Subgroups — #64408⋊#504 #34034679

Order 18688.

834 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627182	C₂	1	2	yes	maximal
2	#22627094	C₂	1	6	yes	maximal, complemented
3	#64408⋊M₃₂	C₂	1	3	yes	maximal, complemented
4	C₂×#64408⋊C₁₆	C₂	1	3	yes	maximal, complemented
5	#131×#64408	C₂	1	1	yes	maximal, characteristic
6	#17989709	C₄	1	4	yes	complemented
7	#17989708	C₄	1	12	yes	complemented
8	#17989706	C₂²	1	6	yes	—
9	C₂×#14556686	C₂²	1	6	yes	—
10	#17989682	C₂²	1	6	yes	—
11	#64408⋊C₁₆	C₂²	1	1	yes	characteristic
12	#64408⋊C₁₆	C₂²	1	3	yes	—
13	#64408⋊C₁₆	C₂²	1	6	yes	—
14	D₇₃⋊#131	C₂²	1	1	yes	characteristic
15	#41×#64408	C₄	1	1	yes	characteristic
16	#17989553	C₂²	1	3	yes	—
17	#17989550	C₄	1	3	yes	—
18	M₁₆×#64408	C₄	1	3	yes	—
19	C₂×C₈×#64408	C₂²	1	3	yes	—
20	#131×D₇₃	C₄	1	1	yes	characteristic
21	#14556688	C₂×C₄	1	12	yes	complemented
22	#14556687	C₂×C₄	1	12	yes	complemented
23	#14556686	C₂×C₄	1	12	yes	complemented
24	#14556686	C₂³	1	2	yes	—
25	#14556686	C₂³	1	6	yes	—
26	C₂×#11924040	C₂×C₄	1	12	yes	complemented
27	#14556679	C₂×C₄	1	2	yes	—
28	#14556678	C₂×C₄	1	6	yes	—
29	D₇₃⋊M₁₆	C₂×C₄	1	3	yes	—
30	C₂×#11924036	C₂³	1	3	yes	—
31	Q₈×#64408	C₈	1	2	yes	—
32	D₄×#64408	C₈	1	6	yes	—
33	#14556670	C₂×C₄	1	3	yes	—
34	C₂×C₄×#64408	C₂×C₄	1	3	yes	—
35	#14556645	C₂×C₄	1	6	yes	—
36	C₈×#64408	C₂×C₄	1	3	yes	—
37	C₈×#64408	C₂×C₄	1	3	yes	—
38	C₈×#64408	C₂³	1	1	yes	characteristic
39	C₈×#64408	C₂³	1	3	yes	—
40	#41×D₇₃	C₂×C₄	1	1	yes	characteristic
41	#14556599	C₈	1	3	yes	—
42	#14556598	C₈	1	1	yes	characteristic
43	#14556597	C₈	1	3	yes	—
44	M₁₆×D₇₃	C₂×C₄	1	3	yes	—
45	C₂×C₈×D₇₃	C₂×C₄	1	3	yes	—
46	#131×C₇₃	C₈	1	1	yes	characteristic
47	#11924040	C₂²×C₄	1	4	yes	—
48	#11924040	C₂²×C₄	1	12	yes	—
49	C₂²×#64408	C₂×C₈	1	6	yes	—
50	#11924038	C₄²	1	6	yes	—
51	#11924037	C₄²	1	6	yes	—
52	#11924036	C₂²×C₄	1	3	yes	—
53	#11924036	C₂²×C₄	1	6	yes	—
54	#11924036	C₂⁴	1	1	yes	characteristic
55	C₂×#10596833	C₂²×C₄	1	6	yes	—
56	C₄⋊#64408	C₂×C₈	1	6	yes	—
57	C₂²⋊#64408	C₂×C₈	1	6	yes	—
58	C₄×#64408	C₂×C₈	1	6	yes	—
59	C₄×#64408	C₂²×C₄	1	1	yes	characteristic
60	C₄×#64408	C₂²×C₄	1	3	yes	—
61	#11924029	C₄²	1	1	yes	characteristic
62	#11924028	C₂×C₈	1	3	yes	—
63	#11924027	C₄²	1	3	yes	—
64	Q₈×D₇₃	C₂×C₈	1	1	yes	characteristic
65	D₄×D₇₃	C₂×C₈	1	3	yes	—
66	C₂×C₄×D₇₃	C₂²×C₄	1	3	yes	—
67	#11924012	C₂×C₈	1	3	yes	—
68	#11924011	C₂×C₈	1	6	yes	—
69	C₂×#10596831	C₂×C₈	1	3	yes	—
70	C₈×D₇₃	C₂×C₈	1	3	yes	—
71	C₈×D₇₃	C₂×C₈	1	3	yes	—
