Subgroups — #33238692

Order 18688.

1114 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22624642	C₂	1	1	yes	maximal, characteristic
2	C₂×#17989004	C₂	1	1	yes	maximal, characteristic, complemented
3	#22611560	C₂	1	6	yes	maximal, complemented
4	C₂×#17988372	C₂	1	6	yes	maximal, complemented
5	C₇₃×#3407	C₂	1	1	yes	maximal, characteristic, complemented
6	#17989423	C₄	1	1	yes	characteristic
7	#17989354	C₄	1	1	yes	characteristic
8	#17989353	C₄	1	6	yes	—
9	#17989352	—	2	4	no	—
10	C₂×#14556597	C₂²	1	6	yes	—
11	#131×D₇₃	—	2	4	no	—
12	#17989004	—	2	4	no	—
13	C₂×#14556531	C₂²	1	3	yes	complemented
14	C₂²×#11924003	C₂²	1	3	yes	complemented
15	#17988372	C₂²	1	4	yes	complemented
16	#17988372	C₂²	1	12	yes	complemented
17	C₂×#131×C₇₃	C₂²	1	1	yes	characteristic
18	#504×C₇₃	—	2	4	no	—
19	C₂×M₃₂×C₇₃	C₂²	1	6	yes	complemented
20	#14556639	—	2	1	no	—
21	#14556638	C₈	1	4	yes	—
22	#14556637	C₈	1	4	yes	—
23	#14556636	—	2	3	no	—
24	C₂×#11924028	C₂×C₄	1	6	yes	—
25	#41×D₇₃	—	2	4	no	—
26	#14556599	—	2	12	no	—
27	#14556598	—	2	4	no	—
28	#14556597	—	2	6	no	—
29	#14556597	—	2	6	no	—
30	#14556597	C₂×C₄	1	12	yes	—
31	#14556597	C₂³	1	4	yes	—
32	C₂×#11924012	C₂×C₄	1	6	yes	—
33	C₂×#11924011	—	2	6	no	—
34	M₁₆×D₇₃	—	2	12	no	—
35	C₂×C₈×D₇₃	—	2	6	no	—
36	#14556531	—	2	6	no	—
37	C₂×#11924003	—	2	6	no	—
38	C₂×#11924003	C₂³	1	1	yes	characteristic
39	C₂×#11924003	C₂³	1	3	yes	—
40	C₂×#41×C₇₃	C₂×C₄	1	1	yes	characteristic
41	#131×C₇₃	D₄	1	4	yes	—
42	#131×C₇₃	D₄	1	4	yes	—
43	C₂×M₁₆×C₇₃	C₂×C₄	1	3	yes	—
44	C₂²×C₈×C₇₃	C₂³	1	3	yes	—
45	M₃₂×C₇₃	—	2	6	no	—
46	M₃₂×C₇₃	C₂³	1	4	yes	—
47	C₂×C₁₆×C₇₃	—	2	6	no	—
48	#11924029	—	2	4	no	—
49	#11924028	—	2	6	no	—
50	#11924028	—	2	6	no	—
51	#11924028	C₂×C₈	1	12	yes	—
52	#11924028	C₂²×C₄	1	4	yes	—
53	#11924027	—	2	12	no	—
54	C₂×Dic₁₄₆	C₂×C₈	1	6	yes	—
55	Q₈×D₇₃	—	2	4	no	—
56	C₂×D₂₉₂	C₂×C₈	1	6	yes	—
57	C₂×C₂²⋊D₇₃	—	2	6	no	—
58	D₄×D₇₃	—	2	12	no	—
59	C₂×C₄×D₇₃	—	2	6	no	—
60	#11924012	—	2	6	no	—
61	#11924012	C₂²×C₄	1	4	yes	—
62	#11924011	—	2	12	no	—
63	#11924011	—	2	4	no	—
64	C₂×#10596831	—	2	6	no	—
65	C₈×D₇₃	—	2	12	no	—
66	C₈×D₇₃	—	2	4	no	—
67	#11924003	—	2	1	no	—
68	#11924003	—	2	3	no	—
69	#41×C₇₃	C₂²⋊C₄	1	4	yes	—
70	#41×C₇₃	C₂×C₈	1	4	yes	—
71	C₂×Q₈×C₇₃	M₁₆	1	1	yes	characteristic
72	C₂×D₄×C₇₃	M₁₆	1	3	yes	—
73	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
74	M₁₆×C₇₃	C₂²⋊C₄	1	12	yes	—
75	C₂×C₈×C₇₃	C₂²×C₄	1	3	yes	—
76	C₂×C₈×C₇₃	C₂×D₄	1	6	yes	—
77	C₂×C₈×C₇₃	C₂×D₄	1	6	yes	—
78	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
79	C₁₆×C₇₃	—	2	4	no	—
