Subgroups — C₂×#22620440 #33955584

Order 18688.

1942 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989142	C₂	1	2	yes	maximal
2	#22620440	C₂	1	16	yes	maximal, direct product factor
3	C₂×#17988990	C₂	1	2	yes	maximal, complemented
4	C₂×#17988987	C₂	1	4	yes	maximal, complemented
5	C₂×#17988985	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17988622	C₂	1	4	yes	maximal
7	C₂×C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#498×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₇₃	C₂²	1	2	yes	—
10	#17989142	—	2	8	no	—
11	#17989142	C₂²	1	16	yes	—
12	C₂×#14556591	C₂²	1	2	yes	—
13	C₂×#14556582	C₂²	1	4	yes	—
14	C₂×#14556571	—	2	2	no	—
15	C₂×#14556567	—	2	2	no	—
16	C₂×Q₈×Dic₇₃	—	2	2	no	—
17	C₂×#14556556	—	2	2	no	—
18	C₂×#14556549	C₂²	1	4	yes	—
19	C₂×C₄²×D₇₃	C₂²	1	1	yes	characteristic
20	#17988990	C₂²	1	16	yes	complemented
21	#17988987	C₂²	1	32	yes	complemented
22	#17988985	C₂²	1	8	yes	complemented
23	#17988622	C₂²	1	32	yes	—
24	C₂×#14556524	C₂²	1	8	yes	—
25	C₂×#14556523	C₂²	1	4	yes	—
26	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
27	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
28	C₄×C₈×D₇₃	C₂²	1	8	yes	—
29	C₂×C₄⋊Q₈×C₇₃	C₂²	1	2	yes	—
30	#498×C₇₃	C₂²	1	8	yes	complemented
31	C₂×#106×C₇₃	C₂²	1	4	yes	—
32	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
33	C₂²×Dic₁₄₆	—	2	2	no	—
34	C₂²×C₄×D₇₃	D₄	1	2	yes	—
35	C₂²×C₄×D₇₃	C₂³	1	1	yes	characteristic
36	C₂²×#10596831	C₂³	1	2	yes	—
37	C₂×C₈×D₇₃	Q₈	1	16	yes	complemented
38	C₂×C₈×D₇₃	C₂³	1	8	yes	—
39	#14556591	—	2	4	no	—
40	#14556591	C₂³	1	8	yes	—
41	#14556582	—	2	8	no	—
42	#14556582	C₂³	1	16	yes	—
43	#14556571	—	2	16	no	—
44	#14556567	—	2	16	no	—
45	Q₈×Dic₇₃	—	2	16	no	—
46	#14556556	—	2	16	no	—
47	#14556549	—	2	8	no	—
48	#14556549	C₂³	1	16	yes	—
49	C₂×C₄⋊Dic₇₃	—	2	4	no	—
50	C₂×C₄⋊Dic₇₃	—	2	2	no	—
51	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
52	C₂×#11924007	—	2	4	no	—
53	C₂×#11924005	—	2	4	no	—
54	C₂×C₄×Dic₇₃	—	2	4	no	—
55	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
56	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
57	C₄²×D₇₃	D₄	1	4	yes	—
58	C₄²×D₇₃	C₂³	1	4	yes	—
59	#14556524	C₂³	1	32	yes	—
60	#14556523	C₂³	1	16	yes	—
61	C₄×#10596831	C₂³	1	4	yes	—
62	C₈×Dic₇₃	C₂³	1	8	yes	—
63	C₂²×Q₈×C₇₃	—	2	2	no	—
64	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
65	C₄⋊Q₈×C₇₃	—	2	4	no	—
66	C₄⋊Q₈×C₇₃	C₂³	1	8	yes	—
67	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
68	C₂×C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
69	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
