Subgroups — #34003934

Order 18688.

494 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556658	C₂	1	2	yes	maximal
2	C₂×#17989428	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#10596833	C₂	1	1	yes	maximal, characteristic
4	#22624753	C₂	1	2	yes	maximal
5	#97×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556658	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	#17989428	—	2	2	no	—
11	C₈×#10596833	—	2	2	no	—
12	C₂×#14556502	C₄	1	1	yes	characteristic
13	C₂×#14556501	C₄	1	1	yes	characteristic
14	C₂×#14556501	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
17	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
18	C₂×#97×C₇₃	C₄	1	1	yes	characteristic
19	C₂²×#10596833	C₂×C₄	1	4	yes	—
20	C₂²×#10596833	C₂×C₄	1	4	yes	—
21	#14556658	—	2	8	no	—
22	C₄×#10596833	—	2	2	no	—
23	C₄×#10596833	—	2	2	no	—
24	C₄×#10596833	—	2	2	no	—
25	C₄×#10596833	—	2	2	no	—
26	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
27	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
32	#14556502	—	2	2	no	—
33	#14556501	C₈	1	4	yes	complemented
34	#14556501	C₈	1	4	yes	complemented
35	C₈×Dic₇₃	—	2	2	no	—
36	C₈×Dic₇₃	D₄	1	2	yes	—
37	C₈×Dic₇₃	Q₈	1	2	yes	—
38	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
41	#97×C₇₃	—	2	2	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	—	2	4	no	—
47	C₂×#10596833	M₁₆	1	8	yes	—
48	C₂²×Dic₇₃	C₄²	1	2	yes	—
49	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
50	C₂×#10596831	—	2	1	no	—
51	C₂×#10596831	—	2	1	no	—
52	C₂×#10596831	C₄⋊C₄	1	2	yes	—
53	C₂×#10596831	C₂×C₈	1	2	yes	—
54	C₂×#10596831	C₂×C₈	1	2	yes	—
55	C₂×#10596831	M₁₆	1	2	yes	—
56	C₄×Dic₇₃	—	2	4	no	—
57	C₄×Dic₇₃	—	2	4	no	—
58	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
59	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
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	1	yes	characteristic
64	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
65	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₇₃	—	2	1	no	—
67	C₂×C₈×C₇₃	—	2	1	no	—
68	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
69	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
70	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
71	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
72	C₄²×C₇₃	—	2	1	no	—
73	C₄²×C₇₃	—	2	1	no	—
74	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₇₃	—	2	1	no	—
76	C₂×C₄×C₇₃	—	2	1	no	—
77	C₂×C₄×C₇₃	—	2	2	no	—
78	C₂×C₄×C₇₃	—	2	2	no	—
79	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
81	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
85	C₂×Dic₇₃	—	2	2	no	—
86	C₂×Dic₇₃	—	2	2	no	—
87	C₂×Dic₇₃	C₄×C₈	1	4	yes	—
88	C₂×Dic₇₃	C₄×D₄	1	2	yes	—
89	C₂×Dic₇₃	C₄×Q₈	1	2	yes	—
90	C₂×Dic₇₃	C₂×M₁₆	1	4	yes	—
91	C₂×Dic₇₃	#131	1	4	yes	—
92	#10596833	—	2	8	no	—
93	#10596833	—	4	4	no	—
94	C₈×C₇₃	—	2	2	no	—
95	C₈×C₇₃	C₄⋊C₈	1	4	yes	complemented
96	#10596831	—	2	2	no	—
97	#10596831	C₄⋊C₈	1	4	yes	complemented
98	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
100	C₂²×C₇₃	C₄×M₁₆	1	2	yes	—
101	C₂²×C₇₃	C₄×M₁₆	1	1	yes	characteristic
102	C₂²×C₇₃	#405	1	2	yes	—
103	C₄×C₇₃	—	4	1	no	—
104	C₄×C₇₃	—	4	1	no	—
105	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
106	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
107	C₄×C₇₃	#431	1	1	yes	characteristic
108	C₄×C₇₃	#431	1	1	yes	characteristic
109	Dic₇₃	—	2	4	no	—
110	Dic₇₃	#435	1	4	yes	—
111	Dic₇₃	#446	1	4	yes	—
112	#8262	—	73	1	no	maximal, Sylow
113	C₂×C₇₃	C₈×M₁₆	1	2	yes	—
114	C₂×C₇₃	#1453	1	1	yes	characteristic
115	C₂×C₇₃	#1455	1	1	yes	characteristic
116	C₂×C₇₃	C₄×C₄⋊C₈	1	1	yes	characteristic
117	C₂×C₇₃	#1771	1	2	yes	—
118	C₄×C₄⋊C₈	—	73	2	no	—
119	C₄×#97	—	73	1	no	—
120	C₂×#322	—	73	1	no	—
