Subgroups — #346×#64408 #34012077

Order 18688.

510 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	#22626132	C₂	1	1	yes	maximal, characteristic
2	C₂×#17989474	C₂	1	2	yes	maximal
3	C₂×C₁₆×#64408	C₂	1	2	yes	maximal
4	C₄×C₈×#64408	C₂	1	1	yes	maximal, characteristic
5	#346×D₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556664	C₂²	1	2	yes	—
7	C₄×#11924036	C₄	1	1	yes	characteristic
8	C₂×C₈×#64408	C₄	1	1	yes	characteristic
9	C₂×C₈×#64408	C₄	1	1	yes	characteristic
10	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
11	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
12	C₄²×#64408	C₄	1	1	yes	characteristic
13	#17989476	C₄	1	2	yes	complemented
14	#17989475	C₄	1	2	yes	complemented
15	#17989474	C₄	1	8	yes	complemented
16	C₁₆×#64408	C₄	1	8	yes	complemented
17	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
18	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
19	#17988363	C₄	1	1	yes	characteristic, complemented
20	#17988362	C₄	1	2	yes	complemented
21	#346×C₇₃	C₄	1	1	yes	characteristic, complemented
22	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
23	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×#64408	C₈	1	4	yes	—
25	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
27	#14556664	C₂×C₄	1	4	yes	—
28	#14556664	C₂×C₄	1	4	yes	—
29	C₂×#11924031	C₂×C₄	1	4	yes	—
30	C₄×#10596833	C₈	1	4	yes	—
31	C₈×#64408	C₈	1	4	yes	—
32	C₈×#64408	C₈	1	4	yes	—
33	C₈×#64408	C₂×C₄	1	4	yes	—
34	C₈×#64408	C₂×C₄	1	4	yes	—
35	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
37	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×#11924003	C₂×C₄	1	2	yes	—
39	C₁₆×D₇₃	C₂×C₄	1	4	yes	—
40	C₁₆×D₇₃	C₂×C₄	1	4	yes	—
41	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
42	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
43	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×#64408	C₂×C₈	1	2	yes	—
47	C₂²×#64408	C₂×C₈	1	2	yes	—
48	#11924036	C₄²	1	2	yes	—
49	#11924036	C₄²	1	2	yes	—
50	#11924036	C₂×C₈	1	2	yes	—
51	#11924036	C₂×C₈	1	2	yes	—
52	C₂×#10596833	C₂×C₈	1	2	yes	—
53	C₂×#10596833	C₂×C₈	1	2	yes	—
54	C₄×#64408	—	2	8	no	—
55	C₄×#64408	C₄²	1	2	yes	—
56	C₄×#64408	C₄²	1	2	yes	—
57	C₄×#64408	C₂×C₈	1	2	yes	—
58	C₄×#64408	C₂×C₈	1	2	yes	—
59	#11924031	C₄²	1	8	yes	complemented
60	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
61	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×#10596831	C₄²	1	1	yes	characteristic
63	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
64	C₈×D₇₃	C₂×C₈	1	2	yes	—
65	C₈×D₇₃	C₂×C₈	1	2	yes	—
66	C₈×D₇₃	C₂²×C₄	1	2	yes	—
67	C₈×D₇₃	C₂²×C₄	1	2	yes	—
68	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
