Subgroups — #34012685

Order 18688.

326 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626270	C₂	1	1	yes	maximal, characteristic
2	#22626139	C₂	1	2	yes	maximal
3	#22625733	C₂	1	1	yes	maximal, characteristic
4	#22624984	C₂	1	2	yes	maximal
5	#346×D₇₃	C₂	1	1	yes	maximal, characteristic
6	#17989643	C₂²	1	2	yes	—
7	D₇₃⋊#108	C₄	1	1	yes	characteristic
8	#17989597	C₄	1	1	yes	characteristic
9	#17989550	C₂²	1	2	yes	—
10	#17989510	C₄	1	2	yes	complemented
11	#17989509	C₄	1	2	yes	complemented
12	#17989474	—	2	2	no	—
13	C₁₆×#64408	—	2	2	no	—
14	#17989458	C₄	1	2	yes	—
15	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
17	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
18	#17988363	C₄	1	1	yes	characteristic, complemented
19	#17988362	C₄	1	1	yes	characteristic, complemented
20	#17988362	C₄	1	1	yes	characteristic, complemented
21	#346×C₇₃	C₄	1	1	yes	characteristic, complemented
22	D₇₃⋊M₁₆	C₂×C₄	1	2	yes	—
23	#14556670	C₂×C₄	1	2	yes	—
24	#14556666	—	2	2	no	—
25	#14556665	C₂×C₄	1	4	yes	—
26	#14556664	—	2	2	no	—
27	#14556662	C₈	1	2	yes	—
28	#14556661	C₈	1	2	yes	—
29	#14556660	C₈	1	2	yes	—
30	#14556659	C₈	1	2	yes	—
31	#14556645	—	2	2	no	—
32	C₈×#64408	—	2	2	no	—
33	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
35	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
36	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
37	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
38	C₁₆×D₇₃	—	2	1	no	—
39	C₁₆×D₇₃	—	2	1	no	—
40	C₁₆×D₇₃	D₄	1	2	yes	—
41	C₁₆×D₇₃	Q₈	1	2	yes	—
42	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
43	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
44	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
48	#11924038	C₂×C₈	1	4	yes	—
49	#11924037	—	2	2	no	—
50	#11924036	—	2	2	no	—
51	C₄⋊#64408	C₂×C₈	1	4	yes	—
52	C₂²⋊#64408	—	2	2	no	—
53	C₄×#64408	—	2	2	no	—
54	#11924031	—	2	4	no	—
55	C₂×C₄×D₇₃	—	2	1	no	—
56	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×#10596831	C₄²	1	1	yes	characteristic
58	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
59	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
60	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
61	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
62	C₈×D₇₃	C₂×C₈	1	1	yes	characteristic
63	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
64	C₈×D₇₃	C₂×Q₈	1	1	yes	characteristic
65	C₈×D₇₃	#41	1	1	yes	characteristic
66	C₈×D₇₃	#41	1	1	yes	characteristic
67	C₄×Dic₇₃	—	2	1	no	—
68	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
69	#11924003	—	2	1	no	—
70	#11924003	C₄⋊C₄	1	2	yes	—
71	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
72	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₇₃	C₄²	1	1	yes	characteristic
74	C₁₆×C₇₃	—	2	1	no	—
75	C₁₆×C₇₃	C₄⋊C₄	1	2	yes	—
76	C₂²×D₇₃	#131	1	1	yes	characteristic
77	C₂×#64408	—	2	4	no	—
78	C₂×C₄×C₇₃	—	2	1	no	—
79	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
80	C₂×Dic₇₃	—	2	1	no	—
81	C₂×Dic₇₃	#131	1	1	yes	characteristic
82	C₄×D₇₃	—	4	2	no	—
83	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
84	C₄×D₇₃	C₄×Q₈	1	1	yes	characteristic
85	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
86	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
87	#10596833	—	2	4	no	—
88	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
89	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
90	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₈×C₇₃	#117	1	1	yes	characteristic
92	#10596831	C₄×C₈	1	1	yes	characteristic
93	#10596831	C₄×C₈	1	1	yes	characteristic
94	#10596831	C₄×D₄	1	1	yes	characteristic
95	#10596831	C₄×Q₈	1	1	yes	characteristic
96	C₂²×C₇₃	#405	1	1	yes	characteristic
97	C₂×D₇₃	—	2	2	no	—
98	C₂×D₇₃	C₈×D₄	1	1	yes	characteristic
99	C₂×D₇₃	C₈×Q₈	1	1	yes	characteristic
100	#64408	—	4	4	no	—
101	C₄×C₇₃	—	4	1	no	—
102	C₄×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
103	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
