Subgroups — #33004007

Order 18688.

284 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22611696	C₂	1	2	yes	maximal
2	C₄×#14556531	C₂	1	1	yes	maximal, characteristic
3	#22609810	C₂	1	1	yes	maximal, characteristic
4	#22608016	C₂	1	2	yes	maximal
5	C₂×#364×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556531	C₂²	1	1	yes	characteristic
7	C₂×#14556531	C₂²	1	1	yes	characteristic
8	C₂×#14556527	C₄	1	1	yes	characteristic
9	#17988625	C₄	1	1	yes	characteristic
10	#17988542	C₂²	1	2	yes	—
11	#17988407	—	2	4	no	—
12	#17988363	—	2	1	no	—
13	C₄×#11924003	—	2	1	no	—
14	#17988347	C₄	1	2	yes	—
15	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	#364×C₇₃	—	2	1	no	—
18	#364×C₇₃	—	2	1	no	—
19	C₂×#11924012	C₂×C₄	1	2	yes	—
20	#14556550	C₂×C₄	1	2	yes	—
21	#14556531	—	2	1	no	—
22	#14556531	—	2	1	no	—
23	#14556531	D₄	1	2	yes	—
24	#14556531	Q₈	1	2	yes	—
25	C₂×#11924003	—	2	1	no	—
26	C₂×#11924003	—	2	1	no	—
27	#14556527	C₈	1	2	yes	—
28	#14556527	C₈	1	2	yes	—
29	#14556525	—	2	1	no	—
30	#14556523	—	2	1	no	—
31	#14556501	—	2	2	no	—
32	C₈×Dic₇₃	—	2	2	no	—
33	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₇₃	—	2	2	no	—
37	M₃₂×C₇₃	—	2	2	no	—
38	C₂×C₁₆×C₇₃	—	2	2	no	—
39	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
40	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
42	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
43	#11924012	—	2	2	no	—
44	#11924012	C₂×C₈	1	4	yes	—
45	C₂×#10596831	—	2	2	no	—
46	C₄⋊Dic₇₃	—	2	2	no	—
47	C₂²⋊Dic₇₃	—	2	2	no	—
48	C₄×Dic₇₃	—	2	2	no	—
49	#11924003	—	2	2	no	—
50	#11924003	—	2	2	no	—
51	C₂²×C₄×C₇₃	—	2	1	no	—
52	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
56	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
57	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
58	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
59	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
64	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
65	C₄²×C₇₃	—	2	1	no	—
66	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
67	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
68	C₁₆×C₇₃	—	4	2	no	—
69	C₂³×C₇₃	#131	1	1	yes	characteristic
70	C₂×C₄×C₇₃	—	2	1	no	—
71	C₂×C₄×C₇₃	—	2	1	no	—
72	C₂×C₄×C₇₃	—	4	2	no	—
73	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
79	C₂×Dic₇₃	—	4	2	no	—
80	C₈×C₇₃	C₄⋊C₈	1	1	yes	characteristic
81	C₈×C₇₃	C₄⋊C₈	1	1	yes	characteristic
82	C₈×C₇₃	C₄⋊C₈	1	1	yes	characteristic
83	C₈×C₇₃	C₄⋊C₈	1	1	yes	characteristic
84	C₈×C₇₃	#121	1	1	yes	characteristic
85	C₈×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
86	C₈×C₇₃	#125	1	1	yes	characteristic
87	C₈×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
88	#10596831	—	2	4	no	—
89	C₂²×C₇₃	—	4	1	no	—
90	C₂²×C₇₃	#424	1	1	yes	characteristic
91	C₂²×C₇₃	#436	1	1	yes	characteristic
92	C₂²×C₇₃	C₈×Q₈	1	1	yes	characteristic
93	C₄×C₇₃	—	2	2	no	—
94	C₄×C₇₃	—	4	1	no	—
95	C₄×C₇₃	#389	1	1	yes	characteristic
96	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
97	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
98	C₄×C₇₃	#446	1	1	yes	characteristic
99	Dic₇₃	—	8	2	no	—
100	#13164	—	73	1	no	maximal, Sylow
101	C₂×C₇₃	—	2	1	no	—
102	C₂×C₇₃	—	2	1	no	—
103	C₂×C₇₃	—	2	1	no	—
104	C₂×C₇₃	#1939	1	1	yes	characteristic
105	#2153	—	73	2	no	—
106	C₂×#364	D₇₃	1	1	yes	characteristic
107	C₄×M₃₂	—	73	1	no	—
108	#1772	—	73	1	no	—
109	#1378	—	73	2	no	—
110	C₇₃	#13164	1	1	yes	characteristic, complemented, Sylow
111	C₂×M₃₂	—	73	1	no	—
112	C₂×M₃₂	—	73	1	no	—
113	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
114	C₂×#108	—	73	1	no	—