72	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
73	C₈×D₇₃	C₂²×C₄	1	3	yes	—
74	#41×C₇₃	C₂×C₈	1	1	yes	characteristic
75	M₁₆×C₇₃	C₂×C₈	1	3	yes	—
76	C₂×C₈×C₇₃	C₂×C₈	1	3	yes	—
77	C₂²×D₇₃	C₂²×C₈	1	3	yes	—
78	C₂×#64408	—	2	12	no	—
79	C₂×#64408	C₂²×C₈	1	6	yes	—
80	C₂×#64408	C₂²×C₈	1	2	yes	—
81	Q₈×C₇₃	C₄×C₈	1	1	yes	characteristic
82	D₄×C₇₃	C₄×C₈	1	3	yes	—
83	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
84	C₂²⋊D₇₃	C₄×C₈	1	6	yes	—
85	C₂×Dic₇₃	C₂×C₄²	1	3	yes	—
86	D₂₉₂	C₄×C₈	1	3	yes	—
87	C₄×D₇₃	C₂×C₄²	1	3	yes	—
88	C₄×D₇₃	C₂²×C₈	1	3	yes	—
89	C₄×D₇₃	C₂²×C₈	1	3	yes	—
90	C₄×D₇₃	C₂³×C₄	1	1	yes	characteristic
91	Dic₁₄₆	C₄×C₈	1	3	yes	—
92	#10596833	C₂×C₄²	1	6	yes	—
93	#10596833	C₂³×C₄	1	2	yes	—
94	C₈×C₇₃	C₂²×C₈	1	3	yes	—
95	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
96	#10596831	C₂²×C₈	1	1	yes	characteristic
97	#10596831	C₂²×C₈	1	3	yes	—
98	C₂²×C₇₃	C₂×C₄×C₈	1	3	yes	—
99	C₂×D₇₃	—	2	6	no	—
100	C₂×D₇₃	C₂×C₄×C₈	1	3	yes	—
101	C₂×D₇₃	C₂³×C₈	1	1	yes	characteristic
102	#64408	—	2	6	no	—
103	#64408	#504	1	4	yes	complemented
104	C₄×C₇₃	C₂×C₄×C₈	1	3	yes	—
105	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
106	Dic₇₃	C₂×C₄×C₈	1	3	yes	—
107	Dic₇₃	C₂²×C₄²	1	1	yes	characteristic
108	C₄×#504	—	73	1	no	maximal, Sylow
109	C₂×C₇₃	—	2	3	no	—
110	C₂×C₇₃	C₂²×C₄×C₈	1	1	yes	characteristic
111	D₇₃	—	2	3	no	—
112	D₇₃	C₂×#504	1	2	yes	—
113	C₂×#504	—	73	2	no	—
114	C₄×#131	—	73	1	no	—
115	#2108	—	73	6	no	—
116	C₄×M₃₂	—	73	3	no	—
117	C₂×C₄×C₁₆	—	73	3	no	—
118	C₇₃	C₄×#504	1	1	yes	characteristic, complemented, Sylow
119	C₂×#131	—	73	1	no	—
120	C₂×#131	—	73	1	no	—
121	C₄×#41	—	73	1	no	—
122	#504	—	73	12	no	—
123	#504	—	73	4	no	—
124	C₂×M₃₂	—	73	6	no	—
125	C₂²×C₁₆	—	73	6	no	—
126	#405	—	73	3	no	—
127	#405	—	73	3	no	—
128	C₄×M₁₆	—	73	3	no	—
129	C₂×C₄×C₈	—	73	3	no	—
130	#346	—	73	6	no	—
131	C₄×C₁₆	—	73	1	no	—
132	C₄×C₁₆	—	73	3	no	—
133	C₄×C₁₆	—	73	6	no	—
134	C₂×#41	—	73	1	no	—
135	#131	—	73	6	no	—
136	#131	—	73	2	no	—
137	#131	—	73	1	no	—
138	#131	—	73	3	no	—
139	#131	—	73	3	no	—
140	#131	#10596842	1	1	yes	characteristic
141	C₂×M₁₆	—	73	3	no	—
142	C₂×M₁₆	—	73	3	no	—
143	C₂²×C₈	—	73	3	no	—
144	C₂²×C₈	—	73	3	no	—
145	C₄×Q₈	—	73	2	no	—
146	C₄×D₄	—	73	6	no	—
147	#117	—	73	3	no	—
148	C₂×C₄²	—	73	3	no	—
149	M₃₂	—	73	12	no	—
150	M₃₂	—	73	12	no	—
151	C₂×C₁₆	—	73	12	no	—
152	C₂×C₁₆	—	73	12	no	—
153	C₂×C₁₆	—	73	2	no	—
154	C₂×C₁₆	—	73	6	no	—
155	#97	—	73	6	no	—
156	C₄×C₈	—	73	1	no	—
157	C₄×C₈	—	73	3	no	—
158	C₄×C₈	—	73	3	no	—
