Subgroups — #32973166

Order 18688.

208 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988408	C₂	1	1	yes	maximal, characteristic
2	#22610103	C₂	1	1	yes	maximal, characteristic
3	C₂×#364×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556531	—	2	1	no	—
5	C₂×#14556527	—	2	1	no	—
6	C₂×#14556527	—	2	1	no	—
7	#17988611	C₄	1	2	yes	—
8	#17988408	—	2	1	no	—
9	#17988408	C₄	1	2	yes	—
10	C₂×M₃₂×C₇₃	—	2	1	no	—
11	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
12	#364×C₇₃	—	2	1	no	—
13	#364×C₇₃	C₄	1	2	yes	—
14	C₂×#11924012	—	2	2	no	—
15	#14556531	—	2	2	no	—
16	#14556531	—	4	1	no	—
17	C₂×#11924003	—	4	1	no	—
18	#14556527	—	4	1	no	—
19	#14556527	—	4	1	no	—
20	#14556527	—	4	1	no	—
21	#14556527	—	4	1	no	—
22	C₄⋊#10596831	—	4	2	no	—
23	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
26	M₃₂×C₇₃	—	2	2	no	—
27	M₃₂×C₇₃	—	4	1	no	—
28	C₂×C₁₆×C₇₃	—	4	1	no	—
29	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
33	#11924012	—	4	2	no	—
34	#11924012	—	4	2	no	—
35	C₂×#10596831	—	4	2	no	—
36	#11924003	—	4	2	no	—
37	C₂²×C₄×C₇₃	—	2	1	no	—
38	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
44	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
45	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
46	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
47	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₇₃	Q₁₆	1	1	yes	characteristic
50	C₂×C₈×C₇₃	Q₁₆	1	1	yes	characteristic
51	C₄²×C₇₃	—	2	1	no	—
52	C₄²×C₇₃	C₄²	1	1	yes	characteristic
53	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
54	C₁₆×C₇₃	—	4	2	no	—
55	C₂³×C₇₃	#101	1	1	yes	characteristic
56	C₂×C₄×C₇₃	—	2	1	no	—
57	C₂×C₄×C₇₃	—	2	1	no	—
58	C₂×C₄×C₇₃	—	4	2	no	—
59	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
60	C₂×C₄×C₇₃	#100	1	1	yes	characteristic
61	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
62	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
63	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
64	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
65	C₈×C₇₃	#102	1	1	yes	characteristic
66	C₈×C₇₃	#102	1	1	yes	characteristic
67	C₈×C₇₃	#103	1	1	yes	characteristic
68	C₈×C₇₃	#103	1	1	yes	characteristic
69	C₈×C₇₃	#106	1	1	yes	characteristic
70	C₈×C₇₃	#106	1	1	yes	characteristic
71	C₈×C₇₃	#107	1	1	yes	characteristic
72	C₈×C₇₃	#107	1	1	yes	characteristic
73	#10596831	—	8	2	no	—
74	C₂²×C₇₃	—	4	1	no	—
75	C₂²×C₇₃	#332	1	1	yes	characteristic
76	C₂²×C₇₃	#332	1	1	yes	characteristic
77	C₂²×C₇₃	#343	1	1	yes	characteristic
78	C₄×C₇₃	—	2	2	no	—
79	C₄×C₇₃	—	4	1	no	—
80	C₄×C₇₃	#331	1	1	yes	characteristic
81	C₄×C₇₃	#333	1	1	yes	characteristic
82	C₄×C₇₃	#340	1	1	yes	characteristic
83	C₄×C₇₃	#340	1	1	yes	characteristic
84	#7272	—	73	1	no	maximal, Sylow
85	C₂×C₇₃	—	2	1	no	—
86	C₂×C₇₃	—	2	1	no	—
87	C₂×C₇₃	—	2	1	no	—
88	C₂×C₇₃	#1296	1	1	yes	characteristic
89	C₂×#364	—	73	1	no	—
90	C₂×#364	D₇₃	1	1	yes	characteristic
91	#1848	—	73	1	no	—
92	C₇₃	#7272	1	1	yes	characteristic, complemented, Sylow
93	C₂×M₃₂	—	2	1	no	—
94	C₂×M₃₂	—	146	1	no	—
95	C₂×#108	—	146	1	no	—
96	C₂×#108	—	146	1	no	—
97	#423	—	73	2	no	—