70	#106×C₇₃	C₂³	1	16	yes	—
71	C₄×C₈×C₇₃	C₂³	1	4	yes	—
72	C₂³×D₇₃	C₂×D₄	1	1	yes	characteristic
73	C₂×Dic₁₄₆	—	2	8	no	—
74	C₂×Dic₁₄₆	—	4	8	no	—
75	C₂²×Dic₇₃	—	2	4	no	—
76	C₂²×Dic₇₃	C₂×D₄	1	2	yes	—
77	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×D₇₃	—	2	8	no	—
79	C₂×C₄×D₇₃	C₂×D₄	1	8	yes	—
80	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
81	C₂×C₄×D₇₃	C₂×Q₈	1	8	yes	—
82	C₂×C₄×D₇₃	C₂⁴	1	2	yes	—
83	C₂×#10596831	C₂×Q₈	1	8	yes	—
84	C₂×#10596831	C₂⁴	1	4	yes	—
85	C₈×D₇₃	C₂×Q₈	1	32	yes	complemented
86	C₄⋊Dic₇₃	—	2	8	no	—
87	C₄⋊Dic₇₃	—	2	4	no	—
88	C₄⋊Dic₇₃	—	2	16	no	—
89	C₄⋊Dic₇₃	C₂⁴	1	8	yes	—
90	#11924007	—	2	16	no	—
91	#11924005	—	2	16	no	—
92	C₄×Dic₇₃	—	2	16	no	—
93	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
94	C₄×Dic₇₃	C₂×D₄	1	8	yes	—
95	C₄×Dic₇₃	C₂⁴	1	2	yes	—
96	C₂×Q₈×C₇₃	—	2	8	no	—
97	C₂×Q₈×C₇₃	—	4	8	no	—
98	C₂²×C₄×C₇₃	—	2	4	no	—
99	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
100	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
101	C₂×C₈×C₇₃	C₂×Q₈	1	8	yes	—
102	C₂×C₈×C₇₃	C₂⁴	1	4	yes	—
103	C₄⋊C₄×C₇₃	—	2	8	no	—
104	C₄⋊C₄×C₇₃	—	2	4	no	—
105	C₄⋊C₄×C₇₃	C₂⁴	1	8	yes	—
106	C₄²×C₇₃	C₂×D₄	1	2	yes	—
107	C₄²×C₇₃	C₂⁴	1	2	yes	—
108	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
109	C₂²×D₇₃	—	2	4	no	—
110	C₂²×D₇₃	C₄⋊Q₈	1	4	yes	—
111	C₂²×D₇₃	C₂²×D₄	1	2	yes	—
112	Q₈×C₇₃	—	4	16	no	—
113	C₂×C₄×C₇₃	—	2	8	no	—
114	C₂×C₄×C₇₃	—	4	8	no	—
115	C₂×C₄×C₇₃	#135	1	8	yes	—
116	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂²×D₄	1	4	yes	—
118	C₂×C₄×C₇₃	C₂²×Q₈	1	4	yes	—
119	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
120	C₂×Dic₇₃	—	2	4	no	—
121	C₂×Dic₇₃	—	2	8	no	—
122	C₂×Dic₇₃	—	4	8	no	—
123	C₂×Dic₇₃	C₄⋊Q₈	1	4	yes	—
124	C₂×Dic₇₃	C₂²×D₄	1	4	yes	—
125	C₂×Dic₇₃	C₂²×D₄	1	2	yes	—
126	C₄×D₇₃	—	2	16	no	—
127	C₄×D₇₃	C₄⋊Q₈	1	8	yes	—
128	C₄×D₇₃	C₂²×Q₈	1	8	yes	—
129	Dic₁₄₆	—	4	16	no	—
130	C₈×C₇₃	C₂²×Q₈	1	8	yes	—
131	#10596831	C₂²×Q₈	1	8	yes	—
132	C₂²×C₇₃	C₂×C₄⋊Q₈	1	2	yes	—
133	C₂²×C₇₃	C₂×#135	1	4	yes	—
134	C₂²×C₇₃	C₂³×D₄	1	1	yes	characteristic
135	C₂×D₇₃	—	2	8	no	—
136	C₂×D₇₃	—	2	4	no	—
137	C₂×D₇₃	C₂×C₄⋊Q₈	1	4	yes	—
138	C₄×C₇₃	—	4	8	no	—
139	C₄×C₇₃	C₂×C₄⋊Q₈	1	2	yes	—
140	C₄×C₇₃	C₂×#135	1	8	yes	—
141	C₄×C₇₃	C₂³×Q₈	1	2	yes	—
142	Dic₇₃	—	2	4	no	—
143	Dic₇₃	—	4	8	no	—
144	Dic₇₃	C₂×C₄⋊Q₈	1	4	yes	—
145	C₂×#3160	—	73	1	no	maximal, Sylow
146	C₂×C₇₃	#3160	1	4	yes	complemented
147	C₂×C₇₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
148	C₂×C₇₃	C₂²×#135	1	2	yes	—
149	D₇₃	—	2	4	no	—
150	C₂×#533	—	73	2	no	—
151	#3160	—	73	16	no	—
152	C₂×#500	—	73	2	no	—
