Subgroups — #33001203

Order 18688.

462 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×#17988391	C₂	1	2	yes	maximal
2	C₂×#17988363	C₂	1	1	yes	maximal, characteristic
3	C₄×#14556525	C₂	1	1	yes	maximal, characteristic
4	#22608022	C₂	1	2	yes	maximal
5	C₂×C₄×Q₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924003	C₂²	1	2	yes	—
7	C₂×#14556525	C₂²	1	2	yes	—
8	C₄×C₄⋊Dic₇₃	—	2	1	no	—
9	#17988391	C₄	1	8	yes	complemented
10	#17988363	—	2	2	no	—
11	#17988350	—	2	1	no	—
12	C₂²×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₇₃	—	2	1	no	—
14	C₄×Q₁₆×C₇₃	—	2	2	no	—
15	C₄×Q₁₆×C₇₃	C₄	1	4	yes	complemented
16	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×#103×C₇₃	—	2	1	no	—
18	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₇₃	—	2	2	no	—
20	C₂×C₄×Dic₇₃	—	4	1	no	—
21	C₂×#11924003	—	2	2	no	—
22	C₂×#11924003	—	2	2	no	—
23	C₂×#11924003	C₂×C₄	1	4	yes	—
24	#14556525	—	2	2	no	—
25	#14556525	C₂×C₄	1	4	yes	—
26	#14556498	—	4	1	no	—
27	C₂²×Q₈×C₇₃	—	2	1	no	—
28	C₂×Q₁₆×C₇₃	—	2	1	no	—
29	C₂×Q₁₆×C₇₃	—	2	2	no	—
30	C₂×Q₁₆×C₇₃	C₂×C₄	1	4	yes	—
31	C₂×Q₁₆×C₇₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
34	C₄×Q₈×C₇₃	—	2	4	no	—
35	C₄×Q₈×C₇₃	—	4	2	no	—
36	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
37	C₂×C₄⋊C₄×C₇₃	—	4	1	no	—
38	C₂×C₄²×C₇₃	—	4	1	no	—
39	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
40	#107×C₇₃	—	2	1	no	—
41	#107×C₇₃	C₂×C₄	1	2	yes	—
42	#103×C₇₃	—	2	4	no	—
43	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₇₃	D₄	1	2	yes	—
45	C₂²×Dic₇₃	—	4	2	no	—
46	C₄⋊Dic₇₃	—	2	4	no	—
47	C₄⋊Dic₇₃	—	4	2	no	—
48	C₄×Dic₇₃	—	8	2	no	—
49	#11924003	—	2	4	no	—
50	C₂×Q₈×C₇₃	—	2	2	no	—
51	C₂×Q₈×C₇₃	—	2	4	no	—
52	C₂×Q₈×C₇₃	—	4	1	no	—
53	C₂×Q₈×C₇₃	—	4	2	no	—
54	C₂²×C₄×C₇₃	—	4	1	no	—
55	C₂²×C₄×C₇₃	—	4	1	no	—
56	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
59	Q₁₆×C₇₃	—	2	2	no	—
60	Q₁₆×C₇₃	—	2	4	no	—
61	Q₁₆×C₇₃	C₄²	1	4	yes	complemented
62	C₂×C₈×C₇₃	—	2	2	no	—
63	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₇₃	#41	1	2	yes	—
68	C₄⋊C₄×C₇₃	—	2	2	no	—
69	C₄⋊C₄×C₇₃	—	4	4	no	—
70	C₄⋊C₄×C₇₃	—	4	1	no	—
71	C₄²×C₇₃	—	4	4	no	—
72	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₇₃	SD₁₆	1	2	yes	—
74	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
75	Q₈×C₇₃	—	2	4	no	—
76	Q₈×C₇₃	—	4	2	no	—
77	Q₈×C₇₃	—	4	4	no	—
78	C₂×C₄×C₇₃	—	2	1	no	—
79	C₂×C₄×C₇₃	—	2	1	no	—
80	C₂×C₄×C₇₃	—	2	2	no	—
81	C₂×C₄×C₇₃	—	4	2	no	—
82	C₂×C₄×C₇₃	—	4	4	no	—
83	C₂×C₄×C₇₃	—	4	2	no	—
84	C₂×C₄×C₇₃	—	8	1	no	—
85	C₂×C₄×C₇₃	—	8	1	no	—
86	C₂×C₄×C₇₃	#102	1	2	yes	—