104	Dic₇₃	—	4	1	no	—
105	Dic₇₃	C₈×D₄	1	1	yes	characteristic
106	Dic₇₃	C₈×Q₈	1	1	yes	characteristic
107	#12665	—	73	1	no	maximal, Sylow
108	C₂×C₇₃	—	2	1	no	—
109	C₂×C₇₃	C₈×C₄⋊C₄	1	1	yes	characteristic
110	D₇₃	—	2	1	no	—
111	D₇₃	#2171	1	2	yes	—
112	C₂×#364	—	73	1	no	—
113	#2108	—	73	2	no	—
114	C₂×#346	—	73	1	no	—
115	#1772	—	73	1	no	—
116	#1379	—	73	2	no	—
117	C₇₃	#12665	1	1	yes	characteristic, complemented, Sylow
118	C₂×M₃₂	—	73	2	no	—
119	C₂²×C₁₆	—	73	1	no	—
120	C₂²×C₁₆	—	73	1	no	—
121	C₂×#108	—	73	1	no	—
122	#427	—	73	1	no	—
123	#405	—	73	2	no	—
124	C₂×C₄×C₈	—	73	1	no	—
125	#364	—	73	2	no	—
126	#364	—	73	2	no	—
127	#346	—	73	1	no	—
128	#346	—	73	1	no	—
129	#346	—	73	1	no	—
130	#346	—	146	2	no	—
131	#346	#64408	1	1	yes	characteristic, complemented
132	C₄×C₁₆	—	146	2	no	—
133	#339	—	73	2	no	—
134	C₂×M₁₆	—	73	2	no	—
135	C₂²×C₈	—	73	1	no	—
136	C₂²×C₈	—	73	1	no	—
137	#117	—	73	2	no	—
138	C₂×C₄²	—	73	1	no	—
139	M₃₂	—	73	4	no	—
140	M₃₂	—	146	2	no	—
141	C₂×C₁₆	—	73	1	no	—
142	C₂×C₁₆	—	73	2	no	—
143	C₂×C₁₆	—	73	2	no	—
144	C₂×C₁₆	—	73	1	no	—
145	C₂×C₁₆	—	146	1	no	—
146	C₂×C₁₆	—	146	1	no	—
147	C₂×C₁₆	—	146	2	no	—
148	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
149	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
150	#108	—	73	2	no	—
151	#108	—	73	2	no	—
152	#107	—	73	2	no	—
153	#106	—	73	2	no	—
154	#97	—	146	2	no	—
155	C₄×C₈	—	73	1	no	—
156	C₄×C₈	—	73	1	no	—
157	C₄×C₈	—	73	1	no	—
158	C₄×C₈	—	146	2	no	—
159	C₄×C₈	C₂×#64408	1	1	yes	characteristic
160	C₂²×C₄	—	73	1	no	—
161	C₂²×C₄	—	146	1	no	—
162	M₁₆	—	73	4	no	—
163	M₁₆	—	146	2	no	—
164	C₂×C₈	—	73	1	no	—
165	C₂×C₈	—	73	1	no	—
166	C₂×C₈	—	73	1	no	—
167	C₂×C₈	—	73	1	no	—
168	C₂×C₈	—	73	1	no	—
169	C₂×C₈	—	73	1	no	—
170	C₂×C₈	—	73	1	no	—
171	C₂×C₈	—	73	1	no	—
172	C₂×C₈	—	73	1	no	—
173	C₂×C₈	—	73	1	no	—
174	C₂×C₈	—	146	2	no	—
175	C₂×C₈	C₄×#64408	1	1	yes	characteristic
176	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
177	C₄⋊C₄	—	73	4	no	—
178	C₂²⋊C₄	—	146	2	no	—
179	C₄²	—	73	1	no	—
180	C₄²	—	146	1	no	—
181	C₄²	—	146	2	no	—
182	C₄²	C₄×#64408	1	1	yes	characteristic
183	C₁₆	—	2	1	no	—
184	C₁₆	—	73	2	no	—
185	C₁₆	—	146	4	no	—
186	C₁₆	—	146	1	no	—
187	C₁₆	C₄⋊#64408	1	2	yes	—
188	C₂³	—	73	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	292	2	no	—
191	C₂×C₄	—	73	1	no	—
192	C₂×C₄	—	73	1	no	—
193	C₂×C₄	—	73	1	no	—
194	C₂×C₄	—	73	1	no	—
195	C₂×C₄	—	73	1	no	—
196	C₂×C₄	—	146	4	no	—
197	C₂×C₄	—	146	1	no	—
198	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
199	C₈	—	73	1	no	—
200	C₈	—	73	1	no	—
201	C₈	—	73	1	no	—
202	C₈	—	73	1	no	—
203	C₈	—	146	4	no	—
204	C₈	C₈×#64408	1	1	yes	characteristic
205	C₈	C₈×#64408	1	1	yes	characteristic
206	C₈	C₂×C₄⋊#64408	1	1	yes	characteristic
207	C₈	#14556670	1	1	yes	characteristic
208	C₂²	—	73	1	no	—
209	C₂²	—	73	1	no	—
210	C₂²	—	146	2	no	—
211	C₂²	#17989550	1	1	yes	characteristic
212	C₄	—	4	1	no	—
213	C₄	—	292	1	no	—
214	C₄	—	292	4	no	—
215	C₄	—	73	1	no	—
216	C₄	—	73	1	no	—
217	C₄	C₄×C₄⋊#64408	1	1	yes	characteristic
218	C₄	C₂×C₈×#64408	1	1	yes	characteristic
219	C₂	—	2	1	no	—
220	C₂	—	73	2	no	—
221	C₂	—	146	1	no	—
222	C₂	C₈×C₄⋊#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×C₄⋊#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×#64408
Fitting	#346×C₇₃	C₄
socle	C₂×C₇₃	C₈×C₄⋊C₄
prime radical	C₂²×D₇₃	#131
cosocle	C₂×C₈×D₇₃	C₂³
hypercenter	#346	#64408
hypocenter	C₇₃	#12665
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #346

Complements

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

N	H ≅ G/N	type
10. #17989510	C₄	semidirect
11. #17989509	C₄	semidirect
18. #17988363	C₄	semidirect
19. #17988362	C₄	semidirect
20. #17988362	C₄	semidirect
21. #346×C₇₃	C₄	semidirect
117. C₇₃	#12665	semidirect
131. #346	#64408	semidirect

Maximal subgroups

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

subgroup	index
#22626270	2
#22626139	2
#22625733	2
#22624984	2
#346×D₇₃	2
#12665	73