Subgroups — #33004088

Order 18688.

286 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988416	C₂	1	2	yes	maximal
2	C₂×C₄⋊#11924003	C₂	1	1	yes	maximal, characteristic
3	C₄×#14556527	C₂	1	1	yes	maximal, characteristic
4	#22608024	C₂	1	2	yes	maximal
5	C₇₃×#1848	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924003	C₂²	1	2	yes	—
7	C₂×#14556527	C₂²	1	1	yes	characteristic
8	C₂×#14556527	C₂²	1	1	yes	characteristic
9	C₄×#11924012	—	2	1	no	—
10	#17988416	—	2	4	no	—
11	C₄⋊#11924003	C₄	1	2	yes	—
12	C₄⋊#11924003	C₄	1	2	yes	—
13	#17988352	—	2	1	no	—
14	C₂×#108×C₇₃	—	2	1	no	—
15	C₂×#108×C₇₃	C₂²	1	2	yes	—
16	#423×C₇₃	—	2	1	no	—
17	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×#11924012	—	2	1	no	—
19	C₂×#11924012	—	2	1	no	—
20	C₂×#11924003	—	2	2	no	—
21	C₂×#11924003	—	2	2	no	—
22	C₂×#11924003	C₂×C₄	1	4	yes	—
23	#14556527	—	2	1	no	—
24	#14556527	—	2	1	no	—
25	#14556527	D₄	1	2	yes	—
26	#14556527	Q₈	1	2	yes	—
27	#14556502	—	4	1	no	—
28	C₄×#10596831	—	4	1	no	—
29	C₂×M₁₆×C₇₃	—	2	2	no	—
30	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
33	#108×C₇₃	—	2	2	no	—
34	#108×C₇₃	—	2	2	no	—
35	#108×C₇₃	—	4	2	no	—
36	C₄⋊C₈×C₇₃	—	4	1	no	—
37	C₄⋊C₈×C₇₃	—	4	1	no	—
38	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
42	#11924012	—	2	2	no	—
43	#11924012	—	2	2	no	—
44	#11924012	—	4	1	no	—
45	#11924012	—	4	1	no	—
46	C₂×#10596831	—	4	1	no	—
47	C₂×#10596831	—	4	1	no	—
48	#11924003	—	2	4	no	—
49	C₂²×C₄×C₇₃	Q₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₇₃	Q₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
52	M₁₆×C₇₃	—	4	2	no	—
53	M₁₆×C₇₃	—	4	2	no	—
54	C₂×C₈×C₇₃	—	2	1	no	—
55	C₂×C₈×C₇₃	—	2	1	no	—
56	C₂×C₈×C₇₃	—	4	2	no	—
57	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
64	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
65	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
66	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
67	C₄²×C₇₃	D₈	1	1	yes	characteristic
68	C₄²×C₇₃	D₈	1	1	yes	characteristic
69	C₂³×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₄×C₇₃	—	2	2	no	—
71	C₂×C₄×C₇₃	—	4	1	no	—
72	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
73	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
74	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
75	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
82	C₈×C₇₃	—	2	1	no	—
83	C₈×C₇₃	—	2	1	no	—
84	C₈×C₇₃	—	8	2	no	—
85	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
86	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
87	C₈×C₇₃	#125	1	1	yes	characteristic
88	C₈×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
89	#10596831	—	4	2	no	—
90	#10596831	—	4	2	no	—
91	C₂²×C₇₃	—	2	1	no	—
92	C₂²×C₇₃	—	2	1	no	—
93	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
94	C₂²×C₇₃	#462	1	1	yes	characteristic
95	C₂²×C₇₃	#477	1	1	yes	characteristic
96	C₄×C₇₃	—	2	1	no	—
97	C₄×C₇₃	—	2	1	no	—
98	C₄×C₇₃	—	2	1	no	—
99	C₄×C₇₃	#368	1	2	yes	complemented
100	C₄×C₇₃	#389	1	1	yes	characteristic
101	C₄×C₇₃	#428	1	1	yes	characteristic
102	C₄×C₇₃	#459	1	1	yes	characteristic
103	C₄×C₇₃	#474	1	1	yes	characteristic
104	#13179	—	73	1	no	maximal, Sylow
105	C₂×C₇₃	—	2	1	no	—
106	C₂×C₇₃	—	2	1	no	—
107	C₂×C₇₃	#1947	1	1	yes	characteristic
108	C₂×C₇₃	C₂×#368	1	1	yes	characteristic
109	C₂×C₇₃	#2161	1	1	yes	characteristic
110	C₂×#368	—	73	2	no	—
111	C₂×C₄⋊C₁₆	—	73	1	no	—
112	#1848	D₇₃	1	1	yes	characteristic
113	C₄×#108	—	73	1	no	—
114	#1381	—	73	2	no	—
115	C₇₃	#13179	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₁₆	—	73	2	no	—
117	C₂×#108	—	2	1	no	—
118	C₂×#108	—	73	1	no	—
119	C₂×#108	—	73	1	no	—
120	C₂×#108	C₂×D₇₃	1	2	yes	—
121	#423	—	2	1	no	—
122	C₄×M₁₆	—	146	1	no	—
123	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
