Subgroups — #32972888

Order 18688.

286 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11924003	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11924003	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊C₄×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924003	—	2	1	no	—
5	C₂²×#11924003	—	2	1	no	—
6	C₄⋊#11924003	—	2	1	no	—
7	C₄⋊#11924003	C₄	1	2	yes	complemented
8	C₄×#11924003	—	2	1	no	—
9	C₄×#11924003	C₄	1	2	yes	complemented
10	C₂×C₄⋊C₈×C₇₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₇₃	—	2	1	no	—
12	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₇₃	C₄	1	1	yes	characteristic
14	#336×C₇₃	—	2	1	no	—
15	C₂×#11924003	—	2	2	no	—
16	C₂×#11924003	—	2	2	no	—
17	C₂×#11924003	—	2	2	no	—
18	C₂×#11924003	—	2	2	no	—
19	C₂×#11924003	—	4	1	no	—
20	C₂×#11924003	—	4	1	no	—
21	C₂²×C₈×C₇₃	—	2	1	no	—
22	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₇₃	C₈	1	2	yes	—
25	C₂×C₄²×C₇₃	—	2	1	no	—
26	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₄⋊C₈×C₇₃	—	2	1	no	—
28	C₄⋊C₈×C₇₃	C₈	1	2	yes	—
29	C₄×C₈×C₇₃	—	4	1	no	—
30	C₄×C₈×C₇₃	—	4	1	no	—
31	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
34	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
35	#95×C₇₃	—	2	1	no	—
36	#11924003	—	2	4	no	—
37	#11924003	—	4	2	no	—
38	C₂²×C₄×C₇₃	—	2	2	no	—
39	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
42	C₂×C₈×C₇₃	—	2	1	no	—
43	C₂×C₈×C₇₃	—	2	1	no	—
44	C₂×C₈×C₇₃	—	2	2	no	—
45	C₂×C₈×C₇₃	—	4	1	no	—
46	C₂×C₈×C₇₃	—	4	1	no	—
47	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
52	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₇₃	SD₁₆	1	1	yes	characteristic
54	C₂×C₈×C₇₃	Q₁₆	1	1	yes	characteristic
55	C₄⋊C₄×C₇₃	—	2	2	no	—
56	C₄⋊C₄×C₇₃	C₁₆	1	4	yes	complemented
57	C₄²×C₇₃	—	4	1	no	—
58	C₄²×C₇₃	—	4	1	no	—
59	C₄²×C₇₃	C₄²	1	1	yes	characteristic
60	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
61	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
63	C₂³×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₇₃	—	2	2	no	—
65	C₂×C₄×C₇₃	—	2	4	no	—
66	C₂×C₄×C₇₃	—	4	2	no	—
67	C₂×C₄×C₇₃	—	4	2	no	—
68	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
70	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
72	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
73	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
74	C₂×C₄×C₇₃	#104	1	1	yes	characteristic
75	C₂×C₄×C₇₃	#104	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
77	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
79	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
80	C₈×C₇₃	—	2	1	no	—
81	C₈×C₇₃	—	2	1	no	—
82	C₈×C₇₃	—	4	2	no	—
83	C₈×C₇₃	#102	1	1	yes	characteristic
84	C₈×C₇₃	#103	1	1	yes	characteristic
85	C₈×C₇₃	#106	1	1	yes	characteristic
86	C₈×C₇₃	#107	1	1	yes	characteristic
87	C₂²×C₇₃	D₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₇₃	Q₈⋊C₈	1	1	yes	characteristic
89	C₂²×C₇₃	#336	1	1	yes	characteristic
90	C₂²×C₇₃	C₂²⋊C₁₆	1	2	yes	—
91	C₂²×C₇₃	#350	1	2	yes	—
92	C₄×C₇₃	—	2	1	no	—
93	C₄×C₇₃	—	2	1	no	—
94	C₄×C₇₃	—	4	4	no	—
95	C₄×C₇₃	D₄⋊C₈	1	1	yes	characteristic
96	C₄×C₇₃	Q₈⋊C₈	1	1	yes	characteristic
97	C₄×C₇₃	#339	1	1	yes	characteristic
98	C₄×C₇₃	#340	1	1	yes	characteristic
99	C₄×C₇₃	C₄×C₁₆	1	1	yes	characteristic
100	C₄×C₇₃	#346	1	1	yes	characteristic
101	C₄×C₇₃	C₄⋊C₁₆	1	2	yes	—
102	#7167	—	73	1	no	maximal, Sylow
103	C₂×C₇₃	#1279	1	1	yes	characteristic
104	C₂×C₇₃	D₄⋊C₁₆	1	2	yes	—
105	C₂×C₇₃	Q₈⋊C₁₆	1	2	yes	—
106	C₂×C₇₃	#1374	1	1	yes	characteristic
107	C₂×C₇₃	#1379	1	1	yes	characteristic
108	C₂×C₄⋊C₁₆	—	73	1	no	—
109	C₂×C₄×C₁₆	—	73	1	no	—
110	C₈×C₄⋊C₄	D₇₃	1	1	yes	characteristic
111	C₇₃	#7167	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₁₆	—	146	1	no	—
113	C₂²×C₁₆	—	146	1	no	—
114	C₂×C₄⋊C₈	Dic₇₃	1	1	yes	characteristic
115	C₂×C₄×C₈	—	2	1	no	—
116	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
