Subgroups — C₂×#22625571 #34018481

Order 18688.

670 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×#17989638	C₂	1	2	yes	maximal
2	#22625571	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17989450	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊#10596833	C₂	1	2	yes	maximal
5	C₂×C₈×#10596833	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic
7	#17989638	—	2	4	no	—
8	#17989638	C₂²	1	8	yes	—
9	C₂×#14556658	—	2	2	no	—
10	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
11	C₂×#14556646	—	2	2	no	—
12	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
13	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
14	#17989450	C₂²	1	4	yes	—
15	Q₈⋊#10596833	C₂²	1	8	yes	—
16	C₈×#10596833	C₂²	1	4	yes	—
17	C₂×Q₈×Dic₇₃	C₂²	1	2	yes	—
18	Q₁₆×Dic₇₃	C₄	1	4	yes	—
19	Q₁₆×Dic₇₃	C₂²	1	4	yes	—
20	C₂×#14556525	C₄	1	1	yes	characteristic
21	C₂×#14556515	C₄	1	2	yes	—
22	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
23	C₂²×Q₁₆×C₇₃	C₄	1	1	yes	characteristic
24	C₂²×#10596833	—	2	2	no	—
25	C₂²×#10596833	—	2	1	no	—
26	C₂²×#10596833	D₄	1	1	yes	characteristic
27	C₂²×#10596833	D₄	1	1	yes	characteristic
28	#14556658	—	2	8	no	—
29	C₄⋊#10596833	—	2	2	no	—
30	C₄⋊#10596833	C₂³	1	4	yes	—
31	#14556646	—	2	8	no	—
32	C₄×#10596833	—	2	1	no	—
33	C₄×#10596833	D₄	1	2	yes	—
34	C₄×#10596833	C₂³	1	2	yes	—
35	C₄×#10596833	C₂³	1	2	yes	—
36	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
37	Q₈×Dic₇₃	—	2	2	no	—
38	Q₈×Dic₇₃	—	2	2	no	—
39	Q₈×Dic₇₃	C₂×C₄	1	4	yes	—
40	Q₈×Dic₇₃	C₂³	1	4	yes	—
41	C₂×C₄⋊Dic₇₃	—	2	2	no	—
42	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
43	C₂×C₄×Dic₇₃	—	2	2	no	—
44	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
45	#14556525	C₂×C₄	1	4	yes	—
46	#14556515	C₂×C₄	1	8	yes	—
47	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
48	C₈×Dic₇₃	C₂³	1	2	yes	—
49	C₂²×Q₈×C₇₃	C₂×C₄	1	2	yes	—
50	C₂×Q₁₆×C₇₃	—	2	4	no	—
51	C₂×Q₁₆×C₇₃	C₂×C₄	1	4	yes	—
52	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂×#10596833	—	2	4	no	—
54	C₂×#10596833	—	2	2	no	—
55	C₂×#10596833	—	2	2	no	—
56	C₂×#10596833	—	2	4	no	—
57	C₂×#10596833	—	4	4	no	—
58	C₂×#10596833	Q₁₆	1	4	yes	complemented
59	C₂×#10596833	C₂×D₄	1	2	yes	—
60	C₂×#10596833	C₂×D₄	1	2	yes	—
61	C₂²×Dic₇₃	—	2	2	no	—
62	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₇₃	#41	1	1	yes	characteristic
64	C₂×#10596831	—	2	2	no	—
65	C₂×#10596831	C₂²×C₄	1	2	yes	—
66	C₄⋊Dic₇₃	—	2	2	no	—
67	C₄⋊Dic₇₃	—	2	4	no	—
68	C₄⋊Dic₇₃	—	2	4	no	—
69	C₄⋊Dic₇₃	C₂²×C₄	1	4	yes	—
70	C₄×Dic₇₃	—	2	4	no	—
71	C₄×Dic₇₃	—	2	4	no	—
72	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
73	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
74	C₄×Dic₇₃	#41	1	1	yes	characteristic
75	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
76	C₂×Q₈×C₇₃	—	2	2	no	—
77	C₂×Q₈×C₇₃	—	2	4	no	—
78	C₂×Q₈×C₇₃	—	2	4	no	—
79	C₂×Q₈×C₇₃	C₂²×C₄	1	4	yes	—
80	C₂²×C₄×C₇₃	—	2	2	no	—
