Subgroups — #33094447

Order 18688.

1062 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988694	C₂	1	4	yes	maximal
2	C₄⋊Q₈×Dic₇₃	C₂	1	2	yes	maximal
3	#22610803	C₂	1	4	yes	maximal
4	#22610734	C₂	1	2	yes	maximal
5	#22610088	C₂	1	1	yes	maximal, characteristic
6	C₄×#14556501	C₂	1	1	yes	maximal, characteristic
7	C₂×#494×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₇₃	C₂²	1	4	yes	—
9	#17988694	—	2	16	no	—
10	C₂×#14556523	C₂²	1	4	yes	—
11	C₂×#14556515	C₂²	1	8	yes	—
12	C₂×#14556501	C₂²	1	2	yes	—
13	C₂×#14556501	C₂²	1	2	yes	—
14	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
15	#17988494	C₂²	1	4	yes	—
16	#17988486	—	2	4	no	—
17	#17988457	C₂²	1	2	yes	—
18	C₄⋊C₄×Dic₇₃	—	2	4	no	—
19	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
20	C₂²×Q₁₆×C₇₃	C₂²	1	4	yes	—
21	C₂×C₄⋊Q₈×C₇₃	C₂²	1	2	yes	—
22	#494×C₇₃	C₄	1	8	yes	complemented
23	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
24	C₂²×#10596831	C₂³	1	2	yes	—
25	Q₈×Dic₇₃	—	2	16	no	—
26	C₂×C₄⋊Dic₇₃	—	2	4	no	—
27	C₂×C₄⋊Dic₇₃	—	2	4	no	—
28	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
29	C₂×C₄×Dic₇₃	—	2	4	no	—
30	C₂×C₄×Dic₇₃	D₄	1	4	yes	—
31	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
32	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
33	#14556523	—	2	8	no	—
34	#14556515	—	2	16	no	—
35	#14556501	—	2	4	no	—
36	#14556501	D₄	1	8	yes	—
37	C₄×#10596831	—	2	2	no	—
38	#14556498	—	2	8	no	—
39	C₂²×Q₈×C₇₃	C₂³	1	4	yes	—
40	C₂×Q₁₆×C₇₃	—	2	8	no	—
41	C₂×Q₁₆×C₇₃	—	2	8	no	—
42	C₂×Q₁₆×C₇₃	C₂×C₄	1	16	yes	—
43	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₇₃	—	2	4	no	—
45	C₄⋊Q₈×C₇₃	C₂×C₄	1	8	yes	—
46	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
47	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
49	C₂²×Dic₇₃	—	2	4	no	—
50	C₂²×Dic₇₃	C₂×D₄	1	2	yes	—
51	C₂²×Dic₇₃	C₂×D₄	1	2	yes	—
52	C₂×#10596831	—	2	2	no	—
53	C₂×#10596831	—	2	2	no	—
54	C₂×#10596831	C₂×D₄	1	4	yes	—
55	C₄⋊Dic₇₃	—	2	8	no	—
56	C₄⋊Dic₇₃	—	4	8	no	—
57	C₄⋊Dic₇₃	—	4	8	no	—
58	C₄×Dic₇₃	—	2	8	no	—
59	C₄×Dic₇₃	—	2	2	no	—
60	C₄×Dic₇₃	—	4	8	no	—
61	C₄×Dic₇₃	C₂×D₄	1	4	yes	—
62	C₂×Q₈×C₇₃	—	2	4	no	—
63	C₂×Q₈×C₇₃	—	2	8	no	—
64	C₂×Q₈×C₇₃	—	2	8	no	—
65	C₂×Q₈×C₇₃	C₂²×C₄	1	8	yes	—
66	C₂²×C₄×C₇₃	—	2	4	no	—
67	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
69	Q₁₆×C₇₃	—	2	32	no	—
70	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
71	C₂×C₈×C₇₃	C₂×D₄	1	4	yes	—
72	C₂×C₈×C₇₃	#41	1	4	yes	—
73	C₄⋊C₄×C₇₃	—	2	16	no	—
