Subgroups — #33004095

Order 18688.

294 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22611720	C₂	1	2	yes	maximal
2	C₂×#17988408	C₂	1	1	yes	maximal, characteristic
3	#22610095	C₂	1	1	yes	maximal, characteristic
4	#22608041	C₂	1	2	yes	maximal
5	C₇₃×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556531	C₂²	1	2	yes	—
7	C₂×#14556525	C₂²	1	2	yes	—
8	C₂×#14556523	—	2	1	no	—
9	#17988453	—	2	1	no	—
10	#17988414	—	2	2	no	—
11	#17988412	—	2	2	no	—
12	#17988408	C₄	1	4	yes	complemented
13	C₂×#108×C₇₃	C₂²	1	1	yes	characteristic
14	#427×C₇₃	C₂²	1	1	yes	characteristic
15	#405×C₇₃	—	2	1	no	—
16	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	#339×C₇₃	—	2	1	no	—
18	C₂×C₄⋊Dic₇₃	—	2	2	no	—
19	#14556531	—	2	2	no	—
20	#14556531	C₂×C₄	1	4	yes	—
21	C₂×#11924003	—	2	2	no	—
22	#14556525	—	2	2	no	—
23	#14556525	—	2	2	no	—
24	#14556523	—	4	1	no	—
25	#14556523	—	4	1	no	—
26	#14556498	—	4	2	no	—
27	C₂×M₁₆×C₇₃	—	2	1	no	—
28	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
30	#117×C₇₃	—	2	1	no	—
31	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
32	#108×C₇₃	—	2	1	no	—
33	#108×C₇₃	Q₈	1	2	yes	—
34	#107×C₇₃	D₄	1	2	yes	—
35	#106×C₇₃	—	2	1	no	—
36	#97×C₇₃	—	4	1	no	—
37	C₄×C₈×C₇₃	—	4	1	no	—
38	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₇₃	Q₈	1	2	yes	—
40	C₂²×Dic₇₃	—	4	2	no	—
41	C₄⋊Dic₇₃	—	4	2	no	—
42	C₄⋊Dic₇₃	—	4	2	no	—
43	#11924003	—	2	4	no	—
44	C₂²×C₄×C₇₃	—	2	1	no	—
45	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
46	M₁₆×C₇₃	—	2	2	no	—
47	M₁₆×C₇₃	—	4	1	no	—
48	C₂×C₈×C₇₃	—	4	1	no	—
49	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
51	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
52	C₂×C₈×C₇₃	Q₁₆	1	2	yes	—
53	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
58	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
59	C₄⋊C₄×C₇₃	—	2	2	no	—
60	C₂²⋊C₄×C₇₃	—	4	1	no	—
61	C₄²×C₇₃	—	2	1	no	—
62	C₄²×C₇₃	—	4	1	no	—
63	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
64	C₂³×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂×C₄×C₇₃	—	2	2	no	—
66	C₂×C₄×C₇₃	—	4	2	no	—
67	C₂×C₄×C₇₃	—	4	2	no	—
68	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
73	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
74	C₂×Dic₇₃	—	8	4	no	—
75	C₂×Dic₇₃	—	8	2	no	—
76	C₈×C₇₃	—	4	2	no	—
77	C₈×C₇₃	#107	1	2	yes	—
78	C₈×C₇₃	#107	1	2	yes	—
79	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
80	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
81	C₈×C₇₃	#125	1	1	yes	characteristic
82	C₈×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
83	C₂²×C₇₃	—	2	2	no	—
84	C₂²×C₇₃	#428	1	1	yes	characteristic
85	C₂²×C₇₃	#459	1	1	yes	characteristic
86	C₂²×C₇₃	#477	1	1	yes	characteristic
87	C₄×C₇₃	—	4	2	no	—
88	C₄×C₇₃	—	8	2	no	—
89	C₄×C₇₃	#389	1	1	yes	characteristic
90	C₄×C₇₃	C₂×#107	1	1	yes	characteristic
91	C₄×C₇₃	#462	1	1	yes	characteristic
92	C₄×C₇₃	#474	1	1	yes	characteristic
93	Dic₇₃	—	16	2	no	—
94	#13180	—	73	1	no	maximal, Sylow
95	C₂×C₇₃	—	2	1	no	—
96	C₂×C₇₃	—	2	1	no	—
97	C₂×C₇₃	#1947	1	1	yes	characteristic
98	C₂×C₇₃	#2212	1	2	yes	—
99	#2159	—	73	2	no	—
100	C₂×#364	—	73	1	no	—
101	#1846	—	73	1	no	—
102	#1772	D₇₃	1	1	yes	characteristic
103	#1387	—	73	2	no	—
104	C₇₃	#13180	1	1	yes	characteristic, complemented, Sylow
105	C₂×M₃₂	—	73	2	no	—
106	C₂×#108	C₂×D₇₃	1	1	yes	characteristic
