Subgroups — C₂×#22626264 #34030058

Order 18688.

798 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556666	C₂	1	2	yes	maximal
2	#22626264	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17989508	C₂	1	2	yes	maximal, complemented
4	C₂×C₄⋊#11924031	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556666	—	2	8	no	—
7	C₂×#14556666	C₂²	1	8	yes	—
8	C₂²×#11924031	C₂²	1	4	yes	—
9	#17989508	C₂²	1	8	yes	complemented
10	C₄⋊#11924031	C₂²	1	8	yes	complemented
11	C₂²×C₈×D₇₃	C₄	1	1	yes	characteristic
12	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×D₇₃	C₄	1	1	yes	characteristic
14	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
17	C₄×C₈×D₇₃	C₄	1	4	yes	—
18	C₄×C₈×D₇₃	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
20	#14556666	—	2	16	no	—
21	C₂×#11924031	—	2	8	no	—
22	C₂×#11924031	C₂³	1	8	yes	—
23	C₂²×C₄×D₇₃	C₈	1	1	yes	characteristic
24	C₂²×C₄×D₇₃	C₈	1	1	yes	characteristic
25	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
27	C₂²×#10596831	C₂³	1	1	yes	characteristic
28	C₂×C₈×D₇₃	—	2	4	no	—
29	C₂×C₈×D₇₃	C₂×C₄	1	4	yes	—
30	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₇₃	D₄	1	4	yes	—
32	C₂×C₈×D₇₃	Q₈	1	4	yes	—
33	C₂×C₈×D₇₃	C₂³	1	2	yes	—
34	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
35	C₂×C₄×Dic₇₃	C₈	1	1	yes	characteristic
36	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
37	C₄²×D₇₃	C₈	1	4	yes	—
38	C₄²×D₇₃	C₂×C₄	1	4	yes	—
39	C₄×#10596831	C₂×C₄	1	2	yes	—
40	C₄×#10596831	C₂³	1	2	yes	—
41	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
42	C₈×Dic₇₃	C₂×C₄	1	4	yes	—
43	C₈×Dic₇₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
48	#11924031	—	2	8	no	—
49	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
53	C₂×C₄×D₇₃	—	2	4	no	—
54	C₂×C₄×D₇₃	—	2	4	no	—
55	C₂×C₄×D₇₃	C₄⋊C₄	1	4	yes	—
56	C₂×C₄×D₇₃	C₂×C₈	1	4	yes	—
57	C₂×C₄×D₇₃	C₂×C₈	1	4	yes	—
58	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
59	C₂×C₄×D₇₃	M₁₆	1	4	yes	—
60	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
61	C₂×#10596831	—	2	2	no	—
62	C₂×#10596831	C₂²×C₄	1	2	yes	—
63	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
64	C₂×#10596831	C₂×D₄	1	2	yes	—
65	C₂×#10596831	C₂×Q₈	1	2	yes	—
66	C₂×#10596831	C₂⁴	1	1	yes	characteristic
67	C₈×D₇₃	—	2	8	no	—
68	C₈×D₇₃	C₄⋊C₄	1	8	yes	—
69	C₈×D₇₃	C₂×D₄	1	4	yes	—
70	C₈×D₇₃	C₂×Q₈	1	4	yes	—
71	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
72	C₄×Dic₇₃	C₂×C₈	1	4	yes	—
73	C₄×Dic₇₃	C₂×C₈	1	4	yes	—
74	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₇₃	—	2	2	no	—
79	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
80	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
82	C₄²×C₇₃	C₂×C₈	1	2	yes	—
83	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
84	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
85	C₂²×D₇₃	—	2	2	no	—
86	C₂²×D₇₃	—	2	2	no	—
87	C₂²×D₇₃	C₄⋊C₈	1	4	yes	—
88	C₂²×D₇₃	C₂²×C₈	1	2	yes	—
89	C₂×C₄×C₇₃	—	2	2	no	—
90	C₂×C₄×C₇₃	M₃₂	1	4	yes	complemented
91	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
93	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
