Subgroups — C₂×#22590462 #32152012

Order 18176.

638 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986415	C₂	1	1	yes	maximal, characteristic
2	C₂×#17986271	C₂	1	2	yes	maximal
3	#22590462	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17985829	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#17985815	C₂	1	2	yes	maximal, complemented
6	C₂²×Q₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14556159	—	2	2	no	—
8	C₂×#14556134	C₂²	1	1	yes	characteristic
9	#17986415	—	2	2	no	—
10	#17986415	C₂²	1	4	yes	—
11	C₂×#14556068	C₂²	1	1	yes	characteristic
12	C₂×#14556067	—	2	1	no	—
13	C₂×C₁₆×D₇₁	—	2	1	no	—
14	#17986271	—	2	4	no	—
15	#17986271	C₂²	1	8	yes	—
16	C₂×#14556064	C₂²	1	2	yes	—
17	C₂×#14556047	—	2	2	no	—
18	#17985829	C₂²	1	4	yes	complemented
19	#17985815	C₂²	1	8	yes	complemented
20	C₂²×Q₁₆×C₇₁	C₂²	1	2	yes	—
21	C₂×Q₃₂×C₇₁	—	2	1	no	—
22	C₂×Q₃₂×C₇₁	—	2	1	no	—
23	C₂×Q₃₂×C₇₁	—	2	2	no	—
24	C₂×Q₃₂×C₇₁	C₂²	1	4	yes	complemented
25	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
26	C₂×#11765865	D₄	1	1	yes	characteristic
27	#14556159	—	2	8	no	—
28	#14556159	—	4	4	no	—
29	C₂×#11765855	—	4	2	no	—
30	C₂×#11765852	—	4	2	no	—
31	#14556134	—	2	1	no	—
32	#14556134	—	2	1	no	—
33	#14556134	D₄	1	2	yes	—
34	#14556134	C₂³	1	2	yes	—
35	C₂×#11765849	—	2	2	no	—
36	C₂×#11765849	D₄	1	1	yes	characteristic
37	C₂×#11765848	—	2	1	no	—
38	C₂×C₈×D₇₁	—	2	1	no	—
39	#14556068	—	2	1	no	—
40	#14556068	C₂³	1	2	yes	—
41	#14556067	—	2	4	no	—
42	C₂×C₇₁⋊C₁₆	—	2	1	no	—
43	C₁₆×D₇₁	—	2	4	no	—
44	#14556064	—	2	2	no	—
45	#14556064	C₂³	1	4	yes	—
46	#14556047	—	2	8	no	—
47	C₂²×Q₈×C₇₁	—	2	2	no	—
48	C₂×Q₁₆×C₇₁	—	2	2	no	—
49	C₂×Q₁₆×C₇₁	—	2	2	no	—
50	C₂×Q₁₆×C₇₁	—	2	2	no	—
51	C₂×Q₁₆×C₇₁	—	2	2	no	—
52	C₂×Q₁₆×C₇₁	—	2	2	no	—
53	C₂×Q₁₆×C₇₁	C₂³	1	4	yes	—
54	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
55	Q₃₂×C₇₁	—	2	4	no	—
56	Q₃₂×C₇₁	—	2	4	no	—
57	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
58	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
59	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
60	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
61	#11765865	—	2	1	no	—
62	#11765865	—	2	1	no	—
63	#11765865	D₈	1	2	yes	—
64	#11765865	C₂×D₄	1	2	yes	—
65	C₂×Dic₁₄₂	D₈	1	1	yes	characteristic
66	C₂×D₂₈₄	D₈	1	1	yes	characteristic
67	C₂×C₂²⋊D₇₁	—	2	1	no	—
68	C₂×C₄×D₇₁	—	2	1	no	—
69	#11765855	—	4	8	no	—
70	#11765852	—	4	8	no	—
71	#11765849	—	2	1	no	—
72	#11765849	—	2	4	no	—
73	#11765849	—	4	2	no	—
74	#11765849	C₂×D₄	1	2	yes	—
