Subgroups — C₂×#22625867 #34021333

Order 18688.

1358 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556660	C₂	1	4	yes	maximal
2	C₂×#17989533	C₂	1	2	yes	maximal
3	#22625867	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊#64408	—	2	4	no	—
6	C₂²×C₄⋊#64408	C₂²	1	4	yes	—
7	C₂×#14556660	—	2	16	no	—
8	C₂×#14556660	C₂²	1	16	yes	—
9	#17989533	—	2	4	no	—
10	#17989533	C₂²	1	8	yes	—
11	C₂²×C₈×D₇₃	C₂²	1	2	yes	—
12	C₂×C₄²×D₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×#10596831	C₄	1	1	yes	characteristic, complemented
14	C₂×C₈×Dic₇₃	C₄	1	2	yes	complemented
15	C₄×C₈×D₇₃	C₄	1	4	yes	complemented
16	C₄×C₈×D₇₃	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic, complemented
18	C₂³×#64408	—	2	4	no	—
19	C₂×C₄⋊#64408	—	2	16	no	—
20	C₂×C₄⋊#64408	—	2	16	no	—
21	C₂×C₄⋊#64408	—	2	4	no	—
22	C₂×C₄⋊#64408	—	4	16	no	—
23	C₂×C₄⋊#64408	C₂³	1	8	yes	—
24	#14556660	—	2	32	no	—
25	C₂²×C₄×D₇₃	D₄	1	2	yes	—
26	C₂²×C₄×D₇₃	C₂³	1	1	yes	characteristic
27	C₂²×#10596831	C₂×C₄	1	2	yes	—
28	C₂×C₈×D₇₃	C₂×C₄	1	4	yes	—
29	C₂×C₈×D₇₃	D₄	1	8	yes	—
30	C₂×C₈×D₇₃	Q₈	1	8	yes	—
31	C₂×C₈×D₇₃	C₂³	1	4	yes	—
32	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₇₃	Q₈	1	2	yes	—
34	C₄²×D₇₃	C₂×C₄	1	2	yes	—
35	C₄²×D₇₃	D₄	1	2	yes	—
36	C₄²×D₇₃	Q₈	1	2	yes	—
37	C₄²×D₇₃	C₂³	1	2	yes	—
38	C₄×#10596831	C₂×C₄	1	4	yes	complemented
39	C₈×Dic₇₃	C₂×C₄	1	8	yes	complemented
40	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	complemented
43	C₂²×#64408	—	2	8	no	—
44	C₂²×#64408	—	4	8	no	—
45	C₂²×#64408	—	4	16	no	—
46	C₄⋊#64408	—	2	16	no	—
47	C₄⋊#64408	—	4	8	no	—
48	C₄⋊#64408	—	4	32	no	—
49	C₂³×D₇₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
51	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×D₇₃	C₄⋊C₄	1	4	yes	—
53	C₂×C₄×D₇₃	D₈	1	8	yes	—
54	C₂×C₄×D₇₃	Q₁₆	1	8	yes	—
55	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×D₇₃	C₂×D₄	1	4	yes	—
57	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×D₇₃	C₂×D₄	1	4	yes	—
59	C₂×C₄×D₇₃	C₂×Q₈	1	4	yes	—
60	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₄×D₇₃	C₂⁴	1	1	yes	characteristic
62	C₂×#10596831	C₄⋊C₄	1	8	yes	complemented
63	C₂×#10596831	C₂²×C₄	1	4	yes	—
64	C₈×D₇₃	C₄⋊C₄	1	16	yes	complemented
65	C₈×D₇₃	C₂×D₄	1	8	yes	—
66	C₈×D₇₃	C₂×Q₈	1	8	yes	—
67	C₄×Dic₇₃	C₄⋊C₄	1	4	yes	—
68	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
69	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
