Subgroups — #33000874

Order 18688.

554 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556527	C₂	1	2	yes	maximal
2	#22608025	C₂	1	4	yes	maximal, complemented
3	C₂²×M₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924012	—	2	2	no	—
5	C₂×#14556527	—	2	2	no	—
6	C₂×#14556527	—	2	4	no	—
7	C₂×#14556527	C₄	1	8	yes	—
8	C₂×#14556527	C₂²	1	4	yes	—
9	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
10	C₂×M₃₂×C₇₃	—	2	4	no	—
11	C₂×M₃₂×C₇₃	C₄	1	4	yes	—
12	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
13	C₂×#11924012	—	2	8	no	—
14	C₂×#11924012	—	2	4	no	—
15	C₂×#11924012	—	4	2	no	—
16	C₂×#11924012	—	4	4	no	—
17	C₂²×#10596831	—	4	2	no	—
18	#14556527	—	2	8	no	—
19	#14556527	—	2	4	no	—
20	#14556527	C₂×C₄	1	8	yes	—
21	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₃	D₄	1	4	yes	—
26	C₂²×C₈×C₇₃	Q₈	1	2	yes	—
27	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
28	M₃₂×C₇₃	—	2	8	no	—
29	C₂×C₁₆×C₇₃	—	2	4	no	—
30	C₂×C₁₆×C₇₃	C₂×C₄	1	4	yes	—
31	#11924012	—	2	8	no	—
32	#11924012	—	4	8	no	—
33	#11924012	—	4	4	no	—
34	C₂×#10596831	—	4	8	no	—
35	C₂×#10596831	—	4	4	no	—
36	C₂⁴×C₇₃	Q₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	C₂²⋊C₄	1	4	yes	—
38	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₇₃	D₈	1	2	yes	—
40	C₂²×C₄×C₇₃	SD₁₆	1	4	yes	—
41	C₂²×C₄×C₇₃	Q₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₄²	1	4	yes	—
44	C₂×C₈×C₇₃	C₂²⋊C₄	1	4	yes	—
45	C₂×C₈×C₇₃	C₂²⋊C₄	1	4	yes	—
46	C₂×C₈×C₇₃	C₄⋊C₄	1	8	yes	—
47	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
49	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
52	C₁₆×C₇₃	—	2	4	no	—
53	C₂³×C₇₃	—	4	2	no	—
54	C₂³×C₇₃	#103	1	4	yes	—
55	C₂³×C₇₃	#107	1	2	yes	—
56	C₂³×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
57	C₂×C₄×C₇₃	#95	1	4	yes	—
58	C₂×C₄×C₇₃	#102	1	4	yes	—
59	C₂×C₄×C₇₃	#102	1	4	yes	—
60	C₂×C₄×C₇₃	#103	1	4	yes	—
61	C₂×C₄×C₇₃	#106	1	4	yes	—
62	C₂×C₄×C₇₃	#107	1	2	yes	—
63	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
67	C₈×C₇₃	#95	1	4	yes	—
68	C₈×C₇₃	C₂×C₄²	1	1	yes	characteristic
69	C₈×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₈×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
71	#10596831	—	4	8	no	—
72	C₂²×C₇₃	—	2	4	no	—
73	C₂²×C₇₃	—	2	2	no	—
74	C₂²×C₇₃	—	4	4	no	—
75	C₂²×C₇₃	#340	1	4	yes	—
76	C₂²×C₇₃	C₂×#103	1	2	yes	—
77	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
78	C₄×C₇₃	#340	1	4	yes	—
79	C₄×C₇₃	C₂×#95	1	1	yes	characteristic
80	C₄×C₇₃	C₂×#102	1	2	yes	—
81	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
82	#12559	—	73	1	no	maximal, Sylow
83	C₂×C₇₃	—	2	4	no	—
84	C₂×C₇₃	—	2	2	no	—
85	C₂×C₇₃	C₂×#340	1	1	yes	characteristic
86	C₂×C₇₃	#2161	1	2	yes	—
87	C₂²×M₃₂	D₇₃	1	1	yes	characteristic
88	C₂²×#108	—	73	2	no	—
89	#1381	—	73	4	no	—
90	C₇₃	#12559	1	1	yes	characteristic, complemented, Sylow
91	C₂²×M₁₆	—	146	2	no	—
92	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
93	C₂×M₃₂	—	2	4	no	—
94	C₂×M₃₂	Dic₇₃	1	4	yes	—
95	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