75	#11765848	—	2	2	no	—
76	#11765848	—	2	2	no	—
77	C₂×C₇₁⋊C₈	—	2	1	no	—
78	C₂×C₇₁⋊C₈	—	4	2	no	—
79	C₈×D₇₁	—	2	2	no	—
80	C₈×D₇₁	—	2	2	no	—
81	C₇₁⋊C₁₆	—	2	2	no	—
82	C₂×Q₈×C₇₁	—	2	4	no	—
83	C₂×Q₈×C₇₁	—	4	2	no	—
84	C₂×Q₈×C₇₁	—	4	2	no	—
85	C₂×Q₈×C₇₁	—	4	2	no	—
86	C₂×Q₈×C₇₁	—	4	2	no	—
87	C₂×Q₈×C₇₁	—	4	2	no	—
88	C₂²×C₄×C₇₁	—	4	2	no	—
89	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
90	Q₁₆×C₇₁	—	2	4	no	—
91	Q₁₆×C₇₁	—	2	4	no	—
92	Q₁₆×C₇₁	—	4	2	no	—
93	Q₁₆×C₇₁	—	4	2	no	—
94	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
98	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
99	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
100	C₁₆×C₇₁	C₂×D₄	1	2	yes	—
101	C₁₆×C₇₁	C₂×D₄	1	2	yes	—
102	C₂³×C₇₁	#135	1	1	yes	characteristic
103	C₂²×D₇₁	—	2	1	no	—
104	Q₈×C₇₁	—	4	4	no	—
105	Q₈×C₇₁	—	4	4	no	—
106	Q₈×C₇₁	—	8	2	no	—
107	Q₈×C₇₁	—	8	2	no	—
108	C₂×C₄×C₇₁	—	4	4	no	—
109	C₂×C₄×C₇₁	—	8	2	no	—
110	C₂×C₄×C₇₁	—	8	2	no	—
111	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
112	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
113	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
116	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
117	C₂²⋊D₇₁	—	2	2	no	—
118	C₂²⋊D₇₁	—	2	2	no	—
119	C₂×Dic₇₁	—	2	1	no	—
120	D₂₈₄	—	2	1	no	—
121	D₂₈₄	C₂×D₈	1	2	yes	—
122	C₄×D₇₁	—	2	2	no	—
123	C₄×D₇₁	—	2	2	no	—
124	Dic₁₄₂	—	2	1	no	—
125	Dic₁₄₂	C₂×D₈	1	2	yes	—
126	C₈×C₇₁	#121	1	1	yes	characteristic
127	C₈×C₇₁	#121	1	1	yes	characteristic
128	C₈×C₇₁	C₂²×D₄	1	1	yes	characteristic
129	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
130	C₇₁⋊C₈	—	2	2	no	—
131	C₇₁⋊C₈	—	4	4	no	—
132	C₂²×C₇₁	—	2	2	no	—
133	C₂²×C₇₁	#466	1	1	yes	characteristic
134	C₂²×C₇₁	#466	1	1	yes	characteristic
135	C₂²×C₇₁	C₂×#135	1	1	yes	characteristic
136	C₂×D₇₁	—	2	2	no	—
137	C₂×D₇₁	—	4	2	no	—
138	C₄×C₇₁	—	8	4	no	—
139	C₄×C₇₁	#466	1	1	yes	characteristic
140	C₄×C₇₁	#466	1	1	yes	characteristic
141	C₄×C₇₁	C₂×#121	1	1	yes	characteristic
142	C₄×C₇₁	C₂²×D₈	1	1	yes	characteristic
143	Dic₇₁	—	2	2	no	—
144	C₂×#2222	—	71	1	no	maximal, Sylow
145	C₂×C₇₁	—	2	1	no	—
146	C₂×C₇₁	—	2	1	no	—
147	C₂×C₇₁	#2222	1	2	yes	complemented
148	C₂×C₇₁	C₂×#466	1	1	yes	characteristic
149	D₇₁	—	4	2	no	—
150	C₂²×Q₃₂	D₇₁	1	1	yes	characteristic, complemented
151	C₂×#504	—	71	1	no	—
152	C₂×#473	—	71	2	no	—
153	#2222	—	71	8	no	—
154	C₂×#368	—	71	1	no	—
155	C₂×#362	—	71	2	no	—
156	C₇₁	C₂×#2222	1	1	yes	characteristic, complemented, Sylow
157	C₂×#137	—	142	2	no	—
158	C₂²×Q₁₆	C₂×D₇₁	1	2	yes	—
