Subgroups — C₂×Q₃₂×#64408 #34030254

Order 18688.

678 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×#64408	C₂	1	2	yes	maximal
2	Q₃₂×#64408	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17989513	C₂	1	1	yes	maximal, characteristic
4	C₂×#17989500	C₂	1	2	yes	maximal
5	C₂×C₁₆×#64408	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₃₂×D₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₈×#64408	—	2	2	no	—
8	C₂×#14556664	C₂²	1	1	yes	characteristic
9	Q₁₆×#64408	—	2	4	no	—
10	Q₁₆×#64408	C₂²	1	8	yes	—
11	C₂×#14556660	C₂²	1	2	yes	—
12	C₂×#14556654	—	2	2	no	—
13	C₂×C₈×#64408	C₂²	1	1	yes	characteristic
14	#17989513	C₂²	1	4	yes	—
15	#17989500	C₂²	1	8	yes	—
16	C₁₆×#64408	C₂²	1	4	yes	—
17	C₂×Q₁₆×D₇₃	C₂²	1	2	yes	—
18	C₂×Dic₅₈₄	C₄	1	1	yes	characteristic, complemented
19	C₂×#14556537	C₄	1	2	yes	complemented
20	Q₃₂×D₇₃	C₄	1	4	yes	complemented
21	Q₃₂×D₇₃	C₂²	1	4	yes	—
22	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
23	C₂×Q₃₂×C₇₃	C₄	1	1	yes	characteristic, complemented
24	C₂×#11924036	D₄	1	1	yes	characteristic
25	Q₈×#64408	—	2	8	no	—
26	Q₈×#64408	—	4	4	no	—
27	C₂×C₄⋊#64408	—	2	2	no	—
28	C₂×C₄⋊#64408	—	4	2	no	—
29	C₂×C₄×#64408	—	4	2	no	—
30	C₂×C₄×#64408	D₄	1	1	yes	characteristic
31	#14556664	—	2	1	no	—
32	#14556664	C₂³	1	2	yes	—
33	C₂×#11924031	—	2	1	no	—
34	#14556660	—	2	2	no	—
35	#14556660	C₂³	1	4	yes	—
36	#14556654	—	2	8	no	—
37	C₈×#64408	D₄	1	2	yes	—
38	C₈×#64408	C₂³	1	2	yes	—
39	C₂×Q₈×D₇₃	—	2	2	no	—
40	C₂×Dic₂₉₂	C₂×C₄	1	2	yes	—
41	C₂×#11924018	—	2	2	no	—
42	Q₁₆×D₇₃	—	2	2	no	—
43	Q₁₆×D₇₃	—	2	2	no	—
44	Q₁₆×D₇₃	C₂×C₄	1	4	yes	—
45	Q₁₆×D₇₃	C₂³	1	4	yes	—
46	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
47	Dic₅₈₄	C₂×C₄	1	4	yes	complemented
48	#14556537	C₂×C₄	1	8	yes	complemented
49	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
50	C₁₆×D₇₃	C₂×C₄	1	2	yes	—
51	C₁₆×D₇₃	C₂³	1	2	yes	—
52	C₂×Q₁₆×C₇₃	C₂×C₄	1	2	yes	—
53	Q₃₂×C₇₃	C₂×C₄	1	4	yes	complemented
54	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
55	C₂²×#64408	—	4	2	no	—
56	C₂²×#64408	D₈	1	1	yes	characteristic
57	C₂²×#64408	D₈	1	1	yes	characteristic
58	#11924036	—	2	1	no	—
59	#11924036	C₂×D₄	1	2	yes	—
60	C₂×#10596833	—	2	1	no	—
61	C₄⋊#64408	—	2	4	no	—
62	C₄⋊#64408	—	4	8	no	—
63	C₄⋊#64408	—	4	2	no	—
64	C₄×#64408	—	4	8	no	—
65	C₄×#64408	D₈	1	2	yes	—
66	C₄×#64408	C₂×D₄	1	2	yes	—