98	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
99	#364	—	2	1	no	—
100	#364	—	73	2	no	—
101	#364	—	146	1	no	—
102	#364	Dic₇₃	1	2	yes	—
103	C₂×M₁₆	—	146	2	no	—
104	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
105	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
106	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
107	M₃₂	—	2	2	no	—
108	M₃₂	—	4	1	no	—
109	M₃₂	—	292	1	no	—
110	M₃₂	—	146	2	no	—
111	C₂×C₁₆	—	4	1	no	—
112	C₂×C₁₆	—	292	1	no	—
113	#108	—	292	1	no	—
114	#108	—	292	1	no	—
115	#108	—	292	1	no	—
116	#108	—	292	1	no	—
117	C₄⋊C₈	—	292	2	no	—
118	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
119	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
121	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	#11924005	1	1	yes	characteristic
124	M₁₆	—	292	2	no	—
125	M₁₆	—	292	2	no	—
126	C₂×C₈	—	292	2	no	—
127	C₂×C₈	#11924005	1	1	yes	characteristic
128	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
129	C₂×C₈	#11924007	1	1	yes	characteristic
130	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
131	C₂×C₈	#11924013	1	1	yes	characteristic
132	C₂×C₈	D₅₈₄	1	1	yes	characteristic
133	C₂×C₈	#11924015	1	1	yes	characteristic
134	C₂×C₈	#11924016	1	1	yes	characteristic
135	C₂×C₈	#11924017	1	1	yes	characteristic
136	C₂×C₈	#11924017	1	1	yes	characteristic
137	C₂×C₈	#11924018	1	1	yes	characteristic
138	C₂×C₈	Dic₂₉₂	1	1	yes	characteristic
139	C₄²	—	2	1	no	—
140	C₄²	C₄×Dic₇₃	1	1	yes	characteristic
141	C₄²	#11924007	1	1	yes	characteristic
142	C₁₆	—	4	2	no	—
143	C₁₆	—	292	2	no	—
144	C₂³	#14556509	1	1	yes	characteristic
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	#14556498	1	1	yes	characteristic
149	C₂×C₄	#14556507	1	1	yes	characteristic
150	C₂×C₄	#14556511	1	1	yes	characteristic
151	C₂×C₄	#14556513	1	1	yes	characteristic
152	C₂×C₄	#14556514	1	1	yes	characteristic
153	C₂×C₄	#14556516	1	1	yes	characteristic
154	C₈	—	584	2	no	—
155	C₈	#14556512	1	1	yes	characteristic
156	C₈	#14556513	1	1	yes	characteristic
157	C₈	#14556515	1	1	yes	characteristic
158	C₈	#14556516	1	1	yes	characteristic
159	C₈	#14556522	1	1	yes	characteristic
160	C₈	#14556523	1	1	yes	characteristic
161	C₈	#14556524	1	1	yes	characteristic
162	C₈	#14556525	1	1	yes	characteristic
163	C₂²	—	4	1	no	—
164	C₂²	#17988332	1	1	yes	characteristic
165	C₂²	#17988334	1	1	yes	characteristic
166	C₂²	#17988355	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	4	1	no	—
169	C₄	#17988330	1	1	yes	characteristic
170	C₄	#17988335	1	1	yes	characteristic
171	C₄	#17988349	1	1	yes	characteristic
172	C₄	#17988350	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	#22607815	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17988335
commutator	C₄²×C₇₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₇₃
Fitting	C₂×#364×C₇₃	C₂
socle	C₂×C₇₃	#1296
prime radical	C₂³×C₇₃	#101
cosocle	C₂×C₄×C₈×C₇₃	C₂²
hypercenter	C₂×#364	D₇₃
hypocenter	C₇₃	#7272
LCS	G ▷ C₄²×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
92. C₇₃	#7272	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988408	2
#22610103	2
C₂×#364×C₇₃	2
#7272	73