Subgroups — C₂×#22611769 #33239213

Order 18688.

662 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989003	C₂	1	2	yes	maximal, complemented
2	C₂×#17988705	C₂	1	1	yes	maximal, characteristic
3	#22611769	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17988408	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#17988370	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14556613	C₄	1	1	yes	characteristic
8	C₂×#14556597	C₂²	1	2	yes	—
9	#17989003	—	2	4	no	—
10	#17989003	C₂²	1	8	yes	complemented
11	C₂×#14556531	C₂²	1	2	yes	—
12	C₂×#14556530	—	2	2	no	—
13	C₂×C₁₆×D₇₃	—	2	2	no	—
14	#17988705	C₄	1	4	yes	—
15	#17988705	C₂²	1	4	yes	—
16	C₂×#14556527	C₄	1	1	yes	characteristic
17	C₂×#14556519	C₄	1	2	yes	—
18	#17988408	C₂²	1	4	yes	complemented
19	#17988370	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	C₄×C₁₆×C₇₃	C₂²	1	4	yes	complemented
24	C₂×#11924028	C₂×C₄	1	2	yes	—
25	#14556613	C₈	1	4	yes	—
26	#14556613	C₂×C₄	1	4	yes	—
27	C₂×Dic₂₉₂	C₈	1	1	yes	characteristic
28	C₂×#11924017	C₈	1	2	yes	—
29	C₂×D₅₈₄	C₈	1	1	yes	characteristic
30	#14556597	—	2	2	no	—
31	#14556597	—	2	2	no	—
32	#14556597	C₂×C₄	1	4	yes	—
33	#14556597	C₂³	1	4	yes	—
34	C₂×#11924012	C₂×C₄	1	2	yes	—
35	C₂×#11924011	—	2	2	no	—
36	C₂×C₈×D₇₃	—	2	2	no	—
37	#14556531	—	2	2	no	—
38	#14556531	C₂³	1	4	yes	—
39	#14556530	—	2	8	no	—
40	C₂×#11924003	—	2	2	no	—
41	C₁₆×D₇₃	—	2	8	no	—
42	#14556527	C₂×C₄	1	4	yes	—
43	#14556519	C₂×C₄	1	8	yes	—
44	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₁₆×C₇₃	—	2	1	no	—
48	C₂×C₁₆×C₇₃	—	2	1	no	—
49	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
50	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
51	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
52	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
53	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
54	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₇₃	C₂³	1	2	yes	—
56	#11924028	—	2	2	no	—
57	#11924028	—	2	2	no	—
58	#11924028	C₂×C₈	1	4	yes	—
59	#11924028	C₂²×C₄	1	4	yes	—
60	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
61	C₂×D₂₉₂	C₂×C₈	1	2	yes	—
62	C₂×C₂²⋊D₇₃	—	2	2	no	—
63	C₂×C₄×D₇₃	—	2	2	no	—
64	Dic₂₉₂	C₂×C₈	1	4	yes	—
65	#11924017	C₂×C₈	1	8	yes	—
66	D₅₈₄	C₂×C₈	1	4	yes	—
67	#11924012	—	2	2	no	—
68	#11924012	C₂²×C₄	1	4	yes	—
69	#11924011	—	2	4	no	—
70	#11924011	—	2	4	no	—
71	C₂×#10596831	—	2	2	no	—
72	C₈×D₇₃	—	2	4	no	—
73	C₈×D₇₃	—	2	4	no	—
74	#11924003	—	2	4	no	—
75	C₂²×C₄×C₇₃	—	2	1	no	—
76	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
78	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
79	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
80	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
85	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
86	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
87	C₄²×C₇₃	—	2	1	no	—
88	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
89	C₁₆×C₇₃	—	2	2	no	—
90	C₁₆×C₇₃	C₂×D₄	1	2	yes	—
91	C₁₆×C₇₃	C₂×D₄	1	2	yes	—
92	C₂³×C₇₃	#131	1	1	yes	characteristic
93	C₂²×D₇₃	—	2	2	no	—
94	C₂×C₄×C₇₃	—	2	4	no	—
95	C₂×C₄×C₇₃	—	2	2	no	—
96	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
101	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
