Subgroups — #34013879

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	#22626272	C₂	1	2	yes	maximal
2	C₂×#17989509	C₂	1	1	yes	maximal, characteristic
3	#22626137	C₂	1	1	yes	maximal, characteristic
4	#22624982	C₂	1	2	yes	maximal
5	#22609813	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556665	C₂²	1	2	yes	—
7	C₂×#14556665	C₂²	1	1	yes	characteristic
8	C₂×#14556665	C₂²	1	1	yes	characteristic
9	#17989509	—	2	4	no	—
10	#17989509	—	2	1	no	—
11	#17989509	—	2	1	no	—
12	#17989476	—	2	1	no	—
13	#17989473	—	2	1	no	—
14	#17988625	C₄	1	1	yes	characteristic
15	#17988545	C₂²	1	2	yes	—
16	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
17	#17988348	C₄	1	2	yes	—
18	C₂×#108×C₇₃	C₄	1	1	yes	characteristic
19	#14556665	—	2	2	no	—
20	#14556665	—	2	2	no	—
21	#14556665	—	2	1	no	—
22	#14556665	—	2	1	no	—
23	#14556665	D₄	1	2	yes	—
24	#14556665	Q₈	1	2	yes	—
25	C₂×#11924031	—	2	1	no	—
26	C₂×#11924031	—	2	1	no	—
27	C₂×#11924031	—	2	2	no	—
28	C₂²×#10596831	C₂³	1	1	yes	characteristic
29	#14556550	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
31	#14556525	—	2	1	no	—
32	#14556523	—	2	1	no	—
33	#14556501	—	2	2	no	—
34	C₈×Dic₇₃	—	2	2	no	—
35	C₈×Dic₇₃	D₄	1	1	yes	characteristic
36	C₈×Dic₇₃	D₄	1	1	yes	characteristic
37	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
38	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
39	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
41	#108×C₇₃	C₈	1	2	yes	—
42	#108×C₇₃	C₈	1	2	yes	—
43	#11924031	—	2	2	no	—
44	#11924031	—	2	2	no	—
45	#11924031	—	4	2	no	—
46	C₂²×Dic₇₃	—	2	1	no	—
47	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
48	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
49	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
50	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
51	C₂×#10596831	#41	1	1	yes	characteristic
52	C₂×#10596831	#41	1	1	yes	characteristic
53	C₄⋊Dic₇₃	—	2	2	no	—
54	C₂²⋊Dic₇₃	—	2	2	no	—
55	C₄×Dic₇₃	—	2	1	no	—
56	C₄×Dic₇₃	—	2	2	no	—
57	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
58	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
60	M₁₆×C₇₃	—	2	2	no	—
61	M₁₆×C₇₃	C₂×C₈	1	4	yes	—
62	C₂×C₈×C₇₃	—	2	2	no	—
63	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
66	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
67	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
68	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
69	C₂³×C₇₃	#131	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
76	C₂×Dic₇₃	—	2	1	no	—
77	C₂×Dic₇₃	—	2	1	no	—
78	C₂×Dic₇₃	—	4	2	no	—
79	C₂×Dic₇₃	—	4	2	no	—
80	C₈×C₇₃	—	2	4	no	—
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	C₄⋊C₈	1	1	yes	characteristic
85	#10596831	#121	1	1	yes	characteristic
86	#10596831	C₂²⋊Q₈	1	1	yes	characteristic
87	#10596831	#125	1	1	yes	characteristic
88	#10596831	C₄⋊Q₈	1	1	yes	characteristic
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	#389	1	1	yes	characteristic
94	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
95	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
96	C₄×C₇₃	#446	1	1	yes	characteristic
97	Dic₇₃	—	2	2	no	—
98	Dic₇₃	—	4	1	no	—
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	—	73	1	no	—
