Subgroups — C₂×#22611818 #33239391

Order 18688.

654 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989019	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988705	C₂	1	2	yes	maximal, complemented
3	#22611818	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17988417	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#17988395	C₂	1	2	yes	maximal, complemented
6	C₂×#346×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14556613	C₂²	1	2	yes	—
8	C₂×#14556597	—	2	2	no	—
9	#17989019	C₄	1	4	yes	complemented
10	#17989019	C₂²	1	4	yes	—
11	C₂×Dic₅₈₄	C₄	1	1	yes	characteristic, complemented
12	C₂×#14556536	C₄	1	2	yes	complemented
13	C₂×D₁₁₆₈	C₄	1	1	yes	characteristic, complemented
14	#17988705	—	2	4	no	—
15	#17988705	C₂²	1	8	yes	complemented
16	C₂×#14556527	C₂²	1	2	yes	—
17	C₂×#14556519	—	2	2	no	—
18	#17988417	C₂²	1	4	yes	complemented
19	#17988395	C₂²	1	8	yes	complemented
20	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
23	#346×C₇₃	C₂²	1	4	yes	complemented
24	C₂×#11924028	—	2	2	no	—
25	#14556613	—	2	2	no	—
26	#14556613	—	2	2	no	—
27	#14556613	C₂×C₄	1	4	yes	—
28	#14556613	C₂³	1	4	yes	—
29	C₂×Dic₂₉₂	C₂×C₄	1	2	yes	—
30	C₂×#11924017	—	2	2	no	—
31	C₂×D₅₈₄	C₂×C₄	1	2	yes	—
32	#14556597	—	2	8	no	—
33	#14556597	—	4	4	no	—
34	C₂×#11924012	—	2	2	no	—
35	C₂×#11924011	—	4	2	no	—
36	C₂×C₈×D₇₃	—	4	2	no	—
37	Dic₅₈₄	C₂×C₄	1	4	yes	complemented
38	#14556536	C₂×C₄	1	8	yes	complemented
39	D₁₁₆₈	C₂×C₄	1	4	yes	complemented
40	#14556527	—	2	2	no	—
41	#14556527	C₂³	1	4	yes	—
42	#14556519	—	2	8	no	—
43	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
46	C₂×C₁₆×C₇₃	—	2	1	no	—
47	C₂×C₁₆×C₇₃	—	2	1	no	—
48	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
51	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
52	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
53	C₄×C₈×C₇₃	D₄	1	2	yes	—
54	C₄×C₈×C₇₃	C₂³	1	2	yes	—
55	#11924028	—	2	4	no	—
56	#11924028	—	2	4	no	—
57	#11924028	—	4	2	no	—
58	#11924028	—	4	2	no	—
59	C₂×Dic₁₄₆	—	2	2	no	—
60	C₂×D₂₉₂	—	2	2	no	—
61	C₂×C₂²⋊D₇₃	—	4	2	no	—
62	C₂×C₄×D₇₃	—	4	2	no	—
63	Dic₂₉₂	—	2	2	no	—
64	Dic₂₉₂	C₂²×C₄	1	4	yes	—
65	#11924017	—	2	4	no	—
66	#11924017	—	2	4	no	—
67	D₅₈₄	—	2	2	no	—
68	D₅₈₄	C₂²×C₄	1	4	yes	—
69	#11924012	—	2	4	no	—
70	#11924012	—	4	2	no	—
71	#11924011	—	4	8	no	—
72	C₂×#10596831	—	4	2	no	—
73	C₈×D₇₃	—	4	8	no	—
74	C₂²×C₄×C₇₃	—	2	1	no	—
75	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₇₃	D₈	1	2	yes	—
77	C₂×C₈×C₇₃	D₈	1	2	yes	—
78	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
83	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
84	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
85	C₄²×C₇₃	—	2	1	no	—
86	C₄²×C₇₃	C₂×D₄	1	2	yes	—
87	C₁₆×C₇₃	—	2	2	no	—
88	C₁₆×C₇₃	C₂²×C₄	1	2	yes	—
89	C₁₆×C₇₃	C₂²×C₄	1	2	yes	—
90	C₂³×C₇₃	#135	1	1	yes	characteristic
91	C₂²×D₇₃	—	4	2	no	—
92	C₂×C₄×C₇₃	—	2	2	no	—
93	C₂×C₄×C₇₃	—	4	2	no	—
94	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
99	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
100	C₂²⋊D₇₃	—	4	4	no	—
101	C₂²⋊D₇₃	—	4	4	no	—
102	C₂×Dic₇₃	—	4	2	no	—
103	D₂₉₂	—	2	4	no	—
104	D₂₉₂	—	4	2	no	—
105	C₄×D₇₃	—	4	4	no	—
106	C₄×D₇₃	—	4	4	no	—
