Subgroups — #33004120

Order 18688.

306 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988409	C₂	1	2	yes	maximal
2	#22611687	C₂	1	1	yes	maximal, characteristic
3	#22610103	C₂	1	1	yes	maximal, characteristic
4	#22608029	C₂	1	2	yes	maximal
5	C₄×#106×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556531	C₂²	1	2	yes	—
7	C₂×#14556527	—	2	1	no	—
8	C₂×#14556527	C₂²	1	2	yes	—
9	#17988611	—	2	1	no	—
10	#17988409	—	2	4	no	—
11	C₄⋊#11924003	C₄	1	2	yes	complemented
12	C₄⋊#11924003	C₄	1	2	yes	complemented
13	C₂×#106×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×#106×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₇₃	—	2	1	no	—
17	#340×C₇₃	—	2	1	no	—
18	C₂×#11924012	—	2	2	no	—
19	#14556531	—	2	2	no	—
20	#14556531	—	2	2	no	—
21	C₂×#11924003	C₂×C₄	1	4	yes	—
22	#14556527	—	2	2	no	—
23	#14556527	—	2	2	no	—
24	#14556527	—	4	2	no	—
25	C₄⋊#10596831	—	4	1	no	—
26	C₄⋊#10596831	—	4	1	no	—
27	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
30	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
31	C₂×C₄²×C₇₃	—	4	1	no	—
32	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
33	#106×C₇₃	—	2	1	no	—
34	#106×C₇₃	—	2	1	no	—
35	#106×C₇₃	D₄	1	2	yes	—
36	#106×C₇₃	Q₈	1	2	yes	—
37	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
40	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
41	#95×C₇₃	—	4	1	no	—
42	#11924012	—	4	2	no	—
43	#11924012	—	4	2	no	—
44	C₂×#10596831	—	4	2	no	—
45	#11924003	—	2	4	no	—
46	C₂²×C₄×C₇₃	—	4	1	no	—
47	C₂²×C₄×C₇₃	—	4	1	no	—
48	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₇₃	—	2	1	no	—
52	C₂×C₈×C₇₃	—	2	1	no	—
53	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
60	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
61	C₄⋊C₄×C₇₃	—	2	2	no	—
62	C₄⋊C₄×C₇₃	—	2	2	no	—
63	C₄⋊C₄×C₇₃	—	4	1	no	—
64	C₄⋊C₄×C₇₃	—	4	1	no	—
65	C₄²×C₇₃	—	8	2	no	—
66	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
67	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
68	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
69	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
70	C₂³×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₄×C₇₃	—	2	2	no	—
72	C₂×C₄×C₇₃	—	4	1	no	—
73	C₂×C₄×C₇₃	—	4	2	no	—
74	C₂×C₄×C₇₃	—	4	2	no	—
75	C₂×C₄×C₇₃	—	8	1	no	—
76	C₂×C₄×C₇₃	—	8	1	no	—
77	C₂×C₄×C₇₃	—	8	1	no	—
78	C₂×C₄×C₇₃	—	8	1	no	—
79	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
80	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
81	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
82	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
83	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
85	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
89	C₈×C₇₃	—	2	1	no	—
90	C₈×C₇₃	—	2	1	no	—
91	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
92	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
93	C₈×C₇₃	#125	1	1	yes	characteristic
94	C₈×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
95	#10596831	—	8	2	no	—
96	C₂²×C₇₃	—	2	1	no	—
97	C₂²×C₇₃	—	2	1	no	—
98	C₂²×C₇₃	C₂×#106	1	1	yes	characteristic
99	C₂²×C₇₃	#461	1	1	yes	characteristic
100	C₂²×C₇₃	#476	1	1	yes	characteristic
101	C₄×C₇₃	—	2	1	no	—
102	C₄×C₇₃	—	2	1	no	—
103	C₄×C₇₃	—	2	1	no	—
104	C₄×C₇₃	—	8	2	no	—
105	C₄×C₇₃	—	8	2	no	—
106	C₄×C₇₃	#365	1	2	yes	complemented
107	C₄×C₇₃	#389	1	1	yes	characteristic
108	C₄×C₇₃	#428	1	1	yes	characteristic
109	C₄×C₇₃	#460	1	1	yes	characteristic
110	C₄×C₇₃	#475	1	1	yes	characteristic
111	#13185	—	73	1	no	maximal, Sylow
112	C₂×C₇₃	—	2	1	no	—
113	C₂×C₇₃	—	2	1	no	—
114	C₂×C₇₃	#1948	1	1	yes	characteristic
115	C₂×C₇₃	C₂×#365	1	1	yes	characteristic
116	C₂×C₇₃	#2155	1	1	yes	characteristic
117	C₂×#365	—	73	2	no	—
118	#2150	—	73	1	no	—
119	#1848	—	73	1	no	—
120	C₄×#106	D₇₃	1	1	yes	characteristic
121	#1383	—	73	2	no	—
122	C₇₃	#13185	1	1	yes	characteristic, complemented, Sylow
123	C₂×M₃₂	—	73	2	no	—
124	C₂×#108	—	73	2	no	—
125	C₂×#108	—	146	1	no	—
126	C₂×#106	C₂×D₇₃	1	1	yes	characteristic
127	C₂×#106	C₂×D₇₃	1	1	yes	characteristic
