Subgroups — #34013319

Order 18688.

382 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989510	C₂	1	2	yes	maximal
2	C₂×#17989509	C₂	1	2	yes	maximal
3	#22626265	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556665	C₂²	1	4	yes	—
6	#17989510	—	2	4	no	—
7	#17989509	—	2	4	no	—
8	#17989508	—	2	4	no	—
9	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
14	C₄²×Dic₇₃	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
16	#14556665	—	2	4	no	—
17	#14556665	—	2	4	no	—
18	C₂×#11924031	—	2	4	no	—
19	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
20	C₂²×#10596831	C₂³	1	1	yes	characteristic
21	C₂×C₄×Dic₇₃	—	2	2	no	—
22	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
24	C₄×#10596831	D₄	1	2	yes	—
25	C₄×#10596831	Q₈	1	2	yes	—
26	C₈×Dic₇₃	—	2	2	no	—
27	C₈×Dic₇₃	C₈	1	2	yes	—
28	C₈×Dic₇₃	C₈	1	2	yes	—
29	C₈×Dic₇₃	D₄	1	2	yes	—
30	C₈×Dic₇₃	D₄	1	2	yes	—
31	C₈×Dic₇₃	Q₈	1	2	yes	—
32	C₈×Dic₇₃	Q₈	1	2	yes	—
33	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₇₃	—	2	2	no	—
37	#11924031	—	4	4	no	—
38	C₂²×Dic₇₃	—	2	1	no	—
39	C₂²×Dic₇₃	M₁₆	1	2	yes	—
40	C₂×#10596831	C₄⋊C₄	1	2	yes	—
41	C₂×#10596831	C₂×C₈	1	2	yes	—
42	C₂×#10596831	M₁₆	1	2	yes	—
43	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
44	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
45	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
46	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
47	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
48	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
49	C₄×Dic₇₃	—	2	8	no	—
50	C₄×Dic₇₃	—	2	1	no	—
51	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
52	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
53	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
54	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₇₃	—	2	1	no	—
56	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₃	—	2	1	no	—
58	C₂×C₈×C₇₃	—	2	1	no	—
59	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
61	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
62	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
63	C₄²×C₇₃	—	2	1	no	—
64	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
65	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
66	C₂×C₄×C₇₃	—	2	4	no	—
67	C₂×C₄×C₇₃	—	2	2	no	—
68	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
74	C₂×Dic₇₃	—	2	4	no	—
75	C₂×Dic₇₃	—	2	2	no	—
76	C₂×Dic₇₃	—	2	4	no	—
77	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
78	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
79	C₈×C₇₃	—	2	2	no	—
80	C₈×C₇₃	C₄⋊C₈	1	4	yes	—
81	#10596831	C₄⋊C₈	1	4	yes	—
82	#10596831	#127	1	1	yes	characteristic
83	#10596831	C₄⋊Q₈	1	1	yes	characteristic
84	#10596831	C₄⋊Q₈	1	1	yes	characteristic
85	#10596831	C₄⋊Q₈	1	1	yes	characteristic
86	C₂²×C₇₃	—	2	2	no	—
87	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
88	C₂²×C₇₃	#423	1	1	yes	characteristic
89	C₂²×C₇₃	#423	1	1	yes	characteristic
90	C₄×C₇₃	—	2	4	no	—
91	C₄×C₇₃	#384	1	1	yes	characteristic
92	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
93	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
94	C₄×C₇₃	#423	1	1	yes	characteristic
95	Dic₇₃	—	2	4	no	—
96	Dic₇₃	—	2	2	no	—
97	Dic₇₃	#364	1	4	yes	complemented
