Subgroups — C₂×#22627040 #34034488

Order 18688.

982 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989700	C₂	1	2	yes	maximal
2	#22627040	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17989671	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×#10596842	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×#64408	C₂	1	1	yes	maximal, characteristic
6	#17989700	C₄	1	8	yes	—
7	#17989700	C₂²	1	8	yes	—
8	C₂×C₄⋊#10596842	C₄	1	4	yes	—
9	C₂×C₂²⋊#10596842	C₄	1	4	yes	—
10	C₂×C₄×#10596842	C₂²	1	4	yes	—
11	#17989671	C₂²	1	8	yes	complemented
12	C₈×#10596842	C₂²	1	8	yes	complemented
13	C₂²×#11924036	C₄	1	1	yes	characteristic
14	C₂²×C₄×#64408	C₂²	1	1	yes	characteristic
15	C₂×C₈×#64408	C₄	1	1	yes	characteristic
16	C₂×C₈×#64408	C₄	1	1	yes	characteristic
17	C₂×C₈×#64408	C₄	1	4	yes	—
18	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
19	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
20	C₂×C₈×#64408	C₂²	1	4	yes	—
21	C₂²×C₈×D₇₃	C₄	1	1	yes	characteristic
22	C₂²×#10596842	C₂×C₄	1	8	yes	—
23	C₄⋊#10596842	C₂×C₄	1	16	yes	—
24	C₂²⋊#10596842	C₂×C₄	1	16	yes	—
25	C₄×#10596842	C₂×C₄	1	8	yes	—
26	C₄×#10596842	C₂³	1	8	yes	—
27	C₂³×#64408	C₂×C₄	1	2	yes	—
28	C₂×#11924036	C₈	1	2	yes	complemented
29	C₂×#11924036	C₈	1	4	yes	complemented
30	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
31	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
32	C₂×#11924036	C₂×C₄	1	4	yes	—
33	C₂²×#10596833	C₈	1	2	yes	complemented
34	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
35	C₂×C₄×#64408	C₂×C₄	1	4	yes	—
36	C₂×C₄×#64408	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
38	C₂×C₄×#64408	C₂³	1	1	yes	characteristic
39	C₂×C₄×#64408	C₂³	1	2	yes	—
40	C₈×#64408	C₂×C₄	1	2	yes	—
41	C₈×#64408	C₂×C₄	1	2	yes	—
42	C₈×#64408	C₂×C₄	1	4	yes	—
43	C₈×#64408	C₂×C₄	1	4	yes	—
44	C₈×#64408	C₂³	1	2	yes	—
45	C₈×#64408	C₂³	1	2	yes	—
46	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×#10596831	C₈	1	1	yes	characteristic, complemented
48	C₂×C₈×D₇₃	C₈	1	1	yes	characteristic, complemented
49	C₂×C₈×D₇₃	C₈	1	1	yes	characteristic, complemented
50	C₂×C₈×D₇₃	C₈	1	4	yes	complemented
51	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₇₃	C₂×C₄	1	4	yes	—
54	C₂²×C₈×C₇₃	C₈	1	1	yes	characteristic, complemented
55	C₂×#10596842	—	2	16	no	—