67	#11924031	—	2	2	no	—
68	C₂×Dic₁₄₆	—	2	2	no	—
69	C₂×Dic₁₄₆	—	4	2	no	—
70	Q₈×D₇₃	—	2	4	no	—
71	Q₈×D₇₃	—	2	4	no	—
72	Q₈×D₇₃	—	4	2	no	—
73	Q₈×D₇₃	—	4	2	no	—
74	C₂×C₄×D₇₃	—	4	2	no	—
75	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
76	Dic₂₉₂	—	2	2	no	—
77	Dic₂₉₂	C₂²×C₄	1	4	yes	—
78	#11924018	—	2	4	no	—
79	#11924018	—	2	4	no	—
80	C₂×#10596831	#41	1	1	yes	characteristic
81	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
82	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
83	C₈×D₇₃	#41	1	1	yes	characteristic
84	C₈×D₇₃	C₂⁴	1	1	yes	characteristic
85	#11924003	C₂²×C₄	1	2	yes	—
86	C₂×Q₈×C₇₃	—	2	2	no	—
87	Q₁₆×C₇₃	—	2	2	no	—
88	Q₁₆×C₇₃	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
90	C₁₆×C₇₃	C₂²×C₄	1	2	yes	—
91	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
92	C₂×#64408	—	2	2	no	—
93	C₂×#64408	—	4	4	no	—
94	C₂×#64408	—	8	4	no	—
95	C₂×#64408	Q₃₂	1	4	yes	direct product factor
96	C₂×#64408	C₂×D₈	1	2	yes	—
97	C₂×#64408	C₂×D₈	1	2	yes	—
98	Q₈×C₇₃	—	2	4	no	—
99	Q₈×C₇₃	—	4	2	no	—
100	C₂×C₄×C₇₃	—	4	2	no	—
101	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
102	C₂×Dic₇₃	—	4	2	no	—
103	C₂×Dic₇₃	#135	1	1	yes	characteristic
104	C₄×D₇₃	—	4	4	no	—
105	C₄×D₇₃	—	4	4	no	—
106	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
107	C₄×D₇₃	C₂×D₈	1	1	yes	characteristic
108	C₄×D₇₃	#135	1	1	yes	characteristic
109	C₄×D₇₃	C₂²×D₄	1	1	yes	characteristic
110	Dic₁₄₆	—	2	4	no	—
111	Dic₁₄₆	—	4	2	no	—
112	Dic₁₄₆	—	4	4	no	—
113	Dic₁₄₆	—	4	4	no	—
114	#10596833	—	2	2	no	—
115	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
116	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
117	#10596831	C₄×D₄	1	1	yes	characteristic
118	#10596831	C₂×#41	1	1	yes	characteristic
119	C₂²×C₇₃	C₄×D₈	1	1	yes	characteristic
120	C₂×D₇₃	C₄×D₈	1	1	yes	characteristic
121	C₂×D₇₃	C₂×Q₃₂	1	2	yes	—
122	C₂×D₇₃	#508	1	2	yes	—
123	C₂×D₇₃	C₂²×D₈	1	1	yes	characteristic
124	#64408	—	2	2	no	—
125	#64408	—	8	4	no	—
126	#64408	C₂×Q₃₂	1	4	yes	direct product factor
127	C₄×C₇₃	—	4	4	no	—
128	C₄×C₇₃	C₄×D₈	1	1	yes	characteristic
129	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
130	Dic₇₃	—	4	4	no	—
131	Dic₇₃	C₄×D₈	1	1	yes	characteristic
132	Dic₇₃	C₂×#135	1	1	yes	characteristic
133	C₂×C₄×Q₃₂	—	73	1	no	maximal, Sylow
134	C₂×C₇₃	C₄×Q₃₂	1	2	yes	complemented
135	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