102	C₂²⋊D₇₃	—	2	4	no	—
103	C₂²⋊D₇₃	—	2	4	no	—
104	C₂×Dic₇₃	—	2	2	no	—
105	D₂₉₂	—	2	2	no	—
106	D₂₉₂	C₂²×C₈	1	4	yes	—
107	C₄×D₇₃	—	2	4	no	—
108	C₄×D₇₃	—	2	4	no	—
109	Dic₁₄₆	—	2	2	no	—
110	Dic₁₄₆	C₂²×C₈	1	4	yes	—
111	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
112	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
113	C₈×C₇₃	C₂²×C₈	1	2	yes	—
114	C₈×C₇₃	C₂²×C₈	1	2	yes	—
115	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
116	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
117	#10596831	—	2	4	no	—
118	C₂²×C₇₃	—	2	2	no	—
119	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
120	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
121	C₂²×C₇₃	C₂×#131	1	1	yes	characteristic
122	C₂×D₇₃	—	2	4	no	—
123	C₂×D₇₃	—	4	4	no	—
124	C₄×C₇₃	—	2	4	no	—
125	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
126	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
127	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
128	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
129	Dic₇₃	—	2	4	no	—
130	C₂×#2170	—	73	1	no	maximal, Sylow
131	C₂×C₇₃	—	2	1	no	—
132	C₂×C₇₃	—	2	1	no	—
133	C₂×C₇₃	#2170	1	2	yes	complemented
134	C₂×C₇₃	C₂×C₈×D₄	1	1	yes	characteristic
135	D₇₃	—	4	4	no	—
136	C₂×#504	—	73	2	no	—
137	C₂×#443	—	73	1	no	—
138	#2170	—	73	8	no	—
139	C₂×#364	—	73	1	no	—
140	C₂×#350	—	73	2	no	—
141	C₂×C₄×C₁₆	D₇₃	1	1	yes	characteristic, complemented
142	C₇₃	C₂×#2170	1	1	yes	characteristic, complemented, Sylow
143	C₂×#135	—	73	1	no	—
144	C₂×#131	—	73	2	no	—
145	#504	—	73	8	no	—
146	#504	—	146	4	no	—
147	C₂×M₃₂	—	73	2	no	—
148	C₂×M₃₂	—	146	2	no	—
149	C₂²×C₁₆	—	146	2	no	—
150	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
151	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
152	#443	—	73	4	no	—
153	#443	—	73	4	no	—
154	C₂×#108	—	73	1	no	—
155	C₂×#104	—	73	2	no	—
156	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
157	#364	—	73	4	no	—
158	#350	—	73	8	no	—
159	C₄×C₁₆	C₂×D₇₃	1	4	yes	complemented
160	C₂×#41	—	73	2	no	—
161	#135	—	73	4	no	—
162	#135	—	73	4	no	—
163	C₂×Q₁₆	—	73	1	no	—
164	C₂×SD₁₆	—	73	2	no	—
165	C₂×D₈	—	73	1	no	—
166	#131	—	73	4	no	—
167	#131	—	73	4	no	—
168	#131	—	146	2	no	—
169	#131	—	146	2	no	—
170	C₂×M₁₆	—	73	2	no	—
171	C₂×M₁₆	—	146	2	no	—
172	C₂²×C₈	—	146	2	no	—
173	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
175	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
176	M₃₂	—	73	4	no	—
177	M₃₂	—	146	2	no	—
178	M₃₂	—	146	8	no	—
179	C₂×C₁₆	—	2	1	no	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	146	2	no	—
182	C₂×C₁₆	—	146	8	no	—
183	C₂×C₁₆	D₂₉₂	1	1	yes	characteristic
184	C₂×C₁₆	D₂₉₂	1	2	yes	—
185	C₂×C₁₆	D₂₉₂	1	1	yes	characteristic
186	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
187	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
188	#108	—	73	4	no	—
189	#104	—	73	8	no	—
190	C₄×C₈	C₄×D₇₃	1	2	yes	—
191	C₄×C₈	C₂²×D₇₃	1	2	yes	—
192	#41	—	73	4	no	—
193	#41	—	73	4	no	—
194	#41	—	146	2	no	—
195	#41	—	146	2	no	—
196	C₂×Q₈	—	73	2	no	—
197	C₂×D₄	—	73	2	no	—
198	C₂×D₄	—	146	2	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	146	2	no	—
201	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
202	Q₁₆	—	73	4	no	—
