Subgroups — #31897568

Order 18176.

538 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	#22596738	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#22586224	C₂	1	4	yes	maximal
4	C₇₁×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556128	—	2	1	no	—
6	C₂×#14556128	—	2	1	no	—
7	C₂×#14556128	C₄	1	2	yes	—
8	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₄×C₇₁⋊C₈	—	2	1	no	—
10	#17985869	—	2	1	no	—
11	#17985869	—	2	1	no	—
12	#17985869	C₄	1	2	yes	—
13	#17985869	C₂²	1	2	yes	—
14	#17985754	—	2	1	no	—
15	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
16	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
17	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
18	C₄³×C₇₁	C₂²	1	1	yes	characteristic
19	C₂²×Dic₁₄₂	—	2	2	no	—
20	C₂²×C₇₁⋊C₈	—	2	2	no	—
21	#14556128	—	4	4	no	—
22	#14556128	—	4	4	no	—
23	#14556128	—	4	2	no	—
24	#14556128	—	4	2	no	—
25	C₂×C₄⋊Dic₇₁	—	2	2	no	—
26	C₂×C₄⋊Dic₇₁	—	2	2	no	—
27	C₂×C₄⋊Dic₇₁	—	2	2	no	—
28	C₂×C₄⋊Dic₇₁	—	2	2	no	—
29	C₂×C₄⋊Dic₇₁	—	2	2	no	—
30	C₂×C₄⋊Dic₇₁	C₂×C₄	1	4	yes	—
31	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
32	C₄⋊(C₇₁⋊C₈)	Q₈	1	4	yes	complemented
33	C₄×C₇₁⋊C₈	—	4	2	no	—
34	C₂²×C₈×C₇₁	—	2	2	no	—
35	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₇₁	—	2	4	no	—
43	C₄⋊C₈×C₇₁	—	4	2	no	—
44	C₄⋊C₈×C₇₁	—	4	2	no	—
45	C₂×Dic₁₄₂	—	4	4	no	—
46	C₂×Dic₁₄₂	—	4	4	no	—
47	C₂×Dic₁₄₂	—	4	4	no	—
48	C₂²×Dic₇₁	—	2	4	no	—
49	C₂×C₇₁⋊C₈	—	2	4	no	—
50	C₂×C₇₁⋊C₈	—	4	2	no	—
51	C₂×C₇₁⋊C₈	—	4	2	no	—
52	C₄⋊Dic₇₁	—	4	4	no	—
53	C₄⋊Dic₇₁	—	4	4	no	—
54	C₄⋊Dic₇₁	—	4	4	no	—
55	C₄⋊Dic₇₁	—	4	2	no	—
56	C₄⋊Dic₇₁	—	4	2	no	—
57	C₄⋊Dic₇₁	—	4	4	no	—
58	C₄⋊Dic₇₁	—	4	4	no	—
59	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₇₁	—	4	2	no	—
67	C₂×C₈×C₇₁	—	4	2	no	—
68	C₂×C₈×C₇₁	—	4	2	no	—
69	C₄²×C₇₁	—	2	2	no	—
70	C₄²×C₇₁	—	2	2	no	—
71	C₄²×C₇₁	—	2	2	no	—
72	C₄²×C₇₁	D₈	1	2	yes	—
73	C₄²×C₇₁	SD₁₆	1	2	yes	—
74	C₄²×C₇₁	SD₁₆	1	2	yes	—
75	C₄²×C₇₁	SD₁₆	1	2	yes	—
76	C₄²×C₇₁	Q₁₆	1	2	yes	—
77	C₄²×C₇₁	Q₁₆	1	2	yes	—
78	C₄²×C₇₁	C₂×Q₈	1	2	yes	—
79	C₄²×C₇₁	#41	1	2	yes	—
80	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₇₁	—	2	1	no	—
82	C₂×C₄×C₇₁	—	2	1	no	—
83	C₂×C₄×C₇₁	—	2	1	no	—
84	C₂×C₄×C₇₁	—	2	1	no	—
85	C₂×C₄×C₇₁	—	2	1	no	—
86	C₂×C₄×C₇₁	—	2	1	no	—
87	C₂×C₄×C₇₁	#100	1	2	yes	—
88	C₂×C₄×C₇₁	#101	1	2	yes	—
89	C₂×C₄×C₇₁	#102	1	2	yes	—
90	C₂×C₄×C₇₁	#102	1	2	yes	—
91	C₂×C₄×C₇₁	#103	1	2	yes	—
92	C₂×C₄×C₇₁	#103	1	2	yes	—
93	C₂×C₄×C₇₁	#103	1	2	yes	—
94	C₂×C₄×C₇₁	#103	1	2	yes	—
95	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
96	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
97	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
98	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
104	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
105	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
106	C₂×Dic₇₁	—	4	4	no	—