74	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
75	C₄²×C₇₃	#41	1	2	yes	—
76	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
77	Q₈×C₇₃	—	2	16	no	—
78	Q₈×C₇₃	—	4	8	no	—
79	C₂×C₄×C₇₃	—	2	8	no	—
80	C₂×C₄×C₇₃	—	4	4	no	—
81	C₂×C₄×C₇₃	—	4	4	no	—
82	C₂×C₄×C₇₃	C₄×D₄	1	4	yes	—
83	C₂×C₄×C₇₃	#137	1	8	yes	—
84	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
86	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
87	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
88	C₂×Dic₇₃	—	2	4	no	—
89	C₂×Dic₇₃	—	2	2	no	—
90	C₂×Dic₇₃	—	2	4	no	—
91	C₂×Dic₇₃	—	4	4	no	—
92	C₂×Dic₇₃	—	4	8	no	—
93	C₂×Dic₇₃	#127	1	4	yes	—
94	C₈×C₇₃	C₄×D₄	1	8	yes	—
95	#10596831	—	2	4	no	—
96	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
97	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
98	C₂²×C₇₃	#532	1	1	yes	characteristic
99	C₂²×C₇₃	C₂×#137	1	4	yes	—
100	C₄×C₇₃	—	4	8	no	—
101	C₄×C₇₃	#441	1	8	yes	complemented
102	C₄×C₇₃	C₂×C₄×D₄	1	2	yes	—
103	C₄×C₇₃	#532	1	2	yes	—
104	Dic₇₃	—	2	4	no	—
105	Dic₇₃	—	4	2	no	—
106	Dic₇₃	—	8	4	no	—
107	#21691	—	73	1	no	maximal, Sylow
108	C₂×C₇₃	C₄×#127	1	1	yes	characteristic
109	C₂×C₇₃	C₂×#441	1	2	yes	—
110	C₂×C₇₃	#3152	1	2	yes	—
111	C₂×C₇₃	#3185	1	2	yes	—
112	C₂×#494	D₇₃	1	1	yes	characteristic
113	C₂×#441	—	73	4	no	—
114	C₄×C₄⋊Q₈	—	73	2	no	—
115	#1971	—	73	4	no	—
116	#1956	—	73	2	no	—
117	#1844	—	73	1	no	—
118	C₄×#97	—	73	1	no	—
119	C₇₃	#21691	1	1	yes	characteristic, complemented, Sylow
120	C₂²×Q₁₆	C₂×D₇₃	1	4	yes	—
121	C₂×C₄⋊Q₈	C₂×D₇₃	1	2	yes	—
122	C₂×C₄×Q₈	—	73	4	no	—
123	#494	Dic₇₃	1	8	yes	complemented
124	#441	—	146	16	no	—
125	C₂×#106	—	73	4	no	—
126	C₂×#103	—	73	8	no	—
127	C₂×#97	—	73	2	no	—
128	C₂×#97	—	73	2	no	—
129	C₂×C₄×C₈	—	73	1	no	—
130	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
131	#391	—	73	4	no	—
132	#389	—	146	4	no	—
133	#384	—	73	2	no	—
134	C₄×C₄⋊C₄	—	146	4	no	—
135	C₄³	—	73	1	no	—
136	C₂²×Q₈	C₂²×D₇₃	1	4	yes	—
137	C₂×Q₁₆	—	2	8	no	—
138	C₂×Q₁₆	—	2	8	no	—
139	C₂×Q₁₆	C₂×Dic₇₃	1	16	yes	—
140	C₂²×C₈	—	73	2	no	—
141	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
142	C₄⋊Q₈	—	2	4	no	—
143	C₄⋊Q₈	C₂×Dic₇₃	1	8	yes	—
144	C₄×Q₈	—	146	16	no	—
145	C₂×C₄⋊C₄	—	2	4	no	—
146	C₂×C₄⋊C₄	—	73	4	no	—
147	C₂×C₄⋊C₄	—	146	4	no	—
148	C₂×C₄⋊C₄	—	146	4	no	—
149	C₂×C₄²	—	73	1	no	—
150	C₂×C₄²	—	73	4	no	—
151	C₂×C₄²	—	73	1	no	—
152	C₂×C₄²	—	146	4	no	—
153	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