87	C₂×C₄×C₇₃	#102	1	2	yes	—
88	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄×C₇₃	#135	1	2	yes	—
92	C₂×Dic₇₃	—	4	4	no	—
93	C₂×Dic₇₃	—	8	2	no	—
94	C₂×Dic₇₃	—	8	2	no	—
95	C₈×C₇₃	—	2	2	no	—
96	C₈×C₇₃	C₂×C₄²	1	1	yes	characteristic
97	C₈×C₇₃	#117	1	1	yes	characteristic
98	C₈×C₇₃	C₄×D₄	1	2	yes	—
99	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
100	C₂²×C₇₃	C₄×D₈	1	2	yes	—
101	C₂²×C₇₃	#509	1	1	yes	characteristic
102	C₂²×C₇₃	#510	1	1	yes	characteristic
103	C₂²×C₇₃	#510	1	2	yes	—
104	C₄×C₇₃	—	2	2	no	—
105	C₄×C₇₃	—	2	2	no	—
106	C₄×C₇₃	—	4	4	no	—
107	C₄×C₇₃	—	8	2	no	—
108	C₄×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
109	C₄×C₇₃	#416	1	1	yes	characteristic
110	C₄×C₇₃	C₄×SD₁₆	1	2	yes	—
111	Dic₇₃	—	8	4	no	—
112	#12638	—	73	1	no	maximal, Sylow
113	C₂×C₇₃	C₄×#102	1	1	yes	characteristic
114	C₂×C₇₃	#2139	1	1	yes	characteristic
115	C₂×C₇₃	#2141	1	1	yes	characteristic
116	C₂×C₇₃	#2175	1	2	yes	—
117	C₂×C₇₃	#2176	1	2	yes	—
118	C₂×C₄×Q₁₆	D₇₃	1	1	yes	characteristic
119	C₂×#358	—	73	2	no	—
120	C₂×#346	—	73	1	no	—
121	C₄×#107	—	73	1	no	—
122	#1380	—	73	2	no	—
123	C₇₃	#12638	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
125	C₂×C₄×Q₈	—	2	1	no	—
126	C₂²×C₁₆	—	73	2	no	—
127	C₄×Q₁₆	—	2	2	no	—
128	C₄×Q₁₆	Dic₇₃	1	4	yes	complemented
129	C₂×#107	—	73	2	no	—
130	C₂×#107	C₂×D₇₃	1	1	yes	characteristic
131	C₂×#103	—	2	1	no	—
132	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
133	C₄×C₄⋊C₄	—	146	1	no	—
134	#358	—	73	8	no	—
135	#346	—	146	2	no	—
136	#340	—	146	1	no	—
137	C₂²×Q₈	—	2	1	no	—
138	C₂×Q₁₆	—	2	1	no	—
139	C₂×Q₁₆	—	2	2	no	—
140	C₂×Q₁₆	C₄×D₇₃	1	4	yes	—
141	C₂×Q₁₆	C₂×Dic₇₃	1	2	yes	—
142	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
143	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
144	C₄×Q₈	—	2	4	no	—
145	C₄×Q₈	—	4	2	no	—
146	C₂×C₄⋊C₄	—	2	1	no	—
147	C₂×C₄⋊C₄	—	4	1	no	—
148	C₂×C₄⋊C₄	—	146	2	no	—
149	C₂×C₄²	—	4	1	no	—
150	C₂×C₄²	—	292	1	no	—
151	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
152	C₂×C₁₆	—	73	4	no	—
153	C₂×C₁₆	—	146	2	no	—
154	C₂×C₁₆	—	146	2	no	—
155	#107	—	2	1	no	—
156	#107	—	73	4	no	—
157	#107	—	146	2	no	—
158	#107	C₂×Dic₇₃	1	2	yes	—
159	#103	—	2	4	no	—
160	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
161	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
162	#95	—	292	1	no	—
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	2	4	no	—
165	C₂×Q₈	—	4	1	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂²×C₄	—	4	1	no	—
168	C₂²×C₄	—	4	1	no	—
169	C₂²×C₄	—	292	2	no	—
