Subgroups — #34034134

Order 18688.

398 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	#22627117	C₂	1	1	yes	maximal, characteristic
2	C₂×#64408⋊C₁₆	C₂	1	2	yes	maximal
3	C₄×#14556686	C₂	1	1	yes	maximal, characteristic
4	#22627002	C₂	1	2	yes	maximal
5	#22625724	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556686	C₂²	1	1	yes	characteristic
7	C₂×#14556686	C₂²	1	2	yes	—
8	C₂×#14556686	C₂²	1	1	yes	characteristic
9	#17989690	C₄	1	2	yes	complemented
10	C₄⋊#11924040	C₄	1	2	yes	complemented
11	#64408⋊C₁₆	—	2	4	no	—
12	C₄×#11924040	C₄	1	2	yes	complemented
13	C₄×#11924040	C₄	1	2	yes	complemented
14	C₄×#11924036	C₂²	1	1	yes	characteristic
15	C₂×C₈×#64408	C₂²	1	2	yes	—
16	C₄×C₄⋊#64408	C₄	1	1	yes	characteristic
17	#17989452	C₄	1	2	yes	—
18	C₄⋊C₈×D₇₃	C₄	1	1	yes	characteristic
19	#14556686	—	2	2	no	—
20	#14556686	—	2	1	no	—
21	#14556686	—	2	2	no	—
22	#14556686	—	2	1	no	—
23	#14556686	D₄	1	2	yes	—
24	#14556686	Q₈	1	2	yes	—
25	C₂×#11924040	C₂×C₄	1	2	yes	—
26	C₂×#11924040	C₂×C₄	1	4	yes	—
27	C₂×#11924040	C₂×C₄	1	2	yes	—
28	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
29	C₂×#11924036	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊#64408	C₈	1	2	yes	—
31	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
32	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
33	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
34	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
35	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
36	C₈×#64408	—	2	2	no	—
37	C₈×#64408	—	2	2	no	—
38	(C₂×#64408)⋊C₄	C₈	1	2	yes	—
39	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
40	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
41	C₄⋊#10596831	C₈	1	1	yes	characteristic
42	#14556520	C₈	1	2	yes	—
43	C₄⋊C₈×C₇₃	C₈	1	1	yes	characteristic
44	#11924040	—	2	2	no	—
45	#11924040	—	2	4	no	—
46	#11924040	C₄⋊C₄	1	4	yes	complemented
47	C₂²×#64408	C₂×C₈	1	4	yes	—
48	#11924036	—	2	1	no	—
49	#11924036	—	2	1	no	—
50	#11924036	C₂×D₄	1	1	yes	characteristic
51	#11924036	C₂×Q₈	1	1	yes	characteristic
52	#11924036	#41	1	1	yes	characteristic
53	#11924036	#41	1	1	yes	characteristic
54	C₂×#10596833	C₄²	1	1	yes	characteristic
55	C₂×#10596833	C₄²	1	1	yes	characteristic
56	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
57	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
58	C₄⋊#64408	C₁₆	1	8	yes	complemented
59	C₄×#64408	—	2	2	no	—
60	C₄×#64408	—	2	2	no	—
61	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
62	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
63	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×#10596831	C₂×C₈	1	2	yes	—
65	C₈×D₇₃	—	2	2	no	—
66	C₈×D₇₃	—	2	2	no	—