124	#368	—	146	4	no	—
125	C₄⋊C₁₆	—	73	2	no	—
126	C₄⋊C₁₆	—	73	2	no	—
127	#341	—	146	1	no	—
128	C₂×M₁₆	—	2	2	no	—
129	C₂×M₁₆	—	146	1	no	—
130	C₂×M₁₆	—	146	1	no	—
131	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
132	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
133	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
134	C₂×C₁₆	—	73	4	no	—
135	C₂×C₁₆	—	146	2	no	—
136	C₂×C₁₆	—	146	2	no	—
137	#108	—	2	2	no	—
138	#108	—	2	2	no	—
139	#108	—	4	2	no	—
140	#108	—	73	2	no	—
141	#108	—	73	2	no	—
142	#108	—	146	1	no	—
143	#108	—	146	1	no	—
144	C₄⋊C₈	—	4	1	no	—
145	C₄⋊C₈	—	4	1	no	—
146	#97	—	292	1	no	—
147	C₄×C₈	—	292	1	no	—
148	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
149	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
150	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
151	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
152	C₂²×C₄	#11924018	1	1	yes	characteristic
153	C₂²×C₄	#11924018	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
155	M₁₆	—	4	2	no	—
156	M₁₆	—	4	2	no	—
157	M₁₆	—	292	1	no	—
158	M₁₆	—	292	1	no	—
159	M₁₆	—	146	2	no	—
160	M₁₆	—	146	2	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	—	292	1	no	—
165	C₂×C₈	—	292	1	no	—
166	C₂×C₈	#11924007	1	1	yes	characteristic
167	C₂×C₈	#11924007	1	1	yes	characteristic
168	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
169	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
170	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
171	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
172	C₂×C₈	#11924027	1	1	yes	characteristic
173	C₂×C₈	#11924029	1	1	yes	characteristic
174	C₄²	#11924007	1	1	yes	characteristic
175	C₄²	#11924007	1	1	yes	characteristic
176	C₄²	#11924013	1	1	yes	characteristic
177	C₄²	#11924013	1	1	yes	characteristic
178	C₁₆	—	146	4	no	—
179	C₂³	C₂×#11924018	1	1	yes	characteristic
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	4	1	no	—
182	C₂×C₄	#14556524	1	1	yes	characteristic
183	C₂×C₄	#14556524	1	1	yes	characteristic
184	C₂×C₄	#14556524	1	1	yes	characteristic
185	C₂×C₄	#14556524	1	1	yes	characteristic
186	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
187	C₂×C₄	#14556562	1	1	yes	characteristic
188	C₂×C₄	#14556567	1	1	yes	characteristic
189	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
190	C₂×C₄	#14556617	1	1	yes	characteristic
191	C₂×C₄	#14556622	1	1	yes	characteristic
192	C₈	—	2	1	no	—
193	C₈	—	2	1	no	—
194	C₈	—	8	2	no	—
195	C₈	—	292	2	no	—
196	C₈	—	292	2	no	—
197	C₈	#14556553	1	1	yes	characteristic
198	C₈	#14556556	1	1	yes	characteristic
199	C₈	#14556582	1	1	yes	characteristic
200	C₈	#14556590	1	1	yes	characteristic
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	C₂×#14556524	1	1	yes	characteristic
204	C₂²	#17988794	1	1	yes	characteristic
205	C₂²	#17988879	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	#17988416	1	2	yes	complemented
210	C₄	#17988483	1	1	yes	characteristic
211	C₄	#17988627	1	1	yes	characteristic
212	C₄	#17988779	1	1	yes	characteristic
213	C₄	#17988864	1	1	yes	characteristic
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	#22610687	1	1	yes	characteristic
217	C₂	C₂×#17988416	1	1	yes	characteristic
218	C₂	#22611725	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11924013
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1848	C₂
socle	C₂²×C₇₃	C₂×#107
prime radical	C₂³×C₇₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1848	D₇₃
hypocenter	C₇₃	#13179
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
99. C₄×C₇₃	#368	semidirect
115. C₇₃	#13179	semidirect
209. C₄	#17988416	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988416	2
C₂×C₄⋊#11924003	2
C₄×#14556527	2
#22608024	2
C₇₃×#1848	2
#13179	73