Subgroups — C₂×#22611719 #33239018

Order 18688.

718 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988625	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14556525	C₂	1	1	yes	maximal, characteristic
3	#22611719	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17988415	C₂	1	2	yes	maximal, complemented
5	C₂×#17988413	C₂	1	2	yes	maximal, complemented
6	C₂²×M₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#14556550	—	2	1	no	—
8	C₂²×C₄⋊Dic₇₃	—	2	1	no	—
9	#17988625	—	2	2	no	—
10	#17988625	C₂²	1	4	yes	—
11	C₂×#14556525	—	2	1	no	—
12	C₂×#14556525	—	2	2	no	—
13	C₂×#14556525	C₂²	1	4	yes	—
14	C₂×#14556525	C₂²	1	2	yes	—
15	C₂×#14556525	C₂²	1	1	yes	characteristic
16	C₂×#14556525	C₂²	1	1	yes	characteristic
17	C₂×#14556523	—	2	1	no	—
18	#17988415	C₂²	1	8	yes	complemented
19	#17988413	C₂²	1	8	yes	complemented
20	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×M₃₂×C₇₃	C₄	1	8	yes	complemented
22	C₂×M₃₂×C₇₃	C₂²	1	4	yes	—
23	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
24	C₂³×Dic₇₃	—	4	1	no	—
25	#14556550	—	2	4	no	—
26	#14556550	—	4	2	no	—
27	C₂×C₄⋊Dic₇₃	—	2	1	no	—
28	C₂×C₄⋊Dic₇₃	—	2	1	no	—
29	C₂×C₄⋊Dic₇₃	—	2	2	no	—
30	C₂×C₄⋊Dic₇₃	—	2	4	no	—
31	C₂×C₄⋊Dic₇₃	—	4	1	no	—
32	C₂×C₄⋊Dic₇₃	—	4	2	no	—
33	C₂×C₂²⋊Dic₇₃	—	4	1	no	—
34	C₂×C₄×Dic₇₃	—	4	1	no	—
35	#14556525	—	2	4	no	—
36	#14556525	—	2	2	no	—
37	#14556525	—	2	1	no	—
38	#14556525	—	2	1	no	—
39	#14556525	C₂³	1	4	yes	—
40	#14556525	C₂³	1	2	yes	—
41	#14556525	C₂³	1	2	yes	—
42	#14556523	—	2	4	no	—
43	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₇₃	D₄	1	2	yes	—
47	C₂²×C₈×C₇₃	Q₈	1	4	yes	—
48	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
50	M₃₂×C₇₃	C₂×C₄	1	16	yes	complemented
51	C₂×C₁₆×C₇₃	C₂×C₄	1	8	yes	complemented
52	C₂×C₁₆×C₇₃	C₂³	1	4	yes	—
53	C₂²×Dic₇₃	—	4	1	no	—
54	C₂²×Dic₇₃	—	4	1	no	—
55	C₂²×Dic₇₃	—	4	4	no	—
56	C₂²×Dic₇₃	—	4	2	no	—
57	C₂²×Dic₇₃	—	8	4	no	—
58	C₄⋊Dic₇₃	—	2	2	no	—
59	C₄⋊Dic₇₃	—	2	2	no	—
60	C₄⋊Dic₇₃	—	2	4	no	—
61	C₄⋊Dic₇₃	—	4	4	no	—
62	C₄⋊Dic₇₃	—	4	2	no	—
63	C₄⋊Dic₇₃	—	4	1	no	—
64	C₄⋊Dic₇₃	—	4	1	no	—
65	C₂²⋊Dic₇₃	—	4	4	no	—
66	C₄×Dic₇₃	—	4	4	no	—
67	C₂⁴×C₇₃	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	D₈	1	2	yes	—
71	C₂²×C₄×C₇₃	Q₁₆	1	4	yes	—
72	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
75	C₂×C₈×C₇₃	C₄⋊C₄	1	8	yes	—
76	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
81	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
82	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
83	C₁₆×C₇₃	C₂²×C₄	1	8	yes	complemented
84	C₂³×C₇₃	—	2	2	no	—
85	C₂³×C₇₃	—	2	2	no	—
86	C₂³×C₇₃	#107	1	4	yes	—
87	C₂³×C₇₃	C₂×D₈	1	2	yes	—
88	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₇₃	#107	1	8	yes	—
90	C₂×C₄×C₇₃	#107	1	4	yes	—
91	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₇₃	C₂×Q₁₆	1	4	yes	—
96	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
97	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×Dic₇₃	—	4	2	no	—
99	C₂×Dic₇₃	—	4	2	no	—
100	C₂×Dic₇₃	—	4	4	no	—
101	C₂×Dic₇₃	—	8	8	no	—
102	C₂×Dic₇₃	—	8	2	no	—
103	C₂×Dic₇₃	—	8	4	no	—
