Subgroups — C₂×#22608816 #33055712

Order 18688.

822 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988665	C₂	1	2	yes	maximal, complemented
2	#22608816	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17988340	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄⋊#10596831	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×#10596831	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₈×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11924012	—	2	2	no	—
8	#17988665	—	2	4	no	—
9	#17988665	C₂²	1	8	yes	complemented
10	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
11	C₂×C₂²⋊#10596831	—	2	2	no	—
12	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
13	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
14	#17988340	C₂²	1	4	yes	complemented
15	D₄⋊#10596831	C₂²	1	8	yes	complemented
16	C₈×#10596831	C₂²	1	4	yes	complemented
17	C₂²×D₈×C₇₃	C₄	1	1	yes	characteristic
18	C₂×C₄×D₄×C₇₃	C₂²	1	2	yes	—
19	C₄×D₈×C₇₃	C₄	1	4	yes	—
20	C₄×D₈×C₇₃	C₂²	1	4	yes	—
21	C₂×#107×C₇₃	C₄	1	1	yes	characteristic
22	C₂×#102×C₇₃	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
24	C₂×#11924012	—	2	8	no	—
25	C₂×#11924012	—	4	8	no	—
26	C₂²×#10596831	—	2	2	no	—
27	C₂²×#10596831	—	2	1	no	—
28	C₂²×#10596831	D₄	1	1	yes	characteristic
29	C₂²×#10596831	D₄	1	1	yes	characteristic
30	C₄⋊#10596831	—	2	2	no	—
31	C₄⋊#10596831	C₂³	1	4	yes	—
32	C₂²⋊#10596831	—	2	8	no	—
33	C₄×#10596831	—	2	1	no	—
34	C₄×#10596831	D₄	1	2	yes	—
35	C₄×#10596831	C₂³	1	2	yes	—
36	C₄×#10596831	C₂³	1	2	yes	—
37	C₂²×D₄×C₇₃	C₂×C₄	1	2	yes	—
38	C₂³×C₄×C₇₃	—	2	2	no	—
39	C₂×D₈×C₇₃	—	2	4	no	—
40	C₂×D₈×C₇₃	C₂×C₄	1	4	yes	—
41	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₄×D₄×C₇₃	—	2	2	no	—
44	C₄×D₄×C₇₃	—	2	2	no	—
45	C₄×D₄×C₇₃	C₂×C₄	1	4	yes	—
46	C₄×D₄×C₇₃	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
49	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
50	#107×C₇₃	C₂×C₄	1	4	yes	—
51	#102×C₇₃	C₂×C₄	1	8	yes	—
52	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₇₃	C₂³	1	2	yes	—
54	#11924012	—	4	16	no	—
55	C₂×#10596831	—	2	4	no	—
56	C₂×#10596831	—	2	2	no	—
57	C₂×#10596831	—	2	2	no	—
58	C₂×#10596831	—	2	4	no	—
59	C₂×#10596831	—	4	4	no	—
60	C₂×#10596831	D₈	1	4	yes	complemented
61	C₂×#10596831	C₂×D₄	1	2	yes	—
62	C₂×#10596831	C₂×D₄	1	2	yes	—
63	C₂⁴×C₇₃	—	2	2	no	—
64	C₂×D₄×C₇₃	—	2	2	no	—
65	C₂×D₄×C₇₃	—	2	4	no	—
66	C₂×D₄×C₇₃	—	2	4	no	—
67	C₂×D₄×C₇₃	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₇₃	—	2	2	no	—
69	C₂²×C₄×C₇₃	—	2	4	no	—
70	C₂²×C₄×C₇₃	—	2	4	no	—
71	C₂²×C₄×C₇₃	—	4	4	no	—
72	C₂²×C₄×C₇₃	—	4	4	no	—
73	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
76	D₈×C₇₃	—	2	8	no	—
77	C₂×C₈×C₇₃	—	2	2	no	—
78	C₂×C₈×C₇₃	M₁₆	1	4	yes	complemented
79	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
81	C₄⋊C₄×C₇₃	—	2	2	no	—
82	C₄⋊C₄×C₇₃	C₂²×C₄	1	4	yes	—
83	C₂²⋊C₄×C₇₃	—	2	4	no	—
84	C₂²⋊C₄×C₇₃	—	2	4	no	—
85	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₇₃	#41	1	1	yes	characteristic
88	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
89	C₂³×C₇₃	—	2	4	no	—
90	C₂³×C₇₃	—	4	4	no	—
91	C₂³×C₇₃	—	4	4	no	—
92	C₂³×C₇₃	—	4	4	no	—
93	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
94	D₄×C₇₃	—	2	4	no	—
95	D₄×C₇₃	—	2	4	no	—
