Subgroups — #33004118

Order 18688.

306 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988411	C₂	1	2	yes	maximal
2	#22611687	C₂	1	1	yes	maximal, characteristic
3	C₄×#14556523	C₂	1	1	yes	maximal, characteristic
4	#22608031	C₂	1	2	yes	maximal
5	C₇₃×#1848	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556531	C₂²	1	2	yes	—
7	C₂×#14556523	C₂²	1	1	yes	characteristic
8	C₂×#14556523	C₂²	1	1	yes	characteristic
9	C₄×C₄⋊Dic₇₃	—	2	1	no	—
10	#17988411	—	2	4	no	—
11	C₄⋊#11924003	C₄	1	2	yes	complemented
12	C₄⋊#11924003	C₄	1	2	yes	complemented
13	#17988350	—	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×C₄⋊Dic₇₃	—	2	1	no	—
19	C₂×C₄⋊Dic₇₃	—	2	1	no	—
20	C₂×C₄×Dic₇₃	—	4	1	no	—
21	#14556531	—	2	2	no	—
22	#14556531	—	2	2	no	—
23	C₂×#11924003	C₂×C₄	1	4	yes	—
24	#14556523	—	2	1	no	—
25	#14556523	—	2	1	no	—
26	#14556523	D₄	1	2	yes	—
27	#14556523	Q₈	1	2	yes	—
28	#14556498	—	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	C₂²×Dic₇₃	—	4	1	no	—
43	C₂²×Dic₇₃	—	4	1	no	—
44	C₄⋊Dic₇₃	—	2	2	no	—
45	C₄⋊Dic₇₃	—	2	2	no	—
46	C₄⋊Dic₇₃	—	4	1	no	—
47	C₄⋊Dic₇₃	—	4	1	no	—
48	C₄×Dic₇₃	—	8	2	no	—
49	#11924003	—	2	4	no	—
50	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
53	M₁₆×C₇₃	—	4	2	no	—
54	M₁₆×C₇₃	—	4	2	no	—
55	C₂×C₈×C₇₃	—	2	1	no	—
56	C₂×C₈×C₇₃	—	2	1	no	—
57	C₂×C₈×C₇₃	—	4	2	no	—
58	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
65	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
66	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
67	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
68	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
69	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
70	C₂³×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₇₃	—	2	2	no	—
72	C₂×C₄×C₇₃	—	4	1	no	—
73	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
74	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
75	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
76	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
83	C₂×Dic₇₃	—	4	2	no	—
84	C₂×Dic₇₃	—	4	2	no	—
85	C₂×Dic₇₃	—	8	1	no	—
86	C₂×Dic₇₃	—	8	1	no	—
87	C₂×Dic₇₃	—	8	1	no	—
88	C₂×Dic₇₃	—	8	1	no	—
89	C₈×C₇₃	—	2	1	no	—
90	C₈×C₇₃	—	2	1	no	—
91	C₈×C₇₃	—	8	2	no	—
92	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
93	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
94	C₈×C₇₃	#125	1	1	yes	characteristic
95	C₈×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
96	C₂²×C₇₃	—	2	1	no	—
97	C₂²×C₇₃	—	2	1	no	—
98	C₂²×C₇₃	C₂×#106	1	1	yes	characteristic
99	C₂²×C₇₃	#461	1	1	yes	characteristic
100	C₂²×C₇₃	#476	1	1	yes	characteristic
101	C₄×C₇₃	—	2	1	no	—
102	C₄×C₇₃	—	2	1	no	—
103	C₄×C₇₃	—	2	1	no	—
104	C₄×C₇₃	#365	1	2	yes	complemented
105	C₄×C₇₃	#389	1	1	yes	characteristic
106	C₄×C₇₃	#428	1	1	yes	characteristic
107	C₄×C₇₃	#460	1	1	yes	characteristic
108	C₄×C₇₃	#475	1	1	yes	characteristic
109	Dic₇₃	—	8	2	no	—
110	Dic₇₃	—	8	2	no	—
111	#13185	—	73	1	no	maximal, Sylow
112	C₂×C₇₃	—	2	1	no	—
113	C₂×C₇₃	—	2	1	no	—
114	C₂×C₇₃	#1948	1	1	yes	characteristic
115	C₂×C₇₃	C₂×#365	1	1	yes	characteristic
116	C₂×C₇₃	#2155	1	1	yes	characteristic
117	C₂×#365	—	73	2	no	—
118	#2150	—	73	1	no	—
119	#1848	D₇₃	1	1	yes	characteristic
120	C₄×#106	—	73	1	no	—
121	#1383	—	73	2	no	—
122	C₇₃	#13185	1	1	yes	characteristic, complemented, Sylow
123	C₂×M₃₂	—	73	2	no	—
124	C₂×#108	—	2	1	no	—
125	C₂×#108	C₂×D₇₃	1	2	yes	—
126	C₂×#106	—	73	1	no	—
127	C₂×#106	—	73	1	no	—
128	#423	—	2	1	no	—
129	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
130	C₄×C₄⋊C₄	—	146	1	no	—
131	#365	—	146	4	no	—
132	C₄⋊C₁₆	—	73	2	no	—
133	C₄⋊C₁₆	—	73	2	no	—
134	#340	—	146	1	no	—
135	C₂×M₁₆	—	2	2	no	—
136	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
