Subgroups — C₂×#22607895 #32995719

Order 18688.

406 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11924003	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988340	C₂	1	1	yes	maximal, characteristic
3	#22607895	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×Q₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924003	—	2	1	no	—
6	C₂×C₄⋊#10596831	—	2	1	no	—
7	C₄×#11924003	—	2	1	no	—
8	C₄×#11924003	C₂²	1	2	yes	—
9	#17988340	—	2	1	no	—
10	#17988340	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₇₃	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₇₃	—	2	1	no	—
13	C₄×Q₁₆×C₇₃	—	2	1	no	—
14	C₄×Q₁₆×C₇₃	—	2	1	no	—
15	C₄×Q₁₆×C₇₃	C₄	1	2	yes	—
16	C₄×Q₁₆×C₇₃	C₂²	1	2	yes	—
17	C₂×#107×C₇₃	C₄	1	1	yes	characteristic
18	C₂×#103×C₇₃	—	2	1	no	—
19	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
20	C₂²×#10596831	—	4	1	no	—
21	C₂×#11924003	—	2	4	no	—
22	C₂×#11924003	—	2	2	no	—
23	C₄⋊#10596831	—	2	2	no	—
24	C₄⋊#10596831	—	4	1	no	—
25	C₂²×Q₈×C₇₃	—	2	1	no	—
26	C₂×Q₁₆×C₇₃	—	2	1	no	—
27	C₂×Q₁₆×C₇₃	—	2	2	no	—
28	C₂×Q₁₆×C₇₃	C₈	1	4	yes	complemented
29	C₂×Q₁₆×C₇₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
32	C₄×Q₈×C₇₃	—	2	2	no	—
33	C₄×Q₈×C₇₃	—	2	2	no	—
34	C₄×Q₈×C₇₃	—	4	1	no	—
35	C₄×Q₈×C₇₃	—	4	1	no	—
36	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
37	C₂×C₄⋊C₄×C₇₃	—	4	1	no	—
38	C₂×C₄²×C₇₃	—	4	1	no	—
39	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
40	#107×C₇₃	—	2	1	no	—
41	#107×C₇₃	C₂×C₄	1	2	yes	—
42	#103×C₇₃	—	2	2	no	—
43	#103×C₇₃	—	2	2	no	—
44	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
48	C₂×#10596831	—	4	2	no	—
49	C₂×#10596831	—	8	2	no	—
50	#11924003	—	2	4	no	—
51	C₂×Q₈×C₇₃	—	2	4	no	—
52	C₂×Q₈×C₇₃	—	2	2	no	—
53	C₂×Q₈×C₇₃	—	4	1	no	—
54	C₂×Q₈×C₇₃	—	4	2	no	—
55	C₂²×C₄×C₇₃	—	4	1	no	—
56	C₂²×C₄×C₇₃	—	4	1	no	—
57	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
60	Q₁₆×C₇₃	—	2	4	no	—
61	Q₁₆×C₇₃	—	2	2	no	—
62	Q₁₆×C₇₃	C₂×C₈	1	4	yes	complemented
63	C₂×C₈×C₇₃	—	2	2	no	—
64	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
67	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
68	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
70	C₄⋊C₄×C₇₃	—	2	2	no	—
71	C₄⋊C₄×C₇₃	—	4	1	no	—
72	C₄⋊C₄×C₇₃	—	4	2	no	—
73	C₄⋊C₄×C₇₃	—	4	2	no	—
74	C₄²×C₇₃	—	4	2	no	—
75	C₄²×C₇₃	—	4	2	no	—
76	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₇₃	D₈	1	1	yes	characteristic
78	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
79	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
80	C₂³×C₇₃	#102	1	1	yes	characteristic
81	Q₈×C₇₃	—	2	4	no	—
82	Q₈×C₇₃	—	4	4	no	—
83	Q₈×C₇₃	—	4	2	no	—
84	C₂×C₄×C₇₃	—	2	2	no	—
85	C₂×C₄×C₇₃	—	4	4	no	—
86	C₂×C₄×C₇₃	—	4	2	no	—
87	C₂×C₄×C₇₃	—	4	2	no	—
88	C₂×C₄×C₇₃	—	8	2	no	—
89	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
90	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
91	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
