Subgroups — C₂×#22625486 #34017993

Order 18688.

854 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989591	C₂	1	2	yes	maximal
2	#22625486	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17989448	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924038	—	2	2	no	—
6	#17989591	—	2	4	no	—
7	#17989591	C₂²	1	8	yes	—
8	C₂×C₄⋊#10596833	—	2	2	no	—
9	C₂×C₄⋊#10596833	C₂²	1	4	yes	—
10	#17989448	C₂²	1	16	yes	complemented
11	C₂²×C₈×D₇₃	C₄	1	2	yes	—
12	C₂×C₄²×D₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
15	C₄×C₈×D₇₃	C₄	1	4	yes	—
16	C₄×C₈×D₇₃	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
18	C₂×#11924038	—	2	8	no	—
19	C₂×#11924038	—	4	8	no	—
20	C₂²×#10596833	—	2	4	no	—
21	C₄⋊#10596833	—	2	8	no	—
22	C₄⋊#10596833	—	2	4	no	—
23	C₄⋊#10596833	C₂³	1	8	yes	—
24	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×D₇₃	Q₈	1	1	yes	characteristic
27	C₂²×#10596831	C₂×C₄	1	2	yes	—
28	C₂×C₈×D₇₃	—	2	8	no	—
29	C₂×C₈×D₇₃	C₂×C₄	1	8	yes	—
30	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
31	C₂×C₄×Dic₇₃	Q₈	1	1	yes	characteristic
32	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
33	C₄²×D₇₃	C₂×C₄	1	2	yes	—
34	C₄²×D₇₃	D₄	1	2	yes	—
35	C₄²×D₇₃	Q₈	1	2	yes	—
36	C₄²×D₇₃	C₂³	1	2	yes	—
37	C₄×#10596831	C₂×C₄	1	4	yes	—
38	C₈×Dic₇₃	C₂×C₄	1	4	yes	—
39	C₈×Dic₇₃	C₂³	1	4	yes	—
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	—
43	#11924038	—	4	16	no	—
44	C₂×#10596833	—	2	8	no	—
45	C₂×#10596833	—	4	8	no	—
46	C₂³×D₇₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×D₇₃	—	2	4	no	—
51	C₂×C₄×D₇₃	—	2	4	no	—
52	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
53	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
54	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
55	C₂×C₄×D₇₃	SD₁₆	1	8	yes	—
56	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
57	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×D₇₃	C₂×Q₈	1	2	yes	—
59	C₂×#10596831	—	2	4	no	—
60	C₂×#10596831	C₂²×C₄	1	4	yes	—
61	C₈×D₇₃	—	2	16	no	—
62	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
63	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
64	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
65	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
66	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
67	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
68	C₄×Dic₇₃	C₂×Q₈	1	2	yes	—
69	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₇₃	M₁₆	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	—	2	2	no	—
79	C₂²×D₇₃	—	2	2	no	—
80	C₂²×D₇₃	#106	1	4	yes	—
81	C₂²×D₇₃	C₂×C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₇₃	#106	1	4	yes	—
83	C₂×C₄×C₇₃	#108	1	4	yes	—
84	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
86	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
