Subgroups — #33239180

Order 18688.

674 conjugacy classes of subgroups.

The subgroup lattice has 228 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	2	yes	maximal, complemented
2	#22618644	C₂	1	1	yes	maximal, characteristic
3	#22611758	C₂	1	8	yes	maximal, complemented
4	#22611687	C₂	1	1	yes	maximal, characteristic, complemented
5	#22611544	C₂	1	2	yes	maximal, complemented
6	C₂×#346×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14556597	C₂²	1	2	yes	—
8	#17989137	C₄	1	1	yes	characteristic
9	#17989003	—	2	8	no	—
10	C₂×#14556531	C₂²	1	2	yes	—
11	C₂×#14556530	C₂²	1	4	yes	—
12	C₂×C₁₆×D₇₃	C₂²	1	4	yes	—
13	#17988666	C₄	1	4	yes	—
14	C₈×D₂₉₂	C₂²	1	4	yes	—
15	#17988611	C₄	1	1	yes	characteristic
16	#17988553	C₄	1	2	yes	—
17	C₄⋊#11924003	C₂²	1	4	yes	—
18	#17988366	C₂²	1	8	yes	—
19	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
22	#346×C₇₃	C₂²	1	4	yes	—
23	C₂×#11924028	C₂×C₄	1	2	yes	—
24	#14556597	—	2	4	no	—
25	#14556597	—	2	4	no	—
26	C₂×#11924012	C₂×C₄	1	2	yes	—
27	C₂×#11924011	C₂×C₄	1	4	yes	—
28	C₂×C₈×D₇₃	C₂³	1	4	yes	—
29	#14556591	C₈	1	1	yes	characteristic
30	#14556588	C₈	1	1	yes	characteristic
31	#14556579	C₈	1	2	yes	—
32	#14556565	C₈	1	4	yes	—
33	C₄×D₂₉₂	C₂×C₄	1	4	yes	—
34	#14556531	—	2	4	no	—
35	#14556530	—	2	8	no	—
36	C₂×#11924003	C₂³	1	4	yes	—
37	C₁₆×D₇₃	—	2	8	no	—
38	C₄⋊#10596831	C₂×C₄	1	4	yes	—
39	#14556503	C₂×C₄	1	8	yes	—
40	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₁₆×C₇₃	—	2	2	no	—
44	C₂×C₁₆×C₇₃	D₄	1	4	yes	complemented
45	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
46	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
47	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₇₃	C₂³	1	2	yes	—
49	#11924028	—	2	4	no	—
50	#11924028	—	2	4	no	—
51	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
52	C₂×D₂₉₂	C₂×C₈	1	2	yes	—
53	C₂×D₂₉₂	C₂×C₈	1	4	yes	—
54	C₂×C₂²⋊D₇₃	C₂×C₈	1	4	yes	—
55	C₂×C₄×D₇₃	C₂²×C₄	1	4	yes	—
56	#11924012	—	2	4	no	—
57	#11924011	—	2	8	no	—
58	C₂×#10596831	C₂²×C₄	1	4	yes	—
59	C₈×D₇₃	—	2	4	no	—
60	C₈×D₇₃	—	2	4	no	—
61	C₄⋊Dic₇₃	C₂×C₈	1	4	yes	—
62	#11924005	C₂×C₈	1	8	yes	—
63	#11924003	—	2	4	no	—
64	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₇₃	—	2	2	no	—
67	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₇₃	#41	1	2	yes	—
71	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
72	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
73	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
74	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
75	C₁₆×C₇₃	—	2	2	no	—
76	C₁₆×C₇₃	C₂×D₄	1	4	yes	—
77	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
78	C₂²×D₇₃	C₂²×C₈	1	4	yes	—
79	C₂×C₄×C₇₃	—	4	2	no	—
80	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
82	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
83	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₇₃	#131	1	2	yes	—
85	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
86	C₂²⋊D₇₃	—	2	8	no	—
87	C₂×Dic₇₃	C₂²×C₈	1	4	yes	—
88	D₂₉₂	—	2	4	no	—
89	D₂₉₂	—	4	4	no	—
90	C₄×D₇₃	—	2	4	no	—
91	C₄×D₇₃	—	2	4	no	—
92	Dic₁₄₆	—	2	4	no	—
93	C₈×C₇₃	—	2	2	no	—
94	C₈×C₇₃	C₄×D₄	1	2	yes	—
95	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
96	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
97	#10596831	—	2	4	no	—
98	C₂²×C₇₃	—	2	2	no	—
99	C₂²×C₇₃	C₈×D₄	1	2	yes	—
100	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
101	C₂×D₇₃	—	2	8	no	—
102	C₂×D₇₃	—	2	4	no	—
103	C₄×C₇₃	—	2	2	no	—
104	C₄×C₇₃	—	2	2	no	—
105	C₄×C₇₃	C₈×D₄	1	2	yes	—
106	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
107	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
108	Dic₇₃	—	2	4	no	—
109	#33435	—	73	1	no	maximal, Sylow
110	C₂×C₇₃	—	2	2	no	—
111	C₂×C₇₃	C₂×C₈×D₄	1	1	yes	characteristic
112	C₂×C₇₃	#3407	1	2	yes	—
113	D₇₃	—	4	4	no	—
114	C₂×#504	—	73	2	no	—
115	#2931	—	73	1	no	—
116	#2168	—	73	8	no	—
117	#2150	—	73	1	no	—
118	#2113	—	73	2	no	—
119	C₂×#346	D₇₃	1	1	yes	characteristic, complemented
120	C₇₃	#33435	1	1	yes	characteristic, complemented, Sylow
