Subgroups — #34003938

Order 18688.

470 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	C₄×C₄⋊#10596833	C₂	1	2	yes	maximal
2	C₂×#17989429	C₂	1	2	yes	maximal
3	#22624753	C₂	1	1	yes	maximal, characteristic
4	#22624753	C₂	1	1	yes	maximal, characteristic
5	#97×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#10596833	C₄	1	4	yes	—
7	C₂×C₄×#10596833	C₂²	1	2	yes	—
8	C₂×C₄×#10596833	C₂²	1	2	yes	—
9	#17989453	C₄	1	4	yes	—
10	#17989429	—	2	4	no	—
11	C₂×#14556502	C₄	1	1	yes	characteristic
12	C₂×#14556501	C₄	1	1	yes	characteristic
13	C₂×#14556501	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
16	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
17	C₂×#97×C₇₃	C₄	1	1	yes	characteristic
18	C₂²×#10596833	C₂×C₄	1	4	yes	—
19	C₂²×#10596833	C₂×C₄	1	4	yes	—
20	C₄⋊#10596833	—	2	8	no	—
21	C₄×#10596833	—	2	2	no	—
22	C₄×#10596833	—	2	2	no	—
23	C₄×#10596833	—	2	2	no	—
24	C₄×#10596833	—	2	2	no	—
25	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
26	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
28	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
31	#14556502	—	2	2	no	—
32	#14556501	—	2	2	no	—
33	#14556501	—	2	2	no	—
34	C₈×Dic₇₃	—	2	2	no	—
35	C₈×Dic₇₃	D₄	1	2	yes	—
36	C₈×Dic₇₃	Q₈	1	2	yes	—
37	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
40	#97×C₇₃	—	2	2	no	—
41	C₂×#10596833	—	2	4	no	—
42	C₂×#10596833	—	2	4	no	—
43	C₂×#10596833	—	2	4	no	—
44	C₂×#10596833	—	2	4	no	—
45	C₂×#10596833	—	2	4	no	—
46	C₂×#10596833	M₁₆	1	8	yes	—
47	C₂²×Dic₇₃	C₄²	1	1	yes	characteristic
48	C₂²×Dic₇₃	C₄²	1	1	yes	characteristic
49	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
50	C₂×#10596831	—	2	1	no	—
51	C₂×#10596831	—	2	1	no	—
52	C₂×#10596831	—	2	1	no	—
53	C₂×#10596831	—	2	1	no	—
54	C₂×#10596831	C₄⋊C₄	1	2	yes	—
55	C₂×#10596831	M₁₆	1	2	yes	—
56	C₄×Dic₇₃	—	2	1	no	—
57	C₄×Dic₇₃	—	2	1	no	—
58	C₄×Dic₇₃	—	2	4	no	—
59	C₄×Dic₇₃	—	2	4	no	—
60	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
61	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
62	C₄×Dic₇₃	#41	1	2	yes	—
63	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
64	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₇₃	—	2	1	no	—
66	C₂×C₈×C₇₃	—	2	1	no	—
67	C₂×C₈×C₇₃	—	2	1	no	—
68	C₂×C₈×C₇₃	—	2	1	no	—
69	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
70	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
71	C₄²×C₇₃	M₁₆	1	2	yes	—
72	C₄²×C₇₃	M₁₆	1	2	yes	—
73	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₇₃	—	2	2	no	—
75	C₂×C₄×C₇₃	—	2	2	no	—
76	C₂×C₄×C₇₃	#97	1	4	yes	—
77	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
83	C₂×Dic₇₃	—	2	2	no	—
84	C₂×Dic₇₃	—	2	1	no	—
85	C₂×Dic₇₃	—	2	1	no	—
86	C₂×Dic₇₃	—	2	2	no	—
87	C₂×Dic₇₃	C₄×D₄	1	2	yes	—
88	C₂×Dic₇₃	C₄×Q₈	1	2	yes	—
89	C₂×Dic₇₃	C₂×M₁₆	1	4	yes	—
90	C₂×Dic₇₃	#131	1	4	yes	—
91	#10596833	—	2	8	no	—
92	#10596833	—	4	4	no	—
93	C₈×C₇₃	—	2	2	no	—
94	C₈×C₇₃	—	2	2	no	—
95	#10596831	—	2	2	no	—
96	#10596831	—	2	2	no	—
97	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
99	C₂²×C₇₃	C₄×M₁₆	1	2	yes	—
100	C₂²×C₇₃	C₄×M₁₆	1	1	yes	characteristic
101	C₂²×C₇₃	#405	1	2	yes	—
102	C₄×C₇₃	—	2	4	no	—
103	C₄×C₇₃	#423	1	1	yes	characteristic
104	C₄×C₇₃	#423	1	1	yes	characteristic
105	C₄×C₇₃	#432	1	1	yes	characteristic
106	C₄×C₇₃	#432	1	1	yes	characteristic
107	Dic₇₃	—	4	1	no	—
108	Dic₇₃	—	4	1	no	—
109	Dic₇₃	#435	1	4	yes	—
110	Dic₇₃	#446	1	4	yes	—
111	#8263	—	73	1	no	maximal, Sylow
112	C₂×C₇₃	#1451	1	2	yes	—
113	C₂×C₇₃	#1453	1	1	yes	characteristic
114	C₂×C₇₃	#1455	1	1	yes	characteristic
115	C₂×C₇₃	#1767	1	1	yes	characteristic
