Subgroups — #31885521

Order 18176.

478 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	#22588457	C₂	1	1	yes	maximal, characteristic
2	#22588086	C₂	1	1	yes	maximal, characteristic
3	#22586166	C₂	1	4	yes	maximal
4	C₄×C₄⋊Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
6	C₂×C₄×C₇₁⋊C₈	—	2	1	no	—
7	#17985754	—	2	1	no	—
8	#17985754	—	2	1	no	—
9	#17985754	C₂²	1	2	yes	—
10	#17985754	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₇₁	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₇₁	C₄	1	2	yes	—
13	#391×C₇₁	C₂²	1	2	yes	—
14	#389×C₇₁	—	2	1	no	—
15	#389×C₇₁	—	2	1	no	—
16	#384×C₇₁	C₄	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₇₁	—	2	1	no	—
18	C₄×C₄⋊C₄×C₇₁	—	2	1	no	—
19	C₄³×C₇₁	C₂²	1	1	yes	characteristic
20	C₂²×C₇₁⋊C₈	—	2	2	no	—
21	C₂²×C₇₁⋊C₈	—	2	2	no	—
22	C₄⋊(C₇₁⋊C₈)	—	4	2	no	—
23	C₄×C₇₁⋊C₈	—	2	4	no	—
24	C₂²×Q₈×C₇₁	C₂×C₄	1	2	yes	—
25	C₄⋊Q₈×C₇₁	—	2	2	no	—
26	C₄⋊Q₈×C₇₁	C₈	1	4	yes	complemented
27	C₄×Q₈×C₇₁	—	4	4	no	—
28	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
29	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
30	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
32	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
33	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
34	C₂×C₄²×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	#95×C₇₁	—	2	2	no	—
43	#95×C₇₁	—	2	2	no	—
44	C₂×C₇₁⋊C₈	—	2	4	no	—
45	C₂×C₇₁⋊C₈	—	4	2	no	—
46	C₂×C₇₁⋊C₈	—	4	2	no	—
47	C₂×C₇₁⋊C₈	—	4	2	no	—
48	C₂×C₇₁⋊C₈	—	4	2	no	—
49	C₂×C₇₁⋊C₈	—	4	2	no	—
50	C₂×Q₈×C₇₁	—	2	2	no	—
51	C₂×Q₈×C₇₁	—	4	2	no	—
52	C₂×Q₈×C₇₁	—	4	2	no	—
53	C₂×Q₈×C₇₁	C₂×C₈	1	4	yes	—
54	C₂²×C₄×C₇₁	—	2	2	no	—
55	C₂²×C₄×C₇₁	—	2	2	no	—
56	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₇₁	—	2	4	no	—
64	C₄⋊C₄×C₇₁	—	2	4	no	—
65	C₄⋊C₄×C₇₁	—	4	2	no	—
66	C₄⋊C₄×C₇₁	—	4	2	no	—
67	C₄⋊C₄×C₇₁	—	4	4	no	—
68	C₄⋊C₄×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	—	4	1	no	—
73	C₄²×C₇₁	—	4	1	no	—
74	C₄²×C₇₁	—	4	4	no	—
75	C₄²×C₇₁	C₂×C₈	1	2	yes	—
76	C₄²×C₇₁	M₁₆	1	2	yes	—
77	C₄²×C₇₁	SD₁₆	1	2	yes	—
78	C₄²×C₇₁	Q₁₆	1	2	yes	—
79	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	Q₈×C₇₁	—	4	4	no	—
81	Q₈×C₇₁	—	4	4	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	—	2	1	no	—
88	C₂×C₄×C₇₁	—	2	1	no	—
89	C₂×C₄×C₇₁	—	2	1	no	—
90	C₂×C₄×C₇₁	—	2	1	no	—
91	C₂×C₄×C₇₁	—	2	1	no	—
92	C₂×C₄×C₇₁	—	2	1	no	—
93	C₂×C₄×C₇₁	—	2	1	no	—
94	C₂×C₄×C₇₁	—	2	4	no	—
95	C₂×C₄×C₇₁	—	4	2	no	—
96	C₂×C₄×C₇₁	—	4	2	no	—
97	C₂×C₄×C₇₁	—	4	2	no	—
98	C₂×C₄×C₇₁	—	4	2	no	—
99	C₂×C₄×C₇₁	—	4	2	no	—
100	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
101	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
102	C₂×C₄×C₇₁	#100	1	2	yes	—
103	C₂×C₄×C₇₁	#103	1	2	yes	—
104	C₂×C₄×C₇₁	#103	1	2	yes	—
105	C₂×C₄×C₇₁	#104	1	2	yes	—
106	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
110	C₂×C₄×C₇₁	#137	1	2	yes	—
111	C₇₁⋊C₈	—	4	4	no	—
112	C₇₁⋊C₈	—	8	2	no	—
113	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
114	C₂²×C₇₁	#410	1	1	yes	characteristic
115	C₂²×C₇₁	C₂×#100	1	1	yes	characteristic
116	C₂²×C₇₁	C₂×#103	1	1	yes	characteristic
117	C₂²×C₇₁	#419	1	1	yes	characteristic
118	C₂²×C₇₁	C₂×#104	1	1	yes	characteristic
119	C₂²×C₇₁	#421	1	1	yes	characteristic
120	C₄×C₇₁	—	2	2	no	—
121	C₄×C₇₁	—	2	1	no	—
122	C₄×C₇₁	—	2	1	no	—
123	C₄×C₇₁	—	2	2	no	—
124	C₄×C₇₁	—	2	2	no	—
125	C₄×C₇₁	—	2	1	no	—
126	C₄×C₇₁	—	2	1	no	—
127	C₄×C₇₁	—	2	2	no	—
128	C₄×C₇₁	—	4	4	no	—
129	C₄×C₇₁	—	8	2	no	—
130	C₄×C₇₁	Q₈⋊C₈	1	4	yes	—
131	#7590	—	71	1	no	maximal, Sylow
132	C₂×C₇₁	#1460	1	1	yes	characteristic
133	C₂×C₇₁	C₂×Q₈⋊C₈	1	1	yes	characteristic
134	C₂×C₇₁	#1479	1	1	yes	characteristic
135	C₂×C₇₁	#1506	1	1	yes	characteristic
136	C₂×C₇₁	#1518	1	1	yes	characteristic
