Subgroups — #31903369

Order 18176.

746 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986037	C₂	1	2	yes	maximal
2	#22588073	C₂	1	2	yes	maximal
3	C₂×#17985756	C₂	1	2	yes	maximal
4	C₂×#405×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556087	C₂²	1	4	yes	—
6	#17986037	—	2	4	no	—
7	#17986037	C₄	1	8	yes	complemented
8	C₂×#14556062	—	2	2	no	—
9	C₂×#14556060	—	2	2	no	—
10	#17985766	—	2	4	no	—
11	#17985756	—	2	4	no	—
12	C₂²×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×#117×C₇₁	C₂²	1	1	yes	characteristic
15	#405×C₇₁	—	2	2	no	—
16	#405×C₇₁	—	2	2	no	—
17	C₂×#97×C₇₁	C₄	1	2	yes	complemented
18	C₂×C₄×C₈×C₇₁	C₄	1	2	yes	complemented
19	#14556087	—	2	4	no	—
20	#14556087	—	2	4	no	—
21	#14556087	—	2	4	no	—
22	#14556087	C₂×C₄	1	8	yes	—
23	C₂×C₄⋊Dic₇₁	—	2	4	no	—
24	C₂×C₂²⋊Dic₇₁	—	2	4	no	—
25	C₂×C₄×Dic₇₁	—	2	4	no	—
26	#14556062	—	2	8	no	—
27	#14556060	—	2	8	no	—
28	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₇₁	—	2	2	no	—
30	C₂×M₁₆×C₇₁	—	2	2	no	—
31	C₂×M₁₆×C₇₁	D₄	1	2	yes	—
32	C₂×M₁₆×C₇₁	Q₈	1	2	yes	—
33	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₁	D₄	1	2	yes	—
38	C₂²×C₈×C₇₁	D₄	1	2	yes	—
39	C₂²×C₈×C₇₁	D₄	1	2	yes	—
40	C₂²×C₈×C₇₁	Q₈	1	2	yes	—
41	#117×C₇₁	—	2	2	no	—
42	#117×C₇₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₇₁	Q₈	1	2	yes	—
44	C₂×C₂²⋊C₄×C₇₁	D₄	1	2	yes	—
45	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
46	#97×C₇₁	—	2	4	no	—
47	C₄×C₈×C₇₁	—	2	2	no	—
48	C₄×C₈×C₇₁	—	2	2	no	—
49	C₂²×Dic₇₁	—	4	4	no	—
50	C₄⋊Dic₇₁	—	2	8	no	—
51	C₄⋊Dic₇₁	—	4	4	no	—
52	C₂²⋊Dic₇₁	—	2	8	no	—
53	C₂²⋊Dic₇₁	—	4	4	no	—
54	C₄×Dic₇₁	—	2	8	no	—
55	C₄×Dic₇₁	—	4	4	no	—
56	C₂⁴×C₇₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₁	C₄⋊C₄	1	4	yes	—
58	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₇₁	#41	1	2	yes	—
67	C₂²×C₄×C₇₁	#41	1	2	yes	—
68	M₁₆×C₇₁	—	2	4	no	—
69	M₁₆×C₇₁	—	2	4	no	—
70	C₂×C₈×C₇₁	—	2	2	no	—
71	C₂×C₈×C₇₁	—	2	2	no	—
72	C₂×C₈×C₇₁	C₄²	1	2	yes	—
73	C₂×C₈×C₇₁	C₄²	1	2	yes	—
74	C₂×C₈×C₇₁	C₂²⋊C₄	1	4	yes	—
75	C₂×C₈×C₇₁	C₂²⋊C₄	1	4	yes	—
76	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
80	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
81	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
82	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
83	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
84	C₄⋊C₄×C₇₁	—	2	4	no	—
85	C₂²⋊C₄×C₇₁	—	2	4	no	—
86	C₄²×C₇₁	—	2	2	no	—
87	C₄²×C₇₁	—	2	2	no	—
88	C₂³×C₇₁	—	4	2	no	—
89	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂³×C₇₁	C₄×D₄	1	2	yes	—
91	C₂³×C₇₁	#121	1	2	yes	—
92	C₂³×C₇₁	#127	1	1	yes	characteristic
93	C₂³×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₇₁	—	2	4	no	—
95	C₂×C₄×C₇₁	—	4	4	no	—
96	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
103	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
104	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
106	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
107	C₂×C₄×C₇₁	#121	1	2	yes	—
108	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₇₁	#124	1	2	yes	—
111	C₂×C₄×C₇₁	#125	1	2	yes	—
