Subgroups — #32972900

Order 18688.

282 conjugacy classes of subgroups.

The subgroup lattice has 196 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×#17988340	C₂	1	1	yes	maximal, characteristic
3	C₄×#107×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924003	—	2	1	no	—
5	C₂×C₄⋊#10596831	—	2	1	no	—
6	C₄×#11924003	—	2	1	no	—
7	C₄×#11924003	C₄	1	2	yes	complemented
8	#17988340	—	2	1	no	—
9	#17988340	C₄	1	2	yes	complemented
10	C₂×#107×C₇₃	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₇₃	—	2	1	no	—
13	#340×C₇₃	—	2	1	no	—
14	C₂²×#10596831	—	4	1	no	—
15	C₂×#11924003	—	2	4	no	—
16	C₂×#11924003	—	2	2	no	—
17	C₄⋊#10596831	—	2	2	no	—
18	C₄⋊#10596831	—	4	1	no	—
19	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
22	C₂×C₄²×C₇₃	—	4	1	no	—
23	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
24	#107×C₇₃	—	2	2	no	—
25	#107×C₇₃	C₈	1	4	yes	complemented
26	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
30	#95×C₇₃	—	4	1	no	—
31	C₂×#10596831	—	4	2	no	—
32	C₂×#10596831	—	8	1	no	—
33	C₂×#10596831	—	8	1	no	—
34	#11924003	—	2	4	no	—
35	C₂²×C₄×C₇₃	—	4	2	no	—
36	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
39	C₂×C₈×C₇₃	—	2	2	no	—
40	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
41	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
45	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
46	C₄⋊C₄×C₇₃	—	2	4	no	—
47	C₄⋊C₄×C₇₃	—	4	2	no	—
48	C₄²×C₇₃	—	8	1	no	—
49	C₄²×C₇₃	—	8	1	no	—
50	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
53	C₄²×C₇₃	Q₁₆	1	1	yes	characteristic
54	C₂³×C₇₃	#102	1	1	yes	characteristic
55	C₂×C₄×C₇₃	—	2	2	no	—
56	C₂×C₄×C₇₃	—	4	4	no	—
57	C₂×C₄×C₇₃	—	8	2	no	—
58	C₂×C₄×C₇₃	—	8	2	no	—
59	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
61	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
62	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
63	C₂×C₄×C₇₃	#103	1	1	yes	characteristic
64	C₂×C₄×C₇₃	#104	1	2	yes	—
65	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
66	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
67	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
68	C₂×C₄×C₇₃	SD₃₂	1	1	yes	characteristic
69	C₂×C₄×C₇₃	SD₃₂	1	1	yes	characteristic
70	C₂×C₄×C₇₃	Q₃₂	1	1	yes	characteristic
71	C₈×C₇₃	—	2	1	no	—
72	C₈×C₇₃	—	2	1	no	—
73	C₈×C₇₃	C₄×C₈	1	1	yes	characteristic
74	C₈×C₇₃	#97	1	1	yes	characteristic
75	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
76	#10596831	—	8	2	no	—
77	C₂²×C₇₃	D₄⋊C₈	1	2	yes	—
78	C₂²×C₇₃	#340	1	1	yes	characteristic
79	C₂²×C₇₃	#357	1	1	yes	characteristic
80	C₂²×C₇₃	#358	1	1	yes	characteristic
81	C₂²×C₇₃	#359	1	2	yes	—
82	C₄×C₇₃	—	2	1	no	—
83	C₄×C₇₃	—	2	1	no	—
84	C₄×C₇₃	—	8	4	no	—
85	C₄×C₇₃	Q₈⋊C₈	1	2	yes	—
86	C₄×C₇₃	#336	1	1	yes	characteristic
87	C₄×C₇₃	#339	1	1	yes	characteristic
88	C₄×C₇₃	#357	1	1	yes	characteristic
89	C₄×C₇₃	#358	1	1	yes	characteristic
90	C₄×C₇₃	#366	1	1	yes	characteristic
91	C₄×C₇₃	#367	1	1	yes	characteristic
92	#7171	—	73	1	no	maximal, Sylow
93	C₂×C₇₃	#1279	1	1	yes	characteristic
94	C₂×C₇₃	#1331	1	2	yes	—
95	C₂×C₇₃	#1332	1	2	yes	—
96	C₂×C₇₃	#1380	1	1	yes	characteristic
97	C₂×C₇₃	#1382	1	1	yes	characteristic
98	C₂×C₄×C₁₆	—	73	1	no	—
99	C₄×#107	D₇₃	1	1	yes	characteristic
100	C₂×#335	—	73	1	no	—
101	C₇₃	#7171	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₁₆	—	146	1	no	—
103	C₂×#107	Dic₇₃	1	2	yes	—
104	C₂×C₄⋊C₈	—	146	1	no	—
105	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
106	C₄×C₄⋊C₄	—	2	1	no	—
107	C₄×C₁₆	—	73	2	no	—
