Subgroups — #33055282

Order 18688.

612 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22619063	C₂	1	2	yes	maximal
2	#22608720	C₂	1	6	yes	maximal, complemented
3	#22608687	C₂	1	3	yes	maximal, complemented
4	#22608679	C₂	1	3	yes	maximal, complemented
5	C₈×#41×C₇₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×#10596831	—	2	2	no	—
7	#17988665	—	2	6	no	—
8	#17988640	C₂²	1	6	yes	—
9	#17988611	C₂²	1	6	yes	—
10	#17988340	C₂²	1	6	yes	—
11	#17988338	C₂²	1	1	yes	characteristic
12	#17988338	C₂²	1	3	yes	—
13	#17988338	C₂²	1	6	yes	—
14	C₄×#41×C₇₃	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₇₃	C₄	1	1	yes	characteristic
16	C₈×Q₈×C₇₃	C₄	1	1	yes	characteristic
17	C₈×D₄×C₇₃	C₄	1	3	yes	—
18	C₈×D₄×C₇₃	C₄	1	3	yes	—
19	#431×C₇₃	C₂²	1	3	yes	—
20	C₂×C₄×C₈×C₇₃	C₂²	1	3	yes	—
21	C₂×#11924012	—	2	6	no	—
22	C₄⋊#10596831	—	2	6	no	—
23	C₄⋊#10596831	C₂³	1	6	yes	—
24	C₄⋊#10596831	C₂³	1	2	yes	—
25	C₂²⋊#10596831	—	2	6	no	—
26	C₄×#10596831	—	2	6	no	—
27	C₂×#41×C₇₃	Q₈	1	1	yes	characteristic
28	C₂²×C₈×C₇₃	C₂×C₄	1	3	yes	—
29	C₂²×C₈×C₇₃	C₂×C₄	1	3	yes	—
30	C₄×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₄×Q₈×C₇₃	D₄	1	1	yes	characteristic
32	C₄×D₄×C₇₃	C₂×C₄	1	3	yes	—
33	C₄×D₄×C₇₃	D₄	1	3	yes	—
34	#117×C₇₃	Q₈	1	3	yes	—
35	C₂×C₄²×C₇₃	C₂³	1	3	yes	—
36	C₄⋊C₈×C₇₃	C₂×C₄	1	3	yes	—
37	C₄⋊C₈×C₇₃	C₂×C₄	1	3	yes	—
38	C₂²⋊C₈×C₇₃	C₂×C₄	1	6	yes	—
39	C₄×C₈×C₇₃	C₂×C₄	1	3	yes	—
40	C₄×C₈×C₇₃	C₂×C₄	1	3	yes	—
41	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
42	C₄×C₈×C₇₃	C₂³	1	3	yes	—
43	#11924012	—	8	6	no	—
44	C₂×#10596831	—	2	2	no	—
45	C₂×#10596831	—	2	6	no	—
46	#41×C₇₃	—	2	4	no	—
47	C₂×Q₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×D₄×C₇₃	C₄⋊C₄	1	3	yes	—
49	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
50	C₂²×C₄×C₇₃	C₂×D₄	1	3	yes	—
51	C₂²×C₄×C₇₃	C₂×Q₈	1	3	yes	—
52	C₂×C₈×C₇₃	—	2	6	no	—
53	C₂×C₈×C₇₃	—	4	3	no	—
54	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₃	C₂²×C₄	1	3	yes	—
56	C₂×C₈×C₇₃	C₂²×C₄	1	3	yes	—
57	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
58	C₄⋊C₄×C₇₃	C₄⋊C₄	1	3	yes	—
59	C₄⋊C₄×C₇₃	C₄⋊C₄	1	3	yes	—
60	C₂²⋊C₄×C₇₃	C₄⋊C₄	1	6	yes	—
61	C₄²×C₇₃	C₂²×C₄	1	3	yes	—
62	C₄²×C₇₃	C₂×D₄	1	3	yes	—
63	C₄²×C₇₃	C₂×Q₈	1	3	yes	—
64	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
