Subgroups — #33006196

Order 18688.

290 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989003	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988421	C₂	1	1	yes	maximal, characteristic, complemented
3	#22608074	C₂	1	4	yes	maximal, complemented
4	C₂×M₆₄×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14556597	C₄	1	1	yes	characteristic
6	#17989003	—	2	1	no	—
7	#17989003	—	2	1	no	—
8	#17989003	C₄	1	2	yes	—
9	#17989003	C₂²	1	2	yes	—
10	C₂×#14556531	C₄	1	1	yes	characteristic
11	C₂×#14556530	—	2	1	no	—
12	C₂×C₁₆×D₇₃	—	2	1	no	—
13	#17988421	—	2	1	no	—
14	#17988421	—	2	1	no	—
15	C₂×#14556497	C₂²	1	2	yes	complemented
16	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	M₆₄×C₇₃	—	2	1	no	—
18	M₆₄×C₇₃	C₂²	1	2	yes	complemented
19	C₂×C₃₂×C₇₃	—	2	1	no	—
20	C₂×#11924028	C₈	1	1	yes	characteristic
21	#14556597	—	2	1	no	—
22	#14556597	—	2	1	no	—
23	#14556597	C₈	1	2	yes	—
24	#14556597	C₂×C₄	1	2	yes	—
25	C₂×#11924012	C₈	1	1	yes	characteristic
26	C₂×#11924011	—	2	1	no	—
27	C₂×C₈×D₇₃	—	2	1	no	—
28	#14556531	—	2	1	no	—
29	#14556531	C₂×C₄	1	2	yes	—
30	#14556530	—	2	2	no	—
31	#14556530	—	2	2	no	—
32	C₂×#11924003	—	2	1	no	—
33	C₁₆×D₇₃	—	2	2	no	—
34	C₁₆×D₇₃	—	2	2	no	—
35	#14556497	—	2	2	no	—
36	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
39	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
40	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
43	C₃₂×C₇₃	—	2	2	no	—
44	#11924028	—	2	1	no	—
45	#11924028	—	2	1	no	—
46	#11924028	C₁₆	1	2	yes	—
47	#11924028	C₂×C₈	1	2	yes	—
48	C₂×Dic₁₄₆	C₁₆	1	1	yes	characteristic
49	C₂×D₂₉₂	C₁₆	1	1	yes	characteristic
50	C₂×C₂²⋊D₇₃	—	2	1	no	—
51	C₂×C₄×D₇₃	—	2	1	no	—
52	#11924012	—	2	1	no	—
53	#11924012	C₂×C₈	1	2	yes	—
54	#11924011	—	2	2	no	—
55	#11924011	—	2	2	no	—
56	C₂×#10596831	—	2	1	no	—
57	C₈×D₇₃	—	2	2	no	—
58	C₈×D₇₃	—	2	2	no	—
59	#11924003	—	2	2	no	—
60	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
61	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
64	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
65	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
66	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
67	C₁₆×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₁₆×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₁₆×C₇₃	C₂×D₄	1	1	yes	characteristic
70	C₁₆×C₇₃	C₂×D₄	1	1	yes	characteristic
71	C₂³×C₇₃	M₃₂	1	1	yes	characteristic
72	C₂²×D₇₃	—	2	1	no	—
73	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
74	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
75	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
76	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
77	C₂×C₄×C₇₃	M₃₂	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
79	C₂²⋊D₇₃	—	2	2	no	—
80	C₂²⋊D₇₃	—	2	2	no	—
81	C₂×Dic₇₃	—	2	1	no	—
82	D₂₉₂	—	2	1	no	—
83	D₂₉₂	C₂×C₁₆	1	2	yes	—
84	C₄×D₇₃	—	2	2	no	—
85	C₄×D₇₃	—	2	2	no	—
86	Dic₁₄₆	—	2	1	no	—
87	Dic₁₄₆	C₂×C₁₆	1	2	yes	—
88	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
89	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
90	C₈×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
92	#10596831	—	2	2	no	—
93	C₂²×C₇₃	—	4	1	no	—
94	C₂²×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
95	C₂²×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
96	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
97	C₂×D₇₃	—	2	2	no	—
98	C₂×D₇₃	—	4	2	no	—
99	C₄×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
100	C₄×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
101	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
102	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
103	Dic₇₃	—	2	2	no	—
104	#13558	—	73	1	no	maximal, Sylow
105	C₂×C₇₃	—	2	1	no	—
106	C₂×C₇₃	—	2	1	no	—
107	C₂×C₇₃	—	2	1	no	—
108	C₂×C₇₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
109	D₇₃	—	4	2	no	—
110	C₂×#504	—	73	1	no	—
111	C₂×M₆₄	—	73	1	no	—
112	C₂×M₆₄	D₇₃	1	1	yes	characteristic, complemented