107	Dic₁₄₆	—	2	4	no	—
108	Dic₁₄₆	—	4	2	no	—
109	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
110	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
111	C₈×C₇₃	C₂×D₈	1	4	yes	—
112	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
113	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
114	#10596831	—	4	4	no	—
115	C₂²×C₇₃	—	2	2	no	—
116	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
117	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
118	C₂²×C₇₃	C₂×#135	1	1	yes	characteristic
119	C₂×D₇₃	—	4	4	no	—
120	C₂×D₇₃	—	8	4	no	—
121	C₄×C₇₃	—	4	2	no	—
122	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
123	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
124	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
125	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
126	Dic₇₃	—	4	4	no	—
127	C₂×#2179	—	73	1	no	maximal, Sylow
128	C₂×C₇₃	—	2	1	no	—
129	C₂×C₇₃	—	2	1	no	—
130	C₂×C₇₃	#2179	1	2	yes	complemented
131	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
132	D₇₃	—	8	4	no	—
133	C₂×#508	—	73	1	no	—
134	C₂×#443	—	73	2	no	—
135	#2179	—	73	8	no	—
136	C₂×#368	—	73	1	no	—
137	C₂×#359	—	73	2	no	—
138	C₂×#346	D₇₃	1	1	yes	characteristic, complemented
139	C₇₃	C₂×#2179	1	1	yes	characteristic, complemented, Sylow
140	C₂×#135	—	73	2	no	—
141	C₂×#131	—	146	2	no	—
142	#508	—	73	4	no	—
143	#508	—	73	4	no	—
144	C₂×Q₃₂	—	73	1	no	—
145	C₂×SD₃₂	—	73	2	no	—
146	C₂×D₁₆	—	73	1	no	—
147	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
148	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
149	#443	—	73	8	no	—
150	#443	—	146	4	no	—
151	C₂×#108	—	73	2	no	—
152	C₂×#104	—	146	2	no	—
153	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
154	#368	—	73	4	no	—
155	#359	—	73	8	no	—
156	#346	C₂×D₇₃	1	4	yes	complemented
157	C₂×#41	—	146	2	no	—
158	#135	—	73	4	no	—
159	#135	—	73	4	no	—
160	#135	—	146	2	no	—
161	#135	—	146	2	no	—
162	C₂×Q₁₆	—	73	2	no	—
163	C₂×SD₁₆	—	146	2	no	—
164	C₂×D₈	—	73	2	no	—
165	#131	—	292	4	no	—
166	#131	—	146	8	no	—
167	C₂×M₁₆	—	292	2	no	—
168	C₂×M₁₆	—	146	2	no	—
169	C₂²×C₈	—	292	2	no	—
170	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
172	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
173	Q₃₂	—	73	4	no	—
174	SD₃₂	—	73	8	no	—
175	D₁₆	—	73	4	no	—
176	C₂×C₁₆	—	2	1	no	—
177	C₂×C₁₆	—	2	1	no	—
178	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
179	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
180	C₂×C₁₆	C₄×D₇₃	1	2	yes	—
181	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
182	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
183	#108	—	73	4	no	—
184	#108	—	146	2	no	—
185	#104	—	146	8	no	—
186	C₄×C₈	D₂₉₂	1	2	yes	—
187	C₄×C₈	C₂²×D₇₃	1	2	yes	—
188	#41	—	292	2	no	—
189	#41	—	292	2	no	—
190	#41	—	146	4	no	—
191	#41	—	146	4	no	—
192	C₂×Q₈	—	146	2	no	—
193	C₂×D₄	—	292	2	no	—
194	C₂×D₄	—	146	2	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	292	2	no	—
197	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
198	Q₁₆	—	73	4	no	—
199	Q₁₆	—	146	2	no	—
200	SD₁₆	—	146	4	no	—
201	SD₁₆	—	146	4	no	—
202	D₈	—	73	4	no	—
203	D₈	—	146	2	no	—
204	M₁₆	—	292	2	no	—
205	M₁₆	—	292	8	no	—
