Subgroups — #33241515

Order 18688.

482 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988989	C₂	1	1	yes	maximal, characteristic
2	#22612214	C₂	1	4	yes	maximal, complemented
3	#22612207	C₂	1	4	yes	maximal, complemented
4	#22611720	C₂	1	4	yes	maximal
5	C₄×#14556531	C₂	1	1	yes	maximal, characteristic
6	C₇₃×#3160	C₂	1	1	yes	maximal, characteristic
7	C₂×#14556583	—	2	1	no	—
8	C₂×#14556531	C₂²	1	2	yes	—
9	#17988989	—	2	1	no	—
10	#17988989	—	2	1	no	—
11	#17988989	C₂²	1	2	yes	—
12	#17988989	C₂²	1	2	yes	—
13	C₂×#14556525	C₂²	1	2	yes	—
14	C₂×#14556523	—	2	1	no	—
15	#17988414	C₂²	1	8	yes	—
16	#17988412	C₂²	1	8	yes	—
17	#17988363	C₂²	1	2	yes	—
18	C₄×#11924003	C₂²	1	2	yes	—
19	#533×C₇₃	—	2	1	no	—
20	#500×C₇₃	C₂²	1	2	yes	—
21	#499×C₇₃	—	2	1	no	—
22	#499×C₇₃	C₂²	1	2	yes	—
23	#498×C₇₃	—	2	1	no	—
24	#427×C₇₃	—	2	1	no	—
25	#427×C₇₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
27	#14556583	—	2	2	no	—
28	#14556583	—	2	2	no	—
29	#14556583	—	4	2	no	—
30	C₂×C₄⋊Dic₇₃	—	2	2	no	—
31	C₂×#11924007	—	4	2	no	—
32	#14556531	Q₈	1	8	yes	complemented
33	C₂×#11924003	C₂³	1	4	yes	—
34	#14556525	—	2	2	no	—
35	#14556525	C₂³	1	4	yes	—
36	#14556523	—	2	2	no	—
37	#14556523	—	2	2	no	—
38	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₇₃	—	2	2	no	—
41	#125×C₇₃	—	2	2	no	—
42	C₂²⋊Q₈×C₇₃	—	4	2	no	—
43	C₄×Q₈×C₇₃	—	4	2	no	—
44	#117×C₇₃	—	2	2	no	—
45	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
46	#107×C₇₃	—	2	2	no	—
47	#107×C₇₃	C₂³	1	4	yes	—
48	#106×C₇₃	—	2	2	no	—
49	#106×C₇₃	—	2	2	no	—
50	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
51	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
54	C₂²×Dic₇₃	—	4	2	no	—
55	C₄⋊Dic₇₃	—	2	4	no	—
56	C₄⋊Dic₇₃	—	4	2	no	—
57	#11924007	—	4	4	no	—
58	#11924007	—	4	4	no	—
59	#11924003	C₂×Q₈	1	8	yes	—
60	C₂×Q₈×C₇₃	—	4	2	no	—
61	C₂²×C₄×C₇₃	D₈	1	2	yes	—
62	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₇₃	Q₁₆	1	4	yes	—
64	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
68	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₇₃	—	2	4	no	—
70	C₄⋊C₄×C₇₃	—	4	4	no	—
71	C₄⋊C₄×C₇₃	—	4	2	no	—
72	C₂²⋊C₄×C₇₃	—	4	2	no	—
73	C₄²×C₇₃	—	4	2	no	—
74	C₄²×C₇₃	D₈	1	2	yes	—
75	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
77	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
78	Q₈×C₇₃	—	8	4	no	—
79	C₂×C₄×C₇₃	—	4	2	no	—
80	C₂×C₄×C₇₃	—	4	4	no	—
81	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
82	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
86	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
87	C₂×Dic₇₃	—	4	4	no	—
88	C₂×Dic₇₃	—	8	4	no	—
89	C₈×C₇₃	C₄⋊Q₈	1	2	yes	—
90	C₈×C₇₃	C₂×Q₁₆	1	2	yes	—
91	C₈×C₇₃	C₂×Q₁₆	1	2	yes	—
92	C₈×C₇₃	C₂²×Q₈	1	2	yes	—
93	C₂²×C₇₃	—	2	2	no	—
94	C₂²×C₇₃	#500	1	2	yes	—
95	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
96	C₄×C₇₃	—	2	2	no	—
97	C₄×C₇₃	—	2	2	no	—
98	C₄×C₇₃	—	8	4	no	—
99	C₄×C₇₃	#500	1	2	yes	—
100	C₄×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
102	Dic₇₃	—	8	4	no	—
103	#33732	—	73	1	no	maximal, Sylow
104	C₂×C₇₃	—	2	2	no	—
105	C₂×C₇₃	C₂×#500	1	1	yes	characteristic
106	C₂×C₇₃	#3414	1	2	yes	—
107	#3160	D₇₃	1	1	yes	characteristic
108	C₂×#499	—	73	1	no	—
109	#2255	—	73	4	no	—
110	#2253	—	73	4	no	—
111	#2159	—	73	4	no	—
112	C₄×M₃₂	—	73	1	no	—
113	C₇₃	#33732	1	1	yes	characteristic, complemented, Sylow
114	#533	—	2	1	no	—