98	#12900	—	73	1	no	maximal, Sylow
99	C₂×C₇₃	—	2	1	no	—
100	C₂×C₇₃	—	2	1	no	—
101	C₂×C₇₃	#1842	1	1	yes	characteristic
102	C₂×C₇₃	C₂×#364	1	2	yes	—
103	C₂×#364	—	73	2	no	—
104	C₂×#364	—	73	2	no	—
105	#2150	—	73	2	no	—
106	C₄²×C₈	—	73	1	no	—
107	C₇₃	#12900	1	1	yes	characteristic, complemented, Sylow
108	C₂×M₃₂	—	73	4	no	—
109	C₂×C₄×C₈	—	73	1	no	—
110	C₂×C₄×C₈	—	73	1	no	—
111	C₂×C₄×C₈	—	73	1	no	—
112	C₂×C₄×C₈	—	73	1	no	—
113	C₂×C₄×C₈	—	73	1	no	—
114	C₂×C₄×C₈	#64408	1	1	yes	characteristic
115	C₄³	—	73	1	no	—
116	#364	—	146	4	no	—
117	#364	—	146	4	no	—
118	C₄⋊C₁₆	—	146	4	no	—
119	C₂²×C₈	—	73	1	no	—
120	C₂²×C₈	—	73	1	no	—
121	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
122	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
123	C₂×C₄²	—	73	1	no	—
124	C₂×C₄²	—	73	1	no	—
125	C₂×C₄²	—	146	2	no	—
126	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
127	M₃₂	—	146	4	no	—
128	M₃₂	—	146	4	no	—
129	C₂×C₁₆	—	146	4	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	73	2	no	—
132	C₄×C₈	—	73	2	no	—
133	C₄×C₈	—	73	2	no	—
134	C₄×C₈	—	73	2	no	—
135	C₄×C₈	—	73	2	no	—
136	C₄×C₈	—	73	2	no	—
137	C₄×C₈	—	73	2	no	—
138	C₄×C₈	—	73	2	no	—
139	C₄×C₈	—	146	2	no	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	73	2	no	—
142	C₂²×C₄	—	146	1	no	—
143	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	73	2	no	—
147	C₂×C₈	—	73	2	no	—
148	C₂×C₈	—	73	2	no	—
149	C₂×C₈	—	73	1	no	—
150	C₂×C₈	—	73	1	no	—
151	C₂×C₈	—	73	1	no	—
152	C₂×C₈	—	73	1	no	—
153	C₂×C₈	—	73	1	no	—
154	C₂×C₈	—	73	1	no	—
155	C₂×C₈	C₄⋊#64408	1	2	yes	—
156	C₂×C₈	C₄⋊#64408	1	2	yes	—
157	C₂×C₈	#11924036	1	2	yes	—
158	C₂×C₈	#11924037	1	2	yes	—
159	C₄²	—	2	1	no	—
160	C₄²	—	73	1	no	—
161	C₄²	—	73	1	no	—
162	C₄²	—	73	2	no	—
163	C₄²	—	73	2	no	—
164	C₄²	—	146	8	no	—
165	C₄²	—	146	1	no	—
166	C₄²	C₄⋊#64408	1	2	yes	—
167	C₁₆	—	292	4	no	—
168	C₂³	D₇₃⋊M₁₆	1	1	yes	characteristic
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	73	2	no	—
172	C₂×C₄	—	73	2	no	—
173	C₂×C₄	—	146	4	no	—
174	C₂×C₄	—	146	4	no	—
175	C₂×C₄	—	146	2	no	—
176	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
179	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
180	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
181	C₂×C₄	D₇₃⋊M₁₆	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	—	73	1	no	—
184	C₈	—	73	1	no	—
185	C₈	—	73	1	no	—
186	C₈	—	73	1	no	—
187	C₈	—	73	4	no	—
188	C₈	#14556658	1	4	yes	—
189	C₂²	—	2	2	no	—
190	C₂²	#17989588	1	1	yes	characteristic
191	C₂²	#17989590	1	1	yes	characteristic
192	C₂²	#17989592	1	1	yes	characteristic
193	C₄	—	2	4	no	—
194	C₄	—	73	4	no	—
195	C₄	—	146	4	no	—
196	C₄	—	146	2	no	—
197	C₄	#17989533	1	1	yes	characteristic
198	C₄	#17989588	1	1	yes	characteristic
199	C₄	C₂×#14556658	1	1	yes	characteristic
200	C₄	#17989590	1	1	yes	characteristic
201	C₂	—	2	1	no	—
202	C₂	—	2	1	no	—
203	C₂	#22625864	1	1	yes	characteristic
204	C₂	#22626270	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄⋊C₈
prime radical	C₂³×C₇₃	C₂×M₁₆
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	#12900
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
97. Dic₇₃	#364	semidirect
107. C₇₃	#12900	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989510	2
C₂×#17989509	2
#22626265	2
C₄×C₈×Dic₇₃	2
#12900	73