Subgroups — #33239172

Order 18688.

690 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989005	C₂	1	2	yes	maximal, complemented
2	#22618644	C₂	1	1	yes	maximal, characteristic
3	#22611762	C₂	1	4	yes	maximal, complemented
4	#22611750	C₂	1	4	yes	maximal, complemented
5	#22611546	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×#11924003	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556597	C₂²	1	2	yes	—
9	#17989137	C₄	1	1	yes	characteristic
10	#17989005	—	2	8	no	—
11	C₂×#14556530	C₂²	1	4	yes	—
12	C₂²×#11924003	C₂²	1	1	yes	characteristic
13	C₂²×#11924003	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₇₃	C₂²	1	4	yes	—
15	#17988666	C₄	1	4	yes	—
16	C₈×D₂₉₂	C₂²	1	4	yes	—
17	#17988611	C₄	1	1	yes	characteristic
18	#17988553	C₄	1	2	yes	—
19	#17988366	C₂²	1	8	yes	—
20	C₄×#11924003	C₂²	1	2	yes	—
21	C₄×#11924003	C₂²	1	2	yes	—
22	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
24	C₄⋊C₁₆×C₇₃	C₂²	1	4	yes	—
25	C₂×#11924028	C₂×C₄	1	2	yes	—
26	#14556597	—	2	4	no	—
27	#14556597	—	2	4	no	—
28	C₂×#11924012	C₂×C₄	1	2	yes	—
29	C₂×#11924011	C₂×C₄	1	4	yes	—
30	C₂×C₈×D₇₃	C₂³	1	4	yes	—
31	#14556591	C₈	1	1	yes	characteristic
32	#14556588	C₈	1	1	yes	characteristic
33	#14556579	C₈	1	2	yes	—
34	#14556565	C₈	1	4	yes	—
35	C₄×D₂₉₂	C₂×C₄	1	4	yes	—
36	#14556530	—	2	8	no	—
37	C₂×#11924003	—	2	2	no	—
38	C₂×#11924003	D₄	1	4	yes	complemented
39	C₂×#11924003	C₂³	1	2	yes	—
40	C₂×#11924003	C₂³	1	1	yes	characteristic
41	C₂×#11924003	C₂³	1	1	yes	characteristic
42	C₁₆×D₇₃	—	2	8	no	—
43	C₄⋊#10596831	C₂×C₄	1	4	yes	—
44	#14556503	C₂×C₄	1	8	yes	—
45	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
48	M₃₂×C₇₃	—	2	4	no	—
49	C₂×C₁₆×C₇₃	C₂³	1	4	yes	—
50	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₇₃	C₂³	1	2	yes	—
52	#11924028	—	2	4	no	—
53	#11924028	—	2	4	no	—
54	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
55	C₂×D₂₉₂	C₂×C₈	1	4	yes	—
56	C₂×D₂₉₂	C₂×C₈	1	2	yes	—
57	C₂×C₂²⋊D₇₃	C₂×C₈	1	4	yes	—
58	C₂×C₄×D₇₃	C₂²×C₄	1	4	yes	—
59	#11924012	—	2	4	no	—
60	#11924011	—	2	8	no	—
61	C₂×#10596831	C₂²×C₄	1	4	yes	—
62	C₈×D₇₃	—	2	4	no	—
63	C₈×D₇₃	—	2	4	no	—
64	C₄⋊Dic₇₃	C₂×C₈	1	4	yes	—
65	#11924005	C₂×C₈	1	8	yes	—
66	#11924003	—	2	2	no	—
67	#11924003	C₂×D₄	1	2	yes	—
68	#11924003	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
70	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₇₃	—	2	2	no	—
72	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₇₃	#41	1	2	yes	—
76	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
77	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
78	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
79	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
80	C₁₆×C₇₃	—	2	4	no	—
81	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
82	C₂²×D₇₃	C₂²×C₈	1	4	yes	—
83	C₂×C₄×C₇₃	—	2	4	no	—
84	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
86	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₇₃	#131	1	2	yes	—
89	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
90	C₂²⋊D₇₃	—	2	8	no	—
91	C₂×Dic₇₃	C₂²×C₈	1	4	yes	—
92	D₂₉₂	—	2	4	no	—
93	D₂₉₂	—	2	8	no	—
94	C₄×D₇₃	—	2	4	no	—
95	C₄×D₇₃	—	2	4	no	—
96	Dic₁₄₆	—	2	4	no	—
97	C₈×C₇₃	—	2	2	no	—
98	C₈×C₇₃	C₄×D₄	1	2	yes	—
99	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
100	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
101	#10596831	—	2	4	no	—
102	C₂²×C₇₃	—	2	2	no	—
103	C₂²×C₇₃	C₈×D₄	1	2	yes	—
104	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
105	C₂×D₇₃	—	2	8	no	—
106	C₂×D₇₃	—	2	4	no	—
107	C₄×C₇₃	—	2	2	no	—
108	C₄×C₇₃	C₈×D₄	1	2	yes	—
109	C₄×C₇₃	#504	1	4	yes	complemented
110	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
111	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
112	Dic₇₃	—	2	4	no	—
113	#33434	—	73	1	no	maximal, Sylow
114	C₂×C₇₃	—	2	2	no	—
115	C₂×C₇₃	C₂×C₈×D₄	1	1	yes	characteristic
116	C₂×C₇₃	C₂×#504	1	2	yes	—
117	D₇₃	—	4	4	no	—
118	C₂×#504	—	73	2	no	—
119	#2931	—	73	1	no	—
120	#2169	—	73	4	no	—
121	D₄×C₁₆	—	73	4	no	—
122	#2150	D₇₃	1	1	yes	characteristic, complemented
