Subgroups — C₂×#22625481 #34017974

Order 18688.

830 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989606	C₂	1	2	yes	maximal
2	#22625481	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17989450	C₂	1	2	yes	maximal, complemented
4	C₂×#17989449	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
6	#17989606	—	2	4	no	—
7	#17989606	C₂²	1	8	yes	—
8	C₂×C₄⋊#10596833	C₂²	1	4	yes	—
9	C₂×#14556648	—	2	2	no	—
10	C₂×C₄×#10596833	—	2	2	no	—
11	#17989450	C₂²	1	8	yes	complemented
12	#17989449	C₂²	1	8	yes	complemented
13	C₂²×C₈×D₇₃	C₄	1	1	yes	characteristic
14	C₂²×C₈×D₇₃	C₄	1	1	yes	characteristic
15	C₂×C₄²×D₇₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
19	C₄×C₈×D₇₃	C₄	1	4	yes	—
20	C₄×C₈×D₇₃	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
22	C₂²×#10596833	—	2	4	no	—
23	C₄⋊#10596833	—	2	4	no	—
24	C₄⋊#10596833	C₂³	1	8	yes	—
25	#14556648	—	2	8	no	—
26	C₄×#10596833	—	2	8	no	—
27	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×D₇₃	Q₈	1	1	yes	characteristic
30	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
31	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×D₇₃	—	2	4	no	—
33	C₂×C₈×D₇₃	C₈	1	8	yes	complemented
34	C₂×C₈×D₇₃	C₂×C₄	1	4	yes	—
35	C₂×C₈×D₇₃	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
37	C₂×C₄×Dic₇₃	Q₈	1	1	yes	characteristic
38	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
39	C₄²×D₇₃	C₂×C₄	1	2	yes	—
40	C₄²×D₇₃	D₄	1	2	yes	—
41	C₄²×D₇₃	Q₈	1	2	yes	—
42	C₄²×D₇₃	C₂³	1	2	yes	—
43	C₄×#10596831	C₂×C₄	1	4	yes	—
44	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
45	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
46	C₈×Dic₇₃	C₂³	1	2	yes	—
47	C₈×Dic₇₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
52	C₂×#10596833	—	2	8	no	—
53	C₂×#10596833	—	4	8	no	—
54	C₂³×D₇₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄×D₇₃	—	2	4	no	—
59	C₂×C₄×D₇₃	—	2	4	no	—
60	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
61	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
62	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
63	C₂×C₄×D₇₃	C₂×C₈	1	4	yes	—
64	C₂×C₄×D₇₃	M₁₆	1	4	yes	—
65	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
66	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
67	C₂×C₄×D₇₃	C₂×Q₈	1	2	yes	—
68	C₂×#10596831	—	2	2	no	—
69	C₂×#10596831	C₂×C₈	1	4	yes	—
70	C₂×#10596831	C₂²×C₄	1	2	yes	—
71	C₂×#10596831	C₂²×C₄	1	2	yes	—
72	C₈×D₇₃	—	2	8	no	—
73	C₈×D₇₃	C₂×C₈	1	16	yes	complemented
74	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
75	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
76	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
77	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
78	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
79	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
80	C₄×Dic₇₃	C₂×Q₈	1	2	yes	—
81	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
82	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₇₃	—	2	2	no	—
86	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
87	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
89	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
90	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
91	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂²×D₇₃	—	2	2	no	—