108	C₄×C₁₆	—	146	1	no	—
109	#340	—	2	1	no	—
110	#335	—	73	2	no	—
111	#335	—	146	1	no	—
112	C₂²×C₈	—	292	1	no	—
113	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
114	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄²	—	4	1	no	—
117	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
118	C₂×C₁₆	—	146	4	no	—
119	C₂×C₁₆	—	146	2	no	—
120	#107	—	2	2	no	—
121	#107	#10596831	1	4	yes	complemented
122	C₄⋊C₈	—	292	1	no	—
123	C₄⋊C₈	—	146	2	no	—
124	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
125	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
126	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
127	C₄×C₈	D₂₉₂	1	1	yes	characteristic
128	#95	—	4	1	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
131	C₂²×C₄	#11924013	1	1	yes	characteristic
132	C₂²×C₄	#11924016	1	1	yes	characteristic
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	292	2	no	—
135	C₂×C₈	—	584	1	no	—
136	C₂×C₈	—	584	1	no	—
137	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
138	C₂×C₈	#11924005	1	1	yes	characteristic
139	C₂×C₈	#11924007	1	1	yes	characteristic
140	C₂×C₈	C₄⋊Dic₇₃	1	1	yes	characteristic
141	C₂×C₈	C₂×#10596831	1	2	yes	—
142	C₂×C₈	#11924012	1	2	yes	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄²	—	8	1	no	—
146	C₄²	—	8	1	no	—
147	C₄²	#11924005	1	1	yes	characteristic
148	C₄²	#11924007	1	1	yes	characteristic
149	C₄²	#11924015	1	1	yes	characteristic
150	C₄²	#11924018	1	1	yes	characteristic
151	C₁₆	—	146	4	no	—
152	C₂³	#14556512	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	#14556498	1	1	yes	characteristic
158	C₂×C₄	C₂²⋊#10596831	1	2	yes	—
159	C₂×C₄	#14556511	1	1	yes	characteristic
160	C₂×C₄	#14556514	1	1	yes	characteristic
161	C₂×C₄	#14556515	1	1	yes	characteristic
162	C₂×C₄	#14556518	1	2	yes	—
163	C₂×C₄	#14556522	1	1	yes	characteristic
164	C₂×C₄	#14556524	1	1	yes	characteristic
165	C₂×C₄	#14556532	1	1	yes	characteristic
166	C₂×C₄	#14556534	1	1	yes	characteristic
167	C₂×C₄	#14556535	1	1	yes	characteristic
168	C₂×C₄	#14556537	1	1	yes	characteristic
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	584	2	no	—
172	C₈	C₄×#10596831	1	1	yes	characteristic
173	C₈	#14556502	1	1	yes	characteristic
174	C₈	C₄⋊#10596831	1	2	yes	—
175	C₂²	D₄⋊#10596831	1	2	yes	—
176	C₂²	#17988349	1	1	yes	characteristic
177	C₂²	#17988388	1	1	yes	characteristic
178	C₂²	#17988391	1	1	yes	characteristic
179	C₂²	#17988394	1	2	yes	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	8	4	no	—
183	C₄	Q₈⋊#10596831	1	2	yes	—
184	C₄	#17988342	1	1	yes	characteristic
185	C₄	#17988346	1	1	yes	characteristic
186	C₄	#17988387	1	1	yes	characteristic
187	C₄	#17988390	1	1	yes	characteristic
188	C₄	#17988412	1	1	yes	characteristic
189	C₄	#17988414	1	1	yes	characteristic
190	C₂	#22607772	1	1	yes	characteristic
191	C₂	#22607892	1	2	yes	—
192	C₂	#22607895	1	2	yes	—
193	C₂	#22608022	1	1	yes	characteristic
194	C₂	#22608026	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11924013
commutator	C₈×C₇₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₃
Fitting	C₄×#107×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	#102
cosocle	C₂×C₄×C₈×C₇₃	C₂²
hypercenter	C₄×#107	D₇₃
hypocenter	C₇₃	#7171
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
7. C₄×#11924003	C₄	semidirect
9. #17988340	C₄	semidirect
25. #107×C₇₃	C₈	semidirect
101. C₇₃	#7171	semidirect
121. #107	#10596831	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×#17988340	2
C₄×#107×C₇₃	2
#7171	73