Subgroups — C₂×#22620454 #33955599

Order 18688.

1902 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989140	C₂	1	2	yes	maximal
2	#22620454	C₂	1	16	yes	maximal, direct product factor
3	C₂×#17988995	C₂	1	4	yes	maximal, complemented
4	C₂×#17988994	C₂	1	4	yes	maximal, complemented
5	C₂×#17988625	C₂	1	1	yes	maximal, characteristic
6	C₂×#17988623	C₂	1	2	yes	maximal
7	C₂²×#14556501	C₂	1	1	yes	maximal, characteristic
8	C₂×#427×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂³×#10596831	C₂²	1	1	yes	characteristic
10	#17989140	—	2	8	no	—
11	#17989140	C₂²	1	16	yes	—
12	C₂×#14556590	C₂²	1	4	yes	—
13	C₂×#14556582	C₂²	1	4	yes	—
14	C₂×#14556568	—	2	4	no	—
15	C₂×C₄×Dic₁₄₆	—	2	4	no	—
16	C₂×#14556550	C₂²	1	2	yes	—
17	C₂²×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
18	#17988995	C₂²	1	32	yes	complemented
19	#17988994	C₂²	1	32	yes	complemented
20	#17988625	C₂²	1	8	yes	—
21	#17988623	C₂²	1	16	yes	—
22	C₂×#14556525	C₂²	1	2	yes	—
23	C₂×#14556524	C₂²	1	4	yes	—
24	C₂×#14556523	C₂²	1	2	yes	—
25	C₂×#14556522	C₂²	1	4	yes	—
26	C₂×#14556501	C₂²	1	2	yes	—
27	C₂×#14556501	C₂²	1	2	yes	—
28	C₂×#14556501	C₂²	1	8	yes	—
29	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
30	C₂×#117×C₇₃	C₂²	1	2	yes	—
31	#427×C₇₃	C₂²	1	8	yes	—
32	C₂×#107×C₇₃	C₂²	1	2	yes	—
33	C₂×#106×C₇₃	C₂²	1	2	yes	—
34	C₂²×Dic₁₄₆	—	2	4	no	—
35	C₂³×Dic₇₃	D₄	1	2	yes	—
36	C₂²×#10596831	Q₈	1	8	yes	complemented
37	C₂²×#10596831	C₂³	1	2	yes	—
38	C₂²×#10596831	C₂³	1	4	yes	—
39	#14556590	—	2	8	no	—
40	#14556590	C₂³	1	16	yes	—
41	#14556582	—	2	8	no	—
42	#14556582	C₂³	1	16	yes	—
43	#14556568	—	2	32	no	—
44	C₄×Dic₁₄₆	—	2	32	no	—
45	#14556550	—	2	4	no	—
46	#14556550	C₂³	1	8	yes	—
47	C₂×C₄⋊Dic₇₃	—	2	4	no	—
48	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
49	C₂×#11924007	—	2	4	no	—
50	C₂×#11924007	—	2	4	no	—
51	C₂×C₂²⋊Dic₇₃	—	2	2	no	—
52	C₂×C₄×Dic₇₃	—	2	2	no	—
53	C₂×C₄×Dic₇₃	D₄	1	4	yes	—
54	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
55	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
56	C₂×C₄×Dic₇₃	C₂³	1	4	yes	—
57	#14556525	C₂³	1	8	yes	—
58	#14556524	C₂³	1	16	yes	—
59	#14556523	C₂³	1	8	yes	—
60	#14556522	C₂³	1	16	yes	—
61	#14556501	C₂³	1	8	yes	—
62	#14556501	C₂³	1	8	yes	—
63	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₇₃	Q₈	1	8	yes	complemented
65	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
68	#117×C₇₃	—	2	4	no	—
69	#117×C₇₃	C₂³	1	8	yes	—
70	C₂×C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
71	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
72	C₂×C₄²×C₇₃	—	2	2	no	—
73	#107×C₇₃	C₂³	1	8	yes	—