116	C₂×C₇₃	#1771	1	2	yes	—
117	C₄×C₄⋊C₈	—	73	2	no	—
118	C₄×#97	—	73	1	no	—
119	C₂×#322	—	73	2	no	—
120	#1273	—	73	1	no	—
121	#1273	—	73	1	no	—
122	C₇₃	#8263	1	1	yes	characteristic, complemented, Sylow
123	C₂×C₄⋊C₈	—	73	4	no	—
124	C₂×#97	—	73	1	no	—
125	C₂×#97	—	73	1	no	—
126	C₂×#97	—	73	1	no	—
127	C₂×#97	#64408	1	1	yes	characteristic
128	C₂×C₄×C₈	—	73	1	no	—
129	C₂×C₄×C₈	—	73	2	no	—
130	C₂×C₄×C₈	—	73	2	no	—
131	C₂×C₄×C₈	—	73	1	no	—
132	C₄³	—	73	1	no	—
133	#336	—	73	4	no	—
134	#322	—	146	4	no	—
135	C₂²×C₈	—	73	4	no	—
136	C₂²×C₈	—	73	1	no	—
137	C₂²×C₈	—	73	1	no	—
138	C₂²×C₈	—	73	4	no	—
139	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
140	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
141	C₂×C₄²	—	73	1	no	—
142	C₂×C₄²	—	73	2	no	—
143	C₂×C₄²	—	73	2	no	—
144	C₂×C₄²	—	73	1	no	—
145	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
146	C₄⋊C₈	—	146	8	no	—
147	#97	—	2	2	no	—
148	#97	—	146	2	no	—
149	#97	—	146	2	no	—
150	#97	—	146	2	no	—
151	C₄×C₈	—	73	2	no	—
152	C₄×C₈	—	73	2	no	—
153	C₄×C₈	—	146	2	no	—
154	C₄×C₈	—	146	2	no	—
155	C₄×C₈	—	146	2	no	—
156	C₄×C₈	—	146	2	no	—
157	C₄×C₈	—	146	2	no	—
158	C₂²×C₄	—	73	2	no	—
159	C₂²×C₄	—	73	1	no	—
160	C₂²×C₄	—	73	1	no	—
161	C₂²×C₄	C₄×#64408	1	2	yes	—
162	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	73	8	no	—
168	C₂×C₈	—	73	2	no	—
169	C₂×C₈	—	73	2	no	—
170	C₂×C₈	—	146	4	no	—
171	C₂×C₈	—	146	4	no	—
172	C₂×C₈	—	146	4	no	—
173	C₂×C₈	—	146	4	no	—
174	C₂×C₈	—	146	4	no	—
175	C₂×C₈	—	146	1	no	—
176	C₂×C₈	—	146	1	no	—
177	C₂×C₈	—	146	1	no	—
178	C₂×C₈	—	146	1	no	—
179	C₂×C₈	C₄⋊#64408	1	2	yes	—
180	C₂×C₈	#11924038	1	2	yes	—
181	C₄²	—	73	1	no	—
182	C₄²	—	73	1	no	—
183	C₄²	—	73	2	no	—
184	C₄²	—	146	4	no	—
185	C₄²	—	146	4	no	—
186	C₄²	—	146	1	no	—
187	C₄²	—	146	1	no	—
188	C₄²	#11924037	1	2	yes	—
189	C₄²	#11924038	1	2	yes	—
190	C₂³	C₂×C₄×#64408	1	1	yes	characteristic
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	73	2	no	—
194	C₂×C₄	—	73	2	no	—
195	C₂×C₄	—	73	4	no	—
196	C₂×C₄	—	73	4	no	—
197	C₂×C₄	—	146	2	no	—
198	C₂×C₄	—	146	1	no	—
199	C₂×C₄	—	146	1	no	—
200	C₂×C₄	—	146	2	no	—
201	C₂×C₄	#14556646	1	4	yes	—
202	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
203	C₂×C₄	#14556670	1	1	yes	characteristic
204	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
205	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
206	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
207	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₈	—	292	4	no	—
211	C₈	—	146	8	no	—
212	C₈	—	146	2	no	—
213	C₈	—	146	2	no	—
214	C₂²	C₄×C₄⋊#64408	1	1	yes	characteristic
215	C₂²	C₂×#14556646	1	1	yes	characteristic
216	C₂²	M₁₆×#64408	1	2	yes	—
217	C₂²	C₄×#11924038	1	1	yes	characteristic
218	C₂²	#17989553	1	2	yes	—
219	C₄	—	2	4	no	—
220	C₄	—	292	1	no	—
221	C₄	—	292	1	no	—
222	C₄	—	73	4	no	—
223	C₄	—	73	4	no	—
224	C₄	#17989590	1	1	yes	characteristic
225	C₄	#17989591	1	1	yes	characteristic
226	C₄	#17989608	1	1	yes	characteristic
227	C₄	#17989609	1	1	yes	characteristic
228	C₂	#22625138	1	2	yes	—
229	C₂	#22625151	1	1	yes	characteristic
230	C₂	#22625155	1	1	yes	characteristic
231	C₂	#22625717	1	1	yes	characteristic
232	C₂	#22625729	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator, socle, prime radical	C₂³×C₇₃	C₂×C₄²
Fitting	C₂×#97×C₇₃	C₄
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×#97	#64408
hypocenter	C₇₃	#8263
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
122. C₇₃	#8263	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#10596833	2
C₂×#17989429	2
#22624753	2
#97×Dic₇₃	2
#8263	73