018323	2
C₈₉×#1567	2
#12022	89