80	C₂³×C₇₃	C₂×M₁₆	1	3	yes	—
81	C₂²×D₇₃	—	2	6	no	—
82	Q₈×C₇₃	C₂²⋊C₈	1	4	yes	—
83	D₄×C₇₃	C₂²⋊C₈	1	12	yes	—
84	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	6	yes	—
85	C₂×C₄×C₇₃	C₂²×C₈	1	6	yes	—
86	C₂×C₄×C₇₃	C₂×M₁₆	1	3	yes	—
87	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
88	C₂²⋊D₇₃	—	2	4	no	—
89	C₂²⋊D₇₃	—	2	12	no	—
90	C₂×Dic₇₃	—	2	6	no	—
91	D₂₉₂	—	2	6	no	—
92	D₂₉₂	C₂²×C₈	1	4	yes	—
93	C₄×D₇₃	—	2	4	no	—
94	C₄×D₇₃	—	2	12	no	—
95	Dic₁₄₆	—	2	6	no	—
96	Dic₁₄₆	C₂²×C₈	1	4	yes	—
97	C₈×C₇₃	C₂×C₂²⋊C₄	1	6	yes	—
98	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
99	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
100	#10596831	—	2	4	no	—
101	C₂²×C₇₃	—	4	3	no	—
102	C₂²×C₇₃	C₂×C₂²⋊C₈	1	6	yes	—
103	C₂²×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
104	C₂×D₇₃	—	2	4	no	—
105	C₂×D₇₃	—	8	6	no	—
106	C₄×C₇₃	C₂×C₂²⋊C₈	1	6	yes	—
107	C₄×C₇₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
108	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
109	Dic₇₃	—	2	4	no	—
110	#33340	—	73	1	no	maximal, Sylow
111	C₂×C₇₃	—	2	6	no	—
112	C₂×C₇₃	—	2	1	no	—
113	C₂×C₇₃	C₂²×C₂²⋊C₈	1	1	yes	characteristic
114	D₇₃	—	4	4	no	—
115	#3573	—	73	1	no	—
116	#3407	D₇₃	1	1	yes	characteristic, complemented
117	C₂×#504	—	73	1	no	—
118	#2117	—	73	6	no	—
119	C₂×#350	—	73	6	no	—
120	C₇₃	#33340	1	1	yes	characteristic, complemented, Sylow
121	#585	—	73	1	no	—
122	#568	—	73	1	no	—
123	#568	—	73	6	no	—
124	#568	—	146	4	no	—
125	C₂×#131	—	73	6	no	—
126	C₂×#131	—	146	4	no	—
127	C₂×#131	C₂×D₇₃	1	1	yes	characteristic
128	#504	—	2	4	no	—
129	#504	—	146	4	no	—
130	C₂×M₃₂	—	73	3	no	—
131	C₂×M₃₂	C₂×D₇₃	1	6	yes	complemented
132	C₂²×C₁₆	—	73	3	no	—
133	#350	—	73	4	no	—
134	#350	—	73	12	no	—
135	2₋¹⁺⁴	—	73	4	no	—
136	2₋¹⁺⁴	—	146	1	no	—
137	H(5,𝔽₂)	—	73	4	no	—
138	H(5,𝔽₂)	—	146	3	no	—
139	C₂×#41	—	73	6	no	—
140	C₂×#41	—	146	4	no	—
141	C₂×#41	C₄×D₇₃	1	1	yes	characteristic
142	#131	—	73	12	no	—
143	#131	—	73	4	no	—
144	#131	—	146	4	no	—
145	#131	—	146	12	no	—
146	#131	—	146	6	no	—
147	#131	—	146	6	no	—
148	#131	D₂₉₂	1	4	yes	—
149	#131	C₂²⋊D₇₃	1	4	yes	—
150	C₂×M₁₆	—	73	6	no	—
151	C₂×M₁₆	—	146	12	no	—
152	C₂×M₁₆	—	146	6	no	—
153	C₂×M₁₆	C₄×D₇₃	1	3	yes	—
154	C₂²×C₈	—	146	6	no	—
155	C₂²×C₈	C₂²×D₇₃	1	3	yes	—
156	M₃₂	—	2	6	no	—
157	M₃₂	—	146	6	no	—
158	M₃₂	C₂²×D₇₃	1	4	yes	—
159	C₂×C₁₆	—	2	6	no	—
160	C₂×C₁₆	—	73	1	no	—
161	C₂×C₁₆	—	73	3	no	—
162	C₂×C₁₆	—	146	6	no	—
163	#41	—	73	12	no	—
164	#41	—	73	4	no	—
165	#41	—	146	4	no	—
166	#41	—	146	12	no	—
167	#41	—	146	6	no	—
168	#41	—	146	6	no	—
169	#41	#11924005	1	4	yes	—