70	C₄×Dic₇₃	C₂×Q₈	1	4	yes	—
71	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₇₃	C₄⋊C₄	1	8	yes	complemented
74	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
75	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
76	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
77	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂²×D₇₃	#127	1	2	yes	—
80	C₂²×D₇₃	C₄⋊Q₈	1	2	yes	—
81	C₂²×D₇₃	C₂×D₈	1	4	yes	—
82	C₂²×D₇₃	C₂×Q₁₆	1	4	yes	—
83	C₂²×D₇₃	C₂²×D₄	1	1	yes	characteristic
84	C₂×#64408	—	4	8	no	—
85	C₂×#64408	—	4	16	no	—
86	C₂×#64408	—	8	16	no	—
87	C₂×C₄×C₇₃	#107	1	8	yes	complemented
88	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
89	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
91	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
92	C₂×Dic₇₃	#107	1	8	yes	complemented
93	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×Dic₇₃	C₂×C₄⋊C₄	1	4	yes	—
95	C₂×Dic₇₃	C₄⋊Q₈	1	4	yes	—
96	C₂×Dic₇₃	C₂²×Q₈	1	1	yes	characteristic
97	C₄×D₇₃	#107	1	16	yes	complemented
98	C₄×D₇₃	C₂×C₄⋊C₄	1	4	yes	—
99	C₄×D₇₃	#127	1	2	yes	—
100	C₄×D₇₃	C₄⋊Q₈	1	4	yes	—
101	C₄×D₇₃	C₄⋊Q₈	1	2	yes	—
102	C₄×D₇₃	C₂×D₈	1	8	yes	—
103	C₄×D₇₃	C₂×Q₁₆	1	8	yes	—
104	C₄×D₇₃	C₂²×D₄	1	2	yes	—
105	C₄×D₇₃	C₂²×Q₈	1	2	yes	—
106	C₈×C₇₃	C₂×C₄⋊C₄	1	8	yes	complemented
107	#10596831	C₂×C₄⋊C₄	1	8	yes	complemented
108	C₂²×C₇₃	#384	1	2	yes	—
109	C₂²×C₇₃	C₂×#107	1	4	yes	—
110	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
111	C₂×D₇₃	#384	1	2	yes	—
112	C₂×D₇₃	C₂×#107	1	4	yes	—
113	C₂×D₇₃	#493	1	4	yes	—
114	C₂×D₇₃	#494	1	4	yes	—
115	C₂×D₇₃	#500	1	8	yes	—
116	C₂×D₇₃	C₂×#127	1	1	yes	characteristic
117	C₂×D₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
118	C₂×D₇₃	C₂²×D₈	1	2	yes	—
119	C₂×D₇₃	C₂²×Q₁₆	1	2	yes	—
120	#64408	—	8	8	no	—
121	C₄×C₇₃	#384	1	2	yes	—
122	C₄×C₇₃	C₂×#107	1	8	yes	complemented
123	C₄×C₇₃	C₂²×C₄⋊C₄	1	2	yes	—
124	Dic₇₃	#384	1	2	yes	—
125	Dic₇₃	C₂×#107	1	8	yes	complemented
126	Dic₇₃	C₂×C₄⋊Q₈	1	2	yes	—
127	C₂×#1845	—	73	1	no	maximal, Sylow
128	C₂×C₇₃	#1845	1	4	yes	complemented
129	C₂×C₇₃	C₂×#384	1	1	yes	characteristic
130	C₂×C₇₃	C₂²×#107	1	2	yes	—
131	D₇₃	#1845	1	4	yes	complemented
132	D₇₃	C₂×#493	1	1	yes	characteristic
133	D₇₃	C₂×#494	1	1	yes	characteristic
134	D₇₃	C₂×#500	1	2	yes	—
135	C₂²×#107	—	73	4	no	—
136	C₂²×C₄×C₈	—	73	1	no	—
137	C₂×#384	—	73	2	no	—
138	#1845	—	73	8	no	—
139	C₇₃	C₂×#1845	1	1	yes	characteristic, complemented, Sylow
