Subgroups — #31896105

Order 18176.

382 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22589103	C₂	1	2	yes	maximal
2	#22588500	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985752	C₂	1	1	yes	maximal, characteristic
4	#22586224	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556062	—	2	1	no	—
7	C₂×#14556062	—	2	1	no	—
8	C₂×#14556059	—	2	2	no	—
9	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
10	#17985869	C₂²	1	2	yes	—
11	#17985761	—	2	2	no	—
12	#17985752	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₇₁	—	2	1	no	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
17	#336×C₇₁	—	2	1	no	—
18	C₂²×C₇₁⋊C₈	—	2	2	no	—
19	C₂×C₄⋊Dic₇₁	—	2	2	no	—
20	C₂×C₄⋊Dic₇₁	—	2	2	no	—
21	C₂×C₄⋊Dic₇₁	—	2	2	no	—
22	#14556062	—	4	2	no	—
23	#14556062	—	4	2	no	—
24	#14556059	—	4	4	no	—
25	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
26	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₇₁	—	2	1	no	—
28	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₇₁	—	2	1	no	—
33	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₇₁	—	2	1	no	—
35	C₄⋊C₈×C₇₁	Q₈	1	2	yes	—
36	C₄×C₈×C₇₁	—	4	2	no	—
37	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₇₁	Q₈	1	2	yes	—
39	#95×C₇₁	—	2	1	no	—
40	C₂²×Dic₇₁	—	4	2	no	—
41	C₂×C₇₁⋊C₈	—	2	4	no	—
42	C₂×C₇₁⋊C₈	—	4	2	no	—
43	C₂×C₇₁⋊C₈	—	4	2	no	—
44	C₄⋊Dic₇₁	—	4	2	no	—
45	C₄⋊Dic₇₁	—	4	2	no	—
46	C₄⋊Dic₇₁	—	4	2	no	—
47	C₄⋊Dic₇₁	—	4	2	no	—
48	C₄⋊Dic₇₁	—	8	2	no	—
49	C₂²×C₄×C₇₁	—	2	1	no	—
50	C₂²×C₄×C₇₁	—	2	1	no	—
51	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
54	C₂×C₈×C₇₁	—	2	1	no	—
55	C₂×C₈×C₇₁	—	2	1	no	—
56	C₂×C₈×C₇₁	—	2	2	no	—
57	C₂×C₈×C₇₁	—	4	1	no	—
58	C₂×C₈×C₇₁	—	4	1	no	—
59	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
61	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
62	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
63	C₄⋊C₄×C₇₁	—	2	1	no	—
64	C₄⋊C₄×C₇₁	—	2	1	no	—
65	C₄⋊C₄×C₇₁	D₈	1	2	yes	—
66	C₄⋊C₄×C₇₁	Q₁₆	1	2	yes	—
67	C₄²×C₇₁	—	4	1	no	—
68	C₄²×C₇₁	—	4	1	no	—
69	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
72	C₄²×C₇₁	#41	1	1	yes	characteristic
73	C₂³×C₇₁	#121	1	1	yes	characteristic
74	C₂×C₄×C₇₁	—	2	1	no	—
75	C₂×C₄×C₇₁	—	2	1	no	—
76	C₂×C₄×C₇₁	—	2	2	no	—
77	C₂×C₄×C₇₁	—	2	2	no	—
78	C₂×C₄×C₇₁	—	4	1	no	—
79	C₂×C₄×C₇₁	—	4	1	no	—
80	C₂×C₄×C₇₁	—	4	1	no	—
81	C₂×C₄×C₇₁	—	4	1	no	—
82	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₇₁	#125	1	1	yes	characteristic
87	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
92	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
93	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
94	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
96	C₂×Dic₇₁	—	8	2	no	—
97	C₂×Dic₇₁	—	8	2	no	—
98	C₂×Dic₇₁	—	8	2	no	—
99	C₈×C₇₁	—	2	2	no	—
100	C₈×C₇₁	—	4	2	no	—
101	C₈×C₇₁	#106	1	4	yes	complemented
102	C₇₁⋊C₈	—	4	4	no	—
103	C₂²×C₇₁	#389	1	1	yes	characteristic
104	C₂²×C₇₁	#460	1	1	yes	characteristic
105	C₂²×C₇₁	#461	1	1	yes	characteristic
106	C₂²×C₇₁	#466	1	1	yes	characteristic
107	C₂²×C₇₁	#467	1	1	yes	characteristic
108	C₂²×C₇₁	#472	1	1	yes	characteristic
109	C₂²×C₇₁	#473	1	1	yes	characteristic
110	C₄×C₇₁	—	2	1	no	—
111	C₄×C₇₁	—	2	1	no	—
112	C₄×C₇₁	—	4	2	no	—
113	C₄×C₇₁	—	4	2	no	—
114	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
115	C₄×C₇₁	#428	1	1	yes	characteristic
116	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
117	C₄×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
118	C₄×C₇₁	#475	1	1	yes	characteristic
119	C₄×C₇₁	#476	1	1	yes	characteristic
120	C₄×C₇₁	#499	1	1	yes	characteristic
121	C₄×C₇₁	#500	1	1	yes	characteristic
122	Dic₇₁	—	16	2	no	—
123	#9615	—	71	1	no	maximal, Sylow
124	C₂×C₇₁	#1668	1	1	yes	characteristic
125	C₂×C₇₁	#1670	1	1	yes	characteristic
126	C₂×C₇₁	#1673	1	1	yes	characteristic
127	C₂×C₇₁	#1675	1	1	yes	characteristic
128	C₂×C₇₁	#1942	1	1	yes	characteristic
129	C₂×C₇₁	#1948	1	1	yes	characteristic
130	C₂×C₇₁	#1952	1	1	yes	characteristic
131	#1948	—	71	2	no	—
132	#1845	—	71	1	no	—
133	C₈×C₄⋊C₄	D₇₁	1	1	yes	characteristic
134	C₂×#335	—	71	1	no	—
135	#1296	—	71	2	no	—
136	C₇₁	#9615	1	1	yes	characteristic, complemented, Sylow
137	C₂×#107	—	142	1	no	—
138	C₂×#107	—	142	1	no	—
139	C₂×#106	—	142	2	no	—
140	C₂×C₄⋊C₈	—	71	2	no	—
141	C₂×C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
142	C₂×C₄×C₈	—	2	1	no	—
143	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