93	C₂²×D₇₃	—	2	2	no	—
94	C₂²×D₇₃	C₄⋊C₈	1	4	yes	—
95	C₂²×D₇₃	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₇₃	—	2	2	no	—
97	C₂×C₄×C₇₃	#108	1	4	yes	—
98	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
102	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
103	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×Dic₇₃	—	2	2	no	—
105	C₂×Dic₇₃	C₄⋊C₈	1	4	yes	—
106	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×Dic₇₃	#135	1	4	yes	—
110	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
111	C₂×Dic₇₃	C₂²×Q₈	1	1	yes	characteristic
112	C₄×D₇₃	—	2	4	no	—
113	C₄×D₇₃	—	2	4	no	—
114	C₄×D₇₃	—	2	4	no	—
115	C₄×D₇₃	—	2	4	no	—
116	C₄×D₇₃	C₄⋊C₈	1	8	yes	—
117	C₄×D₇₃	C₂²×C₈	1	4	yes	—
118	C₄×D₇₃	C₂×M₁₆	1	4	yes	—
119	#10596833	—	4	8	no	—
120	C₈×C₇₃	—	2	2	no	—
121	C₈×C₇₃	C₂²×C₈	1	4	yes	—
122	#10596831	—	2	2	no	—
123	#10596831	C₂²×C₈	1	4	yes	—
124	C₂²×C₇₃	C₂×C₄⋊C₈	1	2	yes	—
125	C₂²×C₇₃	#427	1	2	yes	—
126	C₂²×C₇₃	C₂×#108	1	2	yes	—
127	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₂×D₇₃	—	2	8	no	—
129	C₂×D₇₃	—	2	2	no	—
130	C₂×D₇₃	—	2	2	no	—
131	C₂×D₇₃	C₂×C₄⋊C₈	1	4	yes	—
132	C₄×C₇₃	—	2	2	no	—
133	C₄×C₇₃	C₂×C₄⋊C₈	1	2	yes	—
134	C₄×C₇₃	C₂×#108	1	4	yes	—
135	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
136	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
137	Dic₇₃	—	2	1	no	—
138	Dic₇₃	—	2	1	no	—
139	Dic₇₃	C₂×C₄⋊C₈	1	4	yes	—
140	Dic₇₃	#427	1	2	yes	—
141	Dic₇₃	C₂×#135	1	2	yes	—
142	C₂×#1564	—	73	1	no	maximal, Sylow
143	C₂×C₇₃	#1564	1	4	yes	complemented
144	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
145	C₂×C₇₃	C₂×#427	1	1	yes	characteristic
146	C₂×C₇₃	C₂²×#108	1	1	yes	characteristic
147	D₇₃	—	2	4	no	—
148	C₂×#431	—	73	2	no	—
149	C₂²×C₄×C₈	—	73	1	no	—
150	#1564	—	73	8	no	—
151	C₂×#335	—	73	2	no	—
152	C₂×#334	—	73	2	no	—
153	C₇₃	C₂×#1564	1	1	yes	characteristic, complemented, Sylow
154	C₂³×C₈	—	73	1	no	—
155	C₂³×C₈	—	73	1	no	—
156	C₂²×C₄²	—	73	1	no	—
157	#431	—	73	8	no	—
158	#431	—	146	4	no	—
159	C₂×C₄⋊C₈	—	73	4	no	—
160	C₂×C₂²⋊C₈	—	146	2	no	—
161	C₂×C₄×C₈	—	73	1	no	—
162	C₂×C₄×C₈	—	73	1	no	—
163	C₂×C₄×C₈	—	73	1	no	—
164	C₂×C₄×C₈	—	73	4	no	—
165	C₂×C₄×C₈	—	73	4	no	—
166	C₂×C₄×C₈	—	146	2	no	—
167	C₂×C₄×C₈	#64408	1	1	yes	characteristic
168	#335	—	73	8	no	—
169	#334	—	73	8	no	—
170	C₂³×C₄	—	73	1	no	—
171	C₂³×C₄	—	73	1	no	—
172	C₂³×C₄	—	73	1	no	—
173	C₂²×C₈	—	73	1	no	—
174	C₂²×C₈	—	73	8	no	—
175	C₂²×C₈	—	73	4	no	—
176	C₂²×C₈	—	73	1	no	—
177	C₂²×C₈	—	73	4	no	—
178	C₂²×C₈	—	146	4	no	—
179	C₂²×C₈	—	146	4	no	—
180	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
181	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
182	C₂×C₄²	—	73	1	no	—
183	C₂×C₄²	—	73	1	no	—
184	C₂×C₄²	—	73	1	no	—
185	C₂×C₄²	—	73	2	no	—
186	C₂×C₄²	—	73	2	no	—
187	C₂×C₄²	—	73	2	no	—
188	C₂×C₄²	—	73	2	no	—
189	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
190	C₄⋊C₈	—	73	8	no	—
191	C₄⋊C₈	—	146	4	no	—
192	C₂²⋊C₈	—	146	8	no	—
193	C₄×C₈	—	73	2	no	—
194	C₄×C₈	—	73	2	no	—
195	C₄×C₈	—	73	2	no	—
196	C₄×C₈	—	73	2	no	—
197	C₄×C₈	—	73	4	no	—
198	C₄×C₈	—	146	8	no	—
199	C₄×C₈	C₂×#64408	1	4	yes	—
200	C₂⁴	—	73	1	no	—
201	C₂²×C₄	—	73	1	no	—
202	C₂²×C₄	—	73	1	no	—
203	C₂²×C₄	—	73	4	no	—