115	#427	—	73	1	no	—
116	#405	—	73	2	no	—
117	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
118	#364	—	2	1	no	—
119	#364	—	2	1	no	—
120	#364	—	146	4	no	—
121	#346	—	146	1	no	—
122	C₄×C₁₆	—	146	1	no	—
123	#339	—	73	2	no	—
124	C₂×M₁₆	—	73	2	no	—
125	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
127	#117	—	73	2	no	—
128	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
129	M₃₂	—	2	2	no	—
130	M₃₂	—	2	2	no	—
131	M₃₂	—	73	2	no	—
132	M₃₂	—	73	2	no	—
133	M₃₂	—	146	1	no	—
134	M₃₂	—	146	1	no	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	146	1	no	—
137	C₂×C₁₆	—	146	1	no	—
138	#108	—	73	2	no	—
139	#108	—	73	2	no	—
140	#107	—	146	1	no	—
141	#106	—	146	1	no	—
142	#97	—	146	2	no	—
143	C₄×C₈	—	146	2	no	—
144	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
145	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
146	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
147	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
150	M₁₆	—	73	4	no	—
151	M₁₆	—	146	2	no	—
152	C₂×C₈	—	146	2	no	—
153	C₂×C₈	#11924007	1	1	yes	characteristic
154	C₂×C₈	#11924007	1	1	yes	characteristic
155	C₂×C₈	C₈×D₇₃	1	1	yes	characteristic
156	C₂×C₈	C₈×D₇₃	1	1	yes	characteristic
157	C₂×C₈	#11924011	1	1	yes	characteristic
158	C₂×C₈	#11924011	1	1	yes	characteristic
159	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
160	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
161	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
162	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
163	C₂×C₈	#11924027	1	1	yes	characteristic
164	C₂×C₈	#11924029	1	1	yes	characteristic
165	C₄⋊C₄	—	146	2	no	—
166	C₂²⋊C₄	—	146	2	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	146	2	no	—
169	C₄²	#11924007	1	1	yes	characteristic
170	C₄²	#11924007	1	1	yes	characteristic
171	C₁₆	—	4	2	no	—
172	C₁₆	—	146	2	no	—
173	C₁₆	—	146	2	no	—
174	C₂³	#14556599	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	292	2	no	—
179	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
180	C₂×C₄	#14556553	1	1	yes	characteristic
181	C₂×C₄	#14556556	1	1	yes	characteristic
182	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
183	C₂×C₄	C₂×#11924011	1	1	yes	characteristic
184	C₂×C₄	#14556599	1	1	yes	characteristic
185	C₈	—	146	4	no	—
186	C₈	#14556520	1	1	yes	characteristic
187	C₈	#14556520	1	1	yes	characteristic
188	C₈	#14556520	1	1	yes	characteristic
189	C₈	#14556520	1	1	yes	characteristic
190	C₈	#14556562	1	1	yes	characteristic
191	C₈	#14556567	1	1	yes	characteristic
192	C₈	#14556582	1	1	yes	characteristic
193	C₈	#14556590	1	1	yes	characteristic
194	C₂²	—	4	1	no	—
195	C₂²	#17988615	1	1	yes	characteristic
196	C₂²	#17988664	1	1	yes	characteristic
197	C₂²	#17988711	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	4	1	no	—
200	C₄	—	584	2	no	—
201	C₄	#17988483	1	1	yes	characteristic
202	C₄	C₂×#14556520	1	1	yes	characteristic
203	C₄	#17988652	1	1	yes	characteristic
204	C₄	#17988714	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	#22610647	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14556562
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₂×#364×C₇₃	C₂
socle	C₂×C₇₃	#1939
prime radical	C₂³×C₇₃	#131
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×#364	D₇₃
hypocenter	C₇₃	#13164
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
110. C₇₃	#13164	semidirect

Maximal subgroups

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

subgroup	index
#22611696	2
C₄×#14556531	2
#22609810	2
#22608016	2
C₂×#364×C₇₃	2
#13164	73