Subgroups — #34035061

Order 18688.

2038 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627185	C₂	1	6	yes	maximal, complemented
2	#22627184	C₂	1	9	yes	maximal, complemented
3	#22627182	C₂	1	6	yes	maximal, complemented
4	C₂×#17989708	C₂	1	9	yes	maximal, complemented
5	#22626926	C₂	1	1	yes	maximal, characteristic
6	#17989709	C₂²	1	2	yes	complemented
7	#17989709	C₂²	1	6	yes	complemented
8	#17989708	C₂²	1	18	yes	complemented
9	#17989708	C₂²	1	36	yes	complemented
10	#17989708	C₂²	1	18	yes	complemented
11	C₂×#14556688	C₂²	1	9	yes	complemented
12	#17989706	C₂²	1	18	yes	complemented
13	C₂×#14556687	C₂²	1	18	yes	complemented
14	C₂×#14556686	C₂²	1	9	yes	complemented
15	C₂²×#11924040	C₂²	1	6	yes	complemented
16	#17989666	C₄	1	6	yes	—
17	#17989665	C₄	1	9	yes	—
18	D₇₃⋊#131	C₂²	1	6	yes	—
19	C₂×#14556678	C₂²	1	9	yes	—
20	#17989422	C₄	1	1	yes	characteristic
21	#14556688	C₂³	1	6	yes	—
22	#14556688	C₂³	1	18	yes	—
23	#14556687	C₂³	1	36	yes	—
24	#14556686	C₂³	1	6	yes	—
25	#14556686	C₂³	1	18	yes	—
26	C₂×#11924040	C₂³	1	9	yes	—
27	C₂×#11924040	C₂³	1	18	yes	—
28	C₂×#11924040	C₂³	1	9	yes	—
29	#14556679	C₂×C₄	1	6	yes	—
30	#14556679	C₂³	1	2	yes	—
31	#14556678	C₂×C₄	1	36	yes	—
32	#14556678	C₂×C₄	1	18	yes	—
33	#14556678	C₂³	1	18	yes	—
34	C₂×#11924038	C₂×C₄	1	9	yes	—
35	D₇₃⋊M₁₆	C₂×C₄	1	18	yes	—
36	C₂×#11924037	C₂×C₄	1	18	yes	—
37	C₂×#11924036	C₂³	1	9	yes	—
38	C₂²×#10596833	C₂³	1	6	yes	—
39	#14556638	C₈	1	6	yes	—
40	#14556636	C₈	1	9	yes	—
41	C₂×#11924027	C₂×C₄	1	9	yes	—
42	#41×D₇₃	C₂×C₄	1	6	yes	—
43	H(5,𝔽₂)×C₇₃	C₈	1	1	yes	characteristic
44	#11924040	C₂⁴	1	1	yes	characteristic
45	#11924040	C₂⁴	1	9	yes	—
46	#11924040	C₂⁴	1	6	yes	—
47	#11924038	C₂²×C₄	1	6	yes	—
48	#11924038	C₂²×C₄	1	18	yes	—
49	#11924037	C₂²×C₄	1	36	yes	—
50	#11924036	C₂²×C₄	1	18	yes	—
51	#11924036	C₂⁴	1	6	yes	—
52	C₂×#10596833	C₂²×C₄	1	18	yes	—
53	C₂×#10596833	C₂²×C₄	1	9	yes	—
54	C₂×#10596833	C₂⁴	1	9	yes	—
55	#11924029	C₂²×C₄	1	2	yes	—
56	#11924028	C₂×C₈	1	18	yes	—
57	#11924027	C₂×C₈	1	36	yes	—
58	#11924027	C₂²×C₄	1	18	yes	—