107	#427	C₂×D₇₃	1	1	yes	characteristic
108	C₂×#107	—	73	2	no	—
109	C₂×#106	—	146	1	no	—
110	#405	—	2	1	no	—
111	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
112	#382	—	146	1	no	—
113	#367	—	146	2	no	—
114	#366	—	146	2	no	—
115	#364	—	73	4	no	—
116	#339	—	2	1	no	—
117	C₂×M₁₆	—	2	1	no	—
118	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
120	#117	—	2	1	no	—
121	C₂×C₄⋊C₄	—	146	2	no	—
122	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
123	M₃₂	—	73	4	no	—
124	M₃₂	—	146	2	no	—
125	C₂×C₁₆	—	146	2	no	—
126	#108	—	2	1	no	—
127	#108	Dic₁₄₆	1	2	yes	—
128	#107	—	146	2	no	—
129	#107	—	146	2	no	—
130	#107	D₂₉₂	1	2	yes	—
131	#106	—	2	1	no	—
132	#106	—	292	1	no	—
133	#106	—	292	1	no	—
134	#97	—	4	1	no	—
135	C₄×C₈	—	4	1	no	—
136	C₄×C₈	Dic₁₄₆	1	2	yes	—
137	C₄×C₈	C₄×D₇₃	1	2	yes	—
138	#95	—	292	2	no	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	292	2	no	—
141	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
142	M₁₆	—	2	2	no	—
143	M₁₆	—	4	1	no	—
144	C₂×C₈	—	4	1	no	—
145	C₂×C₈	#11924007	1	2	yes	—
146	C₂×C₈	#11924013	1	1	yes	characteristic
147	C₂×C₈	#11924013	1	1	yes	characteristic
148	C₂×C₈	#11924018	1	2	yes	—
149	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
150	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
151	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
152	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
153	C₂×C₈	#11924028	1	1	yes	characteristic
154	C₂×C₈	#11924028	1	1	yes	characteristic
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	292	2	no	—
157	C₄⋊C₄	—	292	2	no	—
158	C₂²⋊C₄	—	4	1	no	—
159	C₄²	—	2	1	no	—
160	C₄²	—	4	1	no	—
161	C₄²	#11924007	1	2	yes	—
162	C₁₆	—	146	4	no	—
163	C₂³	#14556616	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	584	4	no	—
168	C₂×C₄	—	584	2	no	—
169	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
170	C₂×C₄	#14556565	1	1	yes	characteristic
171	C₂×C₄	#14556570	1	1	yes	characteristic
172	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
173	C₂×C₄	C₂×#11924018	1	1	yes	characteristic
174	C₂×C₄	#14556621	1	1	yes	characteristic
175	C₈	—	4	2	no	—
176	C₈	#14556524	1	2	yes	—
177	C₈	#14556524	1	2	yes	—
178	C₈	C₄×D₂₉₂	1	1	yes	characteristic
179	C₈	C₄×Dic₁₄₆	1	1	yes	characteristic
180	C₈	#14556583	1	1	yes	characteristic
181	C₈	#14556591	1	1	yes	characteristic
182	C₂²	—	2	2	no	—
183	C₂²	#17988630	1	1	yes	characteristic
184	C₂²	#17988782	1	1	yes	characteristic
185	C₂²	#17988882	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	8	2	no	—
188	C₄	—	1168	2	no	—
189	C₄	#17988485	1	1	yes	characteristic
190	C₄	C₂×#14556524	1	1	yes	characteristic
191	C₄	#17988797	1	1	yes	characteristic
192	C₄	#17988867	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#22610689	1	1	yes	characteristic
196	C₂	#22612000	1	2	yes	—

	subgroup	G/H
center	C₂²	#17988782
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1772	C₂
socle	C₂²×C₇₃	#459
prime radical	C₂³×C₇₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1772	D₇₃
hypocenter	C₇₃	#13180
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
12. #17988408	C₄	semidirect
104. C₇₃	#13180	semidirect

Maximal subgroups

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

subgroup	index
#22611720	2
C₂×#17988408	2
#22610095	2
#22608041	2
C₇₃×#1772	2
#13180	73