96	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
97	C₂×Dic₇₃	—	2	2	no	—
98	C₂×Dic₇₃	C₄⋊C₈	1	4	yes	—
99	C₂×Dic₇₃	M₃₂	1	4	yes	complemented
100	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
101	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
102	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
103	C₄×D₇₃	—	2	8	no	—
104	C₄×D₇₃	—	2	8	no	—
105	C₄×D₇₃	C₄⋊C₈	1	8	yes	—
106	C₄×D₇₃	C₂×C₄⋊C₄	1	4	yes	—
107	C₄×D₇₃	C₂×M₁₆	1	4	yes	—
108	C₈×C₇₃	—	2	2	no	—
109	C₈×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
110	#10596831	—	2	2	no	—
111	#10596831	C₂×C₄⋊C₄	1	2	yes	—
112	#10596831	C₂²×D₄	1	1	yes	characteristic
113	#10596831	C₂²×Q₈	1	1	yes	characteristic
114	C₂²×C₇₃	C₂×C₄⋊C₈	1	2	yes	—
115	C₂²×C₇₃	C₂×M₃₂	1	2	yes	—
116	C₂²×C₇₃	C₂×M₃₂	1	2	yes	—
117	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
118	C₂×D₇₃	—	2	2	no	—
119	C₂×D₇₃	—	2	2	no	—
120	C₂×D₇₃	—	2	8	no	—
121	C₂×D₇₃	C₂×C₄⋊C₈	1	4	yes	—
122	C₄×C₇₃	—	2	2	no	—
123	C₄×C₇₃	C₂×C₄⋊C₈	1	2	yes	—
124	C₄×C₇₃	C₂×M₃₂	1	4	yes	complemented
125	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
127	Dic₇₃	—	2	2	no	—
128	Dic₇₃	C₂×C₄⋊C₈	1	4	yes	—
129	Dic₇₃	C₂×M₃₂	1	4	yes	complemented
130	C₂×#2150	—	73	1	no	maximal, Sylow
131	C₂×C₇₃	#2150	1	4	yes	complemented
132	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
133	C₂×C₇₃	C₂²×M₃₂	1	1	yes	characteristic
134	C₂×C₇₃	C₂²×M₃₂	1	1	yes	characteristic
135	D₇₃	—	2	4	no	—
136	C₂²×M₃₂	—	73	2	no	—
137	C₂²×C₄×C₈	—	73	1	no	—
138	#2150	—	73	8	no	—
139	C₂×C₄⋊C₁₆	—	73	2	no	—
140	C₂×C₄⋊C₁₆	—	73	2	no	—
141	C₇₃	C₂×#2150	1	1	yes	characteristic, complemented, Sylow
142	C₂³×C₈	—	73	1	no	—
143	C₂³×C₈	—	73	1	no	—
144	C₂²×C₄²	—	73	1	no	—
145	C₂×M₃₂	—	73	8	no	—
146	C₂×M₃₂	—	146	8	no	—
147	C₂²×C₁₆	—	73	4	no	—
148	C₂×C₄×C₈	—	73	1	no	—
149	C₂×C₄×C₈	—	73	1	no	—
150	C₂×C₄×C₈	—	73	1	no	—
151	C₂×C₄×C₈	—	73	4	no	—
152	C₂×C₄×C₈	—	73	4	no	—
153	C₂×C₄×C₈	#64408	1	1	yes	characteristic
154	C₄⋊C₁₆	—	73	8	no	—
155	C₄⋊C₁₆	—	73	8	no	—
156	C₂³×C₄	—	73	1	no	—
157	C₂³×C₄	—	73	1	no	—
158	C₂³×C₄	—	73	1	no	—
159	C₂²×C₈	—	73	1	no	—
160	C₂²×C₈	—	73	4	no	—
161	C₂²×C₈	—	73	4	no	—
162	C₂²×C₈	—	73	2	no	—
163	C₂²×C₈	—	73	2	no	—
164	C₂²×C₈	—	73	1	no	—
165	C₂²×C₈	—	73	4	no	—
166	C₂²×C₈	—	146	4	no	—
167	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
168	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
169	C₂×C₄²	—	73	1	no	—
170	C₂×C₄²	—	73	1	no	—
171	C₂×C₄²	—	73	1	no	—
172	C₂×C₄²	—	73	4	no	—
173	C₂×C₄²	—	73	4	no	—
174	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
175	M₃₂	—	146	16	no	—
176	C₂×C₁₆	—	73	8	no	—
177	C₂×C₁₆	—	146	8	no	—
178	C₄×C₈	—	73	2	no	—
179	C₄×C₈	—	73	4	no	—
180	C₄×C₈	—	73	2	no	—
181	C₄×C₈	—	73	2	no	—
182	C₄×C₈	—	73	2	no	—
183	C₄×C₈	C₂×#64408	1	4	yes	—
184	C₂⁴	—	73	1	no	—
185	C₂²×C₄	—	73	1	no	—
186	C₂²×C₄	—	73	1	no	—
187	C₂²×C₄	—	73	2	no	—
188	C₂²×C₄	—	73	4	no	—
189	C₂²×C₄	—	73	2	no	—
190	C₂²×C₄	—	73	4	no	—
191	C₂²×C₄	—	73	1	no	—
192	C₂²×C₄	—	73	4	no	—
193	C₂²×C₄	—	73	4	no	—
194	C₂²×C₄	—	146	4	no	—
195	C₂²×C₄	—	146	4	no	—
196	C₂²×C₄	#11924036	1	1	yes	characteristic
