Subgroups — #33002195

Order 18688.

542 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#11924003	C₂	1	2	yes	maximal
2	#22608009	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924003	C₂²	1	4	yes	—
5	C₂²⋊#11924003	—	2	4	no	—
6	C₂³×C₈×C₇₃	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₇₃	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₇₃	—	2	4	no	—
10	C₂×C₄×C₈×C₇₃	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
12	C₂×#11924003	—	2	4	no	—
13	C₂×#11924003	—	2	8	no	—
14	C₂³×C₄×C₇₃	C₈	1	1	yes	characteristic
15	C₂³×C₄×C₇₃	C₈	1	1	yes	characteristic
16	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₇₃	—	2	2	no	—
18	C₂²×C₈×C₇₃	—	2	4	no	—
19	C₂²×C₈×C₇₃	—	2	4	no	—
20	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₇₃	D₄	1	4	yes	—
23	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₇₃	—	2	2	no	—
25	C₂×C₄²×C₇₃	—	2	2	no	—
26	C₂×C₄²×C₇₃	C₈	1	2	yes	—
27	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₇₃	—	2	4	no	—
29	C₄×C₈×C₇₃	—	2	4	no	—
30	#11924003	—	4	4	no	—
31	C₂⁴×C₇₃	C₂×C₈	1	1	yes	characteristic
32	C₂²×C₄×C₇₃	—	2	2	no	—
33	C₂²×C₄×C₇₃	—	2	2	no	—
34	C₂²×C₄×C₇₃	—	2	2	no	—
35	C₂²×C₄×C₇₃	—	2	4	no	—
36	C₂²×C₄×C₇₃	—	2	2	no	—
37	C₂²×C₄×C₇₃	—	2	2	no	—
38	C₂²×C₄×C₇₃	C₁₆	1	4	yes	complemented
39	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
44	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₃	—	2	8	no	—
46	C₂×C₈×C₇₃	—	2	2	no	—
47	C₂×C₈×C₇₃	—	2	2	no	—
48	C₂×C₈×C₇₃	—	2	2	no	—
49	C₂×C₈×C₇₃	—	2	8	no	—
50	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₇₃	M₁₆	1	4	yes	—
52	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₇₃	#41	1	4	yes	—
54	C₄²×C₇₃	—	2	4	no	—
55	C₄²×C₇₃	—	2	2	no	—
56	C₄²×C₇₃	—	2	2	no	—
57	C₂³×C₇₃	—	2	2	no	—
58	C₂³×C₇₃	—	2	2	no	—
59	C₂³×C₇₃	C₂²⋊C₈	1	2	yes	—
60	C₂³×C₇₃	C₂×C₁₆	1	2	yes	—
61	C₂³×C₇₃	M₃₂	1	2	yes	—
62	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
63	C₂×C₄×C₇₃	—	2	8	no	—
64	C₂×C₄×C₇₃	—	2	8	no	—
65	C₂×C₄×C₇₃	—	2	2	no	—
66	C₂×C₄×C₇₃	—	2	2	no	—
67	C₂×C₄×C₇₃	—	2	2	no	—
68	C₂×C₄×C₇₃	—	2	2	no	—
69	C₂×C₄×C₇₃	—	2	2	no	—
70	C₂×C₄×C₇₃	—	2	2	no	—
71	C₂×C₄×C₇₃	—	2	2	no	—
72	C₂×C₄×C₇₃	—	2	4	no	—
73	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
74	C₂×C₄×C₇₃	C₂×C₁₆	1	4	yes	—
75	C₂×C₄×C₇₃	M₃₂	1	4	yes	—
76	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₇₃	#117	1	2	yes	—
78	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
79	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
80	C₈×C₇₃	—	2	2	no	—
81	C₈×C₇₃	—	2	4	no	—
82	C₈×C₇₃	#123	1	4	yes	—
83	C₂²×C₇₃	—	2	2	no	—
84	C₂²×C₇₃	—	2	2	no	—
85	C₂²×C₇₃	—	2	4	no	—
86	C₂²×C₇₃	—	2	4	no	—
87	C₂²×C₇₃	C₂²⋊C₁₆	1	4	yes	—
88	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
89	C₂²×C₇₃	#431	1	2	yes	—
90	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
91	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
92	C₂²×C₇₃	C₂×M₃₂	1	2	yes	—
93	C₄×C₇₃	—	2	2	no	—
94	C₄×C₇₃	—	2	4	no	—
95	C₄×C₇₃	—	2	4	no	—
96	C₄×C₇₃	#381	1	1	yes	characteristic
97	C₄×C₇₃	#407	1	1	yes	characteristic
98	C₄×C₇₃	#432	1	2	yes	—
99	#12844	—	73	1	no	maximal, Sylow
100	C₂×C₇₃	—	2	4	no	—
101	C₂×C₇₃	#1817	1	1	yes	characteristic
102	C₂×C₇₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
103	C₂×C₇₃	#2112	1	1	yes	characteristic
104	C₂×C₇₃	#2162	1	2	yes	—
105	C₂×C₇₃	#2163	1	2	yes	—
106	C₂²×C₄×C₈	D₇₃	1	1	yes	characteristic
107	C₂×C₂²⋊C₁₆	—	73	2	no	—
108	#1374	—	73	4	no	—
109	C₇₃	#12844	1	1	yes	characteristic, complemented, Sylow