144	#384	—	71	2	no	—
145	C₄×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
146	#340	—	142	2	no	—
147	#336	—	2	1	no	—
148	#335	—	71	4	no	—
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	—	142	2	no	—
151	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
152	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
153	C₂×C₄⋊C₄	—	142	2	no	—
154	C₂×C₄⋊C₄	—	142	2	no	—
155	C₂×C₄⋊C₄	—	142	2	no	—
156	C₂×C₄⋊C₄	D₂₈₄	1	1	yes	characteristic
157	C₂×C₄⋊C₄	D₂₈₄	1	1	yes	characteristic
158	C₂×C₄²	—	2	1	no	—
159	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
160	#107	—	284	2	no	—
161	#107	—	284	2	no	—
162	#106	—	284	4	no	—
163	C₄⋊C₈	—	2	1	no	—
164	C₄⋊C₈	—	71	4	no	—
165	C₄⋊C₈	—	142	2	no	—
166	C₄⋊C₈	Dic₁₄₂	1	2	yes	—
167	C₄×C₈	—	4	2	no	—
168	C₄×C₈	Dic₁₄₂	1	2	yes	—
169	C₄×C₈	C₄×D₇₁	1	2	yes	—
170	#95	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	284	2	no	—
174	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
175	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
176	C₂²×C₄	#11765865	1	1	yes	characteristic
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	1	no	—
182	C₂×C₈	—	284	2	no	—
183	C₂×C₈	—	284	2	no	—
184	C₂×C₈	—	142	4	no	—
185	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
186	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
187	C₂×C₈	#11765852	1	2	yes	—
188	C₂×C₈	#11765853	1	2	yes	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	284	2	no	—
192	C₄⋊C₄	—	284	2	no	—
193	C₄⋊C₄	—	284	2	no	—
194	C₄⋊C₄	—	284	2	no	—
195	C₄⋊C₄	—	568	2	no	—
196	C₄⋊C₄	D₅₆₈	1	2	yes	—
197	C₄⋊C₄	Dic₂₈₄	1	2	yes	—
198	C₄²	—	4	1	no	—
199	C₄²	—	4	1	no	—
200	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
201	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
202	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
203	C₄²	#11765865	1	1	yes	characteristic
204	C₂³	#14556102	1	1	yes	characteristic
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	568	2	no	—
214	C₂×C₄	—	568	2	no	—
215	C₂×C₄	—	568	2	no	—
216	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
217	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
218	C₂×C₄	C₄×Dic₁₄₂	1	1	yes	characteristic
219	C₂×C₄	#14556107	1	1	yes	characteristic
220	C₂×C₄	#14556120	1	1	yes	characteristic
221	C₂×C₄	#14556128	1	1	yes	characteristic
222	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
223	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
224	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
225	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
226	C₂×C₄	#14556150	1	1	yes	characteristic
227	C₂×C₄	#14556150	1	1	yes	characteristic
228	C₂×C₄	#14556153	1	1	yes	characteristic
229	C₂×C₄	#14556158	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	4	2	no	—
232	C₈	—	284	4	no	—
233	C₈	#14556059	1	4	yes	complemented
234	C₂²	#17985897	1	1	yes	characteristic
235	C₂²	#17986199	1	1	yes	characteristic
236	C₂²	#17986204	1	1	yes	characteristic
237	C₂²	#17986235	1	1	yes	characteristic
238	C₂²	#17986240	1	1	yes	characteristic
239	C₂²	#17986269	1	1	yes	characteristic
240	C₂²	#17986274	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	1136	2	no	—
246	C₄	C₂×#14556059	1	1	yes	characteristic
247	C₄	#17986042	1	1	yes	characteristic
248	C₄	C₄×D₅₆₈	1	1	yes	characteristic
249	C₄	C₄×Dic₂₈₄	1	1	yes	characteristic
250	C₄	#17986284	1	1	yes	characteristic
251	C₄	#17986289	1	1	yes	characteristic
252	C₄	#17986401	1	1	yes	characteristic
253	C₄	#17986404	1	1	yes	characteristic
254	C₂	#22587700	1	1	yes	characteristic
255	C₂	#22587710	1	1	yes	characteristic
256	C₂	#22587729	1	1	yes	characteristic
257	C₂	#22587739	1	1	yes	characteristic
258	C₂	#22589073	1	1	yes	characteristic
259	C₂	#22589103	1	1	yes	characteristic
260	C₂	#22589125	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556102
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₈×C₄⋊C₄×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	#121
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₈×C₄⋊C₄	D₇₁
hypocenter	C₇₁	#9615
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
12. #17985752	C₄	semidirect
101. C₈×C₇₁	#106	semidirect
136. C₇₁	#9615	semidirect
233. C₈	#14556059	semidirect

Maximal subgroups

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

subgroup	index
#22589103	2
#22588500	2
C₂×#17985752	2
#22586224	2
C₈×C₄⋊C₄×C₇₁	2
#9615	71