Subgroups — #33238596

Order 18688.

806 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988997	C₂	1	2	yes	maximal
2	C₂×#17988985	C₂	1	1	yes	maximal, characteristic
3	#22616052	C₂	1	2	yes	maximal
4	#22610098	C₂	1	4	yes	maximal
5	#22610092	C₂	1	2	yes	maximal
6	#22610086	C₂	1	2	yes	maximal
7	#22609617	C₂	1	1	yes	maximal, characteristic
8	C₂×#499×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#14556591	C₂²	1	2	yes	—
10	#17988997	—	2	8	no	—
11	#17988985	—	2	4	no	—
12	C₂×#14556524	C₂²	1	4	yes	—
13	C₂×#14556523	C₂²	1	2	yes	—
14	C₂×#14556523	C₂²	1	2	yes	—
15	C₂×#14556522	C₂²	1	4	yes	—
16	C₂×#14556502	C₂²	1	1	yes	characteristic
17	C₂×#14556501	C₂²	1	2	yes	—
18	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
19	#17988517	—	2	4	no	—
20	#17988507	—	2	4	no	—
21	#17988456	C₂²	1	4	yes	—
22	#17988452	C₂²	1	4	yes	—
23	#17988433	C₂²	1	1	yes	characteristic
24	C₂×#125×C₇₃	C₂²	1	2	yes	—
25	#499×C₇₃	—	2	4	no	—
26	C₂×#107×C₇₃	C₂²	1	2	yes	—
27	C₂×#106×C₇₃	C₂²	1	1	yes	characteristic
28	C₂×#106×C₇₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
30	C₂²×Dic₁₄₆	—	2	2	no	—
31	C₂²×#10596831	C₂³	1	2	yes	—
32	#14556591	—	2	4	no	—
33	#14556591	—	2	4	no	—
34	C₂×C₄⋊Dic₇₃	—	2	2	no	—
35	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
36	C₂×#11924007	—	2	8	no	—
37	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
38	#14556524	—	2	8	no	—
39	#14556523	—	2	4	no	—
40	#14556523	—	2	4	no	—
41	#14556522	—	2	8	no	—
42	#14556502	D₄	1	4	yes	—
43	#14556501	Q₈	1	8	yes	complemented
44	C₈×Dic₇₃	Q₈	1	8	yes	complemented
45	#14556498	—	2	8	no	—
46	#14556498	D₄	1	4	yes	—
47	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
49	#125×C₇₃	—	2	4	no	—
50	#125×C₇₃	—	2	4	no	—
51	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
52	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
53	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
56	#107×C₇₃	—	2	4	no	—
57	#106×C₇₃	—	2	2	no	—
58	#106×C₇₃	—	2	2	no	—
59	C₄×C₈×C₇₃	D₄	1	4	yes	—
60	C₂×Dic₁₄₆	—	4	4	no	—
61	C₂×Dic₁₄₆	—	8	4	no	—
62	C₂²×Dic₇₃	—	2	4	no	—
63	C₂²×Dic₇₃	C₂×D₄	1	4	yes	—
64	C₂×#10596831	C₂×D₄	1	4	yes	—
65	C₂×#10596831	C₂×Q₈	1	4	yes	—
66	C₂×#10596831	C₂×Q₈	1	4	yes	—
67	C₄⋊Dic₇₃	—	2	4	no	—
68	C₄⋊Dic₇₃	—	2	4	no	—
69	C₄⋊Dic₇₃	—	4	4	no	—
70	#11924007	—	8	8	no	—
71	C₄×Dic₇₃	C₂×Q₈	1	4	yes	—
72	C₄×Dic₇₃	C₂×Q₈	1	4	yes	—
73	C₂²×C₄×C₇₃	—	2	2	no	—
74	C₂²×C₄×C₇₃	—	2	2	no	—
75	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
80	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
81	C₄⋊C₄×C₇₃	—	2	2	no	—
82	C₄⋊C₄×C₇₃	—	2	2	no	—