65	C₂³×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
66	Q₈×C₇₃	#108	1	4	yes	—
67	D₄×C₇₃	—	2	6	no	—
68	C₂×C₄×C₇₃	—	2	3	no	—
69	C₂×C₄×C₇₃	—	2	3	no	—
70	C₂×C₄×C₇₃	—	2	6	no	—
71	C₂×C₄×C₇₃	—	2	6	no	—
72	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
73	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	3	yes	—
75	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
78	C₈×C₇₃	—	2	3	no	—
79	C₈×C₇₃	—	2	1	no	—
80	C₈×C₇₃	—	2	3	no	—
81	C₈×C₇₃	#131	1	2	yes	—
82	#10596831	—	4	2	no	—
83	#10596831	—	4	6	no	—
84	C₂²×C₇₃	—	2	3	no	—
85	C₂²×C₇₃	—	2	3	no	—
86	C₂²×C₇₃	—	2	3	no	—
87	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
88	C₄×C₇₃	—	2	3	no	—
89	C₄×C₇₃	—	2	3	no	—
90	C₄×C₇₃	—	2	1	no	—
91	C₄×C₇₃	C₂×#108	1	6	yes	—
92	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
93	C₄×C₇₃	#568	1	1	yes	characteristic
94	C₄×C₇₃	#577	1	2	yes	—
95	#17835	—	73	1	no	maximal, Sylow
96	C₂×C₇₃	—	4	3	no	—
97	C₂×C₇₃	#2906	1	1	yes	characteristic
98	C₂×C₇₃	#2912	1	1	yes	characteristic
99	C₂×C₇₃	C₂²×#108	1	1	yes	characteristic
100	#2991	—	73	2	no	—
101	C₈×#41	D₇₃	1	1	yes	characteristic
102	#1574	—	73	6	no	—
103	#1566	—	73	3	no	—
104	#1564	—	73	3	no	—
105	C₇₃	#17835	1	1	yes	characteristic, complemented, Sylow
106	C₄×#41	C₂×D₇₃	1	1	yes	characteristic
107	C₈×Q₈	—	146	2	no	—
108	C₈×Q₈	Dic₇₃	1	1	yes	characteristic
109	C₈×Q₈	Dic₇₃	1	1	yes	characteristic
110	#436	—	146	6	no	—
111	C₈×D₄	Dic₇₃	1	3	yes	—
112	C₈×D₄	Dic₇₃	1	3	yes	—
113	#431	—	73	6	no	—
114	#431	C₂×D₇₃	1	3	yes	—
115	#423	—	73	6	no	—
116	C₂×C₄×C₈	C₂×D₇₃	1	3	yes	—
117	#335	—	73	6	no	—
118	#334	—	73	1	no	—
119	#334	—	73	3	no	—
120	#334	—	73	6	no	—
121	C₂×#41	Dic₁₄₆	1	1	yes	characteristic
122	C₂×M₁₆	—	146	6	no	—
123	C₂²×C₈	C₂×Dic₇₃	1	3	yes	—
124	C₂²×C₈	C₂×Dic₇₃	1	3	yes	—
125	C₄×Q₈	D₂₉₂	1	1	yes	characteristic
126	C₄×Q₈	C₂×Dic₇₃	1	1	yes	characteristic
127	C₄×D₄	D₂₉₂	1	3	yes	—
128	C₄×D₄	C₂×Dic₇₃	1	3	yes	—
129	#117	Dic₁₄₆	1	3	yes	—
130	C₂×C₄²	C₂²×D₇₃	1	3	yes	—
131	C₄⋊C₈	—	73	6	no	—
132	C₄⋊C₈	—	73	2	no	—
133	C₄⋊C₈	—	146	6	no	—
134	C₄⋊C₈	C₂×Dic₇₃	1	3	yes	—
135	C₄⋊C₈	C₂×Dic₇₃	1	3	yes	—
136	C₂²⋊C₈	—	146	6	no	—
137	C₂²⋊C₈	C₂×Dic₇₃	1	6	yes	—
138	C₄×C₈	—	146	6	no	—
139	C₄×C₈	C₂×Dic₇₃	1	3	yes	—
140	C₄×C₈	C₂×Dic₇₃	1	3	yes	—
141	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