153	C₂×#499	—	73	4	no	—
154	C₂×#498	—	73	1	no	—
155	C₂×#498	D₇₃	1	1	yes	characteristic, complemented
156	C₂×#427	—	73	4	no	—
157	C₂²×C₄×C₈	—	73	1	no	—
158	C₇₃	C₂×#3160	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₈	—	73	2	no	—
160	#533	—	73	16	no	—
161	#533	—	146	8	no	—
162	C₂×C₄⋊Q₈	—	73	2	no	—
163	C₂×C₄⋊Q₈	C₂×D₇₃	1	2	yes	—
164	C₂×#125	—	73	4	no	—
165	C₂×C₂²⋊Q₈	—	146	2	no	—
166	C₂×C₂²⋊Q₈	—	146	2	no	—
167	C₂×C₄×Q₈	—	146	2	no	—
168	C₂×C₄×Q₈	—	146	2	no	—
169	C₂×#117	—	73	4	no	—
170	C₂²×C₄²	—	73	1	no	—
171	#500	—	73	16	no	—
172	#499	—	73	32	no	—
173	#498	—	73	8	no	—
174	#498	C₂×D₇₃	1	8	yes	complemented
175	#427	—	73	32	no	—
176	C₂×#107	—	73	8	no	—
177	C₂×#106	—	73	4	no	—
178	C₂×#106	C₂×D₇₃	1	4	yes	—
179	C₂×C₄×C₈	—	73	2	no	—
180	C₂×C₄×C₈	—	73	1	no	—
181	C₂×C₄×C₈	—	73	8	no	—
182	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
183	C₂²×Q₈	—	2	2	no	—
184	C₂²×Q₈	—	146	2	no	—
185	C₂³×C₄	—	73	1	no	—
186	C₂³×C₄	—	73	2	no	—
187	C₂²×C₈	—	73	2	no	—
188	C₂²×C₈	—	73	16	no	—
189	C₂²×C₈	—	73	8	no	—
190	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
191	C₄⋊Q₈	—	2	4	no	—
192	C₄⋊Q₈	—	73	8	no	—
193	C₄⋊Q₈	—	146	4	no	—
194	C₄⋊Q₈	C₂²×D₇₃	1	8	yes	—
195	#125	—	73	16	no	—
196	#125	—	146	8	no	—
197	C₂²⋊Q₈	—	146	16	no	—
198	C₂²⋊Q₈	—	146	16	no	—
199	C₄×Q₈	—	146	16	no	—
200	C₄×Q₈	—	146	16	no	—
201	#117	—	73	16	no	—
202	#117	—	146	8	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	73	4	no	—
205	C₂×C₄⋊C₄	—	146	4	no	—
206	C₂×C₄⋊C₄	—	146	4	no	—
207	C₂×C₄⋊C₄	—	146	2	no	—
208	C₂×C₄⋊C₄	C₂²×D₇₃	1	4	yes	—
209	C₂×C₂²⋊C₄	—	146	4	no	—
210	C₂×C₄²	—	73	2	no	—
211	C₂×C₄²	—	73	1	no	—
212	C₂×C₄²	—	73	4	no	—
213	C₂×C₄²	—	73	4	no	—
214	C₂×C₄²	—	146	4	no	—
215	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
216	#107	—	73	32	no	—
217	#106	—	73	16	no	—
218	#106	C₂²×D₇₃	1	16	yes	—
219	C₄×C₈	—	73	8	no	—
220	C₄×C₈	—	73	4	no	—
221	C₄×C₈	C₂²×D₇₃	1	4	yes	—
222	C₂⁴	—	73	1	no	—
223	C₂×Q₈	—	2	8	no	—
224	C₂×Q₈	—	4	8	no	—
225	C₂×Q₈	—	292	8	no	—
226	C₂×Q₈	—	146	8	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	73	1	no	—
229	C₂²×C₄	—	73	2	no	—
230	C₂²×C₄	—	73	8	no	—
231	C₂²×C₄	—	73	2	no	—
232	C₂²×C₄	—	73	2	no	—
233	C₂²×C₄	—	73	8	no	—
234	C₂²×C₄	—	146	4	no	—
235	C₂²×C₄	—	146	8	no	—
236	C₂²×C₄	D₄×D₇₃	1	2	yes	—
237	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
238	C₂×C₈	—	73	8	no	—
239	C₂×C₈	—	73	4	no	—
240	C₂×C₈	—	73	32	no	—
241	C₂×C₈	Q₈×D₇₃	1	8	yes	—
242	C₂×C₈	C₂³×D₇₃	1	4	yes	—
243	C₄⋊C₄	—	2	8	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	73	8	no	—
246	C₄⋊C₄	—	146	16	no	—