170	C₂²×C₄	#11924013	1	1	yes	characteristic
171	C₂²×C₄	#11924013	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
173	Q₁₆	—	2	2	no	—
174	Q₁₆	—	2	4	no	—
175	Q₁₆	C₄×Dic₇₃	1	4	yes	complemented
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	C₂²⋊Dic₇₃	1	2	yes	—
178	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
179	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
180	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
181	C₂×C₈	#11924028	1	2	yes	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	4	4	no	—
184	C₄⋊C₄	—	4	1	no	—
185	C₄⋊C₄	—	292	2	no	—
186	C₄⋊C₄	—	146	4	no	—
187	C₄²	—	4	4	no	—
188	C₄²	—	584	2	no	—
189	C₄²	C₂²⋊Dic₇₃	1	2	yes	—
190	C₄²	#11924016	1	2	yes	—
191	C₁₆	—	146	4	no	—
192	C₂³	C₂×#11924013	1	1	yes	characteristic
193	Q₈	—	2	4	no	—
194	Q₈	—	4	4	no	—
195	Q₈	—	4	2	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	—	8	1	no	—
203	C₂×C₄	—	8	1	no	—
204	C₂×C₄	—	292	4	no	—
205	C₂×C₄	—	584	2	no	—
206	C₂×C₄	—	584	2	no	—
207	C₂×C₄	#14556512	1	2	yes	—
208	C₂×C₄	#14556512	1	2	yes	—
209	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
210	C₂×C₄	C₄×C₂²⋊D₇₃	1	2	yes	—
211	C₂×C₄	C₂×#11924016	1	1	yes	characteristic
212	C₂×C₄	#14556611	1	2	yes	—
213	C₈	—	2	2	no	—
214	C₈	C₂×C₄×Dic₇₃	1	1	yes	characteristic
215	C₈	#14556550	1	1	yes	characteristic
216	C₈	C₄×C₂²⋊D₇₃	1	2	yes	—
217	C₂²	C₂×#14556512	1	1	yes	characteristic
218	C₂²	C₄×#11924013	1	2	yes	—
219	C₂²	#17989022	1	1	yes	characteristic
220	C₂²	#17989027	1	1	yes	characteristic
221	C₂²	#17989028	1	2	yes	—
222	C₄	—	2	2	no	—
223	C₄	—	2	2	no	—
224	C₄	—	4	4	no	—
225	C₄	—	8	2	no	—
226	C₄	—	584	4	no	—
227	C₄	C₄×C₂²⋊Dic₇₃	1	1	yes	characteristic
228	C₄	#17988583	1	1	yes	characteristic
229	C₄	C₄×#11924016	1	2	yes	—
230	C₂	C₄×#14556512	1	1	yes	characteristic
231	C₂	#22611643	1	1	yes	characteristic
232	C₂	#22611651	1	1	yes	characteristic
233	C₂	#22611793	1	2	yes	—
234	C₂	#22611800	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11924013
commutator	C₈×C₇₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂×C₄×Q₁₆×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂×D₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×C₄×Q₁₆	D₇₃
hypocenter	C₇₃	#12638
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
9. #17988391	C₄	semidirect
15. C₄×Q₁₆×C₇₃	C₄	semidirect
61. Q₁₆×C₇₃	C₄²	semidirect
123. C₇₃	#12638	semidirect
128. C₄×Q₁₆	Dic₇₃	semidirect
175. Q₁₆	C₄×Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988391	2
C₂×#17988363	2
C₄×#14556525	2
#22608022	2
C₂×C₄×Q₁₆×C₇₃	2
#12638	73