140	C₂³×C₈	—	73	2	no	—
141	C₂²×C₄⋊C₄	—	73	4	no	—
142	C₂²×C₄⋊C₄	—	146	4	no	—
143	C₂²×C₄²	—	73	1	no	—
144	C₂×#107	—	73	16	no	—
145	C₂×#107	—	146	16	no	—
146	C₂×C₄×C₈	—	73	2	no	—
147	C₂×C₄×C₈	—	73	1	no	—
148	C₂×C₄×C₈	—	73	4	no	—
149	C₂×C₄×C₈	—	73	4	no	—
150	C₂×C₄×C₈	#64408	1	1	yes	characteristic, complemented
151	#384	—	73	8	no	—
152	#384	—	146	4	no	—
153	C₂³×C₄	—	73	2	no	—
154	C₂³×C₄	—	73	1	no	—
155	C₂³×C₄	—	146	4	no	—
156	C₂²×C₈	—	73	2	no	—
157	C₂²×C₈	—	73	8	no	—
158	C₂²×C₈	—	73	8	no	—
159	C₂²×C₈	—	73	4	no	—
160	C₂²×C₈	—	73	4	no	—
161	C₂²×C₈	C₂×#64408	1	2	yes	—
162	C₂×C₄⋊C₄	—	292	16	no	—
163	C₂×C₄⋊C₄	—	73	8	no	—
164	C₂×C₄⋊C₄	—	146	4	no	—
165	C₂×C₄⋊C₄	—	146	16	no	—
166	C₂×C₄⋊C₄	—	146	16	no	—
167	C₂×C₄²	—	73	2	no	—
168	C₂×C₄²	—	73	1	no	—
169	C₂×C₄²	—	73	2	no	—
170	C₂×C₄²	—	73	2	no	—
171	C₂×C₄²	—	73	2	no	—
172	C₂×C₄²	—	73	2	no	—
173	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
174	#107	—	146	32	no	—
175	C₄×C₈	—	73	8	no	—
176	C₄×C₈	—	73	4	no	—
177	C₄×C₈	C₂×#64408	1	4	yes	complemented
178	C₂⁴	—	73	1	no	—
179	C₂²×C₄	—	292	16	no	—
180	C₂²×C₄	—	292	8	no	—
181	C₂²×C₄	—	73	2	no	—
182	C₂²×C₄	—	73	1	no	—
183	C₂²×C₄	—	73	1	no	—
184	C₂²×C₄	—	73	1	no	—
185	C₂²×C₄	—	73	8	no	—
186	C₂²×C₄	—	73	8	no	—
187	C₂²×C₄	—	73	4	no	—
188	C₂²×C₄	—	73	1	no	—
189	C₂²×C₄	—	73	4	no	—
190	C₂²×C₄	—	73	4	no	—
191	C₂²×C₄	—	73	1	no	—
192	C₂²×C₄	—	73	4	no	—
193	C₂²×C₄	—	146	8	no	—
194	C₂²×C₄	C₄⋊#64408	1	2	yes	—
195	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
196	C₂×C₈	—	73	4	no	—
197	C₂×C₈	—	73	8	no	—
198	C₂×C₈	—	73	16	no	—
199	C₂×C₈	—	73	8	no	—
200	C₂×C₈	—	73	8	no	—
201	C₂×C₈	C₄⋊#64408	1	8	yes	complemented
202	C₂×C₈	C₂²×#64408	1	4	yes	—
203	C₄⋊C₄	—	292	32	no	—
204	C₄⋊C₄	—	292	8	no	—
205	C₄⋊C₄	—	146	16	no	—
206	C₄²	—	73	2	no	—
207	C₄²	—	73	2	no	—
208	C₄²	—	73	4	no	—
209	C₄²	—	73	4	no	—
210	C₄²	C₄⋊#64408	1	2	yes	—
211	C₄²	C₂²×#64408	1	2	yes	—
212	C₂³	—	73	1	no	—
213	C₂³	—	73	1	no	—
214	C₂³	—	73	2	no	—
215	C₂³	—	73	2	no	—
216	C₂³	—	73	4	no	—
217	C₂³	—	73	4	no	—
218	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
219	C₂×C₄	—	292	16	no	—
220	C₂×C₄	—	292	8	no	—
221	C₂×C₄	—	584	16	no	—
222	C₂×C₄	—	73	4	no	—
223	C₂×C₄	—	73	1	no	—
224	C₂×C₄	—	73	1	no	—
225	C₂×C₄	—	73	4	no	—
226	C₂×C₄	—	73	8	no	—
227	C₂×C₄	—	73	2	no	—
228	C₂×C₄	—	73	4	no	—
229	C₂×C₄	—	73	2	no	—
230	C₂×C₄	—	73	2	no	—
231	C₂×C₄	—	73	4	no	—