121	C₂×C₈²	—	73	1	no	—
122	#1273	—	73	2	no	—
123	C₇₃	#8262	1	1	yes	characteristic, complemented, Sylow
124	C₂×C₄⋊C₈	—	73	4	no	—
125	C₂×#97	—	73	1	no	—
126	C₂×#97	—	73	1	no	—
127	C₂×#97	—	73	1	no	—
128	C₂×#97	#64408	1	1	yes	characteristic
129	C₂×C₄×C₈	—	73	1	no	—
130	C₂×C₄×C₈	—	73	2	no	—
131	C₂×C₄×C₈	—	73	2	no	—
132	C₂×C₄×C₈	—	73	1	no	—
133	C₄³	—	73	1	no	—
134	#336	—	73	4	no	—
135	#322	—	146	2	no	—
136	C₈²	—	146	2	no	—
137	C₂²×C₈	—	73	1	no	—
138	C₂²×C₈	—	73	4	no	—
139	C₂²×C₈	—	73	1	no	—
140	C₂²×C₈	—	73	4	no	—
141	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
142	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
143	C₂×C₄²	—	73	1	no	—
144	C₂×C₄²	—	73	1	no	—
145	C₂×C₄²	—	73	2	no	—
146	C₂×C₄²	—	73	2	no	—
147	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
148	C₄⋊C₈	—	146	8	no	—
149	#97	—	2	2	no	—
150	#97	—	73	4	no	—
151	#97	—	73	4	no	—
152	#97	—	146	2	no	—
153	C₄×C₈	—	73	2	no	—
154	C₄×C₈	—	73	2	no	—
155	C₄×C₈	—	146	2	no	—
156	C₄×C₈	—	146	2	no	—
157	C₄×C₈	—	146	2	no	—
158	C₄×C₈	—	146	2	no	—
159	C₄×C₈	—	146	2	no	—
160	C₂²×C₄	—	73	2	no	—
161	C₂²×C₄	—	73	2	no	—
162	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
163	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
164	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	73	2	no	—
168	C₂×C₈	—	73	2	no	—
169	C₂×C₈	—	73	2	no	—
170	C₂×C₈	—	73	8	no	—
171	C₂×C₈	—	73	2	no	—
172	C₂×C₈	—	146	4	no	—
173	C₂×C₈	—	146	4	no	—
174	C₂×C₈	—	146	4	no	—
175	C₂×C₈	—	146	4	no	—
176	C₂×C₈	—	146	4	no	—
177	C₂×C₈	—	146	1	no	—
178	C₂×C₈	—	146	1	no	—
179	C₂×C₈	C₄⋊#64408	1	2	yes	—
180	C₂×C₈	#11924036	1	2	yes	—
181	C₂×C₈	#11924036	1	2	yes	—
182	C₂×C₈	#11924037	1	2	yes	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	73	2	no	—
186	C₄²	—	73	2	no	—
187	C₄²	—	73	1	no	—
188	C₄²	—	73	1	no	—
189	C₄²	—	73	2	no	—
190	C₄²	—	146	4	no	—
191	C₄²	—	146	4	no	—
192	C₂³	C₂×C₄×#64408	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	73	2	no	—
198	C₂×C₄	—	73	2	no	—
199	C₂×C₄	—	73	4	no	—
200	C₂×C₄	—	73	4	no	—
201	C₂×C₄	—	73	4	no	—
202	C₂×C₄	—	146	2	no	—
203	C₂×C₄	—	146	2	no	—
204	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
205	C₂×C₄	#14556670	1	1	yes	characteristic
206	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
207	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
208	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
209	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	292	4	no	—
212	C₈	—	73	4	no	—
213	C₈	—	146	8	no	—
214	C₈	—	146	2	no	—
215	C₈	#14556658	1	4	yes	complemented
216	C₂²	C₄×C₄⋊#64408	1	1	yes	characteristic
217	C₂²	C₄×#11924036	1	1	yes	characteristic
218	C₂²	M₁₆×#64408	1	2	yes	—
219	C₂²	C₄×#11924037	1	1	yes	characteristic
220	C₂²	#17989553	1	2	yes	—
221	C₄	—	4	1	no	—
222	C₄	—	4	1	no	—
223	C₄	—	73	4	no	—
224	C₄	—	73	4	no	—
225	C₄	—	146	4	no	—
226	C₄	#17989588	1	1	yes	characteristic
227	C₄	C₂×#14556658	1	1	yes	characteristic
228	C₄	#17989604	1	1	yes	characteristic
229	C₄	#17989605	1	1	yes	characteristic
230	C₂	#22625129	1	2	yes	—
231	C₂	#22625150	1	1	yes	characteristic
232	C₂	#22625154	1	1	yes	characteristic
233	C₂	C₄×#14556658	1	1	yes	characteristic
234	C₂	#22625729	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₂×#97×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₂×C₄²
cosocle	C₂×C₄×Dic₇₃	C₂³
hypercenter	C₂×#97	#64408
hypocenter	C₇₃	#8262
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
33. #14556501	C₈	semidirect
34. #14556501	C₈	semidirect
95. C₈×C₇₃	C₄⋊C₈	semidirect
97. #10596831	C₄⋊C₈	semidirect
123. C₇₃	#8262	semidirect
215. C₈	#14556658	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556658	2
C₂×#17989428	2
C₂×C₈×#10596833	2
#22624753	2
#97×Dic₇₃	2
#8262	73