87	C₂×C₄×C₇₃	C₂×M₁₆	1	4	yes	—
88	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
89	C₂×Dic₇₃	—	2	2	no	—
90	C₂×Dic₇₃	—	2	2	no	—
91	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×Dic₇₃	C₂×SD₁₆	1	4	yes	—
95	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
96	C₂×Dic₇₃	C₂²×Q₈	1	1	yes	characteristic
97	C₄×D₇₃	—	2	4	no	—
98	C₄×D₇₃	—	2	4	no	—
99	C₄×D₇₃	—	2	4	no	—
100	C₄×D₇₃	—	2	4	no	—
101	C₄×D₇₃	#106	1	8	yes	—
102	C₄×D₇₃	C₂×SD₁₆	1	8	yes	—
103	#10596833	—	4	8	no	—
104	C₈×C₇₃	C₂×M₁₆	1	8	yes	complemented
105	#10596831	—	2	4	no	—
106	C₂²×C₇₃	#423	1	2	yes	—
107	C₂²×C₇₃	C₂×#106	1	2	yes	—
108	C₂²×C₇₃	C₂×#108	1	2	yes	—
109	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
110	C₂×D₇₃	—	2	8	no	—
111	C₂×D₇₃	—	2	2	no	—
112	C₂×D₇₃	—	2	2	no	—
113	C₂×D₇₃	C₂×#106	1	4	yes	—
114	C₄×C₇₃	#423	1	2	yes	—
115	C₄×C₇₃	C₂×#106	1	4	yes	—
116	C₄×C₇₃	C₂×#108	1	4	yes	—
117	C₄×C₇₃	C₂²×M₁₆	1	2	yes	—
118	Dic₇₃	—	2	2	no	—
119	Dic₇₃	—	2	1	no	—
120	Dic₇₃	—	2	1	no	—
121	Dic₇₃	C₂×#106	1	2	yes	—
122	Dic₇₃	C₂²×SD₁₆	1	2	yes	—
123	C₂×#1566	—	73	1	no	maximal, Sylow
124	C₂×C₇₃	#1566	1	4	yes	complemented
125	C₂×C₇₃	C₂×#423	1	1	yes	characteristic
126	C₂×C₇₃	C₂²×#106	1	1	yes	characteristic
127	C₂×C₇₃	C₂²×#108	1	1	yes	characteristic
128	D₇₃	—	2	4	no	—
129	C₂×#423	—	73	2	no	—
130	C₂²×C₄×C₈	—	73	1	no	—
131	#1566	—	73	8	no	—
132	C₂×#334	—	73	4	no	—
133	C₇₃	C₂×#1566	1	1	yes	characteristic, complemented, Sylow
134	C₂²×M₁₆	—	146	2	no	—
135	C₂³×C₈	—	73	2	no	—
136	C₂²×C₄²	—	73	1	no	—
137	#423	—	73	8	no	—
138	#423	—	146	4	no	—
139	C₂×C₄⋊C₈	—	73	4	no	—
140	C₂×C₄⋊C₈	—	146	2	no	—
141	C₂×C₄×C₈	—	73	2	no	—
142	C₂×C₄×C₈	—	73	1	no	—
143	C₂×C₄×C₈	—	73	4	no	—
144	C₂×C₄×C₈	—	73	4	no	—
145	C₂×C₄×C₈	#64408	1	1	yes	characteristic
146	#334	—	73	16	no	—
147	C₂³×C₄	—	73	1	no	—
148	C₂³×C₄	—	73	1	no	—
149	C₂³×C₄	—	73	1	no	—
150	C₂×M₁₆	—	292	8	no	—
151	C₂×M₁₆	—	146	8	no	—
152	C₂²×C₈	—	73	2	no	—
153	C₂²×C₈	—	73	8	no	—
154	C₂²×C₈	—	146	4	no	—
155	C₂²×C₈	—	146	8	no	—
156	C₂²×C₈	C₂×#64408	1	2	yes	—
157	C₂×C₄²	—	73	1	no	—
158	C₂×C₄²	—	73	1	no	—
159	C₂×C₄²	—	73	1	no	—
160	C₂×C₄²	—	73	2	no	—
161	C₂×C₄²	—	73	2	no	—
162	C₂×C₄²	—	73	2	no	—
163	C₂×C₄²	—	73	2	no	—
164	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
165	C₄⋊C₈	—	73	8	no	—
166	C₄⋊C₈	—	146	4	no	—
167	C₄⋊C₈	—	146	8	no	—
168	C₄×C₈	—	73	4	no	—
169	C₄×C₈	—	73	4	no	—
170	C₄×C₈	—	73	4	no	—
171	C₄×C₈	C₂×#64408	1	4	yes	—
172	C₂⁴	—	73	1	no	—
173	C₂²×C₄	—	73	1	no	—
174	C₂²×C₄	—	73	8	no	—
175	C₂²×C₄	—	73	2	no	—
176	C₂²×C₄	—	73	2	no	—
177	C₂²×C₄	—	73	1	no	—
178	C₂²×C₄	—	73	1	no	—
179	C₂²×C₄	—	73	2	no	—
180	C₂²×C₄	—	73	2	no	—
181	C₂²×C₄	—	73	2	no	—
182	C₂²×C₄	—	73	2	no	—
183	C₂²×C₄	—	146	4	no	—
184	C₂²×C₄	—	146	4	no	—
185	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