110	C₂³×C₈	Dic₇₃	1	1	yes	characteristic
111	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
112	C₂²×C₄²	Dic₇₃	1	1	yes	characteristic
113	C₂²×C₁₆	—	73	4	no	—
114	C₂×C₄×C₈	—	2	4	no	—
115	C₂×C₄×C₈	Dic₇₃	1	2	yes	—
116	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
117	C₂²⋊C₁₆	—	146	4	no	—
118	C₂³×C₄	#10596831	1	1	yes	characteristic
119	C₂³×C₄	#10596831	1	1	yes	characteristic
120	C₂³×C₄	C₂×Dic₇₃	1	1	yes	characteristic
121	C₂²×C₈	—	2	2	no	—
122	C₂²×C₈	—	2	4	no	—
123	C₂²×C₈	—	2	4	no	—
124	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
125	C₂²×C₈	C₂×Dic₇₃	1	2	yes	—
126	C₂²×C₈	C₂²⋊D₇₃	1	4	yes	—
127	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
128	C₂×C₄²	—	2	2	no	—
129	C₂×C₄²	—	2	2	no	—
130	C₂×C₄²	#10596831	1	2	yes	—
131	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
132	C₂×C₁₆	—	146	4	no	—
133	C₂×C₁₆	—	146	8	no	—
134	C₄×C₈	—	2	4	no	—
135	C₄×C₈	—	2	4	no	—
136	C₂⁴	C₂×#10596831	1	1	yes	characteristic
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	#11924003	1	4	yes	complemented
144	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
145	C₂²×C₄	C₂×#10596831	1	1	yes	characteristic
146	C₂²×C₄	C₂×#10596831	1	2	yes	—
147	C₂²×C₄	C₂×#10596831	1	2	yes	—
148	C₂²×C₄	#11924012	1	2	yes	—
149	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
150	C₂×C₈	—	2	8	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	8	no	—
155	C₂×C₈	C₂²⋊Dic₇₃	1	2	yes	—
156	C₂×C₈	#11924012	1	4	yes	—
157	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
158	C₂×C₈	#11924028	1	4	yes	—
159	C₄²	—	2	4	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₁₆	—	292	4	no	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	C₂²⋊#10596831	1	2	yes	—
166	C₂³	C₂×#11924003	1	2	yes	—
167	C₂³	#14556531	1	2	yes	—
168	C₂³	C₂²×#10596831	1	1	yes	characteristic
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	—	2	8	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	C₂²⋊#10596831	1	2	yes	—
180	C₂×C₄	C₂×#11924003	1	4	yes	—
181	C₂×C₄	#14556531	1	4	yes	—
182	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
183	C₂×C₄	#14556550	1	2	yes	—
184	C₂×C₄	C₂×#11924012	1	1	yes	characteristic
185	C₂×C₄	C₂×#11924012	1	2	yes	—
186	C₈	—	2	2	no	—
187	C₈	—	2	4	no	—
188	C₈	#14556574	1	4	yes	—
189	C₂²	—	2	4	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	2	2	no	—
193	C₂²	C₂²⋊#11924003	1	4	yes	—
194	C₂²	C₂×C₂²⋊#10596831	1	1	yes	characteristic
195	C₂²	#17988640	1	2	yes	—
196	C₂²	C₂²×#11924003	1	1	yes	characteristic
197	C₂²	C₂×#14556531	1	2	yes	—
198	C₂²	C₂×#14556531	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	4	no	—
201	C₄	—	2	4	no	—
202	C₄	#17988450	1	1	yes	characteristic
203	C₄	#17988551	1	1	yes	characteristic
204	C₄	#17988644	1	2	yes	—
205	C₂	—	2	4	no	—
206	C₂	#22609999	1	1	yes	characteristic
207	C₂	C₂×C₂²⋊#11924003	1	1	yes	characteristic
208	C₂	#22611542	1	1	yes	characteristic
209	C₂	#22611730	1	2	yes	—
210	C₂	#22611734	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₃
commutator	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₂²×C₄×C₈×C₇₃	C₂
socle	C₂³×C₇₃	C₂²×C₈
prime radical	C₂⁴×C₇₃	C₂×C₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂²×C₄×C₈	D₇₃
hypocenter	C₇₃	#12844
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22608009	C₂	semidirect
38. C₂²×C₄×C₇₃	C₁₆	semidirect
109. C₇₃	#12844	semidirect
143. C₂²×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×C₂²⋊#11924003	2
#22608009	2
C₂²×C₄×C₈×C₇₃	2
#12844	73