92	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
93	C₂×C₄×C₇₃	#104	1	2	yes	—
94	C₂×C₄×C₇₃	SD₃₂	1	2	yes	—
95	C₂×C₄×C₇₃	Q₃₂	1	2	yes	—
96	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
99	C₈×C₇₃	—	2	1	no	—
100	C₈×C₇₃	—	2	1	no	—
101	C₈×C₇₃	C₂²⋊C₈	1	2	yes	—
102	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
103	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
104	#10596831	—	8	2	no	—
105	C₂²×C₇₃	D₄⋊C₈	1	2	yes	—
106	C₂²×C₇₃	#358	1	2	yes	—
107	C₂²×C₇₃	#359	1	2	yes	—
108	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
109	C₄×C₇₃	—	2	1	no	—
110	C₄×C₇₃	—	2	1	no	—
111	C₄×C₇₃	—	4	4	no	—
112	C₄×C₇₃	—	8	2	no	—
113	C₄×C₇₃	D₄⋊C₈	1	2	yes	—
114	C₄×C₇₃	#358	1	2	yes	—
115	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
116	C₄×C₇₃	C₂×#104	1	1	yes	characteristic
117	C₄×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
118	C₄×C₇₃	C₂×Q₃₂	1	1	yes	characteristic
119	C₂×#1332	—	73	1	no	maximal, Sylow
120	C₂×C₇₃	#1332	1	4	yes	complemented
121	C₂×C₇₃	C₂×D₄⋊C₈	1	1	yes	characteristic
122	C₂×C₇₃	C₂×#358	1	1	yes	characteristic
123	C₂×C₇₃	C₂×#359	1	1	yes	characteristic
124	C₂×C₄×Q₁₆	D₇₃	1	1	yes	characteristic
125	C₂×C₄×C₁₆	—	73	1	no	—
126	C₂×#335	—	73	1	no	—
127	#1332	—	73	4	no	—
128	C₇₃	C₂×#1332	1	1	yes	characteristic, complemented, Sylow
129	C₂²×Q₁₆	Dic₇₃	1	1	yes	characteristic
130	C₂×C₄×Q₈	—	2	1	no	—
131	C₂²×C₁₆	—	146	1	no	—
132	C₄×Q₁₆	—	2	1	no	—
133	C₄×Q₁₆	—	2	1	no	—
134	C₄×Q₁₆	Dic₇₃	1	2	yes	—
135	C₄×Q₁₆	C₂×D₇₃	1	2	yes	—
136	C₂×#107	Dic₇₃	1	1	yes	characteristic
137	C₂×C₄⋊C₈	—	146	1	no	—
138	C₂×#103	—	2	1	no	—
139	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
140	C₄×C₁₆	—	73	2	no	—
141	C₄×C₁₆	—	146	1	no	—
142	#335	—	73	2	no	—
143	#335	—	146	1	no	—
144	C₂²×Q₈	—	2	1	no	—
145	C₂×Q₁₆	—	2	1	no	—
146	C₂×Q₁₆	—	2	2	no	—
147	C₂×Q₁₆	#10596831	1	4	yes	complemented
148	C₂×Q₁₆	C₂×Dic₇₃	1	2	yes	—
149	C₂²×C₈	—	292	1	no	—
150	C₂²×C₈	C₂×Dic₇₃	1	1	yes	characteristic
151	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
152	C₄×Q₈	—	2	2	no	—
153	C₄×Q₈	—	2	2	no	—
154	C₄×Q₈	—	4	1	no	—
155	C₄×Q₈	—	4	1	no	—
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₄⋊C₄	—	4	1	no	—
158	C₂×C₄²	—	4	1	no	—
159	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
160	C₂×C₁₆	—	146	4	no	—
161	C₂×C₁₆	—	146	2	no	—
162	#107	—	2	1	no	—
163	#107	C₂×Dic₇₃	1	2	yes	—
164	C₄⋊C₈	—	292	1	no	—
165	C₄⋊C₈	—	146	2	no	—
166	#103	—	2	2	no	—
167	#103	—	2	2	no	—
168	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
169	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
170	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
171	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂×Q₈	—	4	2	no	—
176	C₂²×C₄	—	4	1	no	—
177	C₂²×C₄	—	4	1	no	—
178	C₂²×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
179	C₂²×C₄	#11924013	1	1	yes	characteristic
180	C₂²×C₄	#11924016	1	1	yes	characteristic