104	C₂×Dic₇₃	—	8	2	no	—
105	C₈×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₈×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
107	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
108	C₈×C₇₃	C₂²×Q₈	1	1	yes	characteristic
109	C₂²×C₇₃	—	2	2	no	—
110	C₂²×C₇₃	—	2	2	no	—
111	C₂²×C₇₃	—	2	8	no	—
112	C₂²×C₇₃	C₂×#107	1	4	yes	—
113	C₂²×C₇₃	C₂×#107	1	2	yes	—
114	C₂²×C₇₃	#509	1	2	yes	—
115	C₂²×C₇₃	#510	1	2	yes	—
116	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
117	C₄×C₇₃	C₂×#107	1	4	yes	—
118	C₄×C₇₃	C₂×#107	1	2	yes	—
119	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
120	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
121	Dic₇₃	—	8	2	no	—
122	Dic₇₃	—	8	4	no	—
123	Dic₇₃	—	8	2	no	—
124	C₂×#2159	—	73	1	no	maximal, Sylow
125	C₂×C₇₃	—	2	4	no	—
126	C₂×C₇₃	#2159	1	4	yes	complemented
127	C₂×C₇₃	C₂²×#107	1	1	yes	characteristic
128	C₂×C₇₃	C₂×#509	1	1	yes	characteristic
129	C₂×C₇₃	C₂×#510	1	1	yes	characteristic
130	C₂²×M₃₂	D₇₃	1	1	yes	characteristic
131	C₂×#427	—	73	1	no	—
132	C₂²×#107	—	73	1	no	—
133	#2159	—	73	8	no	—
134	C₂×#367	—	73	2	no	—
135	C₂×#366	—	73	2	no	—
136	C₇₃	C₂×#2159	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
138	C₂×#117	—	146	1	no	—
139	C₂²×C₄⋊C₄	—	146	1	no	—
140	C₂×M₃₂	Dic₇₃	1	8	yes	complemented
141	C₂×M₃₂	C₂×D₇₃	1	4	yes	—
142	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
143	#427	—	73	4	no	—
144	#427	—	146	2	no	—
145	C₂×#107	—	73	2	no	—
146	C₂×#107	—	73	4	no	—
147	C₂×#107	—	73	1	no	—
148	C₂×#107	—	73	1	no	—
149	C₂×#107	—	146	1	no	—
150	C₂×#107	—	146	2	no	—
151	C₂×#106	—	146	1	no	—
152	#367	—	73	8	no	—
153	#366	—	73	8	no	—
154	C₂³×C₄	—	292	1	no	—
155	C₂³×C₄	D₂₉₂	1	1	yes	characteristic
156	C₂²×C₈	Dic₁₄₆	1	4	yes	—
157	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
158	C₂²×C₈	D₂₉₂	1	2	yes	—
159	C₂²×C₈	C₂×Dic₇₃	1	4	yes	—
160	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
161	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
162	#117	—	292	2	no	—
163	#117	—	146	4	no	—
164	C₂×C₄⋊C₄	—	292	2	no	—
165	C₂×C₄⋊C₄	—	292	1	no	—
166	C₂×C₄⋊C₄	—	146	4	no	—
167	C₂×C₄⋊C₄	—	146	2	no	—
168	C₂×C₄⋊C₄	—	146	1	no	—
169	C₂×C₄⋊C₄	—	146	1	no	—
170	C₂×C₂²⋊C₄	—	292	1	no	—
171	C₂×C₄²	—	292	1	no	—
172	M₃₂	C₂×Dic₇₃	1	16	yes	complemented
173	C₂×C₁₆	C₂×Dic₇₃	1	8	yes	complemented
174	C₂×C₁₆	C₂²×D₇₃	1	4	yes	—
175	#107	—	73	2	no	—
176	#107	—	73	4	no	—
177	#107	—	73	2	no	—
178	#107	—	146	4	no	—
179	#107	—	146	2	no	—
180	#107	—	146	1	no	—
181	#107	—	146	1	no	—
182	#106	—	146	4	no	—
183	C₂⁴	D₅₈₄	1	1	yes	characteristic
184	C₂²×C₄	—	292	4	no	—
185	C₂²×C₄	—	292	1	no	—
186	C₂²×C₄	—	292	1	no	—
187	C₂²×C₄	—	292	2	no	—
188	C₂²×C₄	—	584	4	no	—
189	C₂²×C₄	C₄⋊Dic₇₃	1	4	yes	—
190	C₂²×C₄	D₅₈₄	1	2	yes	—
191	C₂²×C₄	D₅₈₄	1	1	yes	characteristic
192	C₂²×C₄	Dic₂₉₂	1	4	yes	—
193	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
194	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
195	C₂×C₈	C₄⋊Dic₇₃	1	4	yes	—
196	C₂×C₈	C₄⋊Dic₇₃	1	8	yes	—
197	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
198	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