128	#423	—	146	1	no	—
129	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
130	C₄×C₄⋊C₄	—	2	1	no	—
131	#365	—	146	4	no	—
132	C₄⋊C₁₆	—	73	2	no	—
133	C₄⋊C₁₆	—	73	2	no	—
134	#340	—	2	1	no	—
135	C₂×M₁₆	—	146	2	no	—
136	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
138	C₂×C₄⋊C₄	—	2	1	no	—
139	C₂×C₄⋊C₄	—	2	1	no	—
140	C₂×C₄²	—	4	1	no	—
141	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
142	M₃₂	—	146	2	no	—
143	M₃₂	—	146	2	no	—
144	C₂×C₁₆	—	73	4	no	—
145	#108	—	292	2	no	—
146	#108	—	146	2	no	—
147	#108	—	146	2	no	—
148	#106	—	2	1	no	—
149	#106	—	2	1	no	—
150	#106	Dic₁₄₆	1	2	yes	—
151	#106	D₂₉₂	1	2	yes	—
152	C₄⋊C₈	—	292	1	no	—
153	C₄⋊C₈	—	292	1	no	—
154	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
155	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
156	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
157	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
158	#95	—	4	1	no	—
159	C₂²×C₄	—	4	1	no	—
160	C₂²×C₄	—	4	1	no	—
161	C₂²×C₄	#11924016	1	1	yes	characteristic
162	C₂²×C₄	#11924016	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
164	M₁₆	—	292	2	no	—
165	M₁₆	—	292	2	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	292	2	no	—
169	C₂×C₈	#11924007	1	1	yes	characteristic
170	C₂×C₈	#11924007	1	1	yes	characteristic
171	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
172	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
173	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
174	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
175	C₂×C₈	#11924028	1	1	yes	characteristic
176	C₂×C₈	#11924028	1	1	yes	characteristic
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	4	1	no	—
180	C₄⋊C₄	—	4	1	no	—
181	C₄²	—	8	2	no	—
182	C₄²	#11924007	1	1	yes	characteristic
183	C₄²	#11924007	1	1	yes	characteristic
184	C₄²	#11924015	1	1	yes	characteristic
185	C₄²	#11924015	1	1	yes	characteristic
186	C₁₆	—	146	4	no	—
187	C₂³	C₂×#11924016	1	1	yes	characteristic
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	4	1	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	8	1	no	—
193	C₂×C₄	—	8	1	no	—
194	C₂×C₄	—	8	1	no	—
195	C₂×C₄	—	8	1	no	—
196	C₂×C₄	#14556522	1	1	yes	characteristic
197	C₂×C₄	#14556522	1	1	yes	characteristic
198	C₂×C₄	#14556522	1	1	yes	characteristic
199	C₂×C₄	#14556522	1	1	yes	characteristic
200	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
201	C₂×C₄	#14556565	1	1	yes	characteristic
202	C₂×C₄	#14556570	1	1	yes	characteristic
203	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
204	C₂×C₄	#14556616	1	1	yes	characteristic
205	C₂×C₄	#14556621	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	584	2	no	—
209	C₈	C₄×D₂₉₂	1	1	yes	characteristic
210	C₈	C₄×Dic₁₄₆	1	1	yes	characteristic
211	C₈	#14556583	1	1	yes	characteristic
212	C₈	#14556591	1	1	yes	characteristic
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	C₂×#14556522	1	1	yes	characteristic
216	C₂²	#17988792	1	1	yes	characteristic
217	C₂²	#17988877	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	8	2	no	—
222	C₄	—	8	2	no	—
223	C₄	#17988409	1	2	yes	complemented
224	C₄	#17988485	1	1	yes	characteristic
225	C₄	#17988630	1	1	yes	characteristic
226	C₄	#17988787	1	1	yes	characteristic
227	C₄	#17988872	1	1	yes	characteristic
228	C₂	—	2	1	no	—
229	C₂	—	2	1	no	—
230	C₂	#22610694	1	1	yes	characteristic
231	C₂	C₂×#17988409	1	1	yes	characteristic
232	C₂	#22611703	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556522
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₄×#106×C₇₃	C₂
socle	C₂²×C₇₃	C₂×#106
prime radical	C₂³×C₇₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₄×#106	D₇₃
hypocenter	C₇₃	#13185
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
11. C₄⋊#11924003	C₄	semidirect
12. C₄⋊#11924003	C₄	semidirect
106. C₄×C₇₃	#365	semidirect
122. C₇₃	#13185	semidirect
223. C₄	#17988409	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988409	2
#22611687	2
#22610103	2
#22608029	2
C₄×#106×C₇₃	2
#13185	73