69	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
70	#11924003	C₄²	1	4	yes	complemented
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	C₄²	1	4	yes	complemented
75	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
76	C₂×#64408	—	2	4	no	—
77	C₂×#64408	C₄×C₈	1	4	yes	—
78	C₂×#64408	C₄×C₈	1	4	yes	—
79	C₂×#64408	M₃₂	1	8	yes	—
80	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
81	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
82	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
83	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
84	C₄×D₇₃	—	2	4	no	—
85	C₄×D₇₃	C₂×C₄²	1	1	yes	characteristic
86	C₄×D₇₃	C₂×C₄²	1	1	yes	characteristic
87	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
88	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
89	#10596833	C₄×C₈	1	4	yes	—
90	#10596833	C₄×C₈	1	4	yes	—
91	C₈×C₇₃	C₄×C₈	1	2	yes	—
92	C₈×C₇₃	C₂×C₄²	1	2	yes	—
93	#10596831	C₄×C₈	1	2	yes	—
94	#10596831	C₂×C₄²	1	2	yes	—
95	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
96	C₂×D₇₃	C₂×C₄×C₈	1	1	yes	characteristic
97	C₂×D₇₃	C₂×C₄×C₈	1	1	yes	characteristic
98	C₂×D₇₃	C₂×M₃₂	1	4	yes	—
99	#64408	—	2	4	no	—
100	#64408	#346	1	8	yes	direct product factor
101	C₄×C₇₃	—	2	2	no	—
102	C₄×C₇₃	C₄³	1	1	yes	characteristic
103	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
104	Dic₇₃	—	2	2	no	—
105	Dic₇₃	C₂×C₄×C₈	1	1	yes	characteristic
106	Dic₇₃	C₂×C₄×C₈	1	1	yes	characteristic
107	C₄×#346	—	73	1	no	maximal, Sylow
108	C₂×C₇₃	C₄²×C₈	1	1	yes	characteristic
109	C₂×C₇₃	C₄×M₃₂	1	2	yes	—
110	D₇₃	C₂×#346	1	2	yes	—
111	D₇₃	C₄×M₃₂	1	2	yes	—
112	C₂×#346	—	73	2	no	—
113	C₂×#346	—	73	1	no	—
114	C₂×#346	—	73	1	no	—
115	C₂×C₄×C₁₆	—	73	2	no	—
116	C₄²×C₈	—	73	1	no	—
117	C₇₃	C₄×#346	1	1	yes	characteristic, complemented, Sylow
118	C₂²×C₁₆	—	73	2	no	—
119	C₂²×C₁₆	—	73	2	no	—
120	C₂×C₄×C₈	—	73	1	no	—
121	C₂×C₄×C₈	—	73	1	no	—
122	C₂×C₄×C₈	—	73	1	no	—
123	C₂×C₄×C₈	—	73	1	no	—
124	C₂×C₄×C₈	—	73	1	no	—
125	C₂×C₄×C₈	—	73	1	no	—
126	C₄³	—	73	1	no	—
127	#346	—	73	8	no	—
128	#346	—	73	2	no	—
129	#346	—	73	2	no	—
130	#346	—	73	2	no	—
131	#346	—	73	1	no	—
132	#346	#64408	1	1	yes	characteristic, direct product factor
133	C₄×C₁₆	—	73	8	no	—
134	C₂²×C₈	—	73	1	no	—
135	C₂²×C₈	—	73	1	no	—
136	C₂²×C₈	—	73	1	no	—
137	C₂²×C₈	—	73	1	no	—
138	C₂×C₄²	—	73	1	no	—
139	C₂×C₄²	—	73	1	no	—
140	C₂×C₄²	—	73	1	no	—
141	C₂×C₄²	—	73	4	no	—
142	C₂×C₁₆	—	73	4	no	—
143	C₂×C₁₆	—	73	4	no	—
144	C₂×C₁₆	—	73	4	no	—
145	C₂×C₁₆	—	73	2	no	—
146	C₂×C₁₆	—	73	4	no	—
147	C₂×C₁₆	—	73	4	no	—
148	C₂×C₁₆	C₂×#64408	1	2	yes	—
149	C₄×C₈	—	73	4	no	—
150	C₄×C₈	—	73	4	no	—
151	C₄×C₈	—	73	4	no	—