186	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
187	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
188	M₁₆	—	292	16	no	—
189	C₂×C₈	—	292	8	no	—
190	C₂×C₈	—	73	4	no	—
191	C₂×C₈	—	146	8	no	—
192	C₂×C₈	—	146	4	no	—
193	C₂×C₈	—	146	16	no	—
194	C₂×C₈	#11924038	1	8	yes	complemented
195	C₂×C₈	C₂²×#64408	1	4	yes	—
196	C₄²	—	73	1	no	—
197	C₄²	—	73	1	no	—
198	C₄²	—	73	2	no	—
199	C₄²	—	73	2	no	—
200	C₄²	—	73	1	no	—
201	C₄²	—	73	2	no	—
202	C₄²	—	73	1	no	—
203	C₄²	—	73	2	no	—
204	C₄²	C₄⋊#64408	1	2	yes	—
205	C₄²	C₂²×#64408	1	2	yes	—
206	C₂³	—	73	2	no	—
207	C₂³	—	73	4	no	—
208	C₂³	—	146	2	no	—
209	C₂³	—	146	2	no	—
210	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
211	C₂×C₄	—	73	1	no	—
212	C₂×C₄	—	73	1	no	—
213	C₂×C₄	—	73	4	no	—
214	C₂×C₄	—	73	8	no	—
215	C₂×C₄	—	73	8	no	—
216	C₂×C₄	—	73	2	no	—
217	C₂×C₄	—	73	1	no	—
218	C₂×C₄	—	73	1	no	—
219	C₂×C₄	—	146	2	no	—
220	C₂×C₄	—	146	2	no	—
221	C₂×C₄	—	146	4	no	—
222	C₂×C₄	—	146	4	no	—
223	C₂×C₄	—	146	4	no	—
224	C₂×C₄	—	146	4	no	—
225	C₂×C₄	#14556659	1	4	yes	—
226	C₂×C₄	#14556661	1	4	yes	—
227	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
228	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
229	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
230	C₂×C₄	C₂×#11924038	1	4	yes	—
231	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
232	C₈	—	292	8	no	—
233	C₈	—	146	4	no	—
234	C₈	C₂×#11924038	1	8	yes	complemented
235	C₂²	—	73	4	no	—
236	C₂²	—	146	2	no	—
237	C₂²	—	146	2	no	—
238	C₂²	—	146	8	no	—
239	C₂²	#17989591	1	2	yes	—
240	C₂²	C₂×#14556659	1	2	yes	—
241	C₂²	C₂×#14556661	1	2	yes	—
242	C₂²	C₂²×C₄⋊#64408	1	1	yes	characteristic
243	C₄	—	73	2	no	—
244	C₄	—	73	2	no	—
245	C₄	—	146	2	no	—
246	C₄	—	146	1	no	—
247	C₄	—	146	1	no	—
248	C₄	#17989591	1	2	yes	—
249	C₄	C₂×#14556659	1	4	yes	—
250	C₄	C₂×#14556661	1	4	yes	—
251	C₄	C₂²×#11924038	1	2	yes	—
252	C₂	—	146	4	no	—
253	C₂	#22625486	1	4	yes	direct product factor
254	C₂	C₂×#17989591	1	1	yes	characteristic
255	C₂	C₂²×#14556659	1	1	yes	characteristic
256	C₂	C₂²×#14556661	1	1	yes	characteristic

	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	C₄⋊C₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	C₂×#1566
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
2. #22625486	C₂	direct (N × H)
3. C₂×#17989448	C₂	semidirect
10. #17989448	C₂²	semidirect
73. C₂×C₈×C₇₃	M₁₆	semidirect
104. C₈×C₇₃	C₂×M₁₆	semidirect
124. C₂×C₇₃	#1566	semidirect
133. C₇₃	C₂×#1566	semidirect
194. C₂×C₈	#11924038	semidirect
234. C₈	C₂×#11924038	semidirect
253. C₂	#22625486	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×#17989591	2
#22625486	2
C₂×#17989448	2
C₂×C₄×C₈×D₇₃	2
C₂×#1566	73

Subgroups — C2×#22625486 #34017993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22625486 #34017993