56	C₂×#10596842	C₂²×C₄	1	16	yes	—
57	C₂²×#64408	—	2	8	no	—
58	C₂²×#64408	C₄²	1	2	yes	—
59	C₂²×#64408	C₄²	1	4	yes	—
60	C₂²×#64408	C₂²×C₄	1	4	yes	—
61	C₂²×#64408	C₂²×C₄	1	2	yes	—
62	#11924036	C₂×C₈	1	2	yes	—
63	#11924036	C₂×C₈	1	2	yes	—
64	#11924036	C₂×C₈	1	8	yes	complemented
65	#11924036	C₂²×C₄	1	2	yes	—
66	#11924036	C₂²×C₄	1	2	yes	—
67	C₂×#10596833	C₂×C₈	1	2	yes	—
68	C₂×#10596833	C₂×C₈	1	2	yes	—
69	C₂×#10596833	C₂×C₈	1	8	yes	complemented
70	C₄×#64408	C₄²	1	2	yes	—
71	C₄×#64408	C₄²	1	4	yes	—
72	C₄×#64408	C₂²×C₄	1	2	yes	—
73	C₄×#64408	C₂²×C₄	1	4	yes	—
74	C₄×#64408	C₂²×C₄	1	2	yes	—
75	C₄×#64408	C₂²×C₄	1	1	yes	characteristic
76	C₄×#64408	C₂⁴	1	1	yes	characteristic
77	C₂³×D₇₃	C₄²	1	1	yes	characteristic
78	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
79	C₂×C₄×D₇₃	C₄²	1	2	yes	—
80	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
81	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
82	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
84	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
87	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
88	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
89	C₂×#10596831	C₂×C₈	1	4	yes	complemented
90	C₈×D₇₃	C₂×C₈	1	2	yes	—
91	C₈×D₇₃	C₂×C₈	1	2	yes	—
92	C₈×D₇₃	C₂×C₈	1	4	yes	complemented
93	C₈×D₇₃	C₂×C₈	1	4	yes	complemented
94	C₈×D₇₃	C₂²×C₄	1	2	yes	—
95	C₈×D₇₃	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
97	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
98	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
99	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	complemented
100	C₂³×C₇₃	C₄×C₈	1	1	yes	characteristic
101	C₂²×D₇₃	—	2	4	no	—
102	C₂²×D₇₃	C₄×C₈	1	1	yes	characteristic
103	C₂²×D₇₃	C₄×C₈	1	2	yes	—
104	C₂²×D₇₃	C₂×C₄²	1	2	yes	—
105	C₂²×D₇₃	C₂×C₄²	1	1	yes	characteristic
106	C₂×#64408	—	2	4	no	—
107	C₂×#64408	—	2	8	no	—
108	C₂×#64408	—	2	8	no	—
109	C₂×#64408	C₂×C₄²	1	4	yes	—
110	C₂×#64408	C₂×C₄²	1	2	yes	—
111	C₂×#64408	#131	1	8	yes	—
112	C₂×#64408	C₂³×C₄	1	2	yes	—
113	#10596842	—	2	16	no	—
114	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
115	C₂×C₄×C₇₃	C₄×C₈	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
118	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
119	C₂×Dic₇₃	C₄×C₈	1	1	yes	characteristic
120	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
