Subgroups — #33004100

Order 18688.

310 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22611720	C₂	1	2	yes	maximal
2	C₂×#17988408	C₂	1	1	yes	maximal, characteristic
3	#22610091	C₂	1	1	yes	maximal, characteristic
4	#22608041	C₂	1	2	yes	maximal
5	C₇₃×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556531	C₂²	1	2	yes	—
7	C₂×#14556525	—	2	1	no	—
8	C₂×#14556525	C₂²	1	2	yes	—
9	#17988457	—	2	1	no	—
10	#17988414	—	2	2	no	—
11	#17988412	—	2	2	no	—
12	#17988408	C₄	1	4	yes	complemented
13	C₂×#108×C₇₃	C₂²	1	1	yes	characteristic
14	#427×C₇₃	C₂²	1	1	yes	characteristic
15	#405×C₇₃	—	2	1	no	—
16	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	#339×C₇₃	—	2	1	no	—
18	C₂×C₄⋊Dic₇₃	—	2	2	no	—
19	C₂×C₄⋊Dic₇₃	—	4	2	no	—
20	#14556531	—	2	2	no	—
21	#14556531	C₂×C₄	1	4	yes	—
22	C₂×#11924003	—	2	2	no	—
23	#14556525	—	2	2	no	—
24	#14556525	—	2	2	no	—
25	#14556525	—	4	1	no	—
26	#14556525	—	4	1	no	—
27	C₂×M₁₆×C₇₃	—	2	1	no	—
28	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
30	#117×C₇₃	—	2	1	no	—
31	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
32	#108×C₇₃	—	2	1	no	—
33	#108×C₇₃	Q₈	1	2	yes	—
34	#107×C₇₃	D₄	1	1	yes	characteristic
35	#107×C₇₃	D₄	1	1	yes	characteristic
36	#106×C₇₃	—	2	1	no	—
37	#97×C₇₃	—	4	1	no	—
38	C₄×C₈×C₇₃	—	4	1	no	—
39	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₇₃	Q₈	1	2	yes	—
41	C₂²×Dic₇₃	—	4	2	no	—
42	C₄⋊Dic₇₃	—	4	2	no	—
43	C₄⋊Dic₇₃	—	4	2	no	—
44	C₄⋊Dic₇₃	—	8	4	no	—
45	#11924003	—	2	4	no	—
46	C₂²×C₄×C₇₃	—	2	1	no	—
47	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
48	M₁₆×C₇₃	—	2	2	no	—
49	M₁₆×C₇₃	—	4	1	no	—
50	C₂×C₈×C₇₃	—	4	1	no	—
51	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
53	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
54	C₂×C₈×C₇₃	Q₁₆	1	2	yes	—
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	1	no	—
62	C₄⋊C₄×C₇₃	—	2	1	no	—
63	C₂²⋊C₄×C₇₃	—	4	1	no	—
64	C₄²×C₇₃	—	2	1	no	—
65	C₄²×C₇₃	—	4	1	no	—
66	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
67	C₂³×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂×C₄×C₇₃	—	2	2	no	—
69	C₂×C₄×C₇₃	—	2	2	no	—
70	C₂×C₄×C₇₃	—	2	2	no	—
71	C₂×C₄×C₇₃	—	4	1	no	—
72	C₂×C₄×C₇₃	—	4	1	no	—
73	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₇₃	#137	1	1	yes	characteristic
79	C₂×Dic₇₃	—	8	4	no	—
80	C₂×Dic₇₃	—	8	2	no	—
81	C₈×C₇₃	—	4	2	no	—
82	C₈×C₇₃	#107	1	2	yes	—
83	C₈×C₇₃	#107	1	2	yes	—
84	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
85	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
86	C₈×C₇₃	#125	1	1	yes	characteristic
87	C₈×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
88	C₂²×C₇₃	—	2	1	no	—
89	C₂²×C₇₃	—	2	1	no	—
90	C₂²×C₇₃	#428	1	1	yes	characteristic
91	C₂²×C₇₃	#459	1	1	yes	characteristic
92	C₂²×C₇₃	#477	1	1	yes	characteristic
93	C₄×C₇₃	—	2	4	no	—
94	C₄×C₇₃	—	8	1	no	—
95	C₄×C₇₃	—	8	1	no	—
96	C₄×C₇₃	#389	1	1	yes	characteristic
97	C₄×C₇₃	C₂×#107	1	1	yes	characteristic
98	C₄×C₇₃	#462	1	1	yes	characteristic
99	C₄×C₇₃	#474	1	1	yes	characteristic
100	Dic₇₃	—	16	2	no	—
101	#13181	—	73	1	no	maximal, Sylow
102	C₂×C₇₃	—	2	1	no	—
103	C₂×C₇₃	—	2	1	no	—
104	C₂×C₇₃	#1947	1	1	yes	characteristic
105	C₂×C₇₃	#2213	1	1	yes	characteristic
106	C₂×C₇₃	#2214	1	1	yes	characteristic
107	#2159	—	73	2	no	—
108	C₂×#364	—	73	1	no	—
109	#1845	—	73	1	no	—
110	#1772	D₇₃	1	1	yes	characteristic
111	#1387	—	73	2	no	—
112	C₇₃	#13181	1	1	yes	characteristic, complemented, Sylow
113	C₂×M₃₂	—	73	2	no	—
114	C₂×#108	C₂×D₇₃	1	1	yes	characteristic
115	#427	C₂×D₇₃	1	1	yes	characteristic
116	C₂×#107	—	73	2	no	—
117	C₂×#107	—	146	1	no	—
