Subgroups — C₂×#22625489 #34017997

Order 18688.

854 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×#17989591	C₂	1	2	yes	maximal
2	#22625489	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17989450	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	#17989450	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	D₈	1	4	yes	—
56	C₂×C₄×D₇₃	Q₁₆	1	4	yes	—
57	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
58	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×D₇₃	C₂×Q₈	1	2	yes	—
60	C₂×#10596831	—	2	4	no	—
61	C₂×#10596831	C₂²×C₄	1	4	yes	—
62	C₈×D₇₃	—	2	16	no	—
63	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
64	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
65	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
66	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
67	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
68	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
69	C₄×Dic₇₃	C₂×Q₈	1	2	yes	—
70	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
71	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₇₃	M₁₆	1	8	yes	complemented
75	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
76	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
77	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
78	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂²×D₇₃	—	2	2	no	—
80	C₂²×D₇₃	—	2	2	no	—
81	C₂²×D₇₃	#107	1	4	yes	—
82	C₂²×D₇₃	C₂×C₄⋊C₄	1	2	yes	—
83	C₂×C₄×C₇₃	#107	1	4	yes	—
84	C₂×C₄×C₇₃	#108	1	4	yes	—
85	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
87	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
88	C₂×C₄×C₇₃	C₂×M₁₆	1	4	yes	—
89	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
90	C₂×Dic₇₃	—	2	2	no	—
91	C₂×Dic₇₃	—	2	2	no	—
92	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
96	C₂×Dic₇₃	C₂×Q₁₆	1	2	yes	—
97	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×Dic₇₃	C₂²×Q₈	1	1	yes	characteristic
99	C₄×D₇₃	—	2	4	no	—
100	C₄×D₇₃	—	2	4	no	—
101	C₄×D₇₃	—	2	4	no	—
102	C₄×D₇₃	—	2	4	no	—
103	C₄×D₇₃	#107	1	8	yes	—
104	C₄×D₇₃	C₂×D₈	1	4	yes	—
105	C₄×D₇₃	C₂×Q₁₆	1	4	yes	—
106	#10596833	—	4	8	no	—
107	C₈×C₇₃	C₂×M₁₆	1	8	yes	complemented
108	#10596831	—	2	4	no	—
109	C₂²×C₇₃	#423	1	2	yes	—
110	C₂²×C₇₃	C₂×#107	1	2	yes	—
111	C₂²×C₇₃	C₂×#108	1	2	yes	—
112	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
113	C₂×D₇₃	—	2	8	no	—
114	C₂×D₇₃	—	2	2	no	—
115	C₂×D₇₃	—	2	2	no	—
116	C₂×D₇₃	C₂×#107	1	4	yes	—
117	C₄×C₇₃	#423	1	2	yes	—
118	C₄×C₇₃	C₂×#107	1	4	yes	—
119	C₄×C₇₃	C₂×#108	1	4	yes	—
120	C₄×C₇₃	C₂²×M₁₆	1	2	yes	—
121	Dic₇₃	—	2	2	no	—
122	Dic₇₃	—	2	1	no	—
123	Dic₇₃	—	2	1	no	—
124	Dic₇₃	C₂×#107	1	2	yes	—
125	Dic₇₃	C₂²×D₈	1	1	yes	characteristic
126	Dic₇₃	C₂²×Q₁₆	1	1	yes	characteristic
127	C₂×#1567	—	73	1	no	maximal, Sylow
128	C₂×C₇₃	#1567	1	4	yes	complemented
129	C₂×C₇₃	C₂×#423	1	1	yes	characteristic
130	C₂×C₇₃	C₂²×#107	1	1	yes	characteristic
131	C₂×C₇₃	C₂²×#108	1	1	yes	characteristic
132	D₇₃	—	2	4	no	—
133	C₂×#423	—	73	2	no	—
134	C₂²×C₄×C₈	—	73	1	no	—
135	#1567	—	73	8	no	—
136	C₂×#335	—	73	4	no	—
137	C₇₃	C₂×#1567	1	1	yes	characteristic, complemented, Sylow
138	C₂²×M₁₆	—	146	2	no	—
139	C₂³×C₈	—	73	2	no	—
140	C₂²×C₄²	—	73	1	no	—
141	#423	—	73	8	no	—
142	#423	—	146	4	no	—
143	C₂×C₄⋊C₈	—	73	4	no	—
144	C₂×C₄⋊C₈	—	146	2	no	—
145	C₂×C₄×C₈	—	73	2	no	—
146	C₂×C₄×C₈	—	73	1	no	—
147	C₂×C₄×C₈	—	73	4	no	—
148	C₂×C₄×C₈	—	73	4	no	—
149	C₂×C₄×C₈	#64408	1	1	yes	characteristic
150	#335	—	73	16	no	—
151	C₂³×C₄	—	73	1	no	—
152	C₂³×C₄	—	73	1	no	—
153	C₂³×C₄	—	73	1	no	—
154	C₂×M₁₆	—	292	8	no	—
155	C₂×M₁₆	—	146	8	no	—
156	C₂²×C₈	—	73	2	no	—