112	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×Dic₇₁	—	4	8	no	—
115	C₂×Dic₇₁	—	8	8	no	—
116	C₈×C₇₁	—	2	4	no	—
117	C₈×C₇₁	#95	1	2	yes	—
118	C₈×C₇₁	#95	1	4	yes	—
119	C₈×C₇₁	#95	1	2	yes	—
120	C₂²×C₇₁	—	2	4	no	—
121	C₂²×C₇₁	—	2	2	no	—
122	C₂²×C₇₁	—	4	2	no	—
123	C₂²×C₇₁	—	4	1	no	—
124	C₂²×C₇₁	—	4	1	no	—
125	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₇₁	#380	1	1	yes	characteristic
127	C₂²×C₇₁	#384	1	1	yes	characteristic
128	C₂²×C₇₁	#389	1	2	yes	—
129	C₂²×C₇₁	#390	1	2	yes	—
130	C₄×C₇₁	—	4	4	no	—
131	C₄×C₇₁	C₂×#95	1	1	yes	characteristic
132	C₄×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
133	C₄×C₇₁	#380	1	1	yes	characteristic
134	C₄×C₇₁	#382	1	1	yes	characteristic
135	C₄×C₇₁	#389	1	2	yes	—
136	C₄×C₇₁	#391	1	2	yes	—
137	Dic₇₁	—	8	8	no	—
138	#10866	—	71	1	no	maximal, Sylow
139	C₂×C₇₁	—	2	2	no	—
140	C₂×C₇₁	—	2	1	no	—
141	C₂×C₇₁	—	2	1	no	—
142	C₂×C₇₁	—	2	2	no	—
143	C₂×C₇₁	C₄⋊#95	1	1	yes	characteristic
144	C₂×C₇₁	#1812	1	2	yes	—
145	C₂×#427	—	71	2	no	—
146	C₂×#405	D₇₁	1	1	yes	characteristic
147	#1740	—	71	2	no	—
148	C₂×H(3,ℤ/4)	—	71	2	no	—
149	C₇₁	#10866	1	1	yes	characteristic, complemented, Sylow
150	C₂²×M₁₆	C₂×D₇₁	1	1	yes	characteristic
151	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
152	C₂×#117	—	71	4	no	—
153	C₂×#117	C₂×D₇₁	1	1	yes	characteristic
154	#427	—	71	8	no	—
155	#427	—	142	4	no	—
156	C₂×#107	—	142	2	no	—
157	C₂×#106	—	142	2	no	—
158	#405	—	2	2	no	—
159	#405	—	2	2	no	—
160	C₂×#97	Dic₇₁	1	2	yes	complemented
161	C₂×C₄×C₈	Dic₇₁	1	2	yes	complemented
162	#342	—	142	4	no	—
163	H(3,ℤ/4)	—	142	4	no	—
164	C₂³×C₄	C₂²×D₇₁	1	1	yes	characteristic
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	Dic₁₄₂	1	2	yes	—
168	C₂×M₁₆	D₂₈₄	1	2	yes	—
169	C₂²×C₈	Dic₁₄₂	1	2	yes	—
170	C₂²×C₈	C₄×D₇₁	1	4	yes	—
171	C₂²×C₈	D₂₈₄	1	2	yes	—
172	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
173	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
174	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
175	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
176	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
177	#117	—	2	2	no	—
178	#117	—	2	2	no	—
179	#117	—	71	8	no	—
180	#117	—	142	4	no	—
181	#117	—	142	4	no	—
182	#117	—	142	4	no	—
183	C₂×C₄⋊C₄	—	142	4	no	—
184	C₂×C₄⋊C₄	Dic₁₄₂	1	2	yes	—
185	C₂×C₂²⋊C₄	—	142	4	no	—
186	C₂×C₂²⋊C₄	D₂₈₄	1	2	yes	—
187	C₂×C₄²	—	142	4	no	—
188	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
189	#107	—	142	8	no	—
190	#106	—	142	8	no	—
191	#97	—	2	4	no	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	—	2	2	no	—
194	C₂⁴	C₂×D₂₈₄	1	1	yes	characteristic
195	C₂²×C₄	—	284	4	no	—
196	C₂²×C₄	C₄⋊Dic₇₁	1	4	yes	—
197	C₂²×C₄	C₂×C₄×D₇₁	1	2	yes	—
198	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₂²⋊D₇₁	1	2	yes	—
200	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
202	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
203	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
204	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
205	C₂²×C₄	#11765865	1	2	yes	—
206	C₂²×C₄	#11765865	1	2	yes	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	2	4	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
