Subgroups — C₂×#22611768 #33239212

Order 18688.

662 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989004	C₂	1	2	yes	maximal, complemented
2	#22611768	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17988408	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#17988371	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×#11924003	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#443×C₇₃	C₂	1	1	yes	maximal, characteristic
7	#17989004	—	2	4	no	—
8	#17989004	C₂²	1	8	yes	complemented
9	C₂×#14556531	—	2	2	no	—
10	C₂×#14556531	C₂²	1	2	yes	—
11	C₂²×#11924003	—	2	2	no	—
12	C₂²×#11924003	C₂²	1	1	yes	characteristic
13	C₂²×#11924003	C₂²	1	1	yes	characteristic
14	#17988408	C₂²	1	4	yes	complemented
15	#17988371	C₂²	1	8	yes	complemented
16	C₄×#11924003	C₂²	1	4	yes	complemented
17	C₂×#135×C₇₃	C₄	1	1	yes	characteristic
18	C₂×#131×C₇₃	C₂²	1	2	yes	—
19	#443×C₇₃	C₄	1	4	yes	—
20	#443×C₇₃	C₂²	1	4	yes	—
21	C₂×#108×C₇₃	C₄	1	1	yes	characteristic
22	C₂×#104×C₇₃	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
24	#14556531	—	2	8	no	—
25	#14556531	—	2	2	no	—
26	#14556531	C₂³	1	4	yes	—
27	C₂×#11924003	—	2	8	no	—
28	C₂×#11924003	—	2	2	no	—
29	C₂×#11924003	—	2	1	no	—
30	C₂×#11924003	—	2	1	no	—
31	C₂×#11924003	D₄	1	1	yes	characteristic
32	C₂×#11924003	D₄	1	1	yes	characteristic
33	C₂×#11924003	D₄	1	2	yes	—
34	C₂×#11924003	C₂³	1	2	yes	—
35	C₂×#11924003	C₂³	1	2	yes	—
36	C₂×#41×C₇₃	C₂×C₄	1	2	yes	—
37	#135×C₇₃	C₈	1	4	yes	—
38	#135×C₇₃	C₂×C₄	1	4	yes	—
39	C₂×Q₁₆×C₇₃	C₈	1	1	yes	characteristic
40	C₂×SD₁₆×C₇₃	C₈	1	2	yes	—
41	C₂×D₈×C₇₃	C₈	1	1	yes	characteristic
42	#131×C₇₃	—	2	2	no	—
43	#131×C₇₃	—	2	2	no	—
44	#131×C₇₃	C₂×C₄	1	4	yes	—
45	#131×C₇₃	C₂³	1	4	yes	—
46	C₂×M₁₆×C₇₃	—	2	2	no	—
47	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₇₃	—	2	2	no	—
49	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
52	#108×C₇₃	C₂×C₄	1	4	yes	—
53	#104×C₇₃	C₂×C₄	1	8	yes	—
54	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₇₃	C₂³	1	2	yes	—
56	#11924003	—	2	4	no	—
57	#11924003	—	2	2	no	—
58	#11924003	C₂×D₄	1	2	yes	—
59	#11924003	C₂×D₄	1	2	yes	—
60	#41×C₇₃	—	2	2	no	—
61	#41×C₇₃	—	2	2	no	—
62	#41×C₇₃	C₂×C₈	1	4	yes	—
63	#41×C₇₃	C₂²×C₄	1	4	yes	—
64	C₂×Q₈×C₇₃	C₂×C₈	1	2	yes	—
65	C₂×D₄×C₇₃	—	2	2	no	—
66	C₂×D₄×C₇₃	C₂×C₈	1	2	yes	—
67	C₂²×C₄×C₇₃	—	2	1	no	—
68	C₂²×C₄×C₇₃	—	2	2	no	—
69	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
70	Q₁₆×C₇₃	C₂×C₈	1	4	yes	—
71	SD₁₆×C₇₃	C₂×C₈	1	8	yes	—
72	D₈×C₇₃	C₂×C₈	1	4	yes	—
73	M₁₆×C₇₃	—	2	2	no	—
74	M₁₆×C₇₃	—	2	4	no	—
75	M₁₆×C₇₃	—	2	4	no	—
76	M₁₆×C₇₃	C₂²×C₄	1	4	yes	—
77	C₂×C₈×C₇₃	—	2	2	no	—
78	C₂×C₈×C₇₃	—	2	4	no	—
79	C₂×C₈×C₇₃	—	2	4	no	—
80	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
81	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
82	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
83	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
88	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
89	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
90	C₄²×C₇₃	—	2	1	no	—
91	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
92	C₂³×C₇₃	—	2	2	no	—
93	C₂³×C₇₃	#131	1	1	yes	characteristic
94	Q₈×C₇₃	—	2	2	no	—
95	Q₈×C₇₃	C₂²×C₈	1	4	yes	—
