Subgroups — #34030504

Order 18688.

462 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626355	C₂	1	8	yes	maximal, complemented
2	#22626352	C₂	1	4	yes	maximal, complemented
3	C₂×#17989519	C₂	1	2	yes	maximal, complemented
4	C₂×#17989005	C₂	1	1	yes	maximal, characteristic
5	#17989520	C₂²	1	8	yes	complemented
6	#17989519	C₂²	1	4	yes	—
7	#17989519	C₂²	1	4	yes	complemented
8	#17989518	C₂²	1	8	yes	complemented
9	C₂×#14556641	C₂²	1	4	yes	complemented
10	C₂×#14556597	C₄	1	1	yes	characteristic
11	#17989005	C₄	1	4	yes	—
12	#17989005	C₂²	1	4	yes	—
13	C₂×#14556530	C₄	1	2	yes	—
14	C₂²×#11924003	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
16	C₂×M₃₂×C₇₃	C₄	1	1	yes	characteristic
17	#14556641	C₂³	1	8	yes	—
18	C₂×#11924028	C₈	1	1	yes	characteristic
19	#14556597	C₈	1	4	yes	—
20	#14556597	C₂×C₄	1	4	yes	—
21	C₂×#11924012	C₈	1	1	yes	characteristic
22	C₂×#11924011	C₈	1	2	yes	—
23	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
24	#14556530	C₂×C₄	1	8	yes	—
25	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
26	C₂×#11924003	C₂×C₄	1	2	yes	—
27	C₂×#11924003	C₂³	1	1	yes	characteristic
28	C₂×#11924003	C₂³	1	2	yes	—
29	C₁₆×D₇₃	C₂×C₄	1	4	yes	—
30	C₁₆×D₇₃	C₂³	1	4	yes	—
31	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
32	M₃₂×C₇₃	C₂×C₄	1	4	yes	—
33	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
34	#11924028	C₁₆	1	4	yes	—
35	#11924028	C₂×C₈	1	4	yes	—
36	C₂×Dic₁₄₆	C₁₆	1	1	yes	characteristic
37	C₂×D₂₉₂	C₁₆	1	1	yes	characteristic
38	C₂×C₂²⋊D₇₃	C₁₆	1	2	yes	—
39	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
40	#11924012	C₂×C₈	1	4	yes	—
41	#11924011	C₂×C₈	1	8	yes	—
42	C₂×#10596831	C₂×C₈	1	2	yes	—
43	C₈×D₇₃	C₂×C₈	1	4	yes	—
44	C₈×D₇₃	C₂²×C₄	1	4	yes	—
45	#11924003	C₂²×C₄	1	1	yes	characteristic
46	#11924003	C₂²×C₄	1	2	yes	—
47	#11924003	C₂⁴	1	1	yes	characteristic
48	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
53	C₁₆×C₇₃	C₂²×C₄	1	4	yes	—
54	C₂³×C₇₃	C₂×C₁₆	1	1	yes	characteristic
55	C₂²×D₇₃	C₂×C₁₆	1	2	yes	—
56	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
58	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
60	C₂²⋊D₇₃	C₂×C₁₆	1	8	yes	—
61	C₂×Dic₇₃	C₂×C₁₆	1	2	yes	—
62	D₂₉₂	C₂×C₁₆	1	4	yes	—
63	C₄×D₇₃	C₂×C₁₆	1	4	yes	—
64	C₄×D₇₃	C₂²×C₈	1	4	yes	—
65	Dic₁₄₆	C₂×C₁₆	1	4	yes	—
66	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
67	C₈×C₇₃	C₂²×C₈	1	2	yes	—
68	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
69	#10596831	C₂²×C₈	1	4	yes	—
70	C₂²×C₇₃	—	4	1	no	—
71	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
72	C₂²×C₇₃	C₂²×C₁₆	1	2	yes	—
73	C₂×D₇₃	—	4	2	no	—
74	C₂×D₇₃	C₂²×C₁₆	1	4	yes	—
75	C₄×C₇₃	C₂²×C₁₆	1	2	yes	—
76	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
77	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
78	Dic₇₃	C₂²×C₁₆	1	4	yes	—
79	#33974	—	73	1	no	maximal, Sylow
80	C₂×C₇₃	—	2	1	no	—
81	C₂×C₇₃	—	2	2	no	—
82	C₂×C₇₃	C₂³×C₁₆	1	1	yes	characteristic
83	D₇₃	—	2	4	no	—