123	#2113	—	73	2	no	—
124	C₂×C₄×C₁₆	—	73	1	no	—
125	C₇₃	#33434	1	1	yes	characteristic, complemented, Sylow
126	C₂×#131	—	73	2	no	—
127	#532	—	73	1	no	—
128	#504	—	146	8	no	—
129	C₂×M₃₂	—	73	4	no	—
130	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
131	C₂²×C₁₆	—	73	1	no	—
132	C₂²×C₁₆	—	73	1	no	—
133	C₂²×C₁₆	—	73	4	no	—
134	#436	—	73	4	no	—
135	C₈×D₄	—	73	4	no	—
136	#423	—	73	1	no	—
137	#408	—	73	2	no	—
138	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
139	C₄⋊C₁₆	C₂×D₇₃	1	4	yes	—
140	C₂²⋊C₁₆	—	73	8	no	—
141	C₄×C₁₆	—	73	2	no	—
142	C₄×C₁₆	—	73	2	no	—
143	C₂×#41	—	73	2	no	—
144	#131	—	146	4	no	—
145	#131	—	146	4	no	—
146	C₂×M₁₆	—	73	2	no	—
147	C₂×M₁₆	—	73	4	no	—
148	C₂²×C₈	—	73	4	no	—
149	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
151	C₄⋊Q₈	—	73	1	no	—
152	#127	—	73	1	no	—
153	#124	—	73	2	no	—
154	#121	—	73	4	no	—
155	C₄×D₄	—	73	4	no	—
156	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
157	M₃₂	—	2	4	no	—
158	M₃₂	—	146	8	no	—
159	C₂×C₁₆	—	73	4	no	—
160	C₂×C₁₆	—	73	1	no	—
161	C₂×C₁₆	—	73	1	no	—
162	C₂×C₁₆	—	73	2	no	—
163	C₂×C₁₆	—	146	8	no	—
164	C₂×C₁₆	—	146	2	no	—
165	C₂×C₁₆	C₂²×D₇₃	1	4	yes	—
166	C₄⋊C₈	—	73	4	no	—
167	C₂²⋊C₈	—	73	8	no	—
168	C₄×C₈	C₄×D₇₃	1	2	yes	—
169	C₄×C₈	C₂²×D₇₃	1	2	yes	—
170	#41	—	146	4	no	—
171	#41	—	146	4	no	—
172	C₂×Q₈	—	73	2	no	—
173	C₂×D₄	—	73	4	no	—
174	C₂×D₄	—	73	4	no	—
175	C₂×D₄	—	73	2	no	—
176	C₂²×C₄	—	73	4	no	—
177	C₂²×C₄	C₈×D₇₃	1	2	yes	—
178	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
179	M₁₆	—	146	4	no	—
180	M₁₆	—	146	8	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	73	4	no	—
183	C₂×C₈	—	146	4	no	—
184	C₂×C₈	—	146	4	no	—
185	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
186	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
187	C₂×C₈	D₄×D₇₃	1	2	yes	—
188	C₂×C₈	#11924029	1	2	yes	—
189	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
190	C₄⋊C₄	—	73	4	no	—
191	C₂²⋊C₄	—	73	8	no	—
192	C₄²	C₈×D₇₃	1	1	yes	characteristic
193	C₄²	C₈×D₇₃	1	1	yes	characteristic
194	C₄²	C₂×C₄×D₇₃	1	2	yes	—
195	C₁₆	—	2	4	no	—
196	C₁₆	—	73	2	no	—
197	C₁₆	—	73	2	no	—
198	C₁₆	—	146	2	no	—
199	C₂³	—	73	4	no	—
200	C₂³	C₂×C₈×D₇₃	1	1	yes	characteristic
201	Q₈	—	146	4	no	—
202	D₄	—	146	8	no	—
203	D₄	—	146	4	no	—
204	D₄	—	146	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	73	4	no	—
207	C₂×C₄	—	146	4	no	—
208	C₂×C₄	—	146	4	no	—
209	C₂×C₄	#14556553	1	2	yes	—
210	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
211	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
212	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
213	C₂×C₄	#14556599	1	2	yes	—
214	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	146	4	no	—
217	C₈	#14556553	1	2	yes	—
218	C₈	C₂×D₄×D₇₃	1	1	yes	characteristic
219	C₈	C₂×#11924029	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	—	146	8	no	—
222	C₂²	—	146	4	no	—
223	C₂²	#17988652	1	2	yes	—
224	C₂²	C₂²×C₈×D₇₃	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	146	4	no	—
227	C₄	#17988652	1	2	yes	—
228	C₄	#17989005	1	4	yes	complemented
229	C₄	C₂×#14556553	1	1	yes	characteristic
230	C₄	C₂×#14556599	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	—	292	4	no	—
233	C₂	C₂×#17988652	1	1	yes	characteristic
234	C₂	C₂×#17989005	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇₃
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	C₇₃×#2150	C₂
prime radical	#17989137	C₄
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	#2150	D₇₃
hypocenter	C₇₃	#33434
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
1. C₂×#17989005	C₂	semidirect
3. #22611762	C₂	semidirect
4. #22611750	C₂	semidirect
5. #22611546	C₂	semidirect
6. C₂×C₄×#11924003	C₂	semidirect
7. C₇₃×#2150	C₂	semidirect
38. C₂×#11924003	D₄	semidirect
109. C₄×C₇₃	#504	semidirect
122. #2150	D₇₃	semidirect
125. C₇₃	#33434	semidirect
228. C₄	#17989005	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989005	2
#22618644	2
#22611762	2
#22611750	2
#22611546	2
C₂×C₄×#11924003	2
C₇₃×#2150	2
#33434	73