74	#106×C₇₃	C₂³	1	8	yes	—
75	C₂×Dic₁₄₆	—	2	16	no	—
76	C₂×Dic₁₄₆	—	4	16	no	—
77	C₂²×Dic₇₃	—	2	4	no	—
78	C₂²×Dic₇₃	—	4	4	no	—
79	C₂²×Dic₇₃	C₂×D₄	1	4	yes	—
80	C₂²×Dic₇₃	C₂×D₄	1	8	yes	—
81	C₂×#10596831	C₂×Q₈	1	8	yes	—
82	C₂×#10596831	C₂×Q₈	1	16	yes	complemented
83	C₂×#10596831	C₂⁴	1	4	yes	—
84	C₄⋊Dic₇₃	—	2	4	no	—
85	C₄⋊Dic₇₃	—	2	16	no	—
86	C₄⋊Dic₇₃	C₂⁴	1	8	yes	—
87	#11924007	—	2	16	no	—
88	#11924007	—	2	16	no	—
89	C₂²⋊Dic₇₃	—	2	8	no	—
90	C₄×Dic₇₃	—	2	8	no	—
91	C₄×Dic₇₃	C₂×D₄	1	8	yes	—
92	C₄×Dic₇₃	C₂×D₄	1	4	yes	—
93	C₄×Dic₇₃	C₂⁴	1	4	yes	—
94	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₇₃	—	2	4	no	—
96	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₇₃	C₂×Q₈	1	4	yes	—
99	C₂²×C₄×C₇₃	C₂×Q₈	1	4	yes	—
100	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
101	C₂²×C₄×C₇₃	C₂⁴	1	2	yes	—
102	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
103	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
104	C₂×C₈×C₇₃	C₂×Q₈	1	16	yes	complemented
105	C₂×C₈×C₇₃	C₂⁴	1	2	yes	—
106	C₂×C₈×C₇₃	C₂⁴	1	2	yes	—
107	C₄⋊C₄×C₇₃	—	2	4	no	—
108	C₄⋊C₄×C₇₃	C₂⁴	1	8	yes	—
109	C₂²⋊C₄×C₇₃	—	2	8	no	—
110	C₄²×C₇₃	—	2	8	no	—
111	C₂³×C₇₃	—	2	4	no	—
112	C₂³×C₇₃	C₄⋊Q₈	1	4	yes	—
113	C₂³×C₇₃	C₂²×D₄	1	2	yes	—
114	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₃	—	2	8	no	—
116	C₂×C₄×C₇₃	—	4	8	no	—
117	C₂×C₄×C₇₃	C₄⋊Q₈	1	8	yes	—
118	C₂×C₄×C₇₃	C₄⋊Q₈	1	4	yes	—
119	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
120	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
122	C₂×C₄×C₇₃	C₂²×Q₈	1	4	yes	—
123	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
124	C₂×C₄×C₇₃	C₂²×Q₈	1	4	yes	—
125	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
126	C₂×Dic₇₃	—	2	8	no	—
127	C₂×Dic₇₃	—	2	8	no	—
128	C₂×Dic₇₃	—	4	8	no	—
129	C₂×Dic₇₃	—	4	8	no	—
130	C₂×Dic₇₃	C₂²×D₄	1	8	yes	—
131	Dic₁₄₆	—	4	32	no	—
132	C₈×C₇₃	C₂²×Q₈	1	4	yes	—
133	C₈×C₇₃	C₂²×Q₈	1	4	yes	—
134	#10596831	C₂²×Q₈	1	8	yes	—
135	C₂²×C₇₃	—	2	8	no	—
136	C₂²×C₇₃	—	2	4	no	—
137	C₂²×C₇₃	C₂×C₄⋊Q₈	1	4	yes	—
138	C₂²×C₇₃	C₂×C₄⋊Q₈	1	2	yes	—
139	C₂²×C₇₃	#575	1	4	yes	—
140	C₂²×C₇₃	C₂³×D₄	1	1	yes	characteristic
141	C₄×C₇₃	—	4	8	no	—
142	C₄×C₇₃	C₂×C₄⋊Q₈	1	4	yes	—
143	C₄×C₇₃	C₂×C₄⋊Q₈	1	2	yes	—
144	C₄×C₇₃	C₂³×Q₈	1	1	yes	characteristic
145	C₄×C₇₃	C₂³×Q₈	1	1	yes	characteristic
146	Dic₇₃	—	2	8	no	—
147	Dic₇₃	—	4	8	no	—
148	C₂×#3162	—	73	1	no	maximal, Sylow
149	C₂×C₇₃	—	2	4	no	—
150	C₂×C₇₃	#3162	1	4	yes	complemented
151	C₂×C₇₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
152	C₂×C₇₃	C₂×#575	1	2	yes	—
153	C₂×#533	—	73	2	no	—
154	#3162	—	73	16	no	—
155	C₂×#501	—	73	4	no	—
156	C₂×#501	—	73	4	no	—
