Subgroups — #33002136

Order 18688.

396 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556531	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988411	C₂	1	2	yes	maximal, complemented
3	#22609956	C₂	1	1	yes	maximal, characteristic
4	#22608031	C₂	1	2	yes	maximal, complemented
5	C₇₃×#1810	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556531	—	2	4	no	—
7	C₂×#14556531	C₄	1	2	yes	complemented
8	C₂×#14556531	C₂²	1	1	yes	characteristic
9	C₂×#14556531	C₂²	1	1	yes	characteristic
10	C₂²×#11924003	C₄	1	2	yes	complemented
11	C₂×#14556523	C₂²	1	2	yes	—
12	#17988447	—	2	1	no	—
13	#17988411	—	2	4	no	—
14	#17988350	—	2	1	no	—
15	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×#108×C₇₃	C₂²	1	2	yes	—
17	#407×C₇₃	—	2	1	no	—
18	#341×C₇₃	—	2	1	no	—
19	C₂×C₄⋊Dic₇₃	—	2	2	no	—
20	C₂×C₂²⋊Dic₇₃	—	4	1	no	—
21	#14556531	—	2	4	no	—
22	#14556531	—	2	2	no	—
23	#14556531	D₄	1	4	yes	—
24	C₂×#11924003	—	2	4	no	—
25	C₂×#11924003	C₂×C₄	1	2	yes	—
26	C₂×#11924003	C₂×C₄	1	2	yes	—
27	#14556523	—	2	2	no	—
28	#14556523	—	2	2	no	—
29	#14556498	—	4	1	no	—
30	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
31	C₂×M₁₆×C₇₃	—	2	2	no	—
32	C₂²×C₈×C₇₃	—	2	2	no	—
33	C₂²×C₈×C₇₃	—	2	2	no	—
34	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₃	Q₈	1	1	yes	characteristic
38	C₂²×C₈×C₇₃	Q₈	1	1	yes	characteristic
39	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
40	#108×C₇₃	—	2	2	no	—
41	#108×C₇₃	—	2	2	no	—
42	C₂²⋊C₈×C₇₃	—	4	1	no	—
43	C₂²⋊C₈×C₇₃	—	4	1	no	—
44	C₂²×Dic₇₃	—	4	2	no	—
45	C₄⋊Dic₇₃	—	4	2	no	—
46	C₄⋊Dic₇₃	—	4	2	no	—
47	C₂²⋊Dic₇₃	—	8	2	no	—
48	#11924003	—	2	2	no	—
49	#11924003	C₂²⋊C₄	1	4	yes	complemented
50	C₂⁴×C₇₃	SD₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₇₃	—	2	2	no	—
52	C₂²×C₄×C₇₃	—	2	2	no	—
53	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
58	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
59	M₁₆×C₇₃	—	4	2	no	—
60	M₁₆×C₇₃	—	4	2	no	—
61	C₂×C₈×C₇₃	—	2	2	no	—
62	C₂×C₈×C₇₃	—	2	1	no	—
63	C₂×C₈×C₇₃	—	2	1	no	—
64	C₂×C₈×C₇₃	—	2	1	no	—
65	C₂×C₈×C₇₃	—	2	1	no	—
66	C₂×C₈×C₇₃	—	2	2	no	—
67	C₂×C₈×C₇₃	—	2	2	no	—
68	C₂×C₈×C₇₃	—	4	2	no	—
69	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₈×C₇₃	#41	1	2	yes	—
75	C₂³×C₇₃	—	2	2	no	—
76	C₂³×C₇₃	—	2	2	no	—
77	C₂³×C₇₃	—	4	1	no	—
78	C₂³×C₇₃	#106	1	1	yes	characteristic
79	C₂³×C₇₃	#106	1	1	yes	characteristic
80	C₂³×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₇₃	—	2	2	no	—
82	C₂×C₄×C₇₃	—	2	1	no	—
83	C₂×C₄×C₇₃	—	2	1	no	—
84	C₂×C₄×C₇₃	—	2	1	no	—
85	C₂×C₄×C₇₃	—	2	1	no	—
86	C₂×C₄×C₇₃	—	2	2	no	—
87	C₂×C₄×C₇₃	—	2	2	no	—
88	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
89	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
90	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₇₃	#121	1	2	yes	—
92	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₇₃	#135	1	2	yes	—
94	C₂×Dic₇₃	—	8	2	no	—
95	C₂×Dic₇₃	—	8	2	no	—
96	C₂×Dic₇₃	—	8	2	no	—
97	C₈×C₇₃	—	2	1	no	—
98	C₈×C₇₃	—	2	1	no	—
99	C₈×C₇₃	—	8	2	no	—
100	C₈×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₈×C₇₃	#117	1	1	yes	characteristic
102	C₈×C₇₃	C₂²⋊Q₈	1	2	yes	—
103	C₂²×C₇₃	—	2	1	no	—
104	C₂²×C₇₃	—	2	1	no	—
105	C₂²×C₇₃	—	2	1	no	—
106	C₂²×C₇₃	—	2	1	no	—
107	C₂²×C₇₃	—	2	1	no	—
108	C₂²×C₇₃	—	2	1	no	—
109	C₂²×C₇₃	—	2	1	no	—
110	C₂²×C₇₃	—	4	4	no	—
111	C₂²×C₇₃	#365	1	2	yes	complemented
112	C₂²×C₇₃	C₂×#106	1	1	yes	characteristic
113	C₂²×C₇₃	#465	1	2	yes	—
114	C₄×C₇₃	—	2	1	no	—
115	C₄×C₇₃	—	2	1	no	—
116	C₄×C₇₃	#380	1	1	yes	characteristic
117	C₄×C₇₃	#427	1	1	yes	characteristic
118	C₄×C₇₃	#465	1	2	yes	—
119	Dic₇₃	—	16	2	no	—
120	#12831	—	73	1	no	maximal, Sylow
121	C₂×C₇₃	—	2	2	no	—
122	C₂×C₇₃	—	2	2	no	—
123	C₂×C₇₃	—	4	1	no	—
124	C₂×C₇₃	#1807	1	1	yes	characteristic
125	C₂×C₇₃	C₂×#365	1	1	yes	characteristic
126	C₂×C₇₃	#2155	1	1	yes	characteristic
127	C₂²×M₃₂	—	73	1	no	—
128	C₂×#365	—	73	2	no	—
129	#1810	D₇₃	1	1	yes	characteristic