247	C₄⋊C₄	—	146	16	no	—
248	C₄⋊C₄	—	146	4	no	—
249	C₄⋊C₄	—	146	8	no	—
250	C₄⋊C₄	C₂³×D₇₃	1	8	yes	—
251	C₂²⋊C₄	—	146	16	no	—
252	C₄²	—	73	8	no	—
253	C₄²	—	73	2	no	—
254	C₄²	—	73	2	no	—
255	C₄²	—	146	16	no	—
256	C₄²	D₄×D₇₃	1	2	yes	—
257	C₄²	C₂³×D₇₃	1	2	yes	—
258	C₂³	—	73	4	no	—
259	C₂³	—	73	2	no	—
260	C₂³	—	146	4	no	—
261	C₂³	C₂×D₄×D₇₃	1	1	yes	characteristic
262	Q₈	—	4	16	no	—
263	Q₈	—	292	16	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	4	8	no	—
266	C₂×C₄	—	292	8	no	—
267	C₂×C₄	—	73	4	no	—
268	C₂×C₄	—	73	2	no	—
269	C₂×C₄	—	73	8	no	—
270	C₂×C₄	—	73	8	no	—
271	C₂×C₄	—	73	4	no	—
272	C₂×C₄	—	146	8	no	—
273	C₂×C₄	—	146	4	no	—
274	C₂×C₄	—	146	16	no	—
275	C₂×C₄	#14556612	1	8	yes	—
276	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
277	C₂×C₄	C₂×D₄×D₇₃	1	4	yes	—
278	C₂×C₄	C₂×Q₈×D₇₃	1	4	yes	—
279	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
280	C₈	—	73	8	no	—
281	C₈	C₂×Q₈×D₇₃	1	8	yes	—
282	C₂²	—	73	4	no	—
283	C₂²	—	146	8	no	—
284	C₂²	—	146	4	no	—
285	C₂²	C₄⋊Q₈×D₇₃	1	2	yes	—
286	C₂²	C₂×#14556612	1	4	yes	—
287	C₂²	C₂²×D₄×D₇₃	1	1	yes	characteristic
288	C₄	—	4	8	no	—
289	C₄	—	292	8	no	—
290	C₄	—	73	4	no	—
291	C₄	—	146	4	no	—
292	C₄	C₄⋊Q₈×D₇₃	1	2	yes	—
293	C₄	C₂×#14556612	1	8	yes	—
294	C₄	C₂²×Q₈×D₇₃	1	2	yes	—
295	C₂	—	146	4	no	—
296	C₂	#22620440	1	4	yes	direct product factor
297	C₂	C₂×C₄⋊Q₈×D₇₃	1	1	yes	characteristic
298	C₂	C₂²×#14556612	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₇₃
commutator, cosocle	C₂×C₄×C₇₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₃
Fitting	C₂×#498×C₇₃	C₂
socle	C₂³×C₇₃	C₂²×D₄
prime radical	C₂³×D₇₃	C₂×D₄
hypercenter	C₂×#498	D₇₃
hypocenter	C₇₃	C₂×#3160
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

Complements

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

N	H ≅ G/N	type
2. #22620440	C₂	direct (N × H)
3. C₂×#17988990	C₂	semidirect
4. C₂×#17988987	C₂	semidirect
5. C₂×#17988985	C₂	semidirect
8. C₂×#498×C₇₃	C₂	semidirect
20. #17988990	C₂²	semidirect
21. #17988987	C₂²	semidirect
22. #17988985	C₂²	semidirect
30. #498×C₇₃	C₂²	semidirect
37. C₂×C₈×D₇₃	Q₈	semidirect
85. C₈×D₇₃	C₂×Q₈	semidirect
146. C₂×C₇₃	#3160	semidirect
155. C₂×#498	D₇₃	semidirect
158. C₇₃	C₂×#3160	semidirect
174. #498	C₂×D₇₃	semidirect
296. C₂	#22620440	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×#17989142	2
#22620440	2
C₂×#17988990	2
C₂×#17988987	2
C₂×#17988985	2
C₂×#17988622	2
C₂×C₄×C₈×D₇₃	2
C₂×#498×C₇₃	2
C₂×#3160	73

Subgroups — C2×#22620440 #33955584

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22620440 #33955584