Subgroups — #32977281

Order 18688.

350 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	C₄×#14556527	C₂	1	1	yes	maximal, characteristic
2	#22608679	C₂	1	1	yes	maximal, characteristic
3	#22608220	C₂	1	2	yes	maximal
4	#22607768	C₂	1	2	yes	maximal
5	C₂×C₈²×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#17988640	C₂²	1	2	yes	—
7	C₂×#14556527	C₄	1	2	yes	—
8	C₄×#11924012	C₂²	1	2	yes	—
9	#17988352	C₄	1	2	yes	—
10	#17988340	C₄	1	2	yes	—
11	#17988338	C₄	1	2	yes	—
12	#17988309	—	2	2	no	—
13	#17988308	—	2	2	no	—
14	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₈²×C₇₃	C₄	1	2	yes	—
18	C₈²×C₇₃	C₄	1	2	yes	—
19	C₂×#11924012	C₂×C₄	1	4	yes	—
20	#14556527	—	2	4	no	—
21	C₄⋊#10596831	C₂×C₄	1	4	yes	—
22	C₂²⋊#10596831	—	2	2	no	—
23	#14556502	—	2	2	no	—
24	C₄×#10596831	—	2	2	no	—
25	C₄×#10596831	—	2	2	no	—
26	C₂²×C₈×C₇₃	—	2	1	no	—
27	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₃	Q₈	1	2	yes	—
30	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₇₃	—	2	1	no	—
32	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
39	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
40	#11924012	—	2	4	no	—
41	#11924012	—	2	4	no	—
42	C₂×#10596831	—	2	4	no	—
43	C₂×#10596831	—	2	4	no	—
44	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
47	C₂×C₈×C₇₃	—	2	4	no	—
48	C₂×C₈×C₇₃	—	2	2	no	—
49	C₂×C₈×C₇₃	—	2	2	no	—
50	C₂×C₈×C₇₃	—	2	4	no	—
51	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
52	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
55	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₇₃	M₁₆	1	4	yes	—
58	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₇₃	#41	1	1	yes	characteristic
62	C₂³×C₇₃	C₄×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₇₃	—	2	2	no	—
64	C₂×C₄×C₇₃	—	2	2	no	—
65	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
68	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
72	C₂×C₄×C₇₃	#131	1	2	yes	—
73	C₈×C₇₃	—	2	4	no	—
74	C₈×C₇₃	—	2	1	no	—
75	C₈×C₇₃	—	2	1	no	—
76	C₈×C₇₃	—	2	2	no	—
77	C₈×C₇₃	#97	1	2	yes	—
78	C₈×C₇₃	#97	1	2	yes	—
79	C₈×C₇₃	#108	1	4	yes	complemented
80	#10596831	—	4	4	no	—
81	#10596831	—	4	4	no	—
82	C₂²×C₇₃	—	2	1	no	—
83	C₂²×C₇₃	—	2	1	no	—
84	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₇₃	#446	1	2	yes	—
86	C₄×C₇₃	—	2	1	no	—
87	C₄×C₇₃	—	2	1	no	—
88	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
89	C₄×C₇₃	#405	1	1	yes	characteristic
90	C₄×C₇₃	C₂×#108	1	2	yes	—
91	C₄×C₇₃	#436	1	2	yes	—
92	C₄×C₇₃	#443	1	2	yes	—
93	#8293	—	73	1	no	maximal, Sylow
94	C₂×C₇₃	—	2	2	no	—
95	C₂×C₇₃	#1770	1	1	yes	characteristic
96	C₂×C₇₃	#1772	1	1	yes	characteristic
97	C₂×C₇₃	C₄×#108	1	1	yes	characteristic
98	C₄×#108	—	73	1	no	—
99	#1564	—	73	1	no	—
100	#1451	—	73	2	no	—
101	C₂×C₈²	D₇₃	1	1	yes	characteristic
102	#1278	—	73	2	no	—
103	C₇₃	#8293	1	1	yes	characteristic, complemented, Sylow
104	#431	—	73	2	no	—
105	C₂×#108	—	73	2	no	—
106	C₄×M₁₆	—	73	2	no	—
107	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
110	#341	—	73	2	no	—
111	#335	—	73	2	no	—
112	#334	—	73	2	no	—
113	#322	—	146	2	no	—
114	#322	—	146	2	no	—
115	C₈²	Dic₇₃	1	2	yes	—
116	C₈²	Dic₇₃	1	2	yes	—
117	C₂×M₁₆	—	73	4	no	—
118	C₂²×C₈	—	2	1	no	—
119	C₂²×C₈	Dic₁₄₆	1	2	yes	—
120	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
121	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
122	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
123	#108	—	146	4	no	—
124	C₄⋊C₈	—	73	4	no	—
125	C₂²⋊C₈	—	146	2	no	—
126	#97	—	146	2	no	—
127	C₄×C₈	—	2	1	no	—
128	C₄×C₈	—	146	2	no	—
129	C₄×C₈	—	146	2	no	—
130	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
131	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
132	C₄×C₈	D₂₉₂	1	1	yes	characteristic
133	C₄×C₈	D₂₉₂	1	1	yes	characteristic
134	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
135	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
136	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
137	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
138	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
139	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
140	C₂²×C₄	#11924028	1	1	yes	characteristic
141	M₁₆	—	146	4	no	—
142	M₁₆	—	146	4	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	4	no	—
147	C₂×C₈	—	146	4	no	—
148	C₂×C₈	—	146	4	no	—
149	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
150	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
151	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
152	C₂×C₈	C₄×Dic₇₃	1	1	yes	characteristic
153	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
154	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
155	C₂×C₈	#11924011	1	4	yes	—
156	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
157	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
158	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
159	C₄²	#11924028	1	1	yes	characteristic
160	C₂³	C₄×Dic₁₄₆	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
165	C₂×C₄	#14556550	1	1	yes	characteristic
166	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
167	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
168	C₂×C₄	C₄×Dic₁₄₆	1	1	yes	characteristic
169	C₂×C₄	C₂×#11924011	1	2	yes	—
170	C₂×C₄	#14556597	1	2	yes	—
171	C₈	—	2	4	no	—
172	C₈	—	2	2	no	—
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	292	4	no	—
176	C₈	—	292	4	no	—
177	C₈	#14556501	1	2	yes	—
178	C₈	#14556501	1	2	yes	—
179	C₈	#14556527	1	4	yes	complemented
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	C₄×C₄⋊Dic₇₃	1	1	yes	characteristic
183	C₂²	#17988717	1	2	yes	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	C₂×#14556501	1	1	yes	characteristic
187	C₄	#17988542	1	1	yes	characteristic
188	C₄	C₂×#14556527	1	2	yes	—
189	C₄	#17988666	1	2	yes	—
190	C₄	#17988705	1	2	yes	—
191	C₂	—	2	2	no	—
192	C₂	#22609798	1	1	yes	characteristic
193	C₂	#22609810	1	1	yes	characteristic
194	C₂	C₄×#14556527	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×C₈²×C₇₃	C₂
socle	C₂²×C₇₃	C₄×C₄⋊C₄
prime radical	C₂³×C₇₃	C₄×Q₈
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×C₈²	D₇₃
hypocenter	C₇₃	#8293
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
79. C₈×C₇₃	#108	semidirect
103. C₇₃	#8293	semidirect
179. C₈	#14556527	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556527	2
#22608679	2
#22608220	2
#22607768	2
C₂×C₈²×C₇₃	2
#8293	73