136	D₇₃	C₄×Q₃₂	1	2	yes	complemented
137	D₇₃	C₂²×Q₃₂	1	1	yes	characteristic
138	D₇₃	C₂×#508	1	1	yes	characteristic
139	C₂²×Q₃₂	—	73	1	no	—
140	C₂×C₄×Q₁₆	—	73	2	no	—
141	C₄×Q₃₂	—	73	8	no	—
142	C₂×#366	—	73	1	no	—
143	C₂×#358	—	73	2	no	—
144	C₂×C₄×C₁₆	—	73	1	no	—
145	C₇₃	C₂×C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
146	C₂²×Q₁₆	—	73	2	no	—
147	C₂×C₄×Q₈	—	146	2	no	—
148	C₂×Q₃₂	—	73	2	no	—
149	C₂×Q₃₂	—	73	4	no	—
150	C₂×Q₃₂	—	73	4	no	—
151	C₂×Q₃₂	—	73	1	no	—
152	C₂×Q₃₂	#64408	1	1	yes	characteristic, direct product factor
153	C₂²×C₁₆	—	73	1	no	—
154	C₂²×C₁₆	—	73	1	no	—
155	C₄×Q₁₆	—	73	8	no	—
156	C₄×Q₁₆	—	146	4	no	—
157	C₂×#107	—	73	2	no	—
158	C₂×#103	—	146	2	no	—
159	C₂×C₄×C₈	—	73	1	no	—
160	#366	—	73	4	no	—
161	#358	—	73	8	no	—
162	C₄×C₁₆	—	73	4	no	—
163	C₂²×Q₈	—	146	2	no	—
164	C₂×Q₁₆	—	73	4	no	—
165	C₂×Q₁₆	—	73	4	no	—
166	C₂×Q₁₆	—	73	2	no	—
167	C₂×Q₁₆	—	146	2	no	—
168	C₂×Q₁₆	—	146	2	no	—
169	C₂×Q₁₆	—	146	2	no	—
170	C₂×Q₁₆	C₂×#64408	1	2	yes	—
171	C₂²×C₈	—	73	1	no	—
172	C₂²×C₈	—	73	1	no	—
173	C₄×Q₈	—	292	4	no	—
174	C₄×Q₈	—	146	8	no	—
175	C₂×C₄⋊C₄	—	292	2	no	—
176	C₂×C₄⋊C₄	—	146	2	no	—
177	C₂×C₄²	—	292	2	no	—
178	C₂×C₄²	—	73	1	no	—
179	Q₃₂	—	73	8	no	—
180	Q₃₂	—	73	4	no	—
181	Q₃₂	C₂×#64408	1	4	yes	direct product factor
182	C₂×C₁₆	—	73	1	no	—
183	C₂×C₁₆	—	73	2	no	—
184	C₂×C₁₆	—	73	2	no	—
185	C₂×C₁₆	—	73	2	no	—
186	C₂×C₁₆	—	146	1	no	—
187	C₂×C₁₆	—	146	1	no	—
188	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
189	#107	—	73	4	no	—
190	#107	—	146	2	no	—
191	#103	—	146	8	no	—
192	C₄×C₈	—	73	2	no	—
193	C₄×C₈	—	73	2	no	—
194	C₂×Q₈	—	2	2	no	—
195	C₂×Q₈	—	292	2	no	—
196	C₂×Q₈	—	292	2	no	—
197	C₂×Q₈	—	292	2	no	—
198	C₂×Q₈	—	146	4	no	—
199	C₂×Q₈	—	146	4	no	—
200	C₂×Q₈	—	146	2	no	—
201	C₂²×C₄	—	292	2	no	—
202	C₂²×C₄	—	292	2	no	—
203	C₂²×C₄	—	73	1	no	—
204	C₂²×C₄	—	73	1	no	—
205	C₂²×C₄	—	73	1	no	—
206	Q₁₆	—	2	2	no	—
207	Q₁₆	—	73	4	no	—
208	Q₁₆	—	146	4	no	—
209	Q₁₆	—	146	4	no	—
210	Q₁₆	—	146	2	no	—
211	Q₁₆	C₂²×#64408	1	4	yes	—
212	C₂×C₈	—	73	1	no	—
213	C₂×C₈	—	73	1	no	—
214	C₂×C₈	—	73	1	no	—
215	C₂×C₈	—	73	1	no	—
216	C₂×C₈	—	73	1	no	—
217	C₂×C₈	—	73	2	no	—
218	C₂×C₈	—	146	1	no	—
219	C₂×C₈	—	146	1	no	—
220	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
221	C₄⋊C₄	—	292	8	no	—
222	C₄⋊C₄	—	292	2	no	—
223	C₄⋊C₄	—	146	4	no	—
224	C₄²	—	292	8	no	—
