Subgroups — #32977306

Order 18688.

342 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22608679	C₂	1	1	yes	maximal, characteristic
2	#22608218	C₂	1	1	yes	maximal, characteristic
3	M₁₆×#10596831	C₂	1	1	yes	maximal, characteristic
4	C₂×#17988309	C₂	1	1	yes	maximal, characteristic
5	#22607769	C₂	1	1	yes	maximal, characteristic
6	#22607769	C₂	1	1	yes	maximal, characteristic
7	C₄×#108×C₇₃	C₂	1	1	yes	maximal, characteristic
8	#17988640	C₂²	1	1	yes	characteristic
9	#17988640	C₂²	1	1	yes	characteristic
10	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
11	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
12	#17988340	C₄	1	2	yes	—
13	#17988338	C₄	1	2	yes	—
14	#17988309	—	2	1	no	—
15	#17988309	—	2	1	no	—
16	#17988309	C₄	1	2	yes	—
17	#17988309	C₄	1	2	yes	—
18	#17988308	—	2	1	no	—
19	C₈×#10596831	—	2	1	no	—
20	C₂×#108×C₇₃	C₄	1	2	yes	—
21	C₄×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
22	C₄×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
24	#341×C₇₃	C₄	1	2	yes	—
25	C₂²×#10596831	—	2	1	no	—
26	C₂²×#10596831	Q₈	1	2	yes	—
27	C₄⋊#10596831	C₂×C₄	1	2	yes	—
28	C₄⋊#10596831	C₂×C₄	1	2	yes	—
29	C₂²⋊#10596831	—	2	1	no	—
30	C₂²⋊#10596831	—	2	1	no	—
31	C₄×#10596831	—	2	1	no	—
32	C₄×#10596831	—	2	1	no	—
33	C₄×#10596831	—	2	1	no	—
34	C₄×#10596831	C₂×C₄	1	2	yes	—
35	C₄×#10596831	C₂×C₄	1	2	yes	—
36	C₄×#10596831	D₄	1	1	yes	characteristic
37	C₄×#10596831	D₄	1	1	yes	characteristic
38	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
39	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
43	#108×C₇₃	—	2	4	no	—
44	#97×C₇₃	—	2	1	no	—
45	#97×C₇₃	—	2	1	no	—
46	C₄×C₈×C₇₃	—	2	1	no	—
47	C₄×C₈×C₇₃	—	2	1	no	—
48	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂×#10596831	—	2	4	no	—
53	C₂×#10596831	—	2	2	no	—
54	C₂×#10596831	—	2	2	no	—
55	C₂×#10596831	—	2	2	no	—
56	C₂×#10596831	—	4	2	no	—
57	C₂×#10596831	C₄⋊C₄	1	2	yes	—
58	C₂×#10596831	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
62	M₁₆×C₇₃	—	2	2	no	—
63	M₁₆×C₇₃	—	2	2	no	—
64	M₁₆×C₇₃	—	2	2	no	—
65	M₁₆×C₇₃	M₁₆	1	4	yes	—
66	C₂×C₈×C₇₃	—	2	2	no	—
67	C₂×C₈×C₇₃	—	2	2	no	—
68	C₂×C₈×C₇₃	—	2	2	no	—
69	C₂×C₈×C₇₃	—	2	2	no	—
70	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
71	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
72	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
73	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
74	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₇₃	#41	1	1	yes	characteristic
78	C₂³×C₇₃	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₇₃	—	2	2	no	—
80	C₂×C₄×C₇₃	—	2	2	no	—
81	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
87	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₇₃	#131	1	2	yes	—
89	C₈×C₇₃	—	2	1	no	—
90	C₈×C₇₃	—	2	1	no	—
91	C₈×C₇₃	—	2	1	no	—
92	C₈×C₇₃	—	2	1	no	—
93	C₈×C₇₃	—	2	4	no	—
94	C₈×C₇₃	—	4	2	no	—
95	#10596831	—	2	2	no	—
96	#10596831	—	4	2	no	—
97	#10596831	—	4	2	no	—
98	#10596831	—	4	2	no	—
99	#10596831	#108	1	4	yes	complemented
100	C₂²×C₇₃	—	2	1	no	—
101	C₂²×C₇₃	—	2	1	no	—
102	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₇₃	#446	1	2	yes	—
104	C₄×C₇₃	—	2	1	no	—
105	C₄×C₇₃	—	2	1	no	—
106	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
107	C₄×C₇₃	#405	1	1	yes	characteristic
108	C₄×C₇₃	C₂×#108	1	2	yes	—
109	C₄×C₇₃	#435	1	2	yes	—
110	C₄×C₇₃	#444	1	2	yes	—
111	#8297	—	73	1	no	maximal, Sylow
112	C₂×C₇₃	—	2	2	no	—
113	C₂×C₇₃	#1771	1	1	yes	characteristic
114	C₂×C₇₃	#1773	1	1	yes	characteristic
115	C₂×C₇₃	C₄×#108	1	1	yes	characteristic
116	C₄×#108	D₇₃	1	1	yes	characteristic
117	#1564	—	73	1	no	—
118	#1451	—	73	1	no	—
119	C₈×M₁₆	—	73	1	no	—
120	C₂×#322	—	73	1	no	—
121	#1278	—	73	1	no	—
122	#1278	—	73	1	no	—
123	C₇₃	#8297	1	1	yes	characteristic, complemented, Sylow
124	#431	—	73	1	no	—