206	M₁₆	—	146	4	no	—
207	C₂×C₈	—	292	2	no	—
208	C₂×C₈	—	292	8	no	—
209	C₂×C₈	D₅₈₄	1	2	yes	—
210	C₂×C₈	D₅₈₄	1	2	yes	—
211	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
212	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
213	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
214	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
215	C₂×C₈	#11924028	1	1	yes	characteristic
216	C₂×C₈	#11924028	1	1	yes	characteristic
217	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
218	C₄²	—	2	1	no	—
219	C₄²	C₂×D₂₉₂	1	2	yes	—
220	C₁₆	—	2	2	no	—
221	C₁₆	C₂×C₄×D₇₃	1	2	yes	—
222	C₁₆	C₂×C₄×D₇₃	1	2	yes	—
223	C₂³	—	292	2	no	—
224	C₂³	#14556613	1	1	yes	characteristic
225	Q₈	—	292	2	no	—
226	Q₈	—	146	4	no	—
227	D₄	—	292	4	no	—
228	D₄	—	292	4	no	—
229	D₄	—	292	2	no	—
230	D₄	—	146	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	292	2	no	—
234	C₂×C₄	—	292	4	no	—
235	C₂×C₄	—	292	4	no	—
236	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
237	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
238	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
239	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
240	C₂×C₄	#14556613	1	1	yes	characteristic
241	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
242	C₈	—	292	4	no	—
243	C₈	C₄×D₂₉₂	1	1	yes	characteristic
244	C₈	C₄×D₂₉₂	1	1	yes	characteristic
245	C₈	C₂×D₅₈₄	1	4	yes	—
246	C₈	C₂²×C₄×D₇₃	1	1	yes	characteristic
247	C₈	C₂×#11924028	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	292	4	no	—
250	C₂²	—	584	4	no	—
251	C₂²	C₄×D₅₈₄	1	1	yes	characteristic
252	C₂²	C₄×D₅₈₄	1	1	yes	characteristic
253	C₂²	C₂×#14556613	1	1	yes	characteristic
254	C₄	—	4	2	no	—
255	C₄	—	292	4	no	—
256	C₄	C₄×D₅₈₄	1	1	yes	characteristic
257	C₄	C₄×D₅₈₄	1	1	yes	characteristic
258	C₄	C₂×C₄×D₂₉₂	1	1	yes	characteristic
259	C₄	C₂²×D₅₈₄	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	584	4	no	—
263	C₂	#22611818	1	2	yes	direct product factor
264	C₂	C₂×C₄×D₅₈₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₈₄
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#346×C₇₃	C₂
socle	C₂²×C₇₃	C₄×D₈
prime radical	C₂×#17989019	C₂
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₂×#346	D₇₃
hypocenter	C₇₃	C₂×#2179
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. C₂×#17988705	C₂	semidirect
3. #22611818	C₂	direct (N × H)
4. C₂×#17988417	C₂	semidirect
5. C₂×#17988395	C₂	semidirect
6. C₂×#346×C₇₃	C₂	semidirect
9. #17989019	C₄	semidirect
11. C₂×Dic₅₈₄	C₄	semidirect
12. C₂×#14556536	C₄	semidirect
13. C₂×D₁₁₆₈	C₄	semidirect
15. #17988705	C₂²	semidirect
18. #17988417	C₂²	semidirect
19. #17988395	C₂²	semidirect
23. #346×C₇₃	C₂²	semidirect
37. Dic₅₈₄	C₂×C₄	semidirect
38. #14556536	C₂×C₄	semidirect
39. D₁₁₆₈	C₂×C₄	semidirect
130. C₂×C₇₃	#2179	semidirect
138. C₂×#346	D₇₃	semidirect
139. C₇₃	C₂×#2179	semidirect
156. #346	C₂×D₇₃	semidirect
263. C₂	#22611818	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×#17989019	2
C₂×#17988705	2
#22611818	2
C₂×#17988417	2
C₂×#17988395	2
C₂×#346×C₇₃	2
C₂×#2179	73

Subgroups — C2×#22611818 #33239391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22611818 #33239391