83	C₄⋊C₄×C₇₃	—	2	2	no	—
84	C₄⋊C₄×C₇₃	—	2	2	no	—
85	C₄⋊C₄×C₇₃	—	4	4	no	—
86	C₄⋊C₄×C₇₃	—	4	4	no	—
87	C₄²×C₇₃	C₂×D₄	1	2	yes	—
88	C₄²×C₇₃	C₂×D₄	1	2	yes	—
89	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₇₃	—	2	4	no	—
91	C₂×C₄×C₇₃	—	4	2	no	—
92	C₂×C₄×C₇₃	—	4	2	no	—
93	C₂×C₄×C₇₃	—	4	4	no	—
94	C₂×C₄×C₇₃	—	4	4	no	—
95	C₂×C₄×C₇₃	#127	1	4	yes	—
96	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
99	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
100	C₂×Dic₇₃	—	2	8	no	—
101	C₂×Dic₇₃	—	4	4	no	—
102	C₂×Dic₇₃	—	4	8	no	—
103	C₂×Dic₇₃	C₄⋊Q₈	1	8	yes	—
104	Dic₁₄₆	—	8	8	no	—
105	C₈×C₇₃	C₄⋊Q₈	1	4	yes	—
106	C₈×C₇₃	C₄⋊Q₈	1	4	yes	—
107	#10596831	C₄⋊Q₈	1	8	yes	—
108	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
109	C₂²×C₇₃	C₂×C₄⋊Q₈	1	2	yes	—
110	C₂²×C₇₃	#577	1	4	yes	—
111	C₄×C₇₃	—	2	4	no	—
112	C₄×C₇₃	—	8	2	no	—
113	C₄×C₇₃	—	8	2	no	—
114	C₄×C₇₃	C₂×C₄⋊Q₈	1	2	yes	—
115	C₄×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
116	C₄×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
117	Dic₇₃	—	2	8	no	—
118	Dic₇₃	—	8	4	no	—
119	#33318	—	73	1	no	maximal, Sylow
120	C₂×C₇₃	#2868	1	1	yes	characteristic
121	C₂×C₇₃	#3154	1	2	yes	—
122	C₂×C₇₃	#3163	1	4	yes	—
123	C₂×#501	—	73	2	no	—
124	C₂×#499	D₇₃	1	1	yes	characteristic
125	C₂×#498	—	73	1	no	—
126	#2660	—	73	2	no	—
127	#1847	—	73	4	no	—
128	#1846	—	73	2	no	—
129	#1844	—	73	2	no	—
130	#1726	—	73	1	no	—
131	C₇₃	#33318	1	1	yes	characteristic, complemented, Sylow
132	C₂×C₄⋊Q₈	—	73	2	no	—
133	C₂×#125	C₂×D₇₃	1	2	yes	—
134	#501	—	146	8	no	—
135	#499	—	2	4	no	—
136	#498	—	146	4	no	—
137	C₂×#107	—	73	4	no	—
138	C₂×#107	C₂×D₇₃	1	2	yes	—
139	C₂×#106	—	73	2	no	—
140	C₂×#106	—	73	2	no	—
141	C₂×#106	—	73	4	no	—
142	C₂×#106	C₂×D₇₃	1	1	yes	characteristic
143	C₂×#106	C₂×D₇₃	1	1	yes	characteristic
144	C₂×#97	—	73	2	no	—
145	C₂×#97	—	73	1	no	—
146	C₂×C₄×C₈	—	73	2	no	—
147	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
148	#398	—	146	4	no	—
149	#395	—	146	4	no	—
150	#384	—	73	4	no	—
151	#382	—	73	4	no	—
152	#376	—	73	1	no	—
153	C₂²×Q₈	—	146	2	no	—
154	C₂²×C₈	—	73	2	no	—
155	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
156	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
157	C₄⋊Q₈	—	146	4	no	—
158	C₄⋊Q₈	—	146	4	no	—
159	#125	—	2	4	no	—
160	#125	—	2	4	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	73	4	no	—
164	C₂×C₄⋊C₄	—	146	2	no	—
165	C₂×C₄⋊C₄	—	146	8	no	—
166	C₂×C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
167	C₂×C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