96	C₂×#108	—	73	8	no	—
97	C₂×#108	—	73	4	no	—
98	C₂×#108	—	146	2	no	—
99	C₂×#108	—	146	4	no	—
100	C₂³×C₄	D₂₉₂	1	1	yes	characteristic
101	C₂×M₁₆	—	292	2	no	—
102	C₂×M₁₆	—	292	4	no	—
103	C₂×M₁₆	—	146	8	no	—
104	C₂×M₁₆	—	146	4	no	—
105	C₂²×C₈	—	292	2	no	—
106	C₂²×C₈	Dic₁₄₆	1	2	yes	—
107	C₂²×C₈	C₄×D₇₃	1	4	yes	—
108	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
109	C₂²×C₈	C₂×Dic₇₃	1	2	yes	—
110	C₂²×C₈	C₂²⋊D₇₃	1	4	yes	—
111	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
112	M₃₂	—	2	8	no	—
113	C₂×C₁₆	—	2	4	no	—
114	C₂×C₁₆	C₂×Dic₇₃	1	4	yes	—
115	#108	—	73	8	no	—
116	#108	—	146	8	no	—
117	#108	—	146	4	no	—
118	C₂⁴	Dic₂₉₂	1	1	yes	characteristic
119	C₂²×C₄	#11924005	1	4	yes	—
120	C₂²×C₄	C₄⋊Dic₇₃	1	2	yes	—
121	C₂²×C₄	D₅₈₄	1	2	yes	—
122	C₂²×C₄	#11924017	1	4	yes	—
123	C₂²×C₄	Dic₂₉₂	1	1	yes	characteristic
124	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
125	M₁₆	—	292	8	no	—
126	M₁₆	—	292	4	no	—
127	M₁₆	—	146	8	no	—
128	C₂×C₈	—	292	8	no	—
129	C₂×C₈	—	292	4	no	—
130	C₂×C₈	C₄×Dic₇₃	1	4	yes	—
131	C₂×C₈	#11924005	1	4	yes	—
132	C₂×C₈	C₂²⋊Dic₇₃	1	4	yes	—
133	C₂×C₈	#11924007	1	8	yes	—
134	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
135	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
136	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
137	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
138	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
139	C₁₆	—	2	4	no	—
140	C₂³	—	4	2	no	—
141	C₂³	#14556516	1	4	yes	—
142	C₂³	#14556525	1	2	yes	—
143	C₂³	C₂×Dic₂₉₂	1	1	yes	characteristic
144	C₂×C₄	#14556498	1	4	yes	—
145	C₂×C₄	#14556513	1	4	yes	—
146	C₂×C₄	#14556513	1	4	yes	—
147	C₂×C₄	#14556516	1	4	yes	—
148	C₂×C₄	#14556523	1	4	yes	—
149	C₂×C₄	#14556525	1	2	yes	—
150	C₂×C₄	C₂×#11924005	1	2	yes	—
151	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
152	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
153	C₂×C₄	C₂×#11924017	1	2	yes	—
154	C₈	—	292	8	no	—
155	C₈	#14556498	1	4	yes	—
156	C₈	C₂×C₄×Dic₇₃	1	1	yes	characteristic
157	C₈	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
158	C₈	C₂×#11924007	1	2	yes	—
159	C₂²	—	2	4	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	4	4	no	—
162	C₂²	#17988350	1	4	yes	—
163	C₂²	C₂×#14556516	1	2	yes	—
164	C₂²	C₂×#14556525	1	1	yes	characteristic
165	C₄	#17988350	1	4	yes	—
166	C₄	C₂×#14556498	1	1	yes	characteristic
167	C₄	C₂×#14556513	1	2	yes	—
168	C₄	C₂×#14556523	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	—	2	4	no	—
171	C₂	C₂×#17988350	1	1	yes	characteristic
172	C₂	#22611726	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₈₄
commutator	C₈×C₇₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂²×M₃₂×C₇₃	C₂
socle	C₂²×C₇₃	C₂×#107
prime radical	C₂⁴×C₇₃	Q₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂²×M₃₂	D₇₃
hypocenter	C₇₃	#12559
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
2. #22608025	C₂	semidirect
90. C₇₃	#12559	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14556527	2
#22608025	2
C₂²×M₃₂×C₇₃	2
#12559	73