96	D₄×C₇₃	—	2	4	no	—
97	D₄×C₇₃	—	2	4	no	—
98	D₄×C₇₃	—	2	2	no	—
99	D₄×C₇₃	C₂²×C₈	1	4	yes	—
100	C₂×C₄×C₇₃	—	2	4	no	—
101	C₂×C₄×C₇₃	—	2	2	no	—
102	C₂×C₄×C₇₃	—	2	2	no	—
103	C₂×C₄×C₇₃	—	2	4	no	—
104	C₂×C₄×C₇₃	—	2	4	no	—
105	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
110	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
111	C₈×C₇₃	—	2	4	no	—
112	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
113	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
114	C₈×C₇₃	C₂²×C₈	1	2	yes	—
115	C₈×C₇₃	C₂²×C₈	1	2	yes	—
116	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
117	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
118	C₂²×C₇₃	—	2	2	no	—
119	C₂²×C₇₃	—	2	4	no	—
120	C₂²×C₇₃	—	4	4	no	—
121	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
122	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
123	C₂²×C₇₃	C₂×#131	1	1	yes	characteristic
124	C₄×C₇₃	—	2	4	no	—
125	C₄×C₇₃	—	2	4	no	—
126	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
127	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
128	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
129	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
130	C₂×#2170	—	73	1	no	maximal, Sylow
131	C₂×C₇₃	—	2	1	no	—
132	C₂×C₇₃	—	2	1	no	—
133	C₂×C₇₃	—	4	4	no	—
134	C₂×C₇₃	#2170	1	2	yes	complemented
135	C₂×C₇₃	C₂×C₈×D₄	1	1	yes	characteristic
136	C₂×#504	—	73	2	no	—
137	C₂×#443	D₇₃	1	1	yes	characteristic
138	#2170	—	73	8	no	—
139	C₂×#364	—	73	1	no	—
140	C₂×#350	—	73	2	no	—
141	C₂×C₄×C₁₆	—	73	1	no	—
142	C₇₃	C₂×#2170	1	1	yes	characteristic, complemented, Sylow
143	C₂×#135	Dic₇₃	1	1	yes	characteristic
144	C₂×#131	C₂×D₇₃	1	2	yes	—
145	#504	—	73	8	no	—
146	#504	—	146	4	no	—
147	C₂×M₃₂	—	73	2	no	—
148	C₂×M₃₂	—	146	2	no	—
149	C₂²×C₁₆	—	73	1	no	—
150	C₂²×C₁₆	—	73	1	no	—
151	C₂²×C₁₆	—	146	2	no	—
152	#443	Dic₇₃	1	4	yes	—
153	#443	C₂×D₇₃	1	4	yes	—
154	C₂×#108	Dic₇₃	1	1	yes	characteristic
155	C₂×#104	Dic₇₃	1	2	yes	—
156	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
157	#364	—	73	4	no	—
158	#350	—	73	8	no	—
159	C₄×C₁₆	—	73	4	no	—
160	C₂×#41	C₂×Dic₇₃	1	2	yes	—
161	#135	#10596831	1	4	yes	—
162	#135	C₂×Dic₇₃	1	4	yes	—
163	C₂×Q₁₆	#10596831	1	1	yes	characteristic
164	C₂×SD₁₆	#10596831	1	2	yes	—
165	C₂×D₈	#10596831	1	1	yes	characteristic
166	#131	—	2	2	no	—
167	#131	—	2	2	no	—
168	#131	C₂×Dic₇₃	1	4	yes	—
169	#131	C₂²×D₇₃	1	4	yes	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	C₂×Dic₇₃	1	2	yes	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
175	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
176	M₃₂	—	73	4	no	—
177	M₃₂	—	146	8	no	—
178	M₃₂	—	146	2	no	—
179	C₂×C₁₆	—	73	2	no	—
180	C₂×C₁₆	—	73	1	no	—
181	C₂×C₁₆	—	73	1	no	—
182	C₂×C₁₆	—	73	2	no	—
183	C₂×C₁₆	—	73	2	no	—
184	C₂×C₁₆	—	146	8	no	—
185	C₂×C₁₆	—	146	2	no	—
186	C₂×C₁₆	—	146	1	no	—
187	C₂×C₁₆	—	146	1	no	—
188	#108	C₂×Dic₇₃	1	4	yes	—
189	#104	C₂×Dic₇₃	1	8	yes	—
190	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
191	C₄×C₈	C₂²×D₇₃	1	2	yes	—
192	#41	—	2	2	no	—
193	#41	—	2	2	no	—
194	#41	C₂×#10596831	1	4	yes	—
195	#41	C₂²×Dic₇₃	1	4	yes	—
196	C₂×Q₈	C₂×#10596831	1	2	yes	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	C₂×#10596831	1	2	yes	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