232	C₂×C₄	—	73	2	no	—
233	C₂×C₄	—	73	8	no	—
234	C₂×C₄	—	73	16	no	—
235	C₂×C₄	—	73	8	no	—
236	C₂×C₄	#14556660	1	8	yes	complemented
237	C₂×C₄	C₂×C₄⋊#64408	1	4	yes	—
238	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
239	C₂×C₄	C₂×C₄⋊#64408	1	4	yes	—
240	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
241	C₈	—	73	8	no	—
242	C₈	C₂×C₄⋊#64408	1	8	yes	complemented
243	C₂²	—	73	2	no	—
244	C₂²	—	73	2	no	—
245	C₂²	—	73	4	no	—
246	C₂²	—	73	4	no	—
247	C₂²	—	73	4	no	—
248	C₂²	—	73	8	no	—
249	C₂²	—	73	1	no	—
250	C₂²	—	73	1	no	—
251	C₂²	—	73	2	no	—
252	C₂²	#17989533	1	2	yes	—
253	C₂²	C₂×#14556660	1	4	yes	—
254	C₂²	C₂²×C₄⋊#64408	1	1	yes	characteristic
255	C₄	—	584	8	no	—
256	C₄	—	73	8	no	—
257	C₄	—	73	2	no	—
258	C₄	—	73	2	no	—
259	C₄	#17989533	1	2	yes	—
260	C₄	C₂×#14556660	1	8	yes	complemented
261	C₄	C₂²×C₄⋊#64408	1	2	yes	—
262	C₂	—	73	1	no	—
263	C₂	—	73	4	no	—
264	C₂	—	73	1	no	—
265	C₂	—	73	2	no	—
266	C₂	#22625867	1	4	yes	direct product factor
267	C₂	C₂×#17989533	1	1	yes	characteristic
268	C₂	C₂²×#14556660	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle	C₂³×C₇₃	C₂×C₄⋊C₄
prime radical	C₂³×D₇₃	C₂×D₄
cosocle	C₂×C₄×D₇₃	C₂⁴
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	C₂×#1845
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #22625867	C₂	direct (N × H)
13. C₂×C₄×#10596831	C₄	semidirect
14. C₂×C₈×Dic₇₃	C₄	semidirect
15. C₄×C₈×D₇₃	C₄	semidirect
17. C₂×C₄×C₈×C₇₃	C₄	semidirect
38. C₄×#10596831	C₂×C₄	semidirect
39. C₈×Dic₇₃	C₂×C₄	semidirect
42. C₄×C₈×C₇₃	C₂×C₄	semidirect
62. C₂×#10596831	C₄⋊C₄	semidirect
64. C₈×D₇₃	C₄⋊C₄	semidirect
73. C₂×C₈×C₇₃	C₄⋊C₄	semidirect
87. C₂×C₄×C₇₃	#107	semidirect
92. C₂×Dic₇₃	#107	semidirect
97. C₄×D₇₃	#107	semidirect
106. C₈×C₇₃	C₂×C₄⋊C₄	semidirect
107. #10596831	C₂×C₄⋊C₄	semidirect
122. C₄×C₇₃	C₂×#107	semidirect
125. Dic₇₃	C₂×#107	semidirect
128. C₂×C₇₃	#1845	semidirect
131. D₇₃	#1845	semidirect
139. C₇₃	C₂×#1845	semidirect
150. C₂×C₄×C₈	#64408	semidirect
177. C₄×C₈	C₂×#64408	semidirect
201. C₂×C₈	C₄⋊#64408	semidirect
236. C₂×C₄	#14556660	semidirect
242. C₈	C₂×C₄⋊#64408	semidirect
260. C₄	C₂×#14556660	semidirect
266. C₂	#22625867	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#14556660	2
C₂×#17989533	2
#22625867	2
C₂×C₄×C₈×D₇₃	2
C₂×#1845	73

Subgroups — C2×#22625867 #34021333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22625867 #34021333