159	C₄×C₈	—	73	3	no	—
160	#41	—	73	1	no	—
161	#41	—	73	3	no	—
162	#41	—	73	3	no	—
163	#41	C₂×#10596842	1	1	yes	characteristic
164	C₂×Q₈	—	73	1	no	—
165	C₂×D₄	—	73	3	no	—
166	C₂²×C₄	—	73	3	no	—
167	C₂²×C₄	—	73	6	no	—
168	M₁₆	—	73	6	no	—
169	M₁₆	—	73	6	no	—
170	M₁₆	—	73	3	no	—
171	M₁₆	—	73	6	no	—
172	M₁₆	C₂×#10596842	1	3	yes	—
173	C₂×C₈	—	73	6	no	—
174	C₂×C₈	—	73	3	no	—
175	C₂×C₈	—	73	6	no	—
176	C₂×C₈	—	73	1	no	—
177	C₂×C₈	—	73	3	no	—
178	C₂×C₈	—	73	3	no	—
179	C₂×C₈	—	73	3	no	—
180	C₂×C₈	—	73	3	no	—
181	C₂×C₈	—	73	1	no	—
182	C₂×C₈	C₂×#10596842	1	3	yes	—
183	C₄⋊C₄	—	73	6	no	—
184	C₂²⋊C₄	—	73	6	no	—
185	C₄²	—	73	6	no	—
186	C₄²	—	73	1	no	—
187	C₄²	—	73	3	no	—
188	C₁₆	—	73	4	no	—
189	C₁₆	—	73	12	no	—
190	C₂³	—	73	3	no	—
191	Q₈	—	73	3	no	—
192	Q₈	C₄×#10596842	1	1	yes	characteristic
193	D₄	—	73	6	no	—
194	D₄	—	73	3	no	—
195	D₄	C₄×#10596842	1	3	yes	—
196	C₂×C₄	—	73	2	no	—
197	C₂×C₄	—	73	3	no	—
198	C₂×C₄	—	73	3	no	—
199	C₂×C₄	—	73	3	no	—
200	C₂×C₄	—	73	1	no	—
201	C₂×C₄	—	73	3	no	—
202	C₂×C₄	—	73	6	no	—
203	C₂×C₄	—	146	12	no	—
204	C₂×C₄	C₂²×#10596842	1	3	yes	—
205	C₈	—	73	2	no	—
206	C₈	—	73	6	no	—
207	C₈	—	73	1	no	—
208	C₈	—	73	3	no	—
209	C₈	C₂²×#10596842	1	3	yes	—
210	C₈	C₂²×#10596842	1	1	yes	characteristic
211	C₂²	—	73	1	no	—
212	C₂²	—	73	3	no	—
213	C₂²	—	146	6	no	—
214	C₂²	C₂×C₄×#10596842	1	3	yes	—
215	C₄	—	73	4	no	—
216	C₄	—	73	1	no	—
217	C₄	—	73	3	no	—
218	C₄	—	146	6	no	—
219	C₄	C₂×C₄×#10596842	1	3	yes	—
220	C₄	C₂³×#10596842	1	1	yes	characteristic
221	C₂	—	2	3	no	—
222	C₂	—	73	2	no	—
223	C₂	—	146	3	no	—
224	C₂	C₂²×C₄×#10596842	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×#10596842
commutator, socle	C₂×C₇₃	C₂²×C₄×C₈
Frattini	C₄	C₂³×#10596842
Fitting	#131×C₇₃	C₈
prime radical	#41×D₇₃	C₂×C₄
cosocle	#11924036	C₂⁴
hypercenter	#131	#10596842
hypocenter	C₇₃	C₄×#504
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ #131

Complements

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

N	H ≅ G/N	type
2. #22627094	C₂	semidirect
3. #64408⋊M₃₂	C₂	semidirect
4. C₂×#64408⋊C₁₆	C₂	semidirect
6. #17989709	C₄	semidirect
7. #17989708	C₄	semidirect
21. #14556688	C₂×C₄	semidirect
22. #14556687	C₂×C₄	semidirect
23. #14556686	C₂×C₄	semidirect
26. C₂×#11924040	C₂×C₄	semidirect
103. #64408	#504	semidirect
118. C₇₃	C₄×#504	semidirect

Maximal subgroups

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

subgroup	index
#22627182	2
#22627094	2
#64408⋊M₃₂	2
C₂×#64408⋊C₁₆	2
#131×#64408	2
C₄×#504	73