138	C₂×C₄⋊C₄	—	146	1	no	—
139	C₂×C₄⋊C₄	—	146	1	no	—
140	C₂×C₄²	—	292	1	no	—
141	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
142	M₃₂	—	146	2	no	—
143	M₃₂	—	146	2	no	—
144	C₂×C₁₆	—	73	4	no	—
145	#108	—	2	2	no	—
146	#108	—	2	2	no	—
147	#108	—	4	2	no	—
148	#106	—	73	2	no	—
149	#106	—	73	2	no	—
150	#106	—	146	1	no	—
151	#106	—	146	1	no	—
152	C₄⋊C₈	—	4	1	no	—
153	C₄⋊C₈	—	4	1	no	—
154	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
155	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
156	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
157	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
158	#95	—	292	1	no	—
159	C₂²×C₄	—	292	1	no	—
160	C₂²×C₄	—	292	1	no	—
161	C₂²×C₄	#11924015	1	1	yes	characteristic
162	C₂²×C₄	#11924015	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
164	M₁₆	—	4	2	no	—
165	M₁₆	—	4	2	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	#11924007	1	1	yes	characteristic
170	C₂×C₈	#11924007	1	1	yes	characteristic
171	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
172	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
173	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
174	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
175	C₂×C₈	#11924027	1	1	yes	characteristic
176	C₂×C₈	#11924029	1	1	yes	characteristic
177	C₄⋊C₄	—	292	1	no	—
178	C₄⋊C₄	—	292	1	no	—
179	C₄⋊C₄	—	146	2	no	—
180	C₄⋊C₄	—	146	2	no	—
181	C₄²	—	584	2	no	—
182	C₄²	#11924007	1	1	yes	characteristic
183	C₄²	#11924007	1	1	yes	characteristic
184	C₄²	#11924016	1	1	yes	characteristic
185	C₄²	#11924016	1	1	yes	characteristic
186	C₁₆	—	146	4	no	—
187	C₂³	C₂×#11924015	1	1	yes	characteristic
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	292	2	no	—
191	C₂×C₄	—	292	2	no	—
192	C₂×C₄	—	584	1	no	—
193	C₂×C₄	—	584	1	no	—
194	C₂×C₄	—	584	1	no	—
195	C₂×C₄	—	584	1	no	—
196	C₂×C₄	#14556522	1	1	yes	characteristic
197	C₂×C₄	#14556522	1	1	yes	characteristic
198	C₂×C₄	#14556522	1	1	yes	characteristic
199	C₂×C₄	#14556522	1	1	yes	characteristic
200	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
201	C₂×C₄	#14556562	1	1	yes	characteristic
202	C₂×C₄	#14556567	1	1	yes	characteristic
203	C₂×C₄	C₂×#11924016	1	1	yes	characteristic
204	C₂×C₄	#14556617	1	1	yes	characteristic
205	C₂×C₄	#14556622	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	8	2	no	—
209	C₈	#14556553	1	1	yes	characteristic
210	C₈	#14556556	1	1	yes	characteristic
211	C₈	#14556582	1	1	yes	characteristic
212	C₈	#14556590	1	1	yes	characteristic
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	C₂×#14556522	1	1	yes	characteristic
216	C₂²	#17988789	1	1	yes	characteristic
217	C₂²	#17988874	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	584	2	no	—
222	C₄	—	584	2	no	—
223	C₄	#17988411	1	2	yes	complemented
224	C₄	#17988483	1	1	yes	characteristic
225	C₄	#17988627	1	1	yes	characteristic
226	C₄	#17988784	1	1	yes	characteristic
227	C₄	#17988869	1	1	yes	characteristic
228	C₂	—	2	1	no	—
229	C₂	—	2	1	no	—
230	C₂	#22610692	1	1	yes	characteristic
231	C₂	C₂×#17988411	1	1	yes	characteristic
232	C₂	#22611705	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556522
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1848	C₂
socle	C₂²×C₇₃	C₂×#106
prime radical	C₂³×C₇₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1848	D₇₃
hypocenter	C₇₃	#13185
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
11. C₄⋊#11924003	C₄	semidirect
12. C₄⋊#11924003	C₄	semidirect
104. C₄×C₇₃	#365	semidirect
122. C₇₃	#13185	semidirect
223. C₄	#17988411	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988411	2
#22611687	2
C₄×#14556523	2
#22608031	2
C₇₃×#1848	2
#13185	73