197	C₂²×C₄	#11924036	1	1	yes	characteristic
198	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	73	1	no	—
201	C₂×C₈	—	73	1	no	—
202	C₂×C₈	—	73	2	no	—
203	C₂×C₈	—	73	2	no	—
204	C₂×C₈	—	73	8	no	—
205	C₂×C₈	—	73	4	no	—
206	C₂×C₈	—	73	4	no	—
207	C₂×C₈	—	73	2	no	—
208	C₂×C₈	—	146	8	no	—
209	C₂×C₈	—	146	2	no	—
210	C₂×C₈	C₄⋊#64408	1	4	yes	—
211	C₂×C₈	C₂²×#64408	1	2	yes	—
212	C₂×C₈	C₂²×#64408	1	2	yes	—
213	C₄²	—	73	4	no	—
214	C₄²	—	73	4	no	—
215	C₄²	—	73	2	no	—
216	C₄²	—	73	2	no	—
217	C₄²	#11924036	1	2	yes	—
218	C₄²	C₂²×#64408	1	2	yes	—
219	C₁₆	—	146	8	no	—
220	C₂³	—	73	2	no	—
221	C₂³	—	73	4	no	—
222	C₂³	—	146	2	no	—
223	C₂³	—	146	2	no	—
224	C₂³	C₂×#11924036	1	1	yes	characteristic
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	73	4	no	—
227	C₂×C₄	—	73	2	no	—
228	C₂×C₄	—	73	2	no	—
229	C₂×C₄	—	73	4	no	—
230	C₂×C₄	—	73	4	no	—
231	C₂×C₄	—	73	4	no	—
232	C₂×C₄	—	73	8	no	—
233	C₂×C₄	—	73	2	no	—
234	C₂×C₄	—	146	8	no	—
235	C₂×C₄	—	146	2	no	—
236	C₂×C₄	—	146	8	no	—
237	C₂×C₄	#14556666	1	4	yes	complemented
238	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
239	C₂×C₄	C₂×#11924036	1	2	yes	—
240	C₂×C₄	C₂×#11924036	1	2	yes	—
241	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
242	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
243	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	73	1	no	—
246	C₈	—	73	2	no	—
247	C₈	—	73	1	no	—
248	C₈	—	146	2	no	—
249	C₈	C₂×C₄⋊#64408	1	4	yes	—
250	C₂²	—	73	4	no	—
251	C₂²	—	146	2	no	—
252	C₂²	—	146	2	no	—
253	C₂²	—	146	8	no	—
254	C₂²	#17989588	1	2	yes	—
255	C₂²	#17989643	1	2	yes	—
256	C₂²	C₂×#14556666	1	2	yes	—
257	C₂²	C₂²×#11924036	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	73	4	no	—
260	C₄	—	73	4	no	—
261	C₄	—	146	2	no	—
262	C₄	#17989588	1	2	yes	—
263	C₄	C₂×#14556666	1	4	yes	complemented
264	C₄	C₂²×C₄⋊#64408	1	1	yes	characteristic
265	C₄	C₂×D₇₃⋊M₁₆	1	1	yes	characteristic
266	C₂	—	146	4	no	—
267	C₂	#22626264	1	4	yes	direct product factor
268	C₂	C₂×#17989588	1	1	yes	characteristic
269	C₂	C₂×#17989643	1	1	yes	characteristic
270	C₂	C₂²×#14556666	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#64408
commutator	C₂²×C₇₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle	C₂³×C₇₃	C₂²×C₈
prime radical	C₂³×D₇₃	C₂×C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	C₂×#2150
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22626264	C₂	direct (N × H)
3. C₂×#17989508	C₂	semidirect
4. C₂×C₄⋊#11924031	C₂	semidirect
9. #17989508	C₂²	semidirect
10. C₄⋊#11924031	C₂²	semidirect
90. C₂×C₄×C₇₃	M₃₂	semidirect
99. C₂×Dic₇₃	M₃₂	semidirect
124. C₄×C₇₃	C₂×M₃₂	semidirect
129. Dic₇₃	C₂×M₃₂	semidirect
131. C₂×C₇₃	#2150	semidirect
141. C₇₃	C₂×#2150	semidirect
237. C₂×C₄	#14556666	semidirect
263. C₄	C₂×#14556666	semidirect
267. C₂	#22626264	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#14556666	2
#22626264	2
C₂×#17989508	2
C₂×C₄⋊#11924031	2
C₂×C₄×C₈×D₇₃	2
C₂×#2150	73

Subgroups — C2×#22626264 #34030058

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22626264 #34030058