96	D₄×C₇₃	—	4	4	no	—
97	C₂×C₄×C₇₃	—	2	2	no	—
98	C₂×C₄×C₇₃	—	2	4	no	—
99	C₂×C₄×C₇₃	—	4	4	no	—
100	C₂×C₄×C₇₃	—	4	4	no	—
101	C₂×C₄×C₇₃	—	4	4	no	—
102	C₂×C₄×C₇₃	—	4	8	no	—
103	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₇₃	#131	1	2	yes	—
108	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₇₃	#135	1	2	yes	—
110	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
113	C₈×C₇₃	—	2	2	no	—
114	C₈×C₇₃	C₂×M₁₆	1	4	yes	complemented
115	#10596831	—	2	4	no	—
116	#10596831	—	2	2	no	—
117	#10596831	—	4	4	no	—
118	#10596831	C₂×D₈	1	4	yes	complemented
119	C₂²×C₇₃	—	4	4	no	—
120	C₂²×C₇₃	—	4	4	no	—
121	C₂²×C₇₃	—	4	4	no	—
122	C₂²×C₇₃	—	8	8	no	—
123	C₂²×C₇₃	#436	1	2	yes	—
124	C₂²×C₇₃	C₄×D₈	1	2	yes	—
125	C₂²×C₇₃	#443	1	2	yes	—
126	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
127	C₄×C₇₃	—	2	1	no	—
128	C₄×C₇₃	—	2	1	no	—
129	C₄×C₇₃	—	4	4	no	—
130	C₄×C₇₃	#436	1	2	yes	—
131	C₄×C₇₃	C₄×D₈	1	2	yes	—
132	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
133	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
134	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
135	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
136	C₂×#1589	—	73	1	no	maximal, Sylow
137	C₂×C₇₃	—	8	4	no	—
138	C₂×C₇₃	#1589	1	4	yes	complemented
139	C₂×C₇₃	C₂×#436	1	1	yes	characteristic
140	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
141	C₂×C₇₃	C₂×#443	1	1	yes	characteristic
142	C₂×C₄×D₈	D₇₃	1	1	yes	characteristic
143	C₂×#436	—	73	2	no	—
144	#1589	—	73	8	no	—
145	C₂×#335	—	73	1	no	—
146	C₂×D₄⋊C₈	—	73	2	no	—
147	C₂×C₈²	—	73	1	no	—
148	C₇₃	C₂×#1589	1	1	yes	characteristic, complemented, Sylow
149	C₂²×D₈	Dic₇₃	1	1	yes	characteristic
150	C₂²×M₁₆	—	146	2	no	—
151	C₂×C₄×D₄	C₂×D₇₃	1	2	yes	—
152	C₄×D₈	Dic₇₃	1	4	yes	—
153	C₄×D₈	C₂×D₇₃	1	4	yes	—
154	#436	—	73	8	no	—
155	#436	—	146	4	no	—
156	C₂×#107	Dic₇₃	1	1	yes	characteristic
157	C₂×C₄⋊C₈	—	73	2	no	—
158	C₂×#102	Dic₇₃	1	2	yes	—
159	C₂×C₂²⋊C₈	—	146	2	no	—
160	C₂×C₄×C₈	—	73	1	no	—
161	C₂×C₄×C₈	—	73	1	no	—
162	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
163	#335	—	73	4	no	—
164	D₄⋊C₈	—	73	8	no	—
165	C₈²	—	73	4	no	—
166	C₂²×D₄	C₂×Dic₇₃	1	2	yes	—
167	C₂³×C₄	—	2	2	no	—
168	C₂×D₈	—	2	4	no	—
169	C₂×D₈	C₂×Dic₇₃	1	4	yes	—
170	C₂×M₁₆	—	292	8	no	—
171	C₂×M₁₆	—	146	8	no	—
172	C₂²×C₈	—	73	1	no	—
173	C₂²×C₈	—	73	1	no	—
174	C₂²×C₈	—	146	2	no	—
175	C₂²×C₈	—	146	1	no	—
176	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
177	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
178	C₄×D₄	—	2	2	no	—
179	C₄×D₄	—	2	2	no	—
180	C₄×D₄	C₂×Dic₇₃	1	4	yes	—
181	C₄×D₄	C₂²×D₇₃	1	4	yes	—
182	C₂×C₄⋊C₄	C₂×Dic₇₃	1	2	yes	—
183	C₂×C₂²⋊C₄	—	2	2	no	—
184	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
185	#107	C₂×Dic₇₃	1	4	yes	—
186	C₄⋊C₈	—	73	4	no	—
187	C₄⋊C₈	—	146	2	no	—
188	#102	C₂×Dic₇₃	1	8	yes	—
189	C₂²⋊C₈	—	146	8	no	—
190	C₄×C₈	—	73	2	no	—
191	C₄×C₈	—	73	2	no	—
192	C₄×C₈	—	73	2	no	—
193	C₄×C₈	—	146	1	no	—
194	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
195	C₄×C₈	C₂²×D₇₃	1	2	yes	—
196	C₂⁴	—	2	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	C₂²×Dic₇₃	1	4	yes	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
207	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