204	C₂²×C₄	—	73	4	no	—
205	C₂²×C₄	—	73	2	no	—
206	C₂²×C₄	—	73	2	no	—
207	C₂²×C₄	—	73	1	no	—
208	C₂²×C₄	—	73	2	no	—
209	C₂²×C₄	—	73	2	no	—
210	C₂²×C₄	—	73	2	no	—
211	C₂²×C₄	—	73	2	no	—
212	C₂²×C₄	—	146	4	no	—
213	C₂²×C₄	—	146	4	no	—
214	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
215	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
216	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	292	8	no	—
219	C₂×C₈	—	73	4	no	—
220	C₂×C₈	—	73	2	no	—
221	C₂×C₈	—	73	16	no	—
222	C₂×C₈	—	73	2	no	—
223	C₂×C₈	—	146	8	no	—
224	C₂×C₈	—	146	8	no	—
225	C₂×C₈	—	146	2	no	—
226	C₂×C₈	#11924036	1	4	yes	—
227	C₂×C₈	C₂²×#64408	1	2	yes	—
228	C₂×C₈	C₂²×#64408	1	2	yes	—
229	C₄²	—	73	1	no	—
230	C₄²	—	73	1	no	—
231	C₄²	—	73	2	no	—
232	C₄²	—	73	2	no	—
233	C₄²	—	73	2	no	—
234	C₄²	—	73	1	no	—
235	C₄²	—	73	2	no	—
236	C₄²	—	73	1	no	—
237	C₄²	C₄⋊#64408	1	2	yes	—
238	C₄²	C₂²×#64408	1	2	yes	—
239	C₂³	—	73	2	no	—
240	C₂³	—	73	4	no	—
241	C₂³	—	146	2	no	—
242	C₂³	—	146	2	no	—
243	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	73	1	no	—
246	C₂×C₄	—	73	2	no	—
247	C₂×C₄	—	73	1	no	—
248	C₂×C₄	—	73	4	no	—
249	C₂×C₄	—	73	4	no	—
250	C₂×C₄	—	73	4	no	—
251	C₂×C₄	—	73	4	no	—
252	C₂×C₄	—	73	8	no	—
253	C₂×C₄	—	73	1	no	—
254	C₂×C₄	—	73	1	no	—
255	C₂×C₄	—	146	4	no	—
256	C₂×C₄	—	146	4	no	—
257	C₂×C₄	—	146	2	no	—
258	C₂×C₄	—	146	4	no	—
259	C₂×C₄	—	146	4	no	—
260	C₂×C₄	#14556662	1	4	yes	—
261	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
262	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
264	C₂×C₄	C₂×#11924036	1	2	yes	—
265	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
266	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	292	8	no	—
269	C₈	—	73	4	no	—
270	C₈	—	146	2	no	—
271	C₈	C₂×#11924036	1	4	yes	—
272	C₂²	—	73	4	no	—
273	C₂²	—	146	2	no	—
274	C₂²	—	146	2	no	—
275	C₂²	—	146	8	no	—
276	C₂²	#17989588	1	2	yes	—
277	C₂²	#17989597	1	2	yes	—
278	C₂²	C₂×#14556662	1	2	yes	—
279	C₂²	C₂²×C₄⋊#64408	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	73	4	no	—
282	C₄	—	73	2	no	—
283	C₄	—	73	2	no	—
284	C₄	—	146	1	no	—
285	C₄	—	146	1	no	—
286	C₄	#17989588	1	2	yes	—
287	C₄	C₂×#14556662	1	4	yes	—
288	C₄	C₂²×#11924036	1	1	yes	characteristic
289	C₄	C₂×D₇₃⋊M₁₆	1	1	yes	characteristic
290	C₂	—	146	4	no	—
291	C₂	#22625481	1	4	yes	direct product factor
292	C₂	C₂×#17989588	1	1	yes	characteristic
293	C₂	C₂×#17989597	1	1	yes	characteristic
294	C₂	C₂²×#14556662	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄⋊#64408
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle	C₂³×C₇₃	C₂×C₄⋊C₄
prime radical	C₂³×D₇₃	C₄⋊C₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	C₂×#1564
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22625481	C₂	direct (N × H)
3. C₂×#17989450	C₂	semidirect
4. C₂×#17989449	C₂	semidirect
11. #17989450	C₂²	semidirect
12. #17989449	C₂²	semidirect
33. C₂×C₈×D₇₃	C₈	semidirect
73. C₈×D₇₃	C₂×C₈	semidirect
143. C₂×C₇₃	#1564	semidirect
153. C₇₃	C₂×#1564	semidirect
291. C₂	#22625481	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×#17989606	2
#22625481	2
C₂×#17989450	2
C₂×#17989449	2
C₂×C₄×C₈×D₇₃	2
C₂×#1564	73

Subgroups — C2×#22625481 #34017974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22625481 #34017974