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	—	73	2	no	—
114	C₂×#108	#64408	1	1	yes	characteristic
115	#427	—	73	1	no	—
116	#405	—	73	2	no	—
117	C₂×C₄×C₈	—	73	1	no	—
118	#364	—	146	1	no	—
119	#364	—	146	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	C₂×#64408	1	2	yes	—
125	C₂²×C₈	—	73	1	no	—
126	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
127	#117	—	73	2	no	—
128	C₂×C₄²	—	73	1	no	—
129	M₃₂	—	73	2	no	—
130	M₃₂	—	73	2	no	—
131	M₃₂	—	146	2	no	—
132	M₃₂	—	146	2	no	—
133	M₃₂	—	146	1	no	—
134	M₃₂	—	146	1	no	—
135	C₂×C₁₆	—	146	2	no	—
136	C₂×C₁₆	—	146	1	no	—
137	C₂×C₁₆	—	146	1	no	—
138	#108	#10596833	1	2	yes	—
139	#108	#10596833	1	2	yes	—
140	#107	—	146	1	no	—
141	#106	—	146	1	no	—
142	#97	—	146	2	no	—
143	C₄×C₈	—	73	1	no	—
144	C₄×C₈	—	73	1	no	—
145	C₄×C₈	—	73	1	no	—
146	C₄×C₈	—	73	1	no	—
147	C₄×C₈	—	146	2	no	—
148	C₂²×C₄	—	146	1	no	—
149	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	C₂×#10596833	1	4	yes	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	73	1	no	—
154	C₂×C₈	—	73	1	no	—
155	C₂×C₈	—	73	1	no	—
156	C₂×C₈	—	73	1	no	—
157	C₂×C₈	—	73	1	no	—
158	C₂×C₈	—	73	1	no	—
159	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
160	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
161	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
162	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
163	C₂×C₈	#11924038	1	1	yes	characteristic
164	C₂×C₈	#11924038	1	1	yes	characteristic
165	C₄⋊C₄	—	146	2	no	—
166	C₂²⋊C₄	—	146	2	no	—
167	C₄²	—	73	1	no	—
168	C₄²	—	73	1	no	—
169	C₄²	—	146	2	no	—
170	C₄²	—	146	1	no	—
171	C₁₆	—	292	2	no	—
172	C₁₆	—	146	2	no	—
173	C₁₆	—	146	2	no	—
174	C₂³	#14556679	1	1	yes	characteristic
175	C₂×C₄	—	292	2	no	—
176	C₂×C₄	—	292	2	no	—
177	C₂×C₄	—	146	1	no	—
178	C₂×C₄	—	146	1	no	—
179	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
180	C₂×C₄	D₄×#64408	1	1	yes	characteristic
181	C₂×C₄	Q₈×#64408	1	1	yes	characteristic
182	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
183	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
184	C₂×C₄	#14556678	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	—	73	1	no	—
187	C₈	—	73	1	no	—
188	C₈	—	73	1	no	—
189	C₈	—	73	1	no	—
190	C₈	C₄⋊#10596833	1	1	yes	characteristic
191	C₈	C₄⋊#10596833	1	1	yes	characteristic
192	C₈	C₄⋊#10596833	1	1	yes	characteristic
193	C₈	C₄⋊#10596833	1	1	yes	characteristic
194	C₂²	—	4	1	no	—
195	C₂²	#17989594	1	1	yes	characteristic
196	C₂²	#17989621	1	1	yes	characteristic
197	C₂²	Q₈×#10596833	1	1	yes	characteristic
198	C₄	—	292	1	no	—
199	C₄	—	584	2	no	—
200	C₄	—	146	2	no	—
201	C₄	#17989538	1	1	yes	characteristic
202	C₄	C₂×C₄⋊#10596833	1	1	yes	characteristic
203	C₄	D₄×#10596833	1	1	yes	characteristic
204	C₄	#17989637	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	#22626028	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17989538
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×#108×C₇₃	C₄
socle	C₂×C₇₃	#1939
prime radical	C₂³×C₇₃	#131
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂×#108	#64408
hypocenter	C₇₃	#13164
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

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
#22626272	2
C₂×#17989509	2
#22626137	2
#22624982	2
#22609813	2
#13164	73