81	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
82	Q₁₆×C₇₃	—	2	8	no	—
83	C₂×C₈×C₇₃	M₁₆	1	4	yes	—
84	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
85	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
86	Q₈×C₇₃	—	2	4	no	—
87	Q₈×C₇₃	—	2	4	no	—
88	Q₈×C₇₃	—	4	4	no	—
89	C₂×C₄×C₇₃	—	2	4	no	—
90	C₂×C₄×C₇₃	—	4	4	no	—
91	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₇₃	#131	1	2	yes	—
94	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
95	C₂×Dic₇₃	—	2	2	no	—
96	C₂×Dic₇₃	—	2	4	no	—
97	C₂×Dic₇₃	—	4	4	no	—
98	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
99	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
100	C₂×Dic₇₃	C₂×Q₁₆	1	2	yes	—
101	C₂×Dic₇₃	#135	1	2	yes	—
102	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
103	C₂×Dic₇₃	C₂×#41	1	1	yes	characteristic
104	#10596833	—	2	4	no	—
105	#10596833	—	2	2	no	—
106	#10596833	—	4	4	no	—
107	#10596833	C₂×Q₁₆	1	4	yes	complemented
108	C₈×C₇₃	C₂×M₁₆	1	4	yes	—
109	#10596831	—	2	2	no	—
110	C₂²×C₇₃	#436	1	2	yes	—
111	C₂²×C₇₃	C₄×Q₁₆	1	2	yes	—
112	C₂²×C₇₃	#443	1	2	yes	—
113	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
114	C₄×C₇₃	—	4	4	no	—
115	C₄×C₇₃	#436	1	2	yes	—
116	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
117	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
118	Dic₇₃	—	2	1	no	—
119	Dic₇₃	—	2	1	no	—
120	Dic₇₃	—	4	4	no	—
121	Dic₇₃	C₄×Q₁₆	1	2	yes	—
122	Dic₇₃	C₂²×Q₁₆	1	1	yes	characteristic
123	Dic₇₃	C₂×#135	1	1	yes	characteristic
124	C₂×#1591	—	73	1	no	maximal, Sylow
125	C₂×C₇₃	#1591	1	4	yes	complemented
126	C₂×C₇₃	C₂×#436	1	1	yes	characteristic
127	C₂×C₇₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
128	C₂×C₇₃	C₂×#443	1	1	yes	characteristic
129	C₂×#446	—	73	2	no	—
130	C₂×C₄×Q₁₆	—	73	1	no	—
131	#1591	—	73	8	no	—
132	C₂×#335	—	73	1	no	—
133	C₂×Q₈⋊C₈	—	73	2	no	—
134	C₂×C₈²	—	73	1	no	—
135	C₇₃	C₂×#1591	1	1	yes	characteristic, complemented, Sylow
136	C₂²×Q₁₆	#64408	1	1	yes	characteristic
137	C₂×C₄×Q₈	—	73	2	no	—
138	#446	—	73	8	no	—
139	#446	—	146	4	no	—
140	C₄×Q₁₆	—	73	4	no	—
141	C₄×Q₁₆	—	73	4	no	—
142	C₂×#107	—	73	1	no	—
143	C₂×C₄⋊C₈	—	73	2	no	—
144	C₂×C₄⋊C₈	—	146	2	no	—
145	C₂×#103	—	73	2	no	—
146	C₂×#97	—	146	2	no	—
147	C₂×C₄×C₈	—	73	1	no	—
148	C₂×C₄×C₈	—	73	1	no	—
149	C₂×C₄×C₈	—	73	1	no	—
150	#335	—	73	4	no	—
151	Q₈⋊C₈	—	73	8	no	—
152	C₈²	—	73	4	no	—
153	C₂²×Q₈	C₂×#64408	1	2	yes	—
154	C₂×Q₁₆	—	2	4	no	—
155	C₂×Q₁₆	C₂×#64408	1	4	yes	—
156	C₂²×C₈	—	73	1	no	—
157	C₂²×C₈	—	73	1	no	—
158	C₂²×C₈	—	73	1	no	—
159	C₂²×C₈	—	146	2	no	—
160	C₂²×C₈	—	146	1	no	—
161	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
162	C₄×Q₈	—	73	4	no	—
163	C₄×Q₈	—	73	4	no	—
164	C₄×Q₈	—	146	2	no	—
165	C₄×Q₈	—	146	2	no	—
166	C₂×C₄⋊C₄	—	73	2	no	—
167	C₂×C₄⋊C₄	—	146	2	no	—
168	C₂×C₄²	—	73	1	no	—
169	C₂×C₄²	—	146	2	no	—
170	#107	—	73	4	no	—
171	C₄⋊C₈	—	73	4	no	—
172	C₄⋊C₈	—	146	2	no	—
173	C₄⋊C₈	—	146	8	no	—