84	C₂×#504	—	73	1	no	—
85	#2258	—	73	8	no	—
86	C₂×M₆₄	—	73	2	no	—
87	C₂×M₆₄	—	73	4	no	—
88	C₇₃	#33974	1	1	yes	characteristic, complemented, Sylow
89	C₂×#131	—	73	1	no	—
90	#504	—	73	4	no	—
91	#504	—	73	4	no	—
92	C₂×M₃₂	—	73	2	no	—
93	C₂×M₃₂	#64408	1	1	yes	characteristic
94	C₂²×C₁₆	—	73	1	no	—
95	C₂²×C₁₆	—	73	2	no	—
96	M₆₄	—	73	4	no	—
97	M₆₄	—	73	4	no	—
98	M₆₄	—	73	8	no	—
99	C₂×C₃₂	—	73	4	no	—
100	C₂×C₃₂	—	73	8	no	—
101	C₂×#41	—	73	1	no	—
102	#131	—	73	4	no	—
103	#131	—	73	4	no	—
104	C₂×M₁₆	—	73	1	no	—
105	C₂×M₁₆	—	73	2	no	—
106	C₂²×C₈	—	73	2	no	—
107	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
108	M₃₂	—	73	8	no	—
109	M₃₂	C₂×#64408	1	4	yes	—
110	C₂×C₁₆	—	73	2	no	—
111	C₂×C₁₆	—	73	1	no	—
112	C₂×C₁₆	—	73	2	no	—
113	C₂×C₁₆	—	73	1	no	—
114	C₂×C₁₆	—	73	4	no	—
115	C₂×C₁₆	—	73	4	no	—
116	C₂×C₁₆	C₂×#64408	1	2	yes	—
117	C₃₂	—	73	8	no	—
118	#41	—	73	4	no	—
119	#41	—	73	4	no	—
120	C₂×Q₈	—	73	1	no	—
121	C₂×D₄	—	73	2	no	—
122	C₂×D₄	—	73	1	no	—
123	C₂²×C₄	—	73	2	no	—
124	C₂²×C₄	#11924036	1	1	yes	characteristic
125	M₁₆	—	73	4	no	—
126	M₁₆	—	73	8	no	—
127	C₂×C₈	—	73	2	no	—
128	C₂×C₈	—	73	4	no	—
129	C₂×C₈	—	73	4	no	—
130	C₂×C₈	#11924036	1	1	yes	characteristic
131	C₂×C₈	#11924036	1	2	yes	—
132	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
133	C₂×C₈	C₂²×#64408	1	2	yes	—
134	C₁₆	—	73	1	no	—
135	C₁₆	—	73	2	no	—
136	C₁₆	—	73	1	no	—
137	C₁₆	C₂²×#64408	1	4	yes	—
138	C₂³	—	73	2	no	—
139	C₂³	#14556664	1	1	yes	characteristic
140	Q₈	—	73	4	no	—
141	D₄	—	73	8	no	—
142	D₄	—	73	4	no	—
143	C₂×C₄	—	73	2	no	—
144	C₂×C₄	—	73	4	no	—
145	C₂×C₄	—	73	4	no	—
146	C₂×C₄	#14556664	1	1	yes	characteristic
147	C₂×C₄	#14556664	1	2	yes	—
148	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
149	C₂×C₄	C₂×#11924036	1	2	yes	—
150	C₈	—	73	4	no	—
151	C₈	C₂×#11924036	1	1	yes	characteristic
152	C₈	C₂×#11924036	1	2	yes	—
153	C₈	C₂³×#64408	1	1	yes	characteristic
154	C₂²	—	4	1	no	—
155	C₂²	—	292	2	no	—
156	C₂²	—	73	4	no	—
157	C₂²	C₂×#14556664	1	1	yes	characteristic
158	C₂²	C₂×#14556664	1	2	yes	—
159	C₄	—	73	4	no	—
160	C₄	C₂×#14556664	1	1	yes	characteristic
161	C₄	C₂×#14556664	1	2	yes	—
162	C₄	C₂²×#11924036	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	2	no	—
165	C₂	—	146	4	no	—
166	C₂	C₂²×#14556664	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂³×#64408
commutator, socle	C₂×C₇₃	C₂³×C₁₆
Fitting	C₂×M₃₂×C₇₃	C₄
prime radical	C₂×#11924028	C₈
cosocle	#11924003	C₂⁴
hypercenter	C₂×M₃₂	#64408
hypocenter	C₇₃	#33974
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
1. #22626355	C₂	semidirect
2. #22626352	C₂	semidirect
3. C₂×#17989519	C₂	semidirect
5. #17989520	C₂²	semidirect
7. #17989519	C₂²	semidirect
8. #17989518	C₂²	semidirect
9. C₂×#14556641	C₂²	semidirect
88. C₇₃	#33974	semidirect

Maximal subgroups

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

subgroup	index
#22626355	2
#22626352	2
C₂×#17989519	2
C₂×#17989005	2
#33974	73