Subgroups — #31896681

Order 18176.

486 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986404	C₂	1	1	yes	maximal, characteristic
2	#22589263	C₂	1	2	yes	maximal
3	C₂×#17985752	C₂	1	1	yes	maximal, characteristic
4	C₂×#17985744	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556128	C₂²	1	2	yes	—
7	#17986404	—	2	2	no	—
8	#17986404	C₄	1	4	yes	complemented
9	C₂×#14556062	—	2	1	no	—
10	C₂×#14556062	—	2	1	no	—
11	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
12	C₂×#14556051	—	2	4	no	—
13	#17985752	—	2	2	no	—
14	#17985744	—	2	4	no	—
15	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₇₁	—	2	1	no	—
17	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
19	#336×C₇₁	—	2	1	no	—
20	C₂²×Dic₁₄₂	—	2	2	no	—
21	C₂²×C₇₁⋊C₈	—	2	2	no	—
22	#14556128	—	2	2	no	—
23	#14556128	—	2	2	no	—
24	#14556128	—	2	2	no	—
25	#14556128	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊Dic₇₁	—	2	2	no	—
27	C₂×C₄⋊Dic₇₁	—	2	2	no	—
28	#14556062	—	2	4	no	—
29	#14556062	—	2	4	no	—
30	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
31	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
32	#14556051	—	4	8	no	—
33	C₂²×C₈×C₇₁	—	2	1	no	—
34	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₇₁	D₄	1	2	yes	—
37	C₂×C₄²×C₇₁	—	2	1	no	—
38	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₇₁	—	2	1	no	—
40	C₄⋊C₈×C₇₁	D₄	1	2	yes	—
41	C₄×C₈×C₇₁	—	4	2	no	—
42	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₇₁	D₄	1	2	yes	—
44	#95×C₇₁	—	2	1	no	—
45	C₂×Dic₁₄₂	—	2	4	no	—
46	C₂×Dic₁₄₂	—	4	4	no	—
47	C₂×Dic₁₄₂	—	4	4	no	—
48	C₂×Dic₁₄₂	—	4	2	no	—
49	C₂²×Dic₇₁	—	4	2	no	—
50	C₂×C₇₁⋊C₈	—	4	2	no	—
51	C₂×C₇₁⋊C₈	—	4	4	no	—
52	C₄⋊Dic₇₁	—	2	4	no	—
53	C₄⋊Dic₇₁	—	2	4	no	—
54	C₄⋊Dic₇₁	—	4	2	no	—
55	C₄⋊Dic₇₁	—	4	2	no	—
56	C₂²×C₄×C₇₁	—	2	2	no	—
57	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
60	C₂×C₈×C₇₁	—	2	1	no	—
61	C₂×C₈×C₇₁	—	2	1	no	—
62	C₂×C₈×C₇₁	—	2	2	no	—
63	C₂×C₈×C₇₁	—	4	1	no	—
64	C₂×C₈×C₇₁	—	4	1	no	—
65	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
68	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
69	C₄⋊C₄×C₇₁	—	2	2	no	—
70	C₄⋊C₄×C₇₁	D₈	1	4	yes	—
71	C₄²×C₇₁	—	4	1	no	—
72	C₄²×C₇₁	—	4	1	no	—
73	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₇₁	#41	1	1	yes	characteristic
77	C₂³×C₇₁	#121	1	1	yes	characteristic
78	C₂×C₄×C₇₁	—	2	2	no	—
79	C₂×C₄×C₇₁	—	2	4	no	—
80	C₂×C₄×C₇₁	—	4	2	no	—
81	C₂×C₄×C₇₁	—	4	2	no	—
82	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
86	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
87	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
88	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₄×C₇₁	#135	1	2	yes	—
92	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
93	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
94	C₂×Dic₇₁	—	4	4	no	—
95	C₂×Dic₇₁	—	8	4	no	—
96	Dic₁₄₂	—	4	8	no	—
97	Dic₁₄₂	—	8	2	no	—
98	Dic₁₄₂	—	8	2	no	—
99	C₈×C₇₁	—	2	2	no	—
100	C₈×C₇₁	—	4	2	no	—
101	C₈×C₇₁	#103	1	4	yes	complemented
102	C₇₁⋊C₈	—	8	2	no	—
103	C₂²×C₇₁	#390	1	1	yes	characteristic
104	C₂²×C₇₁	#461	1	1	yes	characteristic
105	C₂²×C₇₁	#462	1	1	yes	characteristic
106	C₂²×C₇₁	#466	1	2	yes	—
107	C₂²×C₇₁	#473	1	2	yes	—
108	C₄×C₇₁	—	2	1	no	—
109	C₄×C₇₁	—	2	1	no	—
110	C₄×C₇₁	—	4	4	no	—
111	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
112	C₄×C₇₁	#419	1	1	yes	characteristic
113	C₄×C₇₁	C₄×D₈	1	2	yes	—
114	C₄×C₇₁	#461	1	1	yes	characteristic
115	C₄×C₇₁	#462	1	1	yes	characteristic
116	C₄×C₇₁	#493	1	1	yes	characteristic
117	C₄×C₇₁	#495	1	1	yes	characteristic
118	Dic₇₁	—	8	4	no	—
119	#9731	—	71	1	no	maximal, Sylow
120	C₂×C₇₁	#1673	1	2	yes	—
121	C₂×C₇₁	#1674	1	2	yes	—
122	C₂×C₇₁	#1970	1	1	yes	characteristic
123	C₂×C₇₁	#1979	1	1	yes	characteristic
124	C₂×C₇₁	#1981	1	1	yes	characteristic
125	C₂×#500	—	71	1	no	—
126	#1981	—	71	2	no	—
127	C₈×C₄⋊C₄	D₇₁	1	1	yes	characteristic
128	C₂×#335	—	71	1	no	—
129	C₂×#332	—	71	2	no	—
130	C₇₁	#9731	1	1	yes	characteristic, complemented, Sylow
131	C₂×C₄⋊Q₈	—	71	2	no	—
132	#500	—	71	4	no	—
133	#500	—	142	2	no	—
134	C₂×#107	—	142	1	no	—