157	C₂×#427	—	73	1	no	—
158	C₂×#427	—	73	2	no	—
159	C₂×#427	D₇₃	1	1	yes	characteristic
160	C₂²×#97	—	73	1	no	—
161	C₇₃	C₂×#3162	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₈	—	73	1	no	—
163	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
164	#533	—	73	16	no	—
165	#533	—	146	8	no	—
166	C₂×C₄⋊Q₈	—	73	4	no	—
167	C₂×#125	—	73	4	no	—
168	C₂×C₂²⋊Q₈	—	146	4	no	—
169	C₂×C₄×Q₈	—	146	4	no	—
170	C₂×#117	—	73	2	no	—
171	C₂×#117	C₂×D₇₃	1	2	yes	—
172	C₂²×C₄²	—	73	1	no	—
173	#501	—	73	32	no	—
174	#501	—	73	32	no	—
175	#427	—	73	8	no	—
176	#427	—	73	16	no	—
177	#427	C₂×D₇₃	1	8	yes	—
178	C₂×#107	—	73	2	no	—
179	C₂×#107	—	73	4	no	—
180	C₂×#107	C₂×D₇₃	1	2	yes	—
181	C₂×#106	—	73	2	no	—
182	C₂×#106	—	73	4	no	—
183	C₂×#106	C₂×D₇₃	1	2	yes	—
184	C₂×#97	—	73	2	no	—
185	C₂×#97	—	73	2	no	—
186	C₂×#97	—	73	8	no	—
187	C₂²×Q₈	—	146	4	no	—
188	C₂³×C₄	—	73	2	no	—
189	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
190	C₂²×C₈	—	73	8	no	—
191	C₂²×C₈	—	73	2	no	—
192	C₂²×C₈	—	73	4	no	—
193	C₂²×C₈	Dic₁₄₆	1	8	yes	complemented
194	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₇₃	1	4	yes	—
197	C₄⋊Q₈	—	73	16	no	—
198	C₄⋊Q₈	—	146	8	no	—
199	#125	—	73	16	no	—
200	#125	—	146	8	no	—
201	C₂²⋊Q₈	—	146	32	no	—
202	C₄×Q₈	—	146	32	no	—
203	#117	—	2	4	no	—
204	#117	—	73	8	no	—
205	#117	—	146	4	no	—
206	#117	C₂²×D₇₃	1	8	yes	—
207	C₂×C₄⋊C₄	—	73	4	no	—
208	C₂×C₄⋊C₄	—	146	4	no	—
209	C₂×C₄⋊C₄	—	146	4	no	—
210	C₂×C₄⋊C₄	—	146	4	no	—
211	C₂×C₄⋊C₄	C₂²×D₇₃	1	4	yes	—
212	C₂×C₂²⋊C₄	—	2	2	no	—
213	C₂×C₂²⋊C₄	—	146	2	no	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	—	73	4	no	—
216	C₂×C₄²	—	73	2	no	—
217	C₂×C₄²	—	73	2	no	—
218	C₂×C₄²	—	73	4	no	—
219	C₂×C₄²	—	146	2	no	—
220	#107	—	73	8	no	—
221	#107	—	73	16	no	—
222	#107	C₂²×D₇₃	1	8	yes	—
223	#106	—	73	8	no	—
224	#106	—	73	16	no	—
225	#106	C₂²×D₇₃	1	8	yes	—
226	#97	—	73	8	no	—
227	#97	—	73	8	no	—
228	C₂⁴	D₄×D₇₃	1	1	yes	characteristic
229	C₂×Q₈	—	292	16	no	—
230	C₂×Q₈	—	146	16	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	292	4	no	—
233	C₂²×C₄	—	73	8	no	—
234	C₂²×C₄	—	73	4	no	—
235	C₂²×C₄	—	146	4	no	—
236	C₂²×C₄	D₄×D₇₃	1	2	yes	—
237	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
238	C₂²×C₄	Q₈×D₇₃	1	4	yes	—
239	C₂²×C₄	C₂×Dic₁₄₆	1	4	yes	—
240	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
241	C₂²×C₄	C₂³×D₇₃	1	2	yes	—
242	C₂×C₈	—	73	8	no	—
243	C₂×C₈	—	73	16	no	—
244	C₂×C₈	—	73	4	no	—
245	C₂×C₈	C₂×Dic₁₄₆	1	4	yes	—
246	C₂×C₈	C₂×Dic₁₄₆	1	4	yes	—
247	C₂×C₈	C₂×Dic₁₄₆	1	16	yes	complemented
248	C₂×C₈	C₂³×D₇₃	1	2	yes	—