113	#1401	—	73	4	no	—
114	C₇₃	#13558	1	1	yes	characteristic, complemented, Sylow
115	C₂×#131	—	73	1	no	—
116	#504	—	73	2	no	—
117	#504	—	73	2	no	—
118	#504	—	146	1	no	—
119	#504	—	146	1	no	—
120	C₂×M₃₂	—	73	1	no	—
121	C₂×M₃₂	—	146	1	no	—
122	C₂²×C₁₆	—	146	1	no	—
123	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
124	M₆₄	—	2	1	no	—
125	M₆₄	—	146	1	no	—
126	M₆₄	—	146	1	no	—
127	M₆₄	C₂×D₇₃	1	2	yes	complemented
128	C₂×C₃₂	—	2	1	no	—
129	C₂×C₃₂	—	73	2	no	—
130	C₂×#41	—	73	1	no	—
131	#131	—	73	2	no	—
132	#131	—	73	2	no	—
133	#131	—	146	1	no	—
134	#131	—	146	1	no	—
135	C₂×M₁₆	—	73	1	no	—
136	C₂×M₁₆	—	146	1	no	—
137	C₂²×C₈	—	146	1	no	—
138	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
139	M₃₂	—	73	2	no	—
140	M₃₂	—	146	1	no	—
141	M₃₂	—	146	2	no	—
142	M₃₂	—	146	2	no	—
143	C₂×C₁₆	—	146	1	no	—
144	C₂×C₁₆	—	146	2	no	—
145	C₂×C₁₆	—	146	2	no	—
146	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
147	C₂×C₁₆	D₂₉₂	1	1	yes	characteristic
148	C₂×C₁₆	D₂₉₂	1	1	yes	characteristic
149	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
150	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
151	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
152	C₃₂	—	2	2	no	—
153	C₃₂	—	146	2	no	—
154	#41	—	73	2	no	—
155	#41	—	73	2	no	—
156	#41	—	146	1	no	—
157	#41	—	146	1	no	—
158	C₂×Q₈	—	73	1	no	—
159	C₂×D₄	—	73	1	no	—
160	C₂×D₄	—	146	1	no	—
161	C₂²×C₄	—	146	1	no	—
162	C₂²×C₄	C₈×D₇₃	1	1	yes	characteristic
163	M₁₆	—	73	2	no	—
164	M₁₆	—	146	1	no	—
165	M₁₆	—	146	2	no	—
166	M₁₆	—	146	2	no	—
167	C₂×C₈	—	146	1	no	—
168	C₂×C₈	—	146	2	no	—
169	C₂×C₈	—	146	2	no	—
170	C₂×C₈	#11924005	1	1	yes	characteristic
171	C₂×C₈	#11924005	1	1	yes	characteristic
172	C₂×C₈	C₈×D₇₃	1	1	yes	characteristic
173	C₂×C₈	#11924011	1	1	yes	characteristic
174	C₂×C₈	#11924011	1	1	yes	characteristic
175	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
176	C₁₆	—	146	2	no	—
177	C₁₆	#11924005	1	1	yes	characteristic
178	C₁₆	#11924005	1	1	yes	characteristic
179	C₁₆	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
180	C₁₆	C₂×D₂₉₂	1	1	yes	characteristic
181	C₂³	—	146	1	no	—
182	C₂³	#14556530	1	1	yes	characteristic
183	Q₈	—	73	2	no	—
184	Q₈	—	146	1	no	—
185	D₄	—	73	2	no	—
186	D₄	—	146	2	no	—
187	D₄	—	146	2	no	—
188	D₄	—	146	1	no	—
189	C₂×C₄	—	146	1	no	—
190	C₂×C₄	—	146	2	no	—
191	C₂×C₄	—	146	2	no	—
192	C₂×C₄	#14556503	1	1	yes	characteristic
193	C₂×C₄	#14556503	1	1	yes	characteristic
194	C₂×C₄	C₁₆×D₇₃	1	1	yes	characteristic
195	C₂×C₄	C₁₆×D₇₃	1	1	yes	characteristic
196	C₂×C₄	#14556530	1	1	yes	characteristic
197	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
198	C₈	—	146	2	no	—
199	C₈	#14556503	1	1	yes	characteristic
200	C₈	#14556503	1	1	yes	characteristic
201	C₈	C₂×#11924005	1	1	yes	characteristic
202	C₈	C₂×#11924011	1	1	yes	characteristic
203	C₂²	—	4	1	no	—
204	C₂²	—	292	2	no	—
205	C₂²	—	146	2	no	—
206	C₂²	#17988366	1	1	yes	characteristic
207	C₂²	#17988366	1	1	yes	characteristic
208	C₂²	C₂×#14556530	1	1	yes	characteristic
209	C₄	—	146	2	no	—
210	C₄	#17988366	1	1	yes	characteristic
211	C₄	#17988366	1	1	yes	characteristic
212	C₄	C₂×#14556503	1	1	yes	characteristic
213	C₄	C₂×C₁₆×D₇₃	1	1	yes	characteristic
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	—	292	2	no	—
218	C₂	C₂×#17988366	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂×C₂²⋊D₇₃
commutator	C₄×C₇₃	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting	C₂×M₆₄×C₇₃	C₂
socle	C₂×C₇₃	C₂×C₂²⋊C₁₆
prime radical	C₂×#11924028	C₈
cosocle	C₂×C₁₆×C₇₃	C₂³
hypercenter	C₂×M₆₄	D₇₃
hypocenter	C₇₃	#13558
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
2. C₂×#17988421	C₂	semidirect
3. #22608074	C₂	semidirect
4. C₂×M₆₄×C₇₃	C₂	semidirect
15. C₂×#14556497	C₂²	semidirect
18. M₆₄×C₇₃	C₂²	semidirect
112. C₂×M₆₄	D₇₃	semidirect
114. C₇₃	#13558	semidirect
127. M₆₄	C₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989003	2
C₂×#17988421	2
#22608074	2
C₂×M₆₄×C₇₃	2
#13558	73