122	C₄×D₇₃	C₄×C₈	1	2	yes	—
123	C₄×D₇₃	C₄×C₈	1	2	yes	—
124	C₄×D₇₃	C₄×C₈	1	2	yes	—
125	C₄×D₇₃	C₂×C₄²	1	1	yes	characteristic
126	C₄×D₇₃	C₂×C₄²	1	2	yes	—
127	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
128	C₄×D₇₃	C₂²×C₈	1	2	yes	—
129	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
130	C₄×D₇₃	C₂²×C₈	1	2	yes	—
131	C₄×D₇₃	C₂³×C₄	1	1	yes	characteristic
132	#10596833	C₂²×C₈	1	4	yes	—
133	#10596833	C₂²×C₈	1	4	yes	—
134	C₈×C₇₃	C₂²×C₈	1	2	yes	—
135	C₈×C₇₃	C₂²×C₈	1	2	yes	—
136	#10596831	C₂²×C₈	1	2	yes	—
137	#10596831	C₂²×C₈	1	2	yes	—
138	C₂²×C₇₃	—	2	2	no	—
139	C₂²×C₇₃	C₂×C₄×C₈	1	2	yes	—
140	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
141	C₂×D₇₃	—	2	2	no	—
142	C₂×D₇₃	—	2	4	no	—
143	C₂×D₇₃	—	2	4	no	—
144	C₂×D₇₃	C₂×C₄×C₈	1	2	yes	—
145	C₂×D₇₃	C₂×C₄×C₈	1	1	yes	characteristic
146	C₂×D₇₃	#405	1	4	yes	—
147	C₂×D₇₃	C₂²×C₄²	1	1	yes	characteristic
148	#64408	—	2	4	no	—
149	#64408	#405	1	4	yes	—
150	#64408	C₂×#131	1	4	yes	—
151	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
152	C₄×C₇₃	C₂×C₄×C₈	1	2	yes	—
153	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
154	Dic₇₃	C₂×C₄×C₈	1	1	yes	characteristic
155	Dic₇₃	C₂×C₄×C₈	1	2	yes	—
156	Dic₇₃	C₂³×C₈	1	1	yes	characteristic
157	C₂×#1450	—	73	1	no	maximal, Sylow
158	C₂×C₇₃	—	2	2	no	—
159	C₂×C₇₃	#1450	1	2	yes	complemented
160	C₂×C₇₃	C₂²×C₄×C₈	1	1	yes	characteristic
161	D₇₃	—	2	2	no	—
162	D₇₃	#1450	1	2	yes	complemented
163	D₇₃	C₂×#405	1	2	yes	—
164	C₂×#431	—	73	2	no	—
165	C₂²×C₄×C₈	—	73	1	no	—
166	#1450	—	73	8	no	—
167	C₂×#322	—	73	2	no	—
168	C₂×C₈²	—	73	2	no	—
169	C₇₃	C₂×#1450	1	1	yes	characteristic, complemented, Sylow
170	C₂³×C₈	—	73	1	no	—
171	C₂³×C₈	—	73	1	no	—
172	C₂²×C₄²	—	73	1	no	—
173	#431	—	73	8	no	—
174	#431	—	73	8	no	—
175	C₂×C₄⋊C₈	—	73	4	no	—
176	C₂×C₂²⋊C₈	—	73	4	no	—
177	C₂×C₄×C₈	—	73	1	no	—
178	C₂×C₄×C₈	—	73	1	no	—
179	C₂×C₄×C₈	—	73	4	no	—
180	C₂×C₄×C₈	—	73	1	no	—
181	C₂×C₄×C₈	—	73	1	no	—
182	C₂×C₄×C₈	—	73	4	no	—
183	C₂×C₄×C₈	—	73	4	no	—
184	#322	—	73	8	no	—
185	C₈²	—	73	8	no	—
186	C₂³×C₄	—	73	1	no	—
187	C₂³×C₄	—	73	2	no	—
188	C₂²×C₈	—	73	2	no	—
189	C₂²×C₈	—	73	2	no	—
190	C₂²×C₈	—	73	4	no	—
191	C₂²×C₈	—	73	1	no	—
192	C₂²×C₈	—	73	1	no	—
193	C₂²×C₈	—	73	4	no	—
194	C₂²×C₈	—	73	1	no	—
195	C₂²×C₈	—	73	1	no	—
196	C₂²×C₈	—	73	1	no	—