159	C₂×#131	—	71	1	no	—
160	C₂×Q₃₂	—	2	1	no	—
161	C₂×Q₃₂	—	2	1	no	—
162	C₂×Q₃₂	—	2	2	no	—
163	C₂×Q₃₂	C₂×D₇₁	1	4	yes	complemented
164	#504	—	71	4	no	—
165	#504	—	142	2	no	—
166	C₂×M₃₂	—	71	1	no	—
167	C₂×M₃₂	—	142	1	no	—
168	C₂²×C₁₆	—	142	1	no	—
169	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
170	#473	—	71	8	no	—
171	#473	—	142	4	no	—
172	C₂×#108	—	71	2	no	—
173	C₂×#101	—	142	2	no	—
174	#368	—	71	4	no	—
175	#362	—	71	8	no	—
176	C₂×#41	—	71	1	no	—
177	C₂²×Q₈	—	2	2	no	—
178	#137	—	284	4	no	—
179	#137	—	142	8	no	—
180	C₂×Q₁₆	—	2	2	no	—
181	C₂×Q₁₆	—	2	2	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	—	2	2	no	—
184	C₂×Q₁₆	—	2	2	no	—
185	C₂×Q₁₆	—	284	2	no	—
186	C₂×Q₁₆	C₂²×D₇₁	1	4	yes	—
187	C₂×SD₁₆	—	284	2	no	—
188	#131	—	71	2	no	—
189	#131	—	71	2	no	—
190	#131	—	142	1	no	—
191	#131	—	142	1	no	—
192	C₂×M₁₆	—	71	1	no	—
193	C₂×M₁₆	—	142	2	no	—
194	C₂×M₁₆	—	142	1	no	—
195	C₂²×C₈	—	142	1	no	—
196	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
197	Q₃₂	—	2	4	no	—
198	Q₃₂	—	2	4	no	—
199	M₃₂	—	71	2	no	—
200	M₃₂	—	142	1	no	—
201	M₃₂	—	142	4	no	—
202	C₂×C₁₆	—	142	1	no	—
203	C₂×C₁₆	—	142	4	no	—
204	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
205	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
206	C₂×C₁₆	C₂²⋊D₇₁	1	2	yes	—
207	C₂×C₁₆	C₂²×D₇₁	1	2	yes	—
208	#108	—	71	4	no	—
209	#108	—	142	2	no	—
210	#101	—	142	8	no	—
211	#41	—	71	2	no	—
212	#41	—	71	2	no	—
213	#41	—	142	1	no	—
214	#41	—	142	1	no	—
215	C₂×Q₈	—	2	4	no	—
216	C₂×Q₈	—	4	2	no	—
217	C₂×Q₈	—	4	2	no	—
218	C₂×Q₈	—	4	2	no	—
219	C₂×Q₈	—	4	2	no	—
220	C₂×Q₈	—	4	2	no	—
221	C₂×Q₈	—	71	1	no	—
222	C₂×D₄	—	71	1	no	—
223	C₂×D₄	—	142	1	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	—	142	1	no	—
226	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
227	Q₁₆	—	2	4	no	—
228	Q₁₆	—	2	4	no	—
229	Q₁₆	—	4	2	no	—
230	Q₁₆	—	4	2	no	—
231	Q₁₆	—	284	8	no	—
232	SD₁₆	—	284	8	no	—
233	M₁₆	—	284	2	no	—
234	M₁₆	—	71	2	no	—
235	M₁₆	—	142	4	no	—
236	M₁₆	—	142	1	no	—
237	M₁₆	—	142	2	no	—
238	M₁₆	—	142	2	no	—
239	C₂×C₈	—	284	2	no	—
240	C₂×C₈	—	142	1	no	—
241	C₂×C₈	—	142	2	no	—
242	C₂×C₈	—	142	2	no	—
243	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
244	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
245	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
246	C₂×C₈	#11765864	1	1	yes	characteristic
247	C₂×C₈	#11765864	1	1	yes	characteristic
248	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
249	C₁₆	—	142	2	no	—