137	C₂×C₇₁	#1530	1	1	yes	characteristic
138	C₂×C₇₁	#1548	1	1	yes	characteristic
139	C₄×C₄⋊Q₈	D₇₁	1	1	yes	characteristic
140	#1831	—	71	1	no	—
141	#1744	—	71	1	no	—
142	#1272	—	71	4	no	—
143	C₇₁	#7590	1	1	yes	characteristic, complemented, Sylow
144	C₂×C₄⋊Q₈	Dic₇₁	1	1	yes	characteristic
145	C₂×C₄×Q₈	Dic₇₁	1	2	yes	—
146	C₂×C₄⋊C₈	—	142	1	no	—
147	C₂×C₄×C₈	—	142	1	no	—
148	#391	C₂×D₇₁	1	2	yes	—
149	#389	—	2	1	no	—
150	#389	—	2	1	no	—
151	#384	Dic₇₁	1	1	yes	characteristic
152	C₄×C₄⋊C₄	—	2	1	no	—
153	C₄×C₄⋊C₄	—	2	1	no	—
154	C₄³	C₂×D₇₁	1	1	yes	characteristic
155	#336	—	71	2	no	—
156	#336	—	71	2	no	—
157	#336	—	142	1	no	—
158	#336	—	142	1	no	—
159	C₂²×Q₈	C₂×Dic₇₁	1	2	yes	—
160	C₂²×C₈	—	142	2	no	—
161	C₂²×C₈	—	142	2	no	—
162	C₄⋊Q₈	—	2	2	no	—
163	C₄⋊Q₈	C₇₁⋊C₈	1	4	yes	complemented
164	C₄×Q₈	—	4	4	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	C₂×Dic₇₁	1	2	yes	—
171	C₂×C₄²	—	2	2	no	—
172	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
173	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
174	C₂×C₄²	C₂²⋊D₇₁	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	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
179	C₄⋊C₈	—	284	2	no	—
180	C₄×C₈	—	142	4	no	—
181	#95	—	2	2	no	—
182	#95	—	2	2	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	C₂×C₇₁⋊C₈	1	4	yes	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
190	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
192	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
193	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
194	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
196	C₂×C₈	—	284	2	no	—
197	C₂×C₈	—	284	2	no	—
198	C₂×C₈	—	284	2	no	—
199	C₂×C₈	—	284	2	no	—
200	C₂×C₈	—	284	2	no	—
201	C₂×C₈	—	142	4	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	4	1	no	—
212	C₄²	—	4	1	no	—
213	C₄²	—	4	4	no	—
214	C₄²	C₂×C₇₁⋊C₈	1	2	yes	—
215	C₄²	#11765849	1	2	yes	—
216	C₄²	#11765852	1	2	yes	—
217	C₄²	#11765855	1	2	yes	—
218	C₂³	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
219	Q₈	—	4	4	no	—
220	Q₈	—	4	4	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	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
240	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
241	C₂×C₄	#14556045	1	2	yes	—
242	C₂×C₄	#14556052	1	2	yes	—
243	C₂×C₄	#14556052	1	2	yes	—
244	C₂×C₄	#14556055	1	2	yes	—
245	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
246	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
247	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
248	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
249	C₂×C₄	#14556159	1	2	yes	—
250	C₈	—	284	4	no	—
251	C₈	—	568	2	no	—
252	C₂²	C₂×C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
253	C₂²	#17985973	1	1	yes	characteristic
254	C₂²	C₂×#14556045	1	1	yes	characteristic
255	C₂²	C₂×#14556052	1	1	yes	characteristic
256	C₂²	#17986008	1	1	yes	characteristic
257	C₂²	C₂×#14556055	1	1	yes	characteristic
258	C₂²	#17986016	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	4	no	—
268	C₄	—	8	2	no	—
269	C₄	Q₈⋊(C₇₁⋊C₈)	1	4	yes	—
270	C₂	#22586661	1	1	yes	characteristic
271	C₂	C₂×Q₈⋊(C₇₁⋊C₈)	1	1	yes	characteristic
272	C₂	#22586735	1	1	yes	characteristic
273	C₂	#22586852	1	1	yes	characteristic
274	C₂	#22586904	1	1	yes	characteristic
275	C₂	#22586951	1	1	yes	characteristic
276	C₂	#22587021	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₄×C₄⋊Q₈×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₄×C₄⋊Q₈	D₇₁
hypocenter	C₇₁	#7590
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
26. C₄⋊Q₈×C₇₁	C₈	semidirect
143. C₇₁	#7590	semidirect
163. C₄⋊Q₈	C₇₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22588457	2
#22588086	2
#22586166	2
C₄×C₄⋊Q₈×C₇₁	2
#7590	71