152	C₄×C₈	—	73	4	no	—
153	C₄×C₈	—	73	4	no	—
154	C₄×C₈	—	73	1	no	—
155	C₄×C₈	—	73	1	no	—
156	C₄×C₈	—	73	1	no	—
157	C₄×C₈	C₂×#64408	1	1	yes	characteristic
158	C₂²×C₄	—	73	2	no	—
159	C₂²×C₄	—	73	1	no	—
160	C₂²×C₄	—	73	2	no	—
161	C₂²×C₄	—	73	2	no	—
162	C₂×C₈	—	73	2	no	—
163	C₂×C₈	—	73	2	no	—
164	C₂×C₈	—	73	2	no	—
165	C₂×C₈	—	73	2	no	—
166	C₂×C₈	—	73	2	no	—
167	C₂×C₈	—	73	2	no	—
168	C₂×C₈	—	73	1	no	—
169	C₂×C₈	—	73	1	no	—
170	C₂×C₈	—	73	2	no	—
171	C₂×C₈	—	73	2	no	—
172	C₂×C₈	—	73	2	no	—
173	C₂×C₈	—	73	2	no	—
174	C₂×C₈	C₄×#64408	1	1	yes	characteristic
175	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
176	C₄²	—	73	2	no	—
177	C₄²	—	73	2	no	—
178	C₄²	—	73	2	no	—
179	C₄²	—	73	2	no	—
180	C₄²	—	73	2	no	—
181	C₄²	—	73	1	no	—
182	C₄²	—	146	8	no	—
183	C₄²	C₄×#64408	1	1	yes	characteristic
184	C₁₆	—	73	8	no	—
185	C₁₆	—	73	4	no	—
186	C₁₆	C₄×#64408	1	4	yes	complemented
187	C₂³	—	73	1	no	—
188	C₂×C₄	—	73	4	no	—
189	C₂×C₄	—	73	8	no	—
190	C₂×C₄	—	73	1	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	4	no	—
196	C₂×C₄	—	73	2	no	—
197	C₂×C₄	—	146	4	no	—
198	C₂×C₄	—	146	4	no	—
199	C₂×C₄	C₈×#64408	1	2	yes	—
200	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
201	C₈	—	73	4	no	—
202	C₈	—	73	4	no	—
203	C₈	—	73	2	no	—
204	C₈	—	73	2	no	—
205	C₈	C₈×#64408	1	2	yes	—
206	C₈	C₂×C₄×#64408	1	2	yes	—
207	C₂²	—	73	1	no	—
208	C₂²	—	73	1	no	—
209	C₂²	—	73	4	no	—
210	C₂²	C₂×C₈×#64408	1	1	yes	characteristic
211	C₄	—	2	2	no	—
212	C₄	—	73	8	no	—
213	C₄	—	73	1	no	—
214	C₄	—	73	1	no	—
215	C₄	—	146	4	no	—
216	C₄	—	146	2	no	—
217	C₄	C₄²×#64408	1	1	yes	characteristic
218	C₄	C₂×C₈×#64408	1	1	yes	characteristic
219	C₂	—	73	2	no	—
220	C₂	—	73	2	no	—
221	C₂	C₄×C₈×#64408	1	1	yes	characteristic
222	C₂	M₃₂×#64408	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#64408
commutator	C₂×C₇₃	C₄²×C₈
Fitting	#346×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×C₈
prime radical	C₂²×D₇₃	C₂²×C₈
cosocle	C₂×C₈×D₇₃	C₂³
hypercenter	#346	#64408
hypocenter	C₇₃	C₄×#346
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
13. #17989476	C₄	semidirect
14. #17989475	C₄	semidirect
15. #17989474	C₄	semidirect
16. C₁₆×#64408	C₄	semidirect
19. #17988363	C₄	semidirect
20. #17988362	C₄	semidirect
21. #346×C₇₃	C₄	semidirect
59. #11924031	C₄²	semidirect
70. #11924003	C₄²	semidirect
74. C₁₆×C₇₃	C₄²	semidirect
100. #64408	#346	direct (N × H)
117. C₇₃	C₄×#346	semidirect
132. #346	#64408	direct (N × H)
186. C₁₆	C₄×#64408	semidirect

Maximal subgroups

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

subgroup	index
#22626132	2
C₂×#17989474	2
C₂×C₁₆×#64408	2
C₄×C₈×#64408	2
#346×D₇₃	2
C₄×#346	73