107	C₂×Dic₇₁	—	4	4	no	—
108	C₂×Dic₇₁	—	4	4	no	—
109	Dic₁₄₂	—	8	8	no	—
110	C₈×C₇₁	—	8	2	no	—
111	C₇₁⋊C₈	—	4	4	no	—
112	C₂²×C₇₁	#391	1	1	yes	characteristic
113	C₂²×C₇₁	C₂×#100	1	1	yes	characteristic
114	C₂²×C₇₁	C₂×#101	1	1	yes	characteristic
115	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
116	C₂²×C₇₁	C₂×#103	1	1	yes	characteristic
117	C₂²×C₇₁	C₂×#103	1	1	yes	characteristic
118	C₂²×C₇₁	#418	1	1	yes	characteristic
119	C₄×C₇₁	—	2	2	no	—
120	C₄×C₇₁	—	2	2	no	—
121	C₄×C₇₁	—	2	2	no	—
122	C₄×C₇₁	#332	1	4	yes	—
123	C₄×C₇₁	#333	1	4	yes	—
124	C₄×C₇₁	#476	1	2	yes	—
125	C₄×C₇₁	#477	1	2	yes	—
126	C₄×C₇₁	#486	1	1	yes	characteristic
127	C₄×C₇₁	#487	1	1	yes	characteristic
128	C₄×C₇₁	#498	1	1	yes	characteristic
129	C₄×C₇₁	#500	1	1	yes	characteristic
130	Dic₇₁	—	8	4	no	—
131	#9896	—	71	1	no	maximal, Sylow
132	C₂×C₇₁	C₂×#332	1	1	yes	characteristic
133	C₂×C₇₁	C₂×#333	1	1	yes	characteristic
134	C₂×C₇₁	#1545	1	1	yes	characteristic
135	C₂×C₇₁	#1549	1	1	yes	characteristic
136	C₂×C₇₁	#1987	1	1	yes	characteristic
137	C₂×C₇₁	#1990	1	1	yes	characteristic
138	C₂×C₇₁	#1997	1	1	yes	characteristic
139	#2868	—	71	1	no	—
140	#1842	D₇₁	1	1	yes	characteristic
141	C₄×C₄⋊C₈	—	71	1	no	—
142	#1296	—	71	4	no	—
143	C₇₁	#9896	1	1	yes	characteristic, complemented, Sylow
144	C₂×C₄⋊Q₈	—	71	2	no	—
145	C₂×C₄⋊Q₈	—	142	1	no	—
146	C₂×C₄⋊Q₈	—	142	1	no	—
147	C₂×C₄⋊C₈	—	2	1	no	—
148	C₂×C₄⋊C₈	—	2	1	no	—
149	C₂×C₄⋊C₈	—	71	2	no	—
150	C₂×C₄⋊C₈	C₂×D₇₁	1	2	yes	—
151	C₂×C₄×C₈	—	142	1	no	—
152	#384	—	71	2	no	—
153	#384	—	71	2	no	—
154	#384	—	142	1	no	—
155	#384	—	142	1	no	—
156	C₄³	C₂×D₇₁	1	1	yes	characteristic
157	#336	—	142	1	no	—
158	C₂²×Q₈	—	142	2	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	142	2	no	—
161	C₄⋊Q₈	—	284	4	no	—
162	C₄⋊Q₈	—	284	4	no	—
163	C₄⋊Q₈	—	284	2	no	—
164	C₄⋊Q₈	—	284	2	no	—
165	C₂×C₄⋊C₄	—	71	4	no	—
166	C₂×C₄⋊C₄	—	142	2	no	—
167	C₂×C₄⋊C₄	—	142	2	no	—
168	C₂×C₄⋊C₄	—	142	2	no	—
169	C₂×C₄⋊C₄	—	142	2	no	—
170	C₂×C₄⋊C₄	—	142	2	no	—
171	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
172	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
173	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
174	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
175	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
176	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
177	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
178	C₄⋊C₈	—	2	4	no	—
179	C₄⋊C₈	—	4	2	no	—
180	C₄⋊C₈	—	4	2	no	—
181	C₄⋊C₈	—	71	4	no	—
182	C₄⋊C₈	—	142	2	no	—
183	C₄×C₈	—	284	2	no	—
184	C₂×Q₈	—	284	4	no	—
185	C₂×Q₈	—	284	4	no	—
186	C₂×Q₈	—	284	4	no	—
187	C₂²×C₄	—	142	4	no	—
188	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
189	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
190	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
191	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	284	2	no	—