118	#405	—	2	1	no	—
119	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
120	#384	—	146	1	no	—
121	#367	—	146	2	no	—
122	#366	—	146	2	no	—
123	#364	—	73	4	no	—
124	#339	—	2	1	no	—
125	C₂×M₁₆	—	2	1	no	—
126	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
127	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
128	#117	—	2	1	no	—
129	C₂×C₄⋊C₄	—	292	2	no	—
130	C₂×C₄⋊C₄	—	146	2	no	—
131	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
132	M₃₂	—	73	4	no	—
133	M₃₂	—	146	2	no	—
134	C₂×C₁₆	—	146	2	no	—
135	#108	—	2	1	no	—
136	#108	Dic₁₄₆	1	2	yes	—
137	#107	—	292	1	no	—
138	#107	—	292	1	no	—
139	#107	—	146	2	no	—
140	#107	—	146	2	no	—
141	#107	D₂₉₂	1	1	yes	characteristic
142	#107	D₂₉₂	1	1	yes	characteristic
143	#106	—	2	1	no	—
144	#97	—	4	1	no	—
145	C₄×C₈	—	4	1	no	—
146	C₄×C₈	Dic₁₄₆	1	2	yes	—
147	C₄×C₈	C₄×D₇₃	1	2	yes	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	292	2	no	—
150	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
151	M₁₆	—	2	2	no	—
152	M₁₆	—	4	1	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	#11924007	1	2	yes	—
155	C₂×C₈	#11924013	1	1	yes	characteristic
156	C₂×C₈	#11924013	1	1	yes	characteristic
157	C₂×C₈	#11924018	1	2	yes	—
158	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
159	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
160	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
161	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
162	C₂×C₈	#11924028	1	1	yes	characteristic
163	C₂×C₈	#11924028	1	1	yes	characteristic
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	292	2	no	—
167	C₄⋊C₄	—	292	2	no	—
168	C₄⋊C₄	—	584	4	no	—
169	C₂²⋊C₄	—	4	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	4	1	no	—
172	C₄²	#11924007	1	2	yes	—
173	C₁₆	—	146	4	no	—
174	C₂³	#14556616	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	1	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	—	584	4	no	—
181	C₂×C₄	—	584	2	no	—
182	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
183	C₂×C₄	#14556565	1	1	yes	characteristic
184	C₂×C₄	#14556570	1	1	yes	characteristic
185	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
186	C₂×C₄	C₂×#11924018	1	1	yes	characteristic
187	C₂×C₄	#14556621	1	1	yes	characteristic
188	C₈	—	4	2	no	—
189	C₈	#14556524	1	2	yes	—
190	C₈	#14556524	1	2	yes	—
191	C₈	C₄×D₂₉₂	1	1	yes	characteristic
192	C₈	C₄×Dic₁₄₆	1	1	yes	characteristic
193	C₈	#14556583	1	1	yes	characteristic
194	C₈	#14556591	1	1	yes	characteristic
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	#17988630	1	1	yes	characteristic
198	C₂²	#17988782	1	1	yes	characteristic
199	C₂²	#17988882	1	1	yes	characteristic
200	C₄	—	2	4	no	—
201	C₄	—	8	1	no	—
202	C₄	—	8	1	no	—
203	C₄	—	1168	2	no	—
204	C₄	#17988485	1	1	yes	characteristic
205	C₄	C₂×#14556524	1	1	yes	characteristic
206	C₄	#17988797	1	1	yes	characteristic
207	C₄	#17988867	1	1	yes	characteristic
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	#22610689	1	1	yes	characteristic
211	C₂	#22612007	1	1	yes	characteristic
212	C₂	#22612012	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17988782
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₇₃×#1772	C₂
socle	C₂²×C₇₃	#459
prime radical	C₂³×C₇₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1772	D₇₃
hypocenter	C₇₃	#13181
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
12. #17988408	C₄	semidirect
112. C₇₃	#13181	semidirect

Maximal subgroups

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

subgroup	index
#22611720	2
C₂×#17988408	2
#22610091	2
#22608041	2
C₇₃×#1772	2
#13181	73