197	C₂²×C₈	—	73	1	no	—
198	C₂²×C₈	—	73	1	no	—
199	C₂²×C₈	—	73	4	no	—
200	C₂²×C₈	—	73	4	no	—
201	C₂²×C₈	—	73	8	no	—
202	C₂²×C₈	#10596842	1	1	yes	characteristic, complemented
203	C₂×C₄²	—	73	1	no	—
204	C₂×C₄²	—	73	2	no	—
205	C₂×C₄²	—	73	2	no	—
206	C₂×C₄²	—	73	1	no	—
207	C₂×C₄²	—	73	2	no	—
208	C₂×C₄²	—	73	4	no	—
209	C₄⋊C₈	—	73	16	no	—
210	C₂²⋊C₈	—	73	16	no	—
211	C₄×C₈	—	73	8	no	—
212	C₄×C₈	—	73	8	no	—
213	C₄×C₈	—	73	2	no	—
214	C₄×C₈	—	73	2	no	—
215	C₄×C₈	—	73	2	no	—
216	C₄×C₈	—	73	2	no	—
217	C₄×C₈	—	73	4	no	—
218	C₄×C₈	—	73	4	no	—
219	C₂⁴	—	73	1	no	—
220	C₂²×C₄	—	73	2	no	—
221	C₂²×C₄	—	73	1	no	—
222	C₂²×C₄	—	73	1	no	—
223	C₂²×C₄	—	73	2	no	—
224	C₂²×C₄	—	73	2	no	—
225	C₂²×C₄	—	73	1	no	—
226	C₂²×C₄	—	73	1	no	—
227	C₂²×C₄	—	73	2	no	—
228	C₂²×C₄	—	73	1	no	—
229	C₂²×C₄	—	73	2	no	—
230	C₂²×C₄	—	73	4	no	—
231	C₂²×C₄	—	73	2	no	—
232	C₂²×C₄	—	73	4	no	—
233	C₂²×C₄	—	146	8	no	—
234	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
235	C₂×C₈	—	73	2	no	—
236	C₂×C₈	—	73	2	no	—
237	C₂×C₈	—	73	8	no	—
238	C₂×C₈	—	73	2	no	—
239	C₂×C₈	—	73	2	no	—
240	C₂×C₈	—	73	8	no	—
241	C₂×C₈	—	73	2	no	—
242	C₂×C₈	—	73	2	no	—
243	C₂×C₈	—	73	1	no	—
244	C₂×C₈	—	73	1	no	—
245	C₂×C₈	—	73	4	no	—
246	C₂×C₈	—	73	2	no	—
247	C₂×C₈	—	73	2	no	—
248	C₂×C₈	—	73	4	no	—
249	C₂×C₈	—	73	4	no	—
250	C₂×C₈	—	73	2	no	—
251	C₂×C₈	—	73	2	no	—
252	C₂×C₈	—	73	16	no	—
253	C₂×C₈	—	146	16	no	—
254	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
255	C₂×C₈	C₂×#10596842	1	1	yes	characteristic
256	C₂×C₈	C₂×#10596842	1	4	yes	complemented
257	C₄²	—	73	4	no	—
258	C₄²	—	73	2	no	—
259	C₄²	—	73	2	no	—
260	C₄²	—	73	1	no	—
261	C₄²	—	73	2	no	—
262	C₄²	—	73	4	no	—
263	C₄²	—	73	1	no	—
264	C₂³	—	73	1	no	—
265	C₂³	—	73	2	no	—
266	C₂³	—	73	2	no	—
267	C₂³	—	73	1	no	—
268	C₂³	—	146	4	no	—
269	C₂³	C₄×#10596842	1	1	yes	characteristic
270	C₂×C₄	—	73	2	no	—
271	C₂×C₄	—	73	2	no	—
272	C₂×C₄	—	73	8	no	—
273	C₂×C₄	—	73	2	no	—
274	C₂×C₄	—	73	1	no	—
275	C₂×C₄	—	73	2	no	—
276	C₂×C₄	—	73	1	no	—
277	C₂×C₄	—	73	1	no	—
278	C₂×C₄	—	73	2	no	—
279	C₂×C₄	—	73	1	no	—
280	C₂×C₄	—	73	2	no	—
281	C₂×C₄	—	73	2	no	—
282	C₂×C₄	—	73	2	no	—
283	C₂×C₄	—	73	2	no	—
284	C₂×C₄	—	73	1	no	—
285	C₂×C₄	—	73	1	no	—
286	C₂×C₄	—	73	2	no	—
287	C₂×C₄	—	73	4	no	—
288	C₂×C₄	—	146	8	no	—
289	C₂×C₄	—	146	8	no	—
290	C₂×C₄	—	146	4	no	—