67	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
68	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
69	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
70	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
71	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
72	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
73	C₂×#64408	—	2	4	no	—
74	C₂×#64408	—	2	4	no	—
75	C₂×#64408	C₂×C₁₆	1	8	yes	—
76	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
79	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
80	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
81	C₂×Dic₇₃	C₂×C₄²	1	1	yes	characteristic
82	C₄×D₇₃	—	2	2	no	—
83	C₄×D₇₃	C₂×C₁₆	1	4	yes	—
84	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
85	C₄×D₇₃	C₄×Q₈	1	1	yes	characteristic
86	C₄×D₇₃	#131	1	1	yes	characteristic
87	C₄×D₇₃	#131	1	1	yes	characteristic
88	#10596833	—	2	1	no	—
89	#10596833	—	2	1	no	—
90	#10596833	C₂×C₄⋊C₄	1	1	yes	characteristic
91	#10596833	#117	1	1	yes	characteristic
92	#10596833	C₄×D₄	1	1	yes	characteristic
93	#10596833	C₄×Q₈	1	1	yes	characteristic
94	C₈×C₇₃	—	2	2	no	—
95	#10596831	—	2	2	no	—
96	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
97	C₂×D₇₃	C₈×D₄	1	1	yes	characteristic
98	C₂×D₇₃	C₈×Q₈	1	1	yes	characteristic
99	C₂×D₇₃	C₂²×C₁₆	1	2	yes	—
100	C₂×D₇₃	#504	1	2	yes	—
101	#64408	—	2	8	no	—
102	C₄×C₇₃	—	2	1	no	—
103	C₄×C₇₃	C₄×C₁₆	1	2	yes	complemented
104	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
105	C₄×C₇₃	C₈×Q₈	1	1	yes	characteristic
106	Dic₇₃	—	2	1	no	—
107	Dic₇₃	C₄×C₁₆	1	2	yes	complemented
108	Dic₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
109	Dic₇₃	#405	1	1	yes	characteristic
110	C₁₆×C₄⋊C₄	—	73	1	no	maximal, Sylow
111	C₂×C₇₃	C₈×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₇₃	C₂×C₄×C₁₆	1	1	yes	characteristic
113	C₂×C₇₃	#2108	1	1	yes	characteristic
114	D₇₃	D₄×C₁₆	1	2	yes	—
115	D₇₃	Q₈×C₁₆	1	2	yes	—
116	C₂×C₄⋊C₁₆	—	73	1	no	—
117	C₂×C₄×C₁₆	—	73	1	no	—
118	C₂×C₄×C₁₆	—	73	2	no	—
119	C₈×C₄⋊C₄	—	73	1	no	—
120	#1374	—	73	2	no	—
121	C₇₃	C₁₆×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
122	C₂²×C₁₆	—	73	1	no	—
123	C₂²×C₁₆	—	73	1	no	—
124	C₂²×C₁₆	—	73	2	no	—
125	C₂×C₄⋊C₈	—	73	1	no	—
126	C₂×C₄×C₈	—	73	1	no	—
127	C₂×C₄×C₈	—	73	2	no	—
128	C₄×C₄⋊C₄	—	73	1	no	—
129	C₄⋊C₁₆	—	73	2	no	—
130	C₄⋊C₁₆	—	73	2	no	—
131	C₄×C₁₆	—	73	2	no	—
132	C₄×C₁₆	—	73	2	no	—
133	C₄×C₁₆	—	146	4	no	—
134	#336	—	73	2	no	—
135	C₂²×C₈	—	73	1	no	—
136	C₂²×C₈	—	73	1	no	—
137	C₂²×C₈	—	73	2	no	—
138	C₂×C₄⋊C₄	—	73	2	no	—
139	C₂×C₄²	—	73	1	no	—
140	C₂×C₄²	—	73	2	no	—
141	C₂×C₁₆	—	73	2	no	—
142	C₂×C₁₆	—	73	2	no	—
143	C₂×C₁₆	—	73	2	no	—
144	C₂×C₁₆	—	73	2	no	—