142	C₄×C₈	C₂²×D₇₃	1	3	yes	—
143	#41	—	2	4	no	—
144	C₂×Q₈	C₄⋊Dic₇₃	1	1	yes	characteristic
145	C₂×D₄	C₄⋊Dic₇₃	1	3	yes	—
146	C₂²×C₄	C₂²×Dic₇₃	1	3	yes	—
147	C₂²×C₄	C₂×D₂₉₂	1	3	yes	—
148	C₂²×C₄	C₂×Dic₁₄₆	1	3	yes	—
149	M₁₆	—	584	6	no	—
150	C₂×C₈	—	2	6	no	—
151	C₂×C₈	—	4	3	no	—
152	C₂×C₈	—	146	2	no	—
153	C₂×C₈	—	146	6	no	—
154	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
155	C₂×C₈	C₂²×Dic₇₃	1	3	yes	—
156	C₂×C₈	C₂²×Dic₇₃	1	3	yes	—
157	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
158	C₄⋊C₄	C₄⋊Dic₇₃	1	3	yes	—
159	C₄⋊C₄	C₄⋊Dic₇₃	1	3	yes	—
160	C₂²⋊C₄	C₄⋊Dic₇₃	1	6	yes	—
161	C₄²	C₂²×Dic₇₃	1	3	yes	—
162	C₄²	C₂×D₂₉₂	1	3	yes	—
163	C₄²	C₂×Dic₁₄₆	1	3	yes	—
164	C₄²	C₂³×D₇₃	1	1	yes	characteristic
165	C₂³	C₂×C₄⋊Dic₇₃	1	3	yes	—
166	Q₈	#14556527	1	4	yes	—
167	D₄	—	2	6	no	—
168	C₂×C₄	—	2	6	no	—
169	C₂×C₄	—	2	6	no	—
170	C₂×C₄	—	2	3	no	—
171	C₂×C₄	—	2	3	no	—
172	C₂×C₄	C₂×C₄⋊Dic₇₃	1	3	yes	—
173	C₂×C₄	C₂×C₄⋊Dic₇₃	1	3	yes	—
174	C₂×C₄	C₂×C₄⋊Dic₇₃	1	3	yes	—
175	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
176	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
177	C₂×C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
178	C₈	—	2	3	no	—
179	C₈	—	2	1	no	—
180	C₈	—	2	3	no	—
181	C₈	—	292	6	no	—
182	C₈	—	292	2	no	—
183	C₈	#14556598	1	2	yes	—
184	C₂²	—	2	3	no	—
185	C₂²	—	2	3	no	—
186	C₂²	—	2	3	no	—
187	C₂²	C₂²×C₄⋊Dic₇₃	1	1	yes	characteristic
188	C₄	—	2	3	no	—
189	C₄	—	2	3	no	—
190	C₄	—	2	1	no	—
191	C₄	C₂×#14556527	1	6	yes	—
192	C₄	C₂×#14556598	1	1	yes	characteristic
193	C₄	#17989354	1	1	yes	characteristic
194	C₄	#17989399	1	2	yes	—
195	C₂	—	4	3	no	—
196	C₂	#22618499	1	1	yes	characteristic
197	C₂	#22618526	1	1	yes	characteristic
198	C₂	C₂²×#14556527	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₈×#41×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₇₃	Q₈
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	C₈×#41	D₇₃
hypocenter	C₇₃	#17835
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. #22608720	C₂	semidirect
3. #22608687	C₂	semidirect
4. #22608679	C₂	semidirect
105. C₇₃	#17835	semidirect

Maximal subgroups

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

subgroup	index
#22619063	2
#22608720	2
#22608687	2
#22608679	2
C₈×#41×C₇₃	2
#17835	73