130	#1807	—	73	1	no	—
131	#1383	—	73	2	no	—
132	C₇₃	#12831	1	1	yes	characteristic, complemented, Sylow
133	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
134	C₂×M₃₂	—	73	1	no	—
135	C₂×M₃₂	—	73	2	no	—
136	C₂×M₃₂	—	73	1	no	—
137	C₂×M₃₂	—	146	4	no	—
138	C₂²×C₁₆	—	73	2	no	—
139	C₂×#108	C₂×D₇₃	1	2	yes	—
140	C₂×#106	—	73	2	no	—
141	#407	—	2	1	no	—
142	#380	—	146	1	no	—
143	#365	—	146	4	no	—
144	#341	—	2	1	no	—
145	#340	—	146	1	no	—
146	C₂³×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
147	C₂×M₁₆	—	2	2	no	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	Dic₁₄₆	1	1	yes	characteristic
151	C₂²×C₈	Dic₁₄₆	1	1	yes	characteristic
152	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
153	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
154	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
156	C₂×C₄⋊C₄	—	146	2	no	—
157	C₂×C₂²⋊C₄	—	292	1	no	—
158	M₃₂	—	73	4	no	—
159	M₃₂	—	146	4	no	—
160	M₃₂	—	146	2	no	—
161	C₂×C₁₆	—	73	2	no	—
162	C₂×C₁₆	—	73	2	no	—
163	C₂×C₁₆	—	146	4	no	—
164	#108	—	2	2	no	—
165	#108	—	2	2	no	—
166	#106	—	146	2	no	—
167	#106	—	146	2	no	—
168	C₂²⋊C₈	—	4	1	no	—
169	C₂²⋊C₈	—	4	1	no	—
170	#95	—	292	1	no	—
171	C₂⁴	#11924015	1	1	yes	characteristic
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	292	2	no	—
175	C₂²×C₄	#11924007	1	1	yes	characteristic
176	C₂²×C₄	#11924007	1	1	yes	characteristic
177	C₂²×C₄	#11924015	1	1	yes	characteristic
178	C₂²×C₄	#11924016	1	1	yes	characteristic
179	C₂²×C₄	#11924016	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
181	M₁₆	—	4	2	no	—
182	M₁₆	—	4	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	#11924007	1	1	yes	characteristic
192	C₂×C₈	#11924007	1	1	yes	characteristic
193	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
194	C₂×C₈	D₄×D₇₃	1	2	yes	—
195	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
196	C₂×C₈	#11924027	1	2	yes	—
197	C₄⋊C₄	—	292	2	no	—
198	C₄⋊C₄	—	292	2	no	—
199	C₂²⋊C₄	—	584	2	no	—
200	C₁₆	—	73	4	no	—
201	C₁₆	—	146	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	4	1	no	—
205	C₂³	#14556522	1	1	yes	characteristic
206	C₂³	#14556522	1	1	yes	characteristic
207	C₂³	C₂×#11924015	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	584	2	no	—
216	C₂×C₄	—	584	2	no	—
217	C₂×C₄	—	584	2	no	—
218	C₂×C₄	#14556522	1	1	yes	characteristic
219	C₂×C₄	#14556522	1	1	yes	characteristic
220	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
221	C₂×C₄	#14556563	1	2	yes	—
222	C₂×C₄	C₂×#11924016	1	1	yes	characteristic
223	C₂×C₄	#14556611	1	2	yes	—
224	C₈	—	2	1	no	—
225	C₈	—	2	1	no	—
226	C₈	—	8	2	no	—
227	C₈	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
228	C₈	#14556547	1	1	yes	characteristic
229	C₈	#14556568	1	2	yes	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	4	4	no	—
238	C₂²	#17988411	1	2	yes	complemented
239	C₂²	C₂×#14556522	1	1	yes	characteristic
240	C₂²	#17988818	1	2	yes	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	1168	2	no	—
244	C₄	#17988445	1	1	yes	characteristic
245	C₄	#17988623	1	1	yes	characteristic
246	C₄	#17988814	1	2	yes	—
247	C₂	—	2	2	no	—
248	C₂	—	2	2	no	—
249	C₂	—	4	1	no	—
250	C₂	#22609954	1	1	yes	characteristic
251	C₂	C₂×#17988411	1	1	yes	characteristic
252	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×#1810	C₂
socle	C₂²×C₇₃	C₂×#106
prime radical	C₂⁴×C₇₃	SD₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1810	D₇₃
hypocenter	C₇₃	#12831
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

Complements

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

N	H ≅ G/N	type
2. C₂×#17988411	C₂	semidirect
4. #22608031	C₂	semidirect
7. C₂×#14556531	C₄	semidirect
10. C₂²×#11924003	C₄	semidirect
49. #11924003	C₂²⋊C₄	semidirect
111. C₂²×C₇₃	#365	semidirect
132. C₇₃	#12831	semidirect
238. C₂²	#17988411	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14556531	2
C₂×#17988411	2
#22609956	2
#22608031	2
C₇₃×#1810	2
#12831	73