225	C₄²	—	73	2	no	—
226	C₄²	—	73	2	no	—
227	C₁₆	—	73	2	no	—
228	C₁₆	—	146	2	no	—
229	C₁₆	C₂²×#64408	1	2	yes	—
230	C₂³	—	73	1	no	—
231	Q₈	—	2	4	no	—
232	Q₈	—	4	2	no	—
233	Q₈	—	292	4	no	—
234	Q₈	—	292	4	no	—
235	Q₈	—	292	2	no	—
236	Q₈	—	146	4	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	292	4	no	—
239	C₂×C₄	—	292	4	no	—
240	C₂×C₄	—	292	2	no	—
241	C₂×C₄	—	292	4	no	—
242	C₂×C₄	—	584	4	no	—
243	C₂×C₄	—	73	2	no	—
244	C₂×C₄	—	73	4	no	—
245	C₂×C₄	—	73	1	no	—
246	C₂×C₄	—	73	1	no	—
247	C₂×C₄	—	73	1	no	—
248	C₂×C₄	—	73	1	no	—
249	C₂×C₄	—	73	1	no	—
250	C₂×C₄	—	73	2	no	—
251	C₂×C₄	—	146	2	no	—
252	C₂×C₄	D₄×#64408	1	1	yes	characteristic
253	C₈	—	73	1	no	—
254	C₈	—	73	1	no	—
255	C₈	—	146	2	no	—
256	C₈	D₄×#64408	1	1	yes	characteristic
257	C₈	C₂³×#64408	1	1	yes	characteristic
258	C₂²	—	73	2	no	—
259	C₂²	—	73	2	no	—
260	C₂²	—	73	1	no	—
261	C₂²	—	73	1	no	—
262	C₂²	D₈×#64408	1	1	yes	characteristic
263	C₄	—	4	4	no	—
264	C₄	—	292	4	no	—
265	C₄	—	584	4	no	—
266	C₄	—	73	4	no	—
267	C₄	—	73	1	no	—
268	C₄	—	73	1	no	—
269	C₄	—	146	2	no	—
270	C₄	D₈×#64408	1	1	yes	characteristic
271	C₄	C₂×D₄×#64408	1	1	yes	characteristic
272	C₂	—	73	1	no	—
273	C₂	—	73	1	no	—
274	C₂	—	73	2	no	—
275	C₂	Q₃₂×#64408	1	2	yes	direct product factor
276	C₂	C₂×D₈×#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×#64408
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₈	C₂³×#64408
Fitting	C₂×Q₃₂×C₇₃	C₄
socle	C₂²×C₇₃	C₄×D₈
prime radical	C₂²×D₇₃	C₂×D₈
cosocle	C₈×D₇₃	C₂⁴
hypercenter	C₂×Q₃₂	#64408
hypocenter	C₇₃	C₂×C₄×Q₃₂
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
2. Q₃₂×#64408	C₂	direct (N × H)
18. C₂×Dic₅₈₄	C₄	semidirect
19. C₂×#14556537	C₄	semidirect
20. Q₃₂×D₇₃	C₄	semidirect
23. C₂×Q₃₂×C₇₃	C₄	semidirect
47. Dic₅₈₄	C₂×C₄	semidirect
48. #14556537	C₂×C₄	semidirect
53. Q₃₂×C₇₃	C₂×C₄	semidirect
95. C₂×#64408	Q₃₂	direct (N × H)
126. #64408	C₂×Q₃₂	direct (N × H)
134. C₂×C₇₃	C₄×Q₃₂	semidirect
136. D₇₃	C₄×Q₃₂	semidirect
145. C₇₃	C₂×C₄×Q₃₂	semidirect
152. C₂×Q₃₂	#64408	direct (N × H)
181. Q₃₂	C₂×#64408	direct (N × H)
275. C₂	Q₃₂×#64408	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×Q₁₆×#64408	2
Q₃₂×#64408	2
C₂×#17989513	2
C₂×#17989500	2
C₂×C₁₆×#64408	2
C₂×Q₃₂×D₇₃	2
C₂×C₄×Q₃₂	73

Subgroups — C2×Q32×#64408 #34030254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₃₂×#64408 #34030254