59	C₂×Dic₁₄₆	C₂×C₈	1	9	yes	—
60	Q₈×D₇₃	C₂×C₈	1	6	yes	—
61	C₂×C₂²⋊D₇₃	C₂×C₈	1	18	yes	—
62	D₄×D₇₃	C₂×C₈	1	18	yes	—
63	C₂²×Dic₇₃	C₂²×C₄	1	6	yes	—
64	C₂×C₄×D₇₃	C₂²×C₄	1	9	yes	—
65	#41×C₇₃	C₂×C₈	1	6	yes	—
66	C₂×D₄×C₇₃	C₂×C₈	1	9	yes	—
67	C₂³×C₇₃	C₂²×C₈	1	6	yes	—
68	C₂²×D₇₃	C₂²×C₈	1	9	yes	—
69	Q₈×C₇₃	C₂²×C₈	1	2	yes	—
70	D₄×C₇₃	C₂²×C₈	1	18	yes	—
71	C₂×C₄×C₇₃	C₂²×C₈	1	9	yes	—
72	C₂²⋊D₇₃	C₂²×C₈	1	36	yes	—
73	C₂×Dic₇₃	C₂²×C₈	1	18	yes	—
74	C₂×Dic₇₃	C₂³×C₄	1	9	yes	—
75	D₂₉₂	C₂²×C₈	1	6	yes	—
76	C₄×D₇₃	C₂²×C₈	1	18	yes	—
77	C₄×D₇₃	C₂³×C₄	1	6	yes	—
78	Dic₁₄₆	C₂²×C₈	1	18	yes	—
79	#10596833	C₂³×C₄	1	9	yes	—
80	#10596833	C₂³×C₄	1	6	yes	—
81	#10596833	C₂⁵	1	1	yes	characteristic
82	C₂²×C₇₃	—	4	6	no	—
83	C₂²×C₇₃	C₂³×C₈	1	9	yes	—
84	C₂×D₇₃	—	4	9	no	—
85	C₂×D₇₃	C₂³×C₈	1	6	yes	—
86	C₄×C₇₃	C₂³×C₈	1	6	yes	—
87	Dic₇₃	C₂³×C₈	1	9	yes	—
88	Dic₇₃	C₂⁴×C₄	1	1	yes	characteristic
89	#62624	—	73	1	no	maximal, Sylow
90	C₂×C₇₃	—	2	9	no	—
91	C₂×C₇₃	C₂⁴×C₈	1	1	yes	characteristic
92	D₇₃	—	2	6	no	—
93	#3573	—	73	1	no	—
94	#3407	—	73	9	no	—
95	#3407	—	73	6	no	—
96	C₂×#504	—	73	9	no	—
97	C₂×#504	—	73	6	no	—
98	C₇₃	#62624	1	1	yes	characteristic, complemented, Sylow
99	#585	—	73	1	no	—
100	#568	—	73	9	no	—
101	#568	—	73	6	no	—
102	C₂×#131	—	73	9	no	—
103	C₂×#131	—	73	6	no	—
104	#504	—	73	18	no	—
105	#504	—	73	36	no	—
106	#504	—	73	18	no	—
107	#504	—	73	2	no	—
108	#504	—	73	6	no	—
109	C₂×M₃₂	—	73	18	no	—
110	C₂×M₃₂	—	73	18	no	—
111	C₂×M₃₂	—	73	9	no	—
112	C₂²×C₁₆	—	73	6	no	—
113	C₂²×C₁₆	—	73	9	no	—
114	2₋¹⁺⁴	—	73	6	no	—
115	H(5,𝔽₂)	—	73	9	no	—
116	H(5,𝔽₂)	#10596842	1	1	yes	characteristic
117	C₂×#41	—	73	6	no	—
118	C₂×#41	—	73	9	no	—
119	#131	—	73	36	no	—
120	#131	—	73	18	no	—
121	#131	—	73	18	no	—
122	#131	—	73	6	no	—
123	#131	—	73	2	no	—
124	C₂×M₁₆	—	73	18	no	—
125	C₂×M₁₆	—	73	18	no	—
126	C₂×M₁₆	—	73	9	no	—
127	C₂²×C₈	—	73	6	no	—
128	C₂²×C₈	—	73	9	no	—
129	M₃₂	—	73	36	no	—
130	M₃₂	—	73	6	no	—
131	M₃₂	—	73	18	no	—
132	C₂×C₁₆	—	73	9	no	—
133	C₂×C₁₆	—	73	18	no	—
134	C₂×C₁₆	—	73	9	no	—
135	C₂×C₁₆	—	73	6	no	—