212	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
213	C₂×C₈	(C₂×D₇₁)⋊C₄	1	4	yes	—
214	C₂×C₈	(C₂×D₇₁)⋊C₄	1	4	yes	—
215	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
216	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
217	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
218	C₂×C₈	Dic₇₁⋊C₄	1	4	yes	—
219	C₂×C₈	Dic₇₁⋊C₄	1	4	yes	—
220	C₂×C₈	C₄⋊Dic₇₁	1	4	yes	—
221	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
222	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	284	4	no	—
225	C₄⋊C₄	—	142	8	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	—	284	4	no	—
228	C₂²⋊C₄	—	142	8	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	284	4	no	—
232	C₄²	—	142	8	no	—
233	C₂³	—	4	2	no	—
234	C₂³	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
235	C₂³	C₄×D₂₈₄	1	2	yes	—
236	C₂³	#14556102	1	2	yes	—
237	C₂³	#14556125	1	1	yes	characteristic
238	C₂³	#14556128	1	1	yes	characteristic
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	284	8	no	—
242	C₂×C₄	—	568	8	no	—
243	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
244	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	2	yes	—
245	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
246	C₂×C₄	C₂×Dic₇₁⋊C₄	1	2	yes	—
247	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
249	C₂×C₄	#14556087	1	1	yes	characteristic
250	C₂×C₄	#14556087	1	1	yes	characteristic
251	C₂×C₄	C₄×D₂₈₄	1	2	yes	—
252	C₂×C₄	C₄×Dic₁₄₂	1	2	yes	—
253	C₂×C₄	C₄×Dic₁₄₂	1	2	yes	—
254	C₂×C₄	#14556102	1	2	yes	—
255	C₂×C₄	#14556107	1	2	yes	—
256	C₂×C₄	#14556107	1	2	yes	—
257	C₂×C₄	#14556116	1	2	yes	—
258	C₂×C₄	#14556120	1	2	yes	—
259	C₂×C₄	#14556128	1	1	yes	characteristic
260	C₂×C₄	#14556128	1	1	yes	characteristic
261	C₈	—	2	4	no	—
262	C₈	C₇₁⋊#95	1	2	yes	—
263	C₈	C₇₁⋊#95	1	4	yes	—
264	C₈	C₇₁⋊#95	1	2	yes	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	C₄×C₄⋊Dic₇₁	1	1	yes	characteristic
271	C₂²	#17985859	1	1	yes	characteristic
272	C₂²	#17985869	1	1	yes	characteristic
273	C₂²	#17985897	1	2	yes	—
274	C₂²	#17985901	1	2	yes	—
275	C₄	—	4	4	no	—
276	C₄	—	568	8	no	—
277	C₄	C₂×C₇₁⋊#95	1	1	yes	characteristic
278	C₄	C₄×C₄⋊Dic₇₁	1	1	yes	characteristic
279	C₄	#17985859	1	1	yes	characteristic
280	C₄	#17985865	1	1	yes	characteristic
281	C₄	#17985897	1	2	yes	—
282	C₄	#17985905	1	2	yes	—
283	C₂	—	2	2	no	—
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	2	no	—
287	C₂	#22586570	1	1	yes	characteristic
288	C₂	#22588386	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14556087
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₇₁
Fitting	C₂×#405×C₇₁	C₂
socle	C₂²×C₇₁	#380
prime radical	C₂⁴×C₇₁	C₂×D₄
cosocle	C₂³×C₄×C₇₁	C₂³
hypercenter	C₂×#405	D₇₁
hypocenter	C₇₁	#10866
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
7. #17986037	C₄	semidirect
17. C₂×#97×C₇₁	C₄	semidirect
18. C₂×C₄×C₈×C₇₁	C₄	semidirect
149. C₇₁	#10866	semidirect
160. C₂×#97	Dic₇₁	semidirect
161. C₂×C₄×C₈	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986037	2
#22588073	2
C₂×#17985756	2
C₂×#405×C₇₁	2
#10866	71