157	C₂²×C₈	—	73	8	no	—
158	C₂²×C₈	—	146	4	no	—
159	C₂²×C₈	—	146	8	no	—
160	C₂²×C₈	C₂×#64408	1	2	yes	—
161	C₂×C₄²	—	73	1	no	—
162	C₂×C₄²	—	73	1	no	—
163	C₂×C₄²	—	73	1	no	—
164	C₂×C₄²	—	73	2	no	—
165	C₂×C₄²	—	73	2	no	—
166	C₂×C₄²	—	73	2	no	—
167	C₂×C₄²	—	73	2	no	—
168	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
169	C₄⋊C₈	—	73	8	no	—
170	C₄⋊C₈	—	146	4	no	—
171	C₄⋊C₈	—	146	8	no	—
172	C₄×C₈	—	73	4	no	—
173	C₄×C₈	—	73	4	no	—
174	C₄×C₈	—	73	4	no	—
175	C₄×C₈	C₂×#64408	1	4	yes	—
176	C₂⁴	—	73	1	no	—
177	C₂²×C₄	—	73	1	no	—
178	C₂²×C₄	—	73	4	no	—
179	C₂²×C₄	—	73	4	no	—
180	C₂²×C₄	—	73	2	no	—
181	C₂²×C₄	—	73	2	no	—
182	C₂²×C₄	—	73	1	no	—
183	C₂²×C₄	—	73	1	no	—
184	C₂²×C₄	—	73	2	no	—
185	C₂²×C₄	—	73	2	no	—
186	C₂²×C₄	—	73	2	no	—
187	C₂²×C₄	—	73	2	no	—
188	C₂²×C₄	—	146	4	no	—
189	C₂²×C₄	—	146	4	no	—
190	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
191	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
192	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
193	M₁₆	—	292	16	no	—
194	C₂×C₈	—	292	8	no	—
195	C₂×C₈	—	73	4	no	—
196	C₂×C₈	—	146	8	no	—
197	C₂×C₈	—	146	4	no	—
198	C₂×C₈	—	146	16	no	—
199	C₂×C₈	#11924038	1	8	yes	complemented
200	C₂×C₈	C₂²×#64408	1	4	yes	—
201	C₄²	—	73	1	no	—
202	C₄²	—	73	1	no	—
203	C₄²	—	73	2	no	—
204	C₄²	—	73	2	no	—
205	C₄²	—	73	1	no	—
206	C₄²	—	73	2	no	—
207	C₄²	—	73	1	no	—
208	C₄²	—	73	2	no	—
209	C₄²	C₄⋊#64408	1	2	yes	—
210	C₄²	C₂²×#64408	1	2	yes	—
211	C₂³	—	73	2	no	—
212	C₂³	—	73	4	no	—
213	C₂³	—	146	2	no	—
214	C₂³	—	146	2	no	—
215	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
216	C₂×C₄	—	73	1	no	—
217	C₂×C₄	—	73	1	no	—
218	C₂×C₄	—	73	2	no	—
219	C₂×C₄	—	73	2	no	—
220	C₂×C₄	—	73	4	no	—
221	C₂×C₄	—	73	8	no	—
222	C₂×C₄	—	73	4	no	—
223	C₂×C₄	—	73	2	no	—
224	C₂×C₄	—	73	1	no	—
225	C₂×C₄	—	73	1	no	—
226	C₂×C₄	—	146	2	no	—
227	C₂×C₄	—	146	2	no	—
228	C₂×C₄	—	146	4	no	—
229	C₂×C₄	—	146	4	no	—
230	C₂×C₄	—	146	4	no	—
231	C₂×C₄	—	146	4	no	—
232	C₂×C₄	#14556660	1	4	yes	—
233	C₂×C₄	#14556662	1	4	yes	—
234	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
235	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
236	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
237	C₂×C₄	C₂×#11924038	1	4	yes	—
238	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
239	C₈	—	292	8	no	—
240	C₈	—	146	4	no	—
241	C₈	C₂×#11924038	1	8	yes	complemented
242	C₂²	—	73	4	no	—
243	C₂²	—	146	2	no	—
244	C₂²	—	146	2	no	—
245	C₂²	—	146	8	no	—
246	C₂²	#17989591	1	2	yes	—
247	C₂²	C₂×#14556660	1	2	yes	—
248	C₂²	C₂×#14556662	1	2	yes	—
249	C₂²	C₂²×C₄⋊#64408	1	1	yes	characteristic
250	C₄	—	73	1	no	—
251	C₄	—	73	1	no	—
252	C₄	—	73	2	no	—
253	C₄	—	146	2	no	—
254	C₄	—	146	1	no	—
255	C₄	—	146	1	no	—
256	C₄	#17989591	1	2	yes	—
257	C₄	C₂×#14556660	1	4	yes	—
258	C₄	C₂×#14556662	1	4	yes	—
259	C₄	C₂²×#11924038	1	2	yes	—
260	C₂	—	146	4	no	—
261	C₂	#22625489	1	4	yes	direct product factor
262	C₂	C₂×#17989591	1	1	yes	characteristic
263	C₂	C₂²×#14556660	1	1	yes	characteristic
264	C₂	C₂²×#14556662	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×#1567
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. #22625489	C₂	direct (N × H)
3. C₂×#17989450	C₂	semidirect
10. #17989450	C₂²	semidirect
74. C₂×C₈×C₇₃	M₁₆	semidirect
107. C₈×C₇₃	C₂×M₁₆	semidirect
128. C₂×C₇₃	#1567	semidirect
137. C₇₃	C₂×#1567	semidirect
199. C₂×C₈	#11924038	semidirect
241. C₈	C₂×#11924038	semidirect
261. C₂	#22625489	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
#22625489	2
C₂×#17989450	2
C₂×C₄×C₈×D₇₃	2
C₂×#1567	73

Subgroups — C2×#22625489 #34017997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22625489 #34017997