181	Q₁₆	—	2	4	no	—
182	Q₁₆	—	2	2	no	—
183	Q₁₆	C₂×#10596831	1	4	yes	complemented
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	292	2	no	—
186	C₂×C₈	—	584	2	no	—
187	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
188	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
189	C₂×C₈	C₂×#10596831	1	2	yes	—
190	C₂×C₈	#11924012	1	2	yes	—
191	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
192	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	4	1	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄²	—	4	2	no	—
198	C₄²	—	4	2	no	—
199	C₄²	C₂²⋊Dic₇₃	1	1	yes	characteristic
200	C₄²	#11924013	1	1	yes	characteristic
201	C₄²	#11924016	1	1	yes	characteristic
202	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
203	C₁₆	—	146	4	no	—
204	C₂³	#14556512	1	1	yes	characteristic
205	Q₈	—	2	4	no	—
206	Q₈	—	4	4	no	—
207	Q₈	—	4	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	8	2	no	—
213	C₂×C₄	C₂²⋊#10596831	1	2	yes	—
214	C₂×C₄	#14556512	1	1	yes	characteristic
215	C₂×C₄	#14556512	1	1	yes	characteristic
216	C₂×C₄	#14556512	1	1	yes	characteristic
217	C₂×C₄	#14556518	1	2	yes	—
218	C₂×C₄	#14556534	1	2	yes	—
219	C₂×C₄	#14556537	1	2	yes	—
220	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
221	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
222	C₂×C₄	C₂×#11924016	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	584	2	no	—
226	C₈	C₂²⋊#10596831	1	2	yes	—
227	C₈	C₂²×#10596831	1	1	yes	characteristic
228	C₈	C₂×#11924012	1	1	yes	characteristic
229	C₂²	D₄⋊#10596831	1	2	yes	—
230	C₂²	#17988391	1	2	yes	—
231	C₂²	#17988394	1	2	yes	—
232	C₂²	C₂×#14556512	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	4	no	—
236	C₄	—	8	2	no	—
237	C₄	D₄⋊#10596831	1	2	yes	—
238	C₄	#17988391	1	2	yes	—
239	C₄	C₂×C₂²⋊#10596831	1	1	yes	characteristic
240	C₄	C₂×#14556518	1	1	yes	characteristic
241	C₄	C₂×#14556534	1	1	yes	characteristic
242	C₄	C₂×#14556537	1	1	yes	characteristic
243	C₂	#22607895	1	4	yes	direct product factor
244	C₂	C₂×D₄⋊#10596831	1	1	yes	characteristic
245	C₂	C₂×#17988391	1	1	yes	characteristic
246	C₂	C₂×#17988394	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11924013
commutator	C₈×C₇₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₂×C₄×Q₁₆×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	#102
cosocle	C₄×C₈×C₇₃	C₂³
hypercenter	C₂×C₄×Q₁₆	D₇₃
hypocenter	C₇₃	C₂×#1332
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #22607895	C₂	direct (N × H)
28. C₂×Q₁₆×C₇₃	C₈	semidirect
62. Q₁₆×C₇₃	C₂×C₈	semidirect
120. C₂×C₇₃	#1332	semidirect
128. C₇₃	C₂×#1332	semidirect
147. C₂×Q₁₆	#10596831	semidirect
183. Q₁₆	C₂×#10596831	semidirect
243. C₂	#22607895	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×C₄×#11924003	2
C₂×#17988340	2
#22607895	2
C₂×C₄×Q₁₆×C₇₃	2
C₂×#1332	73

Subgroups — C2×#22607895 #32995719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22607895 #32995719