136	C₂×C₁₆	—	73	18	no	—
137	#41	—	73	2	no	—
138	#41	—	73	36	no	—
139	#41	—	73	18	no	—
140	#41	—	73	18	no	—
141	#41	C₂×#10596842	1	6	yes	—
142	C₂×Q₈	—	73	6	no	—
143	C₂×Q₈	—	73	9	no	—
144	C₂×D₄	—	73	18	no	—
145	C₂×D₄	—	73	18	no	—
146	C₂×D₄	C₂×#10596842	1	9	yes	—
147	C₂²×C₄	—	73	6	no	—
148	C₂²×C₄	—	73	9	no	—
149	M₁₆	—	73	36	no	—
150	M₁₆	—	73	6	no	—
151	M₁₆	—	73	18	no	—
152	C₂×C₈	—	73	18	no	—
153	C₂×C₈	—	73	9	no	—
154	C₂×C₈	—	73	9	no	—
155	C₂×C₈	—	73	18	no	—
156	C₂×C₈	—	73	6	no	—
157	C₁₆	—	73	1	no	—
158	C₁₆	—	73	9	no	—
159	C₁₆	—	73	6	no	—
160	C₂³	—	73	9	no	—
161	C₂³	C₂²×#10596842	1	6	yes	—
162	Q₈	—	73	18	no	—
163	Q₈	C₂²×#10596842	1	2	yes	—
164	D₄	—	73	36	no	—
165	D₄	—	73	6	no	—
166	D₄	C₂²×#10596842	1	18	yes	—
167	C₂×C₄	—	73	9	no	—
168	C₂×C₄	—	73	18	no	—
169	C₂×C₄	—	73	6	no	—
170	C₂×C₄	—	73	18	no	—
171	C₂×C₄	C₂²×#10596842	1	9	yes	—
172	C₈	—	73	1	no	—
173	C₈	—	73	9	no	—
174	C₈	—	73	6	no	—
175	C₂²	—	4	6	no	—
176	C₂²	—	292	9	no	—
177	C₂²	—	73	6	no	—
178	C₂²	C₂³×#10596842	1	9	yes	—
179	C₄	—	73	1	no	—
180	C₄	—	73	9	no	—
181	C₄	C₂³×#10596842	1	6	yes	—
182	C₂	—	2	9	no	—
183	C₂	—	146	6	no	—
184	C₂	C₂⁴×#10596842	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×#10596842
commutator, socle	C₂×C₇₃	C₂⁴×C₈
Fitting	H(5,𝔽₂)×C₇₃	C₈
prime radical	#17989422	C₄
cosocle	#10596833	C₂⁵
hypercenter	H(5,𝔽₂)	#10596842
hypocenter	C₇₃	#62624
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. #22627185	C₂	semidirect
2. #22627184	C₂	semidirect
3. #22627182	C₂	semidirect
4. C₂×#17989708	C₂	semidirect
6. #17989709	C₂²	semidirect
7. #17989709	C₂²	semidirect
8. #17989708	C₂²	semidirect
9. #17989708	C₂²	semidirect
10. #17989708	C₂²	semidirect
11. C₂×#14556688	C₂²	semidirect
12. #17989706	C₂²	semidirect
13. C₂×#14556687	C₂²	semidirect
14. C₂×#14556686	C₂²	semidirect
15. C₂²×#11924040	C₂²	semidirect
98. C₇₃	#62624	semidirect

Maximal subgroups

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

subgroup	index
#22627185	2
#22627184	2
#22627182	2
C₂×#17989708	2
#22626926	2
#62624	73