125	#431	—	73	1	no	—
126	C₂×#108	Dic₇₃	1	2	yes	—
127	C₄×M₁₆	C₂×D₇₃	1	1	yes	characteristic
128	C₄×M₁₆	C₂×D₇₃	1	1	yes	characteristic
129	C₂×C₄×C₈	—	73	1	no	—
130	C₂×C₄×C₈	—	73	1	no	—
131	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
132	#341	Dic₇₃	1	2	yes	—
133	#335	—	73	2	no	—
134	#334	—	73	2	no	—
135	#322	—	73	2	no	—
136	#322	—	73	2	no	—
137	#322	—	146	1	no	—
138	#322	—	146	1	no	—
139	#322	—	146	1	no	—
140	C₈²	—	146	1	no	—
141	C₂×M₁₆	C₂×Dic₇₃	1	2	yes	—
142	C₂×M₁₆	C₂×Dic₇₃	1	2	yes	—
143	C₂²×C₈	—	73	2	no	—
144	C₂²×C₈	—	146	1	no	—
145	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
146	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
147	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
148	#108	—	2	4	no	—
149	C₄⋊C₈	—	73	2	no	—
150	C₄⋊C₈	—	73	2	no	—
151	C₂²⋊C₈	—	146	1	no	—
152	C₂²⋊C₈	—	146	1	no	—
153	#97	—	2	1	no	—
154	#97	—	2	1	no	—
155	C₄×C₈	—	2	1	no	—
156	C₄×C₈	—	2	1	no	—
157	C₄×C₈	—	73	2	no	—
158	C₄×C₈	—	73	2	no	—
159	C₄×C₈	—	73	1	no	—
160	C₄×C₈	—	73	1	no	—
161	C₄×C₈	—	146	1	no	—
162	C₄×C₈	—	146	1	no	—
163	C₄×C₈	—	146	1	no	—
164	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
165	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
166	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
167	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
168	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
169	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
170	C₂²×C₄	#11924029	1	1	yes	characteristic
171	M₁₆	—	2	2	no	—
172	M₁₆	—	2	2	no	—
173	M₁₆	—	2	2	no	—
174	M₁₆	#11924012	1	4	yes	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	292	2	no	—
180	C₂×C₈	—	73	2	no	—
181	C₂×C₈	—	73	2	no	—
182	C₂×C₈	—	146	2	no	—
183	C₂×C₈	—	146	2	no	—
184	C₂×C₈	—	146	2	no	—
185	C₂×C₈	—	146	4	no	—
186	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
187	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
188	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
189	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
190	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
191	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
192	C₄²	D₄×D₇₃	1	1	yes	characteristic
193	C₄²	#11924027	1	1	yes	characteristic
194	C₂³	Q₈×Dic₇₃	1	1	yes	characteristic
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
198	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
199	C₂×C₄	#14556549	1	1	yes	characteristic
200	C₂×C₄	#14556553	1	1	yes	characteristic
201	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
202	C₂×C₄	#14556556	1	1	yes	characteristic
203	C₂×C₄	C₂×#11924012	1	2	yes	—
204	C₂×C₄	#14556598	1	2	yes	—
205	C₈	—	2	4	no	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	4	2	no	—
211	C₈	—	292	2	no	—
212	C₈	—	292	2	no	—
213	C₈	—	292	2	no	—
214	C₈	—	73	4	no	—
215	C₈	—	146	2	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	C₄⋊C₄×Dic₇₃	1	1	yes	characteristic
219	C₂²	#17988718	1	2	yes	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	C₄×#11924012	1	1	yes	characteristic
223	C₄	#17988543	1	1	yes	characteristic
224	C₄	#108×D₇₃	1	2	yes	—
225	C₄	#17988659	1	2	yes	—
226	C₄	#17988708	1	2	yes	—
227	C₂	—	2	2	no	—
228	C₂	#22609806	1	1	yes	characteristic
229	C₂	#22609818	1	1	yes	characteristic
230	C₂	#108×Dic₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₄×#108×C₇₃	C₂
socle	C₂²×C₇₃	C₄×C₄⋊C₄
prime radical	C₂³×C₇₃	C₄×Q₈
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₄×#108	D₇₃
hypocenter	C₇₃	#8297
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
99. #10596831	#108	semidirect
123. C₇₃	#8297	semidirect

Maximal subgroups

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

subgroup	index
#22608679	2
#22608218	2
M₁₆×#10596831	2
C₂×#17988309	2
#22607769	2
C₄×#108×C₇₃	2
#8297	73