117	C₄×C₄⋊C₄	Dic₇₃	1	1	yes	characteristic
118	C₄⋊C₁₆	—	73	2	no	—
119	C₄⋊C₁₆	—	146	1	no	—
120	C₄×C₁₆	—	73	2	no	—
121	C₄×C₁₆	—	146	1	no	—
122	#336	—	2	1	no	—
123	C₂²×C₈	—	2	1	no	—
124	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
125	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
126	C₂×C₄⋊C₄	#10596831	1	2	yes	—
127	C₂×C₄²	—	2	1	no	—
128	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
129	C₂×C₁₆	—	292	1	no	—
130	C₂×C₁₆	—	292	1	no	—
131	C₂×C₁₆	—	146	2	no	—
132	C₂×C₁₆	—	146	2	no	—
133	C₂×C₁₆	—	146	2	no	—
134	C₂×C₁₆	—	146	2	no	—
135	C₄⋊C₈	—	2	1	no	—
136	C₄⋊C₈	#10596831	1	2	yes	—
137	C₄×C₈	—	4	1	no	—
138	C₄×C₈	—	4	1	no	—
139	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
140	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
141	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
142	C₄×C₈	D₂₉₂	1	1	yes	characteristic
143	#95	—	2	1	no	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
146	C₂²×C₄	C₂×#10596831	1	1	yes	characteristic
147	C₂²×C₄	#11924012	1	1	yes	characteristic
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	#11924005	1	1	yes	characteristic
154	C₂×C₈	#11924005	1	1	yes	characteristic
155	C₂×C₈	#11924007	1	1	yes	characteristic
156	C₂×C₈	#11924007	1	1	yes	characteristic
157	C₂×C₈	#11924013	1	1	yes	characteristic
158	C₂×C₈	#11924015	1	1	yes	characteristic
159	C₂×C₈	#11924016	1	1	yes	characteristic
160	C₂×C₈	#11924018	1	1	yes	characteristic
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	#11924003	1	4	yes	complemented
163	C₄²	—	4	1	no	—
164	C₄²	—	4	1	no	—
165	C₄²	C₄×Dic₇₃	1	1	yes	characteristic
166	C₄²	C₄⋊Dic₇₃	1	1	yes	characteristic
167	C₄²	C₂×#10596831	1	1	yes	characteristic
168	C₄²	#11924012	1	1	yes	characteristic
169	C₁₆	—	292	2	no	—
170	C₁₆	—	146	4	no	—
171	C₂³	C₂²⋊#10596831	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	#14556498	1	1	yes	characteristic
177	C₂×C₄	C₄×#10596831	1	1	yes	characteristic
178	C₂×C₄	#14556502	1	1	yes	characteristic
179	C₂×C₄	C₂²⋊#10596831	1	1	yes	characteristic
180	C₂×C₄	#14556512	1	1	yes	characteristic
181	C₂×C₄	#14556515	1	1	yes	characteristic
182	C₂×C₄	#14556518	1	1	yes	characteristic
183	C₂×C₄	#14556518	1	1	yes	characteristic
184	C₂×C₄	C₄⋊#10596831	1	1	yes	characteristic
185	C₂×C₄	C₄⋊#10596831	1	1	yes	characteristic
186	C₂×C₄	C₂×#11924003	1	2	yes	—
187	C₂×C₄	#14556531	1	2	yes	—
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	4	2	no	—
191	C₈	#14556511	1	1	yes	characteristic
192	C₈	#14556514	1	1	yes	characteristic
193	C₈	#14556522	1	1	yes	characteristic
194	C₈	#14556524	1	1	yes	characteristic
195	C₂²	D₄⋊#10596831	1	1	yes	characteristic
196	C₂²	Q₈⋊#10596831	1	1	yes	characteristic
197	C₂²	#17988342	1	1	yes	characteristic
198	C₂²	C₂²⋊#11924003	1	2	yes	—
199	C₂²	#17988371	1	2	yes	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	4	4	no	—
203	C₄	D₄⋊#10596831	1	1	yes	characteristic
204	C₄	Q₈⋊#10596831	1	1	yes	characteristic
205	C₄	#17988346	1	1	yes	characteristic
206	C₄	#17988349	1	1	yes	characteristic
207	C₄	C₄×#11924003	1	1	yes	characteristic
208	C₄	#17988363	1	1	yes	characteristic
209	C₄	C₄⋊#11924003	1	2	yes	—
210	C₂	#22607772	1	1	yes	characteristic
211	C₂	D₄⋊#11924003	1	2	yes	—
212	C₂	Q₈⋊#11924003	1	2	yes	—
213	C₂	#22608009	1	1	yes	characteristic
214	C₂	#22608020	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₇₃
commutator	C₄×C₇₃	C₄×C₁₆
Frattini	C₂×C₄×C₈	C₂×D₇₃
Fitting	C₈×C₄⋊C₄×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₇₃	C₂²
hypercenter	C₈×C₄⋊C₄	D₇₃
hypocenter	C₇₃	#7167
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
7. C₄⋊#11924003	C₄	semidirect
9. C₄×#11924003	C₄	semidirect
56. C₄⋊C₄×C₇₃	C₁₆	semidirect
111. C₇₃	#7167	semidirect
162. C₄⋊C₄	#11924003	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11924003	2
C₂×C₄×#11924003	2
C₈×C₄⋊C₄×C₇₃	2
#7167	73