250	C₁₆	C₂×C₂²⋊D₇₁	1	2	yes	—
251	C₁₆	C₂×C₂²⋊D₇₁	1	2	yes	—
252	C₂³	—	142	1	no	—
253	C₂³	#14556147	1	1	yes	characteristic
254	Q₈	—	4	4	no	—
255	Q₈	—	4	4	no	—
256	Q₈	—	8	2	no	—
257	Q₈	—	8	2	no	—
258	Q₈	—	71	2	no	—
259	Q₈	—	142	1	no	—
260	D₄	—	71	2	no	—
261	D₄	—	142	2	no	—
262	D₄	—	142	2	no	—
263	D₄	—	142	1	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	8	2	no	—
266	C₂×C₄	—	8	2	no	—
267	C₂×C₄	—	142	1	no	—
268	C₂×C₄	—	142	2	no	—
269	C₂×C₄	—	142	2	no	—
270	C₂×C₄	#14556099	1	1	yes	characteristic
271	C₂×C₄	#14556099	1	1	yes	characteristic
272	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
273	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
274	C₂×C₄	#14556147	1	1	yes	characteristic
275	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
276	C₈	—	284	4	no	—
277	C₈	—	142	2	no	—
278	C₈	#14556099	1	1	yes	characteristic
279	C₈	#14556099	1	1	yes	characteristic
280	C₈	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
281	C₈	C₂×#11765864	1	1	yes	characteristic
282	C₂²	—	2	2	no	—
283	C₂²	—	284	2	no	—
284	C₂²	—	142	2	no	—
285	C₂²	#17986232	1	1	yes	characteristic
286	C₂²	#17986232	1	1	yes	characteristic
287	C₂²	C₂×#14556147	1	1	yes	characteristic
288	C₄	—	8	4	no	—
289	C₄	—	142	2	no	—
290	C₄	#17986232	1	1	yes	characteristic
291	C₄	#17986232	1	1	yes	characteristic
292	C₄	C₂×#14556099	1	1	yes	characteristic
293	C₄	C₂×D₈×D₇₁	1	1	yes	characteristic
294	C₂	—	2	1	no	—
295	C₂	—	2	1	no	—
296	C₂	—	284	2	no	—
297	C₂	#22590462	1	2	yes	direct product factor
298	C₂	C₂×#17986232	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17986232
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₂²×Q₃₂×C₇₁	C₂
socle	C₂²×C₇₁	#466
prime radical	C₂×#11765865	D₄
hypercenter	C₂²×Q₃₂	D₇₁
hypocenter	C₇₁	C₂×#2222
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×Q₃₂

Complements

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

N	H ≅ G/N	type
3. #22590462	C₂	direct (N × H)
4. C₂×#17985829	C₂	semidirect
5. C₂×#17985815	C₂	semidirect
6. C₂²×Q₃₂×C₇₁	C₂	semidirect
18. #17985829	C₂²	semidirect
19. #17985815	C₂²	semidirect
24. C₂×Q₃₂×C₇₁	C₂²	semidirect
147. C₂×C₇₁	#2222	semidirect
150. C₂²×Q₃₂	D₇₁	semidirect
156. C₇₁	C₂×#2222	semidirect
163. C₂×Q₃₂	C₂×D₇₁	semidirect
297. C₂	#22590462	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×#17986415	2
C₂×#17986271	2
#22590462	2
C₂×#17985829	2
C₂×#17985815	2
C₂²×Q₃₂×C₇₁	2
C₂×#2222	71

Subgroups — C2×#22590462 #32152012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22590462 #32152012