115	C₂×#125	—	146	1	no	—
116	C₂×M₃₂	—	73	2	no	—
117	#500	C₂×D₇₃	1	2	yes	—
118	#499	—	2	1	no	—
119	#499	—	73	2	no	—
120	#499	—	73	2	no	—
121	#499	—	146	1	no	—
122	#499	—	146	1	no	—
123	#499	C₂×D₇₃	1	2	yes	—
124	#498	—	2	1	no	—
125	#427	—	2	1	no	—
126	#427	C₂×D₇₃	1	2	yes	—
127	C₂×#107	—	73	2	no	—
128	C₂×#106	—	146	1	no	—
129	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
130	#367	—	73	8	no	—
131	#366	—	73	8	no	—
132	#346	—	73	2	no	—
133	C₄×C₁₆	—	73	2	no	—
134	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
136	C₄⋊Q₈	—	2	2	no	—
137	#125	—	2	2	no	—
138	#125	—	292	2	no	—
139	#125	—	146	2	no	—
140	#125	—	146	2	no	—
141	C₂²⋊Q₈	—	4	2	no	—
142	C₄×Q₈	—	4	2	no	—
143	#117	—	2	2	no	—
144	C₂×C₄⋊C₄	—	292	2	no	—
145	C₂×C₄⋊C₄	—	146	2	no	—
146	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
147	M₃₂	—	73	8	no	—
148	C₂×C₁₆	—	73	4	no	—
149	#107	—	2	2	no	—
150	#107	—	73	4	no	—
151	#107	—	146	2	no	—
152	#107	C₂²×D₇₃	1	4	yes	—
153	#106	—	2	2	no	—
154	#106	—	2	2	no	—
155	#106	—	146	2	no	—
156	#106	—	146	2	no	—
157	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
159	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
160	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
161	C₂×Q₈	—	4	2	no	—
162	C₂²×C₄	—	292	2	no	—
163	C₂²×C₄	#11924013	1	2	yes	—
164	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
165	C₂×C₈	#11924018	1	4	yes	—
166	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
167	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
168	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
169	C₂×C₈	Q₈×D₇₃	1	4	yes	—
170	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	292	4	no	—
175	C₄⋊C₄	—	292	4	no	—
176	C₄⋊C₄	—	292	2	no	—
177	C₄⋊C₄	—	146	4	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₄²	—	4	2	no	—
180	C₄²	#11924013	1	2	yes	—
181	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
182	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
183	C₁₆	—	73	8	no	—
184	C₂³	C₂×#11924013	1	1	yes	characteristic
185	Q₈	—	8	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	292	4	no	—
189	C₂×C₄	—	584	4	no	—
190	C₂×C₄	#14556589	1	2	yes	—
191	C₂×C₄	C₂×#11924013	1	2	yes	—
192	C₂×C₄	C₂×#11924013	1	2	yes	—
193	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
194	C₂×C₄	C₂×#11924018	1	2	yes	—
195	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
196	C₈	#14556589	1	2	yes	—
197	C₈	C₂×#11924018	1	2	yes	—
198	C₈	C₂×#11924018	1	2	yes	—
199	C₈	C₂×Q₈×D₇₃	1	2	yes	—
200	C₂²	—	2	2	no	—
201	C₂²	#17988990	1	2	yes	—
202	C₂²	C₂²×#11924013	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	8	4	no	—
206	C₄	—	584	4	no	—
207	C₄	#17988990	1	2	yes	—
208	C₄	C₂×#14556589	1	1	yes	characteristic
209	C₄	C₂²×#11924018	1	1	yes	characteristic
210	C₂	—	2	2	no	—
211	C₂	C₂×#17988990	1	1	yes	characteristic
212	C₂	#22623065	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#11924013
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#3160	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂³×C₇₃	C₂×D₈
hypercenter	#3160	D₇₃
hypocenter	C₇₃	#33732
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #22612214	C₂	semidirect
3. #22612207	C₂	semidirect
32. #14556531	Q₈	semidirect
113. C₇₃	#33732	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988989	2
#22612214	2
#22612207	2
#22611720	2
C₄×#14556531	2
C₇₃×#3160	2
#33732	73