249	C₂×C₈	C₂³×D₇₃	1	2	yes	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	73	8	no	—
252	C₄⋊C₄	—	146	16	no	—
253	C₄⋊C₄	—	146	16	no	—
254	C₄⋊C₄	—	146	16	no	—
255	C₄⋊C₄	—	146	4	no	—
256	C₄⋊C₄	C₂³×D₇₃	1	8	yes	—
257	C₂²⋊C₄	—	2	8	no	—
258	C₂²⋊C₄	—	146	8	no	—
259	C₄²	—	2	8	no	—
260	C₄²	—	73	8	no	—
261	C₄²	—	73	4	no	—
262	C₄²	—	73	4	no	—
263	C₄²	—	146	8	no	—
264	C₂³	—	2	4	no	—
265	C₂³	#14556590	1	4	yes	—
266	C₂³	C₂×D₄×D₇₃	1	2	yes	—
267	C₂³	C₂×D₄×D₇₃	1	1	yes	characteristic
268	Q₈	—	292	32	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	4	8	no	—
271	C₂×C₄	—	292	8	no	—
272	C₂×C₄	—	292	8	no	—
273	C₂×C₄	—	73	8	no	—
274	C₂×C₄	—	146	8	no	—
275	C₂×C₄	—	146	8	no	—
276	C₂×C₄	#14556590	1	8	yes	—
277	C₂×C₄	#14556590	1	4	yes	—
278	C₂×C₄	C₂×D₄×D₇₃	1	2	yes	—
279	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
280	C₂×C₄	C₂×Q₈×D₇₃	1	4	yes	—
281	C₂×C₄	C₂×Q₈×D₇₃	1	2	yes	—
282	C₂×C₄	C₂²×Dic₁₄₆	1	2	yes	—
283	C₂×C₄	C₂²×Dic₁₄₆	1	4	yes	—
284	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
285	C₈	—	73	8	no	—
286	C₈	C₂²×Dic₁₄₆	1	4	yes	—
287	C₈	C₂²×Dic₁₄₆	1	4	yes	—
288	C₂²	—	2	8	no	—
289	C₂²	—	2	4	no	—
290	C₂²	C₂×#14556590	1	4	yes	—
291	C₂²	C₂×#14556590	1	2	yes	—
292	C₂²	#17989384	1	4	yes	—
293	C₂²	C₂²×D₄×D₇₃	1	1	yes	characteristic
294	C₄	—	4	8	no	—
295	C₄	—	292	8	no	—
296	C₄	—	146	8	no	—
297	C₄	C₂×#14556590	1	2	yes	—
298	C₄	C₂×#14556590	1	4	yes	—
299	C₄	C₂²×Q₈×D₇₃	1	1	yes	characteristic
300	C₄	C₂³×Dic₁₄₆	1	1	yes	characteristic
301	C₂	—	2	4	no	—
302	C₂	#22620454	1	4	yes	direct product factor
303	C₂	C₂²×#14556590	1	1	yes	characteristic
304	C₂	C₂×#17989384	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₃
commutator, cosocle	C₂×C₄×C₇₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₃
Fitting	C₂×#427×C₇₃	C₂
socle	C₂³×C₇₃	C₂²×D₄
prime radical	C₂⁴×C₇₃	C₂×D₄
hypercenter	C₂×#427	D₇₃
hypocenter	C₇₃	C₂×#3162
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
2. #22620454	C₂	direct (N × H)
3. C₂×#17988995	C₂	semidirect
4. C₂×#17988994	C₂	semidirect
18. #17988995	C₂²	semidirect
19. #17988994	C₂²	semidirect
36. C₂²×#10596831	Q₈	semidirect
64. C₂²×C₈×C₇₃	Q₈	semidirect
82. C₂×#10596831	C₂×Q₈	semidirect
104. C₂×C₈×C₇₃	C₂×Q₈	semidirect
150. C₂×C₇₃	#3162	semidirect
161. C₇₃	C₂×#3162	semidirect
193. C₂²×C₈	Dic₁₄₆	semidirect
247. C₂×C₈	C₂×Dic₁₄₆	semidirect
302. C₂	#22620454	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×#17989140	2
#22620454	2
C₂×#17988995	2
C₂×#17988994	2
C₂×#17988625	2
C₂×#17988623	2
C₂²×#14556501	2
C₂×#427×C₇₃	2
C₂×#3162	73

Subgroups — C2×#22620454 #33955599

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22620454 #33955599