135	C₂×#107	—	142	1	no	—
136	C₂×C₄⋊C₈	—	71	2	no	—
137	C₂×C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
138	C₂×#103	—	142	4	no	—
139	C₂×C₄×C₈	—	2	1	no	—
140	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
141	C₄×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
142	#336	—	2	1	no	—
143	#335	—	142	2	no	—
144	#332	—	142	4	no	—
145	C₂²×Q₈	—	142	2	no	—
146	C₂²×C₈	—	2	1	no	—
147	C₂²×C₈	—	142	2	no	—
148	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
149	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
150	C₄⋊Q₈	—	71	4	no	—
151	C₄⋊Q₈	—	142	2	no	—
152	C₄⋊Q₈	—	142	2	no	—
153	C₄⋊Q₈	—	142	2	no	—
154	C₂×C₄⋊C₄	—	142	2	no	—
155	C₂×C₄⋊C₄	—	142	2	no	—
156	C₂×C₄⋊C₄	D₂₈₄	1	2	yes	—
157	C₂×C₄²	—	2	1	no	—
158	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
159	#107	—	142	4	no	—
160	#107	—	142	4	no	—
161	C₄⋊C₈	—	2	1	no	—
162	C₄⋊C₈	—	142	2	no	—
163	C₄⋊C₈	—	142	2	no	—
164	C₄⋊C₈	D₂₈₄	1	2	yes	—
165	#103	—	284	8	no	—
166	C₄×C₈	—	4	2	no	—
167	C₄×C₈	C₄×D₇₁	1	2	yes	—
168	C₄×C₈	D₂₈₄	1	2	yes	—
169	#95	—	2	1	no	—
170	C₂×Q₈	—	284	2	no	—
171	C₂×Q₈	—	284	4	no	—
172	C₂×Q₈	—	284	4	no	—
173	C₂×Q₈	—	142	4	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	284	2	no	—
176	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
177	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
178	C₂²×C₄	#11765865	1	1	yes	characteristic
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	4	1	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	—	284	2	no	—
185	C₂×C₈	—	284	4	no	—
186	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
187	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
188	C₂×C₈	#11765853	1	2	yes	—
189	C₂×C₈	#11765855	1	2	yes	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	284	2	no	—
192	C₄⋊C₄	—	284	2	no	—
193	C₄⋊C₄	—	142	4	no	—
194	C₄⋊C₄	—	142	4	no	—
195	C₄⋊C₄	D₅₆₈	1	4	yes	—
196	C₄²	—	4	1	no	—
197	C₄²	—	4	1	no	—
198	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
199	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
200	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
201	C₄²	#11765865	1	1	yes	characteristic
202	C₂³	#14556102	1	1	yes	characteristic
203	Q₈	—	284	8	no	—
204	Q₈	—	568	2	no	—
205	Q₈	—	568	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	284	4	no	—
211	C₂×C₄	—	568	4	no	—
212	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
213	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
214	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
215	C₂×C₄	#14556102	1	1	yes	characteristic
216	C₂×C₄	#14556116	1	1	yes	characteristic
217	C₂×C₄	#14556125	1	1	yes	characteristic
218	C₂×C₄	C₂×D₅₆₈	1	2	yes	—
219	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
220	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
221	C₂×C₄	#14556150	1	2	yes	—
222	C₂×C₄	#14556158	1	1	yes	characteristic
223	C₂×C₄	#14556158	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	4	2	no	—
226	C₈	—	568	2	no	—
227	C₈	#14556051	1	4	yes	complemented
228	C₂²	#17985901	1	1	yes	characteristic
229	C₂²	#17986204	1	1	yes	characteristic
230	C₂²	#17986209	1	1	yes	characteristic
231	C₂²	#17986235	1	2	yes	—
232	C₂²	#17986274	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	4	no	—
236	C₄	—	568	4	no	—
237	C₄	C₂×#14556051	1	1	yes	characteristic
238	C₄	#17986007	1	1	yes	characteristic
239	C₄	C₄×D₅₆₈	1	2	yes	—
240	C₄	#17986204	1	1	yes	characteristic
241	C₄	#17986209	1	1	yes	characteristic
242	C₄	#17986376	1	1	yes	characteristic
243	C₄	#17986383	1	1	yes	characteristic
244	C₂	#22587729	1	2	yes	—
245	C₂	#22587734	1	2	yes	—
246	C₂	#22589206	1	1	yes	characteristic
247	C₂	#22589255	1	1	yes	characteristic
248	C₂	#22589263	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	#9731
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
8. #17986404	C₄	semidirect
101. C₈×C₇₁	#103	semidirect
130. C₇₁	#9731	semidirect
227. C₈	#14556051	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986404	2
#22589263	2
C₂×#17985752	2
C₂×#17985744	2
C₈×C₄⋊C₄×C₇₁	2
#9731	71