202	Q₁₆	C₂×#10596831	1	4	yes	—
203	SD₁₆	C₂×#10596831	1	8	yes	—
204	D₈	C₂×#10596831	1	4	yes	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	4	no	—
207	M₁₆	—	2	4	no	—
208	M₁₆	C₂²×Dic₇₃	1	4	yes	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	C₂×#10596831	1	1	yes	characteristic
213	C₂×C₈	C₂×#10596831	1	1	yes	characteristic
214	C₂×C₈	C₂×#10596831	1	2	yes	—
215	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
216	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
217	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
218	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
219	C₂×C₈	#11924027	1	1	yes	characteristic
220	C₂×C₈	#11924027	1	1	yes	characteristic
221	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
222	C₄²	—	2	1	no	—
223	C₄²	C₂²×Dic₇₃	1	2	yes	—
224	C₁₆	—	73	2	no	—
225	C₁₆	—	73	2	no	—
226	C₁₆	—	146	4	no	—
227	C₁₆	—	146	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	#14556598	1	1	yes	characteristic
230	Q₈	—	2	2	no	—
231	Q₈	C₂²×#10596831	1	4	yes	—
232	D₄	—	2	4	no	—
233	D₄	—	2	4	no	—
234	D₄	—	2	2	no	—
235	D₄	C₂²×#10596831	1	4	yes	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
242	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
243	C₂×C₄	C₂²×#10596831	1	1	yes	characteristic
244	C₂×C₄	C₂²×#10596831	1	1	yes	characteristic
245	C₂×C₄	#14556598	1	1	yes	characteristic
246	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
247	C₈	—	2	4	no	—
248	C₈	D₄×Dic₇₃	1	1	yes	characteristic
249	C₈	D₄×Dic₇₃	1	1	yes	characteristic
250	C₈	C₂²×#10596831	1	2	yes	—
251	C₈	C₂²×#10596831	1	2	yes	—
252	C₈	C₂×D₄×D₇₃	1	1	yes	characteristic
253	C₈	C₂×#11924027	1	1	yes	characteristic
254	C₂²	—	2	2	no	—
255	C₂²	—	2	4	no	—
256	C₂²	—	4	4	no	—
257	C₂²	D₄×#10596831	1	1	yes	characteristic
258	C₂²	D₄×#10596831	1	1	yes	characteristic
259	C₂²	C₂×#14556598	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	2	4	no	—
262	C₄	D₄×#10596831	1	1	yes	characteristic
263	C₄	D₄×#10596831	1	1	yes	characteristic
264	C₄	C₂×D₄×Dic₇₃	1	1	yes	characteristic
265	C₄	C₂³×#10596831	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	4	4	no	—
269	C₂	#22611768	1	2	yes	direct product factor
270	C₂	C₂×D₄×#10596831	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇₃
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#443×C₇₃	C₂
socle	C₂²×C₇₃	C₈×D₄
prime radical	C₂×#135×C₇₃	C₄
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₂×#443	D₇₃
hypocenter	C₇₃	C₂×#2170
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×#17989004	C₂	semidirect
2. #22611768	C₂	direct (N × H)
3. C₂×#17988408	C₂	semidirect
4. C₂×#17988371	C₂	semidirect
5. C₂×C₄×#11924003	C₂	semidirect
8. #17989004	C₂²	semidirect
14. #17988408	C₂²	semidirect
15. #17988371	C₂²	semidirect
16. C₄×#11924003	C₂²	semidirect
134. C₂×C₇₃	#2170	semidirect
142. C₇₃	C₂×#2170	semidirect
269. C₂	#22611768	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17989004	2
#22611768	2
C₂×#17988408	2
C₂×#17988371	2
C₂×C₄×#11924003	2
C₂×#443×C₇₃	2
C₂×#2170	73

Subgroups — C2×#22611768 #33239212

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22611768 #33239212