199	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
200	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
201	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
202	C₂×C₈	C₂×Dic₁₄₆	1	4	yes	—
203	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
204	C₄⋊C₄	—	292	4	no	—
205	C₄⋊C₄	—	292	2	no	—
206	C₄⋊C₄	—	292	1	no	—
207	C₄⋊C₄	—	292	1	no	—
208	C₄⋊C₄	—	146	2	no	—
209	C₄⋊C₄	—	146	2	no	—
210	C₄⋊C₄	—	146	4	no	—
211	C₂²⋊C₄	—	292	4	no	—
212	C₄²	—	292	4	no	—
213	C₁₆	C₂²×Dic₇₃	1	8	yes	complemented
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	#14556525	1	4	yes	—
217	C₂³	C₂×D₅₈₄	1	2	yes	—
218	C₂³	C₂×D₅₈₄	1	1	yes	characteristic
219	C₂×C₄	—	292	2	no	—
220	C₂×C₄	—	292	2	no	—
221	C₂×C₄	—	292	4	no	—
222	C₂×C₄	—	584	8	no	—
223	C₂×C₄	—	584	4	no	—
224	C₂×C₄	—	584	2	no	—
225	C₂×C₄	—	584	2	no	—
226	C₂×C₄	#14556525	1	8	yes	—
227	C₂×C₄	#14556525	1	4	yes	—
228	C₂×C₄	C₂×C₄⋊Dic₇₃	1	4	yes	—
229	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
230	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
231	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
232	C₂×C₄	C₂×Dic₂₉₂	1	4	yes	—
233	C₂×C₄	C₂×Dic₂₉₂	1	2	yes	—
234	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
235	C₈	C₂×C₄⋊Dic₇₃	1	2	yes	—
236	C₈	C₂×C₄⋊Dic₇₃	1	4	yes	—
237	C₈	C₂³×Dic₇₃	1	1	yes	characteristic
238	C₈	C₂²×Dic₁₄₆	1	1	yes	characteristic
239	C₂²	—	2	8	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	C₂×#14556525	1	4	yes	—
243	C₂²	C₂×#14556525	1	2	yes	—
244	C₂²	#17989025	1	2	yes	—
245	C₂²	#17989030	1	2	yes	—
246	C₂²	C₂²×D₅₈₄	1	1	yes	characteristic
247	C₄	—	584	4	no	—
248	C₄	—	584	2	no	—
249	C₄	—	584	2	no	—
250	C₄	C₂×#14556525	1	4	yes	—
251	C₄	C₂×#14556525	1	2	yes	—
252	C₄	C₂²×C₄⋊Dic₇₃	1	1	yes	characteristic
253	C₄	C₂²×Dic₂₉₂	1	1	yes	characteristic
254	C₂	—	2	4	no	—
255	C₂	#22611719	1	4	yes	direct product factor
256	C₂	C₂²×#14556525	1	1	yes	characteristic
257	C₂	C₂×#17989025	1	1	yes	characteristic
258	C₂	C₂×#17989030	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₈₄
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂²×M₃₂×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₈
prime radical	C₂⁴×C₇₃	D₈
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₂²×M₃₂	D₇₃
hypocenter	C₇₃	C₂×#2159
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #22611719	C₂	direct (N × H)
4. C₂×#17988415	C₂	semidirect
5. C₂×#17988413	C₂	semidirect
18. #17988415	C₂²	semidirect
19. #17988413	C₂²	semidirect
21. C₂×M₃₂×C₇₃	C₄	semidirect
50. M₃₂×C₇₃	C₂×C₄	semidirect
51. C₂×C₁₆×C₇₃	C₂×C₄	semidirect
83. C₁₆×C₇₃	C₂²×C₄	semidirect
126. C₂×C₇₃	#2159	semidirect
136. C₇₃	C₂×#2159	semidirect
140. C₂×M₃₂	Dic₇₃	semidirect
172. M₃₂	C₂×Dic₇₃	semidirect
173. C₂×C₁₆	C₂×Dic₇₃	semidirect
213. C₁₆	C₂²×Dic₇₃	semidirect
255. C₂	#22611719	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×#17988625	2
C₂²×#14556525	2
#22611719	2
C₂×#17988415	2
C₂×#17988413	2
C₂²×M₃₂×C₇₃	2
C₂×#2159	73

Subgroups — C2×#22611719 #33239018

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22611719 #33239018