208	C₂²×C₄	#11924027	1	1	yes	characteristic
209	D₈	—	2	8	no	—
210	M₁₆	—	292	16	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	292	4	no	—
213	C₂×C₈	—	73	4	no	—
214	C₂×C₈	—	73	2	no	—
215	C₂×C₈	—	73	2	no	—
216	C₂×C₈	—	146	4	no	—
217	C₂×C₈	—	146	4	no	—
218	C₂×C₈	—	146	2	no	—
219	C₂×C₈	—	146	2	no	—
220	C₂×C₈	#11924012	1	4	yes	complemented
221	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
222	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	C₂²×Dic₇₃	1	4	yes	—
225	C₂²⋊C₄	—	2	4	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
228	C₄²	D₄×D₇₃	1	1	yes	characteristic
229	C₄²	#11924027	1	1	yes	characteristic
230	C₄²	C₂³×D₇₃	1	1	yes	characteristic
231	C₂³	—	2	4	no	—
232	C₂³	—	4	4	no	—
233	C₂³	—	4	4	no	—
234	C₂³	—	4	4	no	—
235	C₂³	D₄×Dic₇₃	1	1	yes	characteristic
236	D₄	—	2	4	no	—
237	D₄	—	2	4	no	—
238	D₄	—	4	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	8	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
246	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
247	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
248	C₂×C₄	C₂×#11924012	1	2	yes	—
249	C₂×C₄	#14556598	1	2	yes	—
250	C₂×C₄	D₈×D₇₃	1	2	yes	—
251	C₂×C₄	#14556610	1	2	yes	—
252	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
253	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
254	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	292	4	no	—
257	C₈	—	73	4	no	—
258	C₈	—	146	4	no	—
259	C₈	—	146	2	no	—
260	C₈	C₂×#11924012	1	4	yes	complemented
261	C₂²	—	4	4	no	—
262	C₂²	—	4	4	no	—
263	C₂²	—	4	4	no	—
264	C₂²	—	8	8	no	—
265	C₂²	#17988665	1	2	yes	—
266	C₂²	D₈×Dic₇₃	1	2	yes	—
267	C₂²	#17988704	1	2	yes	—
268	C₂²	C₂×D₄×Dic₇₃	1	1	yes	characteristic
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	4	4	no	—
272	C₄	#17988665	1	2	yes	—
273	C₄	D₈×Dic₇₃	1	2	yes	—
274	C₄	C₂²×#11924012	1	1	yes	characteristic
275	C₄	C₂×#14556598	1	1	yes	characteristic
276	C₄	C₂×D₈×D₇₃	1	1	yes	characteristic
277	C₄	C₂×#14556610	1	1	yes	characteristic
278	C₂	—	8	4	no	—
279	C₂	#22608816	1	4	yes	direct product factor
280	C₂	C₂×#17988665	1	1	yes	characteristic
281	C₂	C₂×D₈×Dic₇₃	1	1	yes	characteristic
282	C₂	C₂×#17988704	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₂×C₄×D₈×C₇₃	C₂
socle	C₂³×C₇₃	C₄×D₄
prime radical	C₂²×D₈×C₇₃	C₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	C₂×C₄×D₈	D₇₃
hypocenter	C₇₃	C₂×#1589
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×#17988665	C₂	semidirect
2. #22608816	C₂	direct (N × H)
3. C₂×#17988340	C₂	semidirect
4. C₂×D₄⋊#10596831	C₂	semidirect
5. C₂×C₈×#10596831	C₂	semidirect
9. #17988665	C₂²	semidirect
14. #17988340	C₂²	semidirect
15. D₄⋊#10596831	C₂²	semidirect
16. C₈×#10596831	C₂²	semidirect
60. C₂×#10596831	D₈	semidirect
78. C₂×C₈×C₇₃	M₁₆	semidirect
114. C₈×C₇₃	C₂×M₁₆	semidirect
118. #10596831	C₂×D₈	semidirect
138. C₂×C₇₃	#1589	semidirect
148. C₇₃	C₂×#1589	semidirect
220. C₂×C₈	#11924012	semidirect
260. C₈	C₂×#11924012	semidirect
279. C₂	#22608816	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×#17988665	2
#22608816	2
C₂×#17988340	2
C₂×D₄⋊#10596831	2
C₂×C₈×#10596831	2
C₂×C₄×D₈×C₇₃	2
C₂×#1589	73

Subgroups — C2×#22608816 #33055712

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22608816 #33055712