154	#106	—	146	8	no	—
155	#103	—	146	16	no	—
156	#97	—	73	8	no	—
157	#97	—	146	4	no	—
158	C₄×C₈	—	146	2	no	—
159	C₄×C₈	C₂×Dic₇₃	1	4	yes	—
160	#95	—	146	8	no	—
161	C₂×Q₈	—	2	4	no	—
162	C₂×Q₈	—	2	8	no	—
163	C₂×Q₈	—	2	8	no	—
164	C₂×Q₈	C₂²×Dic₇₃	1	8	yes	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	73	2	no	—
167	C₂²×C₄	—	73	2	no	—
168	C₂²×C₄	—	146	4	no	—
169	C₂²×C₄	D₄×D₇₃	1	2	yes	—
170	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
171	Q₁₆	—	2	32	no	—
172	C₂×C₈	—	73	4	no	—
173	C₂×C₈	—	146	2	no	—
174	C₂×C₈	—	146	2	no	—
175	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
176	C₂×C₈	D₄×D₇₃	1	4	yes	—
177	C₂×C₈	#11924027	1	4	yes	—
178	C₄⋊C₄	—	2	16	no	—
179	C₄⋊C₄	—	292	8	no	—
180	C₄⋊C₄	—	292	8	no	—
181	C₄⋊C₄	—	146	8	no	—
182	C₄²	—	292	8	no	—
183	C₄²	—	73	4	no	—
184	C₄²	—	146	8	no	—
185	C₄²	—	146	2	no	—
186	C₄²	C₂²×Dic₇₃	1	2	yes	—
187	C₄²	#11924027	1	2	yes	—
188	C₂³	C₂×D₄×D₇₃	1	1	yes	characteristic
189	Q₈	—	2	16	no	—
190	Q₈	—	4	8	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	—	292	4	no	—
195	C₂×C₄	—	292	8	no	—
196	C₂×C₄	—	73	4	no	—
197	C₂×C₄	—	146	4	no	—
198	C₂×C₄	—	146	2	no	—
199	C₂×C₄	—	146	4	no	—
200	C₂×C₄	D₄×Dic₇₃	1	4	yes	—
201	C₂×C₄	#14556623	1	8	yes	—
202	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
203	C₂×C₄	C₂×D₄×D₇₃	1	2	yes	—
204	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
205	C₂×C₄	C₂×#11924027	1	2	yes	—
206	C₈	—	146	4	no	—
207	C₈	D₄×Dic₇₃	1	8	yes	—
208	C₂²	C₂×D₄×Dic₇₃	1	1	yes	characteristic
209	C₂²	#127×D₇₃	1	1	yes	characteristic
210	C₂²	#17989135	1	1	yes	characteristic
211	C₂²	C₂×#14556623	1	4	yes	—
212	C₄	—	4	8	no	—
213	C₄	—	292	2	no	—
214	C₄	—	584	4	no	—
215	C₄	—	146	4	no	—
216	C₄	#17988694	1	8	yes	complemented
217	C₄	C₂×D₄×Dic₇₃	1	2	yes	—
218	C₄	#17989135	1	2	yes	—
219	C₂	#127×Dic₇₃	1	1	yes	characteristic
220	C₂	C₂×#17988694	1	2	yes	—
221	C₂	#22620389	1	2	yes	—
222	C₂	#22620674	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂×#494×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂²×D₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×#494	D₇₃
hypocenter	C₇₃	#21691
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
22. #494×C₇₃	C₄	semidirect
101. C₄×C₇₃	#441	semidirect
119. C₇₃	#21691	semidirect
123. #494	Dic₇₃	semidirect
216. C₄	#17988694	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988694	2
C₄⋊Q₈×Dic₇₃	2
#22610803	2
#22610734	2
#22610088	2
C₄×#14556501	2
C₂×#494×C₇₃	2
#21691	73