145	C₂×C₁₆	—	73	4	no	—
146	C₂×C₁₆	—	146	2	no	—
147	C₂×C₁₆	—	146	1	no	—
148	C₂×C₁₆	—	146	1	no	—
149	C₂×C₁₆	—	146	2	no	—
150	C₄⋊C₈	—	73	2	no	—
151	C₄⋊C₈	—	73	1	no	—
152	C₄⋊C₈	#10596842	1	1	yes	characteristic
153	C₄×C₈	—	73	1	no	—
154	C₄×C₈	—	73	1	no	—
155	C₄×C₈	—	73	1	no	—
156	C₄×C₈	—	73	1	no	—
157	C₄×C₈	—	146	2	no	—
158	C₄×C₈	—	146	2	no	—
159	#95	—	73	2	no	—
160	C₂²×C₄	—	73	1	no	—
161	C₂²×C₄	—	73	1	no	—
162	C₂²×C₄	—	73	4	no	—
163	C₂²×C₄	—	73	1	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	2	no	—
173	C₂×C₈	—	146	2	no	—
174	C₂×C₈	—	146	2	no	—
175	C₂×C₈	—	146	1	no	—
176	C₂×C₈	—	146	1	no	—
177	C₂×C₈	C₂×#10596842	1	2	yes	—
178	C₄⋊C₄	—	73	8	no	—
179	C₄²	—	73	1	no	—
180	C₄²	—	73	1	no	—
181	C₄²	—	73	1	no	—
182	C₄²	—	146	2	no	—
183	C₄²	—	146	2	no	—
184	C₄²	C₂×#10596842	1	1	yes	characteristic
185	C₁₆	—	73	4	no	—
186	C₁₆	—	146	4	no	—
187	C₁₆	—	146	2	no	—
188	C₂³	—	73	1	no	—
189	C₂×C₄	—	73	1	no	—
190	C₂×C₄	—	73	1	no	—
191	C₂×C₄	—	73	4	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	—	73	8	no	—
197	C₂×C₄	—	73	1	no	—
198	C₂×C₄	—	146	2	no	—
199	C₂×C₄	—	146	4	no	—
200	C₂×C₄	—	146	4	no	—
201	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
202	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
203	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	73	1	no	—
206	C₈	—	73	1	no	—
207	C₈	—	73	1	no	—
208	C₈	—	73	1	no	—
209	C₈	—	146	1	no	—
210	C₈	—	146	1	no	—
211	C₈	—	146	2	no	—
212	C₂²	—	73	2	no	—
213	C₂²	—	73	2	no	—
214	C₂²	—	73	1	no	—
215	C₂²	—	73	1	no	—
216	C₂²	C₂×C₄×#10596842	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	73	2	no	—
219	C₄	—	73	1	no	—
220	C₄	—	73	1	no	—
221	C₄	—	146	8	no	—
222	C₄	—	146	1	no	—
223	C₄	#64408⋊C₁₆	1	2	yes	complemented
224	C₄	D₄×#10596842	1	1	yes	characteristic
225	C₄	Q₈×#10596842	1	1	yes	characteristic
226	C₂	—	73	2	no	—
227	C₂	—	73	2	no	—
228	C₂	C₄⋊C₄×#10596842	1	1	yes	characteristic
229	C₂	C₂×#64408⋊C₁₆	1	1	yes	characteristic
230	C₂	#22627094	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10596842
commutator	C₂×C₇₃	C₂×C₄×C₁₆
Fitting	C₄⋊C₈×C₇₃	C₈
socle	C₂²×C₇₃	C₂×C₄×C₈
prime radical	C₂²×D₇₃	C₂²×C₈
cosocle	C₂×#11924036	C₂³
hypercenter	C₄⋊C₈	#10596842
hypocenter	C₇₃	C₁₆×C₄⋊C₄
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
9. #17989690	C₄	semidirect
10. C₄⋊#11924040	C₄	semidirect
12. C₄×#11924040	C₄	semidirect
13. C₄×#11924040	C₄	semidirect
46. #11924040	C₄⋊C₄	semidirect
58. C₄⋊#64408	C₁₆	semidirect
103. C₄×C₇₃	C₄×C₁₆	semidirect
107. Dic₇₃	C₄×C₁₆	semidirect
121. C₇₃	C₁₆×C₄⋊C₄	semidirect
223. C₄	#64408⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#22627117	2
C₂×#64408⋊C₁₆	2
C₄×#14556686	2
#22627002	2
#22625724	2
C₁₆×C₄⋊C₄	73