199	C₂×C₈	—	284	2	no	—
200	C₂×C₈	—	142	4	no	—
201	C₄⋊C₄	—	284	4	no	—
202	C₄⋊C₄	—	284	4	no	—
203	C₄⋊C₄	—	284	4	no	—
204	C₄⋊C₄	—	284	4	no	—
205	C₄⋊C₄	—	284	2	no	—
206	C₄⋊C₄	—	284	2	no	—
207	C₄⋊C₄	—	284	4	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	#11765850	1	2	yes	—
212	C₄²	#11765852	1	2	yes	—
213	C₄²	#11765853	1	2	yes	—
214	C₄²	#11765854	1	2	yes	—
215	C₄²	#11765855	1	2	yes	—
216	C₄²	Dic₂₈₄	1	2	yes	—
217	C₄²	Q₈×D₇₁	1	2	yes	—
218	C₄²	#11765864	1	2	yes	—
219	C₂³	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
220	Q₈	—	568	8	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	284	4	no	—
228	C₂×C₄	—	284	4	no	—
229	C₂×C₄	—	284	4	no	—
230	C₂×C₄	#14556044	1	2	yes	—
231	C₂×C₄	#14556046	1	2	yes	—
232	C₂×C₄	#14556048	1	2	yes	—
233	C₂×C₄	#14556048	1	2	yes	—
234	C₂×C₄	#14556051	1	2	yes	—
235	C₂×C₄	#14556051	1	2	yes	—
236	C₂×C₄	#14556053	1	2	yes	—
237	C₂×C₄	#14556053	1	2	yes	—
238	C₂×C₄	#14556093	1	2	yes	—
239	C₂×C₄	#14556108	1	2	yes	—
240	C₂×C₄	#14556115	1	1	yes	characteristic
241	C₂×C₄	#14556126	1	1	yes	characteristic
242	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
243	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
244	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
245	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
246	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
247	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
248	C₂×C₄	#14556156	1	2	yes	—
249	C₈	—	8	2	no	—
250	C₈	—	284	4	no	—
251	C₂²	#17985904	1	1	yes	characteristic
252	C₂²	C₂×#14556044	1	1	yes	characteristic
253	C₂²	C₂×#14556046	1	1	yes	characteristic
254	C₂²	C₂×#14556048	1	1	yes	characteristic
255	C₂²	C₂×#14556051	1	1	yes	characteristic
256	C₂²	C₂×#14556053	1	1	yes	characteristic
257	C₂²	#17986005	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	568	4	no	—
262	C₄	#17985746	1	4	yes	—
263	C₄	#17985747	1	4	yes	—
264	C₄	#17986290	1	2	yes	—
265	C₄	#17986295	1	2	yes	—
266	C₄	#17986345	1	1	yes	characteristic
267	C₄	#17986349	1	1	yes	characteristic
268	C₄	#17986395	1	1	yes	characteristic
269	C₄	#17986402	1	1	yes	characteristic
270	C₂	C₂×#17985746	1	1	yes	characteristic
271	C₂	C₂×#17985747	1	1	yes	characteristic
272	C₂	#22587009	1	1	yes	characteristic
273	C₂	#22587026	1	1	yes	characteristic
274	C₂	#22589287	1	1	yes	characteristic
275	C₂	#22589300	1	1	yes	characteristic
276	C₂	#22589328	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₇₁)⋊C₄
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1842	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1842	D₇₁
hypocenter	C₇₁	#9896
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
32. C₄⋊(C₇₁⋊C₈)	Q₈	semidirect
143. C₇₁	#9896	semidirect

Maximal subgroups

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

subgroup	index
#22596738	2
C₄×C₄⋊(C₇₁⋊C₈)	2
#22586224	2
C₇₁×#1842	2
#9896	71