174	#103	—	73	8	no	—
175	#97	—	146	8	no	—
176	C₄×C₈	—	73	2	no	—
177	C₄×C₈	—	73	2	no	—
178	C₄×C₈	—	73	2	no	—
179	C₄×C₈	—	73	2	no	—
180	C₄×C₈	—	73	2	no	—
181	C₄×C₈	—	146	1	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×Q₈	—	2	4	no	—
184	C₂×Q₈	—	2	4	no	—
185	C₂×Q₈	C₂²×#64408	1	4	yes	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	73	1	no	—
188	C₂²×C₄	—	73	1	no	—
189	C₂²×C₄	—	146	2	no	—
190	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
191	Q₁₆	—	2	8	no	—
192	C₂×C₈	—	292	4	no	—
193	C₂×C₈	—	73	4	no	—
194	C₂×C₈	—	73	2	no	—
195	C₂×C₈	—	73	2	no	—
196	C₂×C₈	—	73	2	no	—
197	C₂×C₈	—	146	4	no	—
198	C₂×C₈	—	146	2	no	—
199	C₂×C₈	—	146	2	no	—
200	C₂×C₈	—	146	4	no	—
201	C₂×C₈	—	146	2	no	—
202	C₂×C₈	#11924037	1	4	yes	—
203	C₂×C₈	C₂²×#64408	1	2	yes	—
204	C₄⋊C₄	—	73	4	no	—
205	C₄⋊C₄	—	146	4	no	—
206	C₄⋊C₄	—	146	4	no	—
207	C₄⋊C₄	—	146	2	no	—
208	C₄²	—	73	1	no	—
209	C₄²	—	73	1	no	—
210	C₄²	—	73	1	no	—
211	C₄²	—	73	1	no	—
212	C₄²	—	146	4	no	—
213	C₄²	—	146	4	no	—
214	C₂³	D₄×#64408	1	1	yes	characteristic
215	Q₈	—	2	4	no	—
216	Q₈	—	2	4	no	—
217	Q₈	—	4	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	292	4	no	—
221	C₂×C₄	—	73	1	no	—
222	C₂×C₄	—	73	1	no	—
223	C₂×C₄	—	73	2	no	—
224	C₂×C₄	—	73	2	no	—
225	C₂×C₄	—	73	1	no	—
226	C₂×C₄	—	73	1	no	—
227	C₂×C₄	—	146	4	no	—
228	C₂×C₄	—	146	2	no	—
229	C₂×C₄	D₄×#64408	1	1	yes	characteristic
230	C₂×C₄	C₂×#11924037	1	2	yes	—
231	C₂×C₄	#14556678	1	2	yes	—
232	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
233	C₈	—	292	4	no	—
234	C₈	—	73	4	no	—
235	C₈	—	146	2	no	—
236	C₈	—	146	4	no	—
237	C₈	—	146	2	no	—
238	C₈	C₂×#11924037	1	4	yes	—
239	C₂²	#17989622	1	2	yes	—
240	C₂²	Q₁₆×#64408	1	2	yes	—
241	C₂²	#17989631	1	2	yes	—
242	C₂²	C₂×D₄×#64408	1	1	yes	characteristic
243	C₄	—	4	4	no	—
244	C₄	—	292	4	no	—
245	C₄	—	73	1	no	—
246	C₄	—	73	1	no	—
247	C₄	—	73	2	no	—
248	C₄	—	146	1	no	—
249	C₄	—	146	1	no	—
250	C₄	#17989622	1	2	yes	—
251	C₄	C₂²×#11924037	1	1	yes	characteristic
252	C₄	C₂×#14556678	1	1	yes	characteristic
253	C₂	#22625571	1	4	yes	direct product factor
254	C₂	C₂×#17989622	1	1	yes	characteristic
255	C₂	C₂×Q₁₆×#64408	1	1	yes	characteristic
256	C₂	C₂×#17989631	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#64408
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₂²×Q₁₆×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	C₂²×Q₁₆	#64408
hypocenter	C₇₃	C₂×#1591
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #22625571	C₂	direct (N × H)
58. C₂×#10596833	Q₁₆	semidirect
107. #10596833	C₂×Q₁₆	semidirect
125. C₂×C₇₃	#1591	semidirect
135. C₇₃	C₂×#1591	semidirect
253. C₂	#22625571	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×#17989638	2
#22625571	2
C₂×#17989450	2
C₂×Q₈⋊#10596833	2
C₂×C₈×#10596833	2
C₂×Q₁₆×Dic₇₃	2
C₂×#1591	73

Subgroups — C2×#22625571 #34018481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22625571 #34018481