291	C₂×C₄	C₄×#10596842	1	2	yes	—
292	C₂×C₄	C₄×#10596842	1	1	yes	characteristic
293	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
294	C₂×C₄	C₂²×#10596842	1	2	yes	—
295	C₈	—	73	4	no	—
296	C₈	—	73	4	no	—
297	C₈	—	73	2	no	—
298	C₈	—	73	2	no	—
299	C₈	—	146	16	no	—
300	C₈	C₂²×#10596842	1	2	yes	—
301	C₈	C₂²×#10596842	1	2	yes	—
302	C₂²	—	2	2	no	—
303	C₂²	—	73	1	no	—
304	C₂²	—	73	1	no	—
305	C₂²	—	73	4	no	—
306	C₂²	—	73	2	no	—
307	C₂²	—	146	4	no	—
308	C₂²	—	146	4	no	—
309	C₂²	—	146	2	no	—
310	C₂²	C₂×C₄×#10596842	1	2	yes	—
311	C₂²	C₂×C₄×#10596842	1	1	yes	characteristic
312	C₄	—	73	4	no	—
313	C₄	—	73	4	no	—
314	C₄	—	73	1	no	—
315	C₄	—	73	2	no	—
316	C₄	—	73	1	no	—
317	C₄	—	146	4	no	—
318	C₄	C₂×C₄×#10596842	1	1	yes	characteristic
319	C₄	C₂×C₄×#10596842	1	2	yes	—
320	C₄	C₂³×#10596842	1	1	yes	characteristic
321	C₂	—	2	2	no	—
322	C₂	—	73	2	no	—
323	C₂	—	73	2	no	—
324	C₂	—	146	2	no	—
325	C₂	#22627040	1	2	yes	direct product factor
326	C₂	C₂²×C₄×#10596842	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×#10596842
commutator	C₂×C₇₃	C₂²×C₄×C₈
Frattini	C₄	C₂³×#10596842
Fitting	C₂²×C₈×C₇₃	C₈
socle	C₂²×C₇₃	C₂×C₄×C₈
prime radical	C₂³×D₇₃	C₄²
cosocle	C₄×#64408	C₂⁴
hypercenter	C₂²×C₈	#10596842
hypocenter	C₇₃	C₂×#1450
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. #22627040	C₂	direct (N × H)
3. C₂×#17989671	C₂	semidirect
4. C₂×C₈×#10596842	C₂	semidirect
11. #17989671	C₂²	semidirect
12. C₈×#10596842	C₂²	semidirect
28. C₂×#11924036	C₈	semidirect
29. C₂×#11924036	C₈	semidirect
33. C₂²×#10596833	C₈	semidirect
47. C₂²×#10596831	C₈	semidirect
48. C₂×C₈×D₇₃	C₈	semidirect
49. C₂×C₈×D₇₃	C₈	semidirect
50. C₂×C₈×D₇₃	C₈	semidirect
54. C₂²×C₈×C₇₃	C₈	semidirect
64. #11924036	C₂×C₈	semidirect
69. C₂×#10596833	C₂×C₈	semidirect
89. C₂×#10596831	C₂×C₈	semidirect
92. C₈×D₇₃	C₂×C₈	semidirect
93. C₈×D₇₃	C₂×C₈	semidirect
99. C₂×C₈×C₇₃	C₂×C₈	semidirect
159. C₂×C₇₃	#1450	semidirect
162. D₇₃	#1450	semidirect
169. C₇₃	C₂×#1450	semidirect
202. C₂²×C₈	#10596842	semidirect
256. C₂×C₈	C₂×#10596842	semidirect
325. C₂	#22627040	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×#17989700	2
#22627040	2
C₂×#17989671	2
C₂×C₈×#10596842	2
C₂²×C₈×#64408	2
C₂×#1450	73

Subgroups — C2×#22627040 #34034488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22627040 #34034488