121	C₂×#131	—	73	2	no	—
122	#532	—	73	1	no	—
123	#504	—	146	8	no	—
124	C₂×M₃₂	—	73	2	no	—
125	C₂×M₃₂	—	73	4	no	—
126	C₂²×C₁₆	—	73	4	no	—
127	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
128	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
129	#436	—	73	4	no	—
130	C₈×D₄	—	73	4	no	—
131	#423	—	73	1	no	—
132	#408	—	73	2	no	—
133	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
134	C₄⋊C₁₆	—	73	4	no	—
135	C₂²⋊C₁₆	—	73	8	no	—
136	#346	C₂×D₇₃	1	4	yes	—
137	C₂×#41	—	73	2	no	—
138	#131	—	146	4	no	—
139	#131	—	146	4	no	—
140	C₂×M₁₆	—	73	2	no	—
141	C₂×M₁₆	—	73	4	no	—
142	C₂²×C₈	—	73	4	no	—
143	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
144	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
145	C₄⋊Q₈	—	73	1	no	—
146	#127	—	73	1	no	—
147	#124	—	73	2	no	—
148	#121	—	73	4	no	—
149	C₄×D₄	—	73	4	no	—
150	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
151	M₃₂	—	146	4	no	—
152	M₃₂	—	146	8	no	—
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	—	73	4	no	—
155	C₂×C₁₆	—	146	8	no	—
156	C₂×C₁₆	D₂₉₂	1	4	yes	complemented
157	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
158	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
159	C₄⋊C₈	—	73	4	no	—
160	C₂²⋊C₈	—	73	8	no	—
161	C₄×C₈	C₄×D₇₃	1	2	yes	—
162	C₄×C₈	C₂²×D₇₃	1	2	yes	—
163	#41	—	146	4	no	—
164	#41	—	146	4	no	—
165	C₂×Q₈	—	73	2	no	—
166	C₂×D₄	—	73	4	no	—
167	C₂×D₄	—	73	4	no	—
168	C₂×D₄	—	73	2	no	—
169	C₂²×C₄	—	73	4	no	—
170	C₂²×C₄	C₈×D₇₃	1	2	yes	—
171	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
172	M₁₆	—	146	4	no	—
173	M₁₆	—	146	8	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	73	4	no	—
176	C₂×C₈	—	146	4	no	—
177	C₂×C₈	—	146	4	no	—
178	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
179	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
180	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
181	C₂×C₈	#11924028	1	2	yes	—
182	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
183	C₄⋊C₄	—	73	4	no	—
184	C₂²⋊C₄	—	73	8	no	—
185	C₄²	C₈×D₇₃	1	1	yes	characteristic
186	C₄²	C₈×D₇₃	1	1	yes	characteristic
187	C₄²	C₂×C₄×D₇₃	1	2	yes	—
188	C₁₆	—	2	2	no	—
189	C₁₆	—	146	4	no	—
190	C₁₆	C₂×D₂₉₂	1	4	yes	—
191	C₂³	—	73	4	no	—
192	C₂³	C₂×C₈×D₇₃	1	1	yes	characteristic
193	Q₈	—	146	4	no	—
194	D₄	—	292	4	no	—
195	D₄	—	146	8	no	—
196	D₄	—	146	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	73	4	no	—
199	C₂×C₄	—	146	4	no	—
200	C₂×C₄	—	146	4	no	—
201	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
202	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
203	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
204	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
205	C₂×C₄	#14556597	1	2	yes	—
206	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	146	4	no	—
209	C₈	C₄×D₂₉₂	1	2	yes	—
210	C₈	C₂²×D₂₉₂	1	1	yes	characteristic
211	C₈	C₂×#11924028	1	1	yes	characteristic
212	C₂²	—	2	2	no	—
213	C₂²	—	146	8	no	—
214	C₂²	—	146	4	no	—
215	C₂²	C₈×D₂₉₂	1	2	yes	—
216	C₂²	C₂²×C₈×D₇₃	1	1	yes	characteristic
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	146	4	no	—
220	C₄	C₈×D₂₉₂	1	2	yes	—
221	C₄	C₂×C₄×D₂₉₂	1	1	yes	characteristic
222	C₄	C₂×#14556597	1	1	yes	characteristic
223	C₂	—	2	2	no	—
224	C₂	—	292	4	no	—
225	C₂	C₂×C₈×D₂₉₂	1	1	yes	characteristic
226	C₂	#22623033	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇₃
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	C₂×#346×C₇₃	C₂
prime radical	#17989137	C₄
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₂×#346	D₇₃
hypocenter	C₇₃	#33435
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
1. C₂×#17989003	C₂	semidirect
3. #22611758	C₂	semidirect
4. #22611687	C₂	semidirect
5. #22611544	C₂	semidirect
6. C₂×#346×C₇₃	C₂	semidirect
44. C₂×C₁₆×C₇₃	D₄	semidirect
119. C₂×#346	D₇₃	semidirect
120. C₇₃	#33435	semidirect
156. C₂×C₁₆	D₂₉₂	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
#22618644	2
#22611758	2
#22611687	2
#22611544	2
C₂×#346×C₇₃	2
#33435	73