203	SD₁₆	—	73	8	no	—
204	D₈	—	73	4	no	—
205	M₁₆	—	73	4	no	—
206	M₁₆	—	146	2	no	—
207	M₁₆	—	146	4	no	—
208	M₁₆	—	146	4	no	—
209	C₂×C₈	—	146	2	no	—
210	C₂×C₈	—	146	4	no	—
211	C₂×C₈	—	146	4	no	—
212	C₂×C₈	C₈×D₇₃	1	1	yes	characteristic
213	C₂×C₈	C₈×D₇₃	1	1	yes	characteristic
214	C₂×C₈	C₈×D₇₃	1	2	yes	—
215	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
216	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
217	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
218	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
219	C₂×C₈	#11924028	1	1	yes	characteristic
220	C₂×C₈	#11924028	1	1	yes	characteristic
221	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
222	C₄²	—	2	1	no	—
223	C₄²	C₂×C₄×D₇₃	1	2	yes	—
224	C₁₆	—	2	2	no	—
225	C₁₆	—	146	4	no	—
226	C₁₆	C₂×D₂₉₂	1	2	yes	—
227	C₁₆	C₂×D₂₉₂	1	2	yes	—
228	C₂³	—	146	2	no	—
229	C₂³	#14556597	1	1	yes	characteristic
230	Q₈	—	73	4	no	—
231	Q₈	—	146	2	no	—
232	D₄	—	73	4	no	—
233	D₄	—	146	4	no	—
234	D₄	—	146	4	no	—
235	D₄	—	146	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	146	2	no	—
239	C₂×C₄	—	146	4	no	—
240	C₂×C₄	—	146	4	no	—
241	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
242	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
243	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
244	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
245	C₂×C₄	#14556597	1	1	yes	characteristic
246	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
247	C₈	—	146	4	no	—
248	C₈	C₄×D₂₉₂	1	1	yes	characteristic
249	C₈	C₄×D₂₉₂	1	1	yes	characteristic
250	C₈	C₂×C₈×D₇₃	1	2	yes	—
251	C₈	C₂×C₈×D₇₃	1	2	yes	—
252	C₈	C₂²×D₂₉₂	1	1	yes	characteristic
253	C₈	C₂×#11924028	1	1	yes	characteristic
254	C₂²	—	2	2	no	—
255	C₂²	—	292	4	no	—
256	C₂²	—	146	4	no	—
257	C₂²	C₈×D₂₉₂	1	1	yes	characteristic
258	C₂²	C₈×D₂₉₂	1	1	yes	characteristic
259	C₂²	C₂×#14556597	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	146	4	no	—
262	C₄	C₈×D₂₉₂	1	1	yes	characteristic
263	C₄	C₈×D₂₉₂	1	1	yes	characteristic
264	C₄	C₂×C₄×D₂₉₂	1	1	yes	characteristic
265	C₄	C₂²×C₈×D₇₃	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	292	4	no	—
269	C₂	#22611769	1	2	yes	direct product factor
270	C₂	C₂×C₈×D₂₉₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	D₂₉₂
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×C₄×C₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₈×D₄
prime radical	C₂×#14556613	C₄
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₇₃
hypocenter	C₇₃	C₂×#2170
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#17989003	C₂	semidirect
3. #22611769	C₂	direct (N × H)
4. C₂×#17988408	C₂	semidirect
5. C₂×#17988370	C₂	semidirect
6. C₂×C₄×C₁₆×C₇₃	C₂	semidirect
10. #17989003	C₂²	semidirect
18. #17988408	C₂²	semidirect
19. #17988370	C₂²	semidirect
23. C₄×C₁₆×C₇₃	C₂²	semidirect
133. C₂×C₇₃	#2170	semidirect
141. C₂×C₄×C₁₆	D₇₃	semidirect
142. C₇₃	C₂×#2170	semidirect
159. C₄×C₁₆	C₂×D₇₃	semidirect
269. C₂	#22611769	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×#17989003	2
C₂×#17988705	2
#22611769	2
C₂×#17988408	2
C₂×#17988370	2
C₂×C₄×C₁₆×C₇₃	2
C₂×#2170	73

Subgroups — C2×#22611769 #33239213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22611769 #33239213