170	#41	C₈×D₇₃	1	4	yes	—
171	C₂×Q₈	—	73	6	no	—
172	C₂×Q₈	—	146	4	no	—
173	C₂×Q₈	#11924011	1	1	yes	characteristic
174	C₂×D₄	—	73	6	no	—
175	C₂×D₄	—	146	6	no	—
176	C₂×D₄	—	146	12	no	—
177	C₂×D₄	#11924011	1	3	yes	—
178	C₂²×C₄	—	146	6	no	—
179	C₂²×C₄	C₂×C₄×D₇₃	1	3	yes	—
180	M₁₆	—	73	4	no	—
181	M₁₆	—	146	6	no	—
182	M₁₆	—	146	12	no	—
183	M₁₆	—	146	4	no	—
184	M₁₆	#11924005	1	12	yes	—
185	C₂×C₈	—	146	6	no	—
186	C₂×C₈	—	146	12	no	—
187	C₂×C₈	—	146	4	no	—
188	C₂×C₈	C₂×C₄×D₇₃	1	3	yes	—
189	C₂×C₈	C₂×C₂²⋊D₇₃	1	6	yes	—
190	C₂×C₈	C₂×D₂₉₂	1	6	yes	—
191	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
192	C₁₆	—	2	4	no	—
193	C₁₆	—	146	1	no	—
194	C₁₆	—	146	3	no	—
195	C₂³	—	146	6	no	—
196	C₂³	C₂×#11924011	1	3	yes	—
197	Q₈	—	73	4	no	—
198	Q₈	—	146	6	no	—
199	Q₈	#14556503	1	4	yes	—
200	D₄	—	73	4	no	—
201	D₄	—	146	12	no	—
202	D₄	—	146	4	no	—
203	D₄	—	146	6	no	—
204	D₄	#14556503	1	12	yes	—
205	C₂×C₄	—	146	6	no	—
206	C₂×C₄	—	146	4	no	—
207	C₂×C₄	—	146	12	no	—
208	C₂×C₄	C₂×#11924005	1	6	yes	—
209	C₂×C₄	C₂×C₈×D₇₃	1	6	yes	—
210	C₂×C₄	C₂×#11924011	1	3	yes	—
211	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
212	C₈	—	146	4	no	—
213	C₈	C₂×#11924005	1	6	yes	—
214	C₈	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
215	C₈	C₂²×D₂₉₂	1	1	yes	characteristic
216	C₂²	—	4	3	no	—
217	C₂²	—	584	6	no	—
218	C₂²	—	146	4	no	—
219	C₂²	C₂×#14556503	1	6	yes	—
220	C₂²	C₂²×#11924011	1	1	yes	characteristic
221	C₄	—	146	4	no	—
222	C₄	C₂×#14556503	1	6	yes	—
223	C₄	C₂²×#11924005	1	1	yes	characteristic
224	C₄	C₂²×C₈×D₇₃	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	2	6	no	—
227	C₂	—	292	4	no	—
228	C₂	C₂²×#14556503	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×C₂²⋊D₇₃
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#3407	C₂
socle	C₂×C₇₃	C₂²×C₂²⋊C₈
prime radical	#17989423	C₄
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	#3407	D₇₃
hypocenter	C₇₃	#33340
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂×#131 ◁ #3407

Complements

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

N	H ≅ G/N	type
2. C₂×#17989004	C₂	semidirect
3. #22611560	C₂	semidirect
4. C₂×#17988372	C₂	semidirect
5. C₇₃×#3407	C₂	semidirect
13. C₂×#14556531	C₂²	semidirect
14. C₂²×#11924003	C₂²	semidirect
15. #17988372	C₂²	semidirect
16. #17988372	C₂²	semidirect
19. C₂×M₃₂×C₇₃	C₂²	semidirect
116. #3407	D₇₃	semidirect
120. C₇₃	#33340	semidirect
131. C₂×M₃₂	C₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22624642	2
C₂×#17989004	2
#22611560	2
C₂×#17988372	2
C₇₃×#3407	2
#33340	73