168	C₂×C₄²	—	73	2	no	—
169	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
170	#107	—	2	4	no	—
171	#107	—	146	8	no	—
172	#106	—	2	2	no	—
173	#106	—	2	2	no	—
174	#106	—	146	8	no	—
175	#106	—	146	4	no	—
176	#106	—	146	4	no	—
177	#97	—	73	8	no	—
178	#97	—	73	4	no	—
179	C₄×C₈	—	73	8	no	—
180	C₄×C₈	C₂²⋊D₇₃	1	4	yes	—
181	#95	—	73	4	no	—
182	#95	—	146	8	no	—
183	C₂×Q₈	—	292	4	no	—
184	C₂×Q₈	—	584	4	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	73	4	no	—
188	C₂²×C₄	—	146	4	no	—
189	C₂²×C₄	D₄×D₇₃	1	2	yes	—
190	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
191	C₂×C₈	—	73	4	no	—
192	C₂×C₈	—	73	4	no	—
193	C₂×C₈	—	73	4	no	—
194	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
195	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
196	C₂×C₈	Q₈×D₇₃	1	4	yes	—
197	C₂×C₈	Q₈×D₇₃	1	4	yes	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	4	4	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₄⋊C₄	—	292	4	no	—
205	C₄⋊C₄	—	584	8	no	—
206	C₄⋊C₄	—	146	4	no	—
207	C₄⋊C₄	—	146	4	no	—
208	C₄²	—	73	4	no	—
209	C₄²	—	73	4	no	—
210	C₄²	D₄×D₇₃	1	2	yes	—
211	C₄²	C₂×C₂²⋊D₇₃	1	2	yes	—
212	C₂³	C₂×D₄×D₇₃	1	1	yes	characteristic
213	Q₈	—	584	8	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	292	4	no	—
220	C₂×C₄	—	292	8	no	—
221	C₂×C₄	—	73	8	no	—
222	C₂×C₄	—	146	8	no	—
223	C₂×C₄	#14556587	1	4	yes	—
224	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
225	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
226	C₂×C₄	C₂×Q₈×D₇₃	1	2	yes	—
227	C₂×C₄	C₂×Q₈×D₇₃	1	2	yes	—
228	C₈	—	73	8	no	—
229	C₈	#14556589	1	4	yes	—
230	C₈	#14556589	1	4	yes	—
231	C₂²	C₂×#14556587	1	1	yes	characteristic
232	C₂²	C₄⋊Q₈×D₇₃	1	2	yes	—
233	C₂²	#17989394	1	4	yes	—
234	C₄	—	2	4	no	—
235	C₄	—	8	2	no	—
236	C₄	—	8	2	no	—
237	C₄	—	584	4	no	—
238	C₄	—	146	8	no	—
239	C₄	C₄⋊Q₈×D₇₃	1	2	yes	—
240	C₄	C₂×#14556589	1	1	yes	characteristic
241	C₄	C₂×#14556589	1	1	yes	characteristic
242	C₂	#22618327	1	1	yes	characteristic
243	C₂	#22620408	1	2	yes	—
244	C₂	#22620460	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₇₃
commutator, cosocle	C₂²×C₄×C₇₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂×#499×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂²×D₄
hypercenter	C₂×#499	D₇₃
hypocenter	C₇₃	#33318
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
43. #14556501	Q₈	semidirect
44. C₈×Dic₇₃	Q₈	semidirect
131. C₇₃	#33318	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988997	2
C₂×#17988985	2
#22616052	2
#22610098	2
#22610092	2
#22610086	2
#22609617	2
C₂×#499×C₇₃	2
#33318	73