Subgroups — #32016351

Order 18176.

638 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11765849	C₂	1	2	yes	maximal
2	#22589116	C₂	1	2	yes	maximal
3	#22589116	C₂	1	4	yes	maximal
4	#22588515	C₂	1	2	yes	maximal
5	#22588512	C₂	1	1	yes	maximal, characteristic
6	#22588246	C₂	1	2	yes	maximal
7	C₄×#14556064	C₂	1	1	yes	maximal, characteristic
8	C₂×#446×C₇₁	C₂	1	1	yes	maximal, characteristic
9	#17986130	—	2	2	no	—
10	#17986130	—	2	4	no	—
11	Q₈×C₇₁⋊C₈	—	2	2	no	—
12	C₂×#14556064	C₂²	1	1	yes	characteristic
13	C₂×#14556064	C₂²	1	2	yes	—
14	C₂×#14556064	C₂²	1	2	yes	—
15	C₂×#14556064	C₂²	1	4	yes	—
16	C₂×#14556064	C₂²	1	1	yes	characteristic
17	#17986023	C₂²	1	2	yes	—
18	#17986023	C₂²	1	4	yes	—
19	C₄×#11765849	C₂²	1	2	yes	—
20	C₄×#11765849	C₂²	1	4	yes	—
21	#17985764	C₂²	1	2	yes	—
22	#17985764	C₂²	1	4	yes	—
23	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
24	#446×C₇₁	C₄	1	4	yes	—
25	#446×C₇₁	C₄	1	4	yes	—
26	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
28	C₂×#97×C₇₁	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
30	C₂×#11765849	C₂³	1	2	yes	—
31	C₂×#11765849	C₂³	1	4	yes	—
32	C₂×#11765849	C₂³	1	2	yes	—
33	#14556064	—	2	2	no	—
34	#14556064	—	2	4	no	—
35	#14556064	—	4	2	no	—
36	#14556064	—	4	4	no	—
37	#14556064	Q₈	1	4	yes	complemented
38	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
39	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
40	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
41	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
42	#14556039	—	2	4	no	—
43	#14556039	—	2	2	no	—
44	C₄×C₇₁⋊C₈	—	2	2	no	—
45	C₄×C₇₁⋊C₈	—	2	4	no	—
46	C₂²×Q₈×C₇₁	Q₈	1	1	yes	characteristic
47	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
49	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
50	C₄×Q₈×C₇₁	C₂×C₄	1	2	yes	—
51	C₄×Q₈×C₇₁	C₂×C₄	1	2	yes	—
52	C₄×Q₈×C₇₁	D₄	1	2	yes	—
53	C₄×Q₈×C₇₁	D₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₇₁	Q₈	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₇₁	Q₈	1	2	yes	—
56	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
58	C₄⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
59	C₄⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
60	C₄⋊C₈×C₇₁	C₂×C₄	1	8	yes	—
61	#97×C₇₁	C₂×C₄	1	8	yes	—
62	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
64	#11765849	—	2	2	no	—
65	#11765849	—	2	2	no	—
66	#11765849	—	2	4	no	—
67	#11765849	—	2	4	no	—
68	#11765849	—	2	2	no	—
69	#11765849	C₂×Q₈	1	4	yes	—
70	C₂×C₇₁⋊C₈	—	2	2	no	—
71	C₂×C₇₁⋊C₈	—	2	2	no	—
72	C₂×C₇₁⋊C₈	—	2	4	no	—
73	C₂×Q₈×C₇₁	—	4	2	no	—
74	C₂×Q₈×C₇₁	C₄⋊C₄	1	2	yes	—
75	C₂×Q₈×C₇₁	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₇₁	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₇₁	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₇₁	—	2	1	no	—
82	C₂×C₈×C₇₁	—	2	2	no	—
83	C₂×C₈×C₇₁	—	2	1	no	—
84	C₂×C₈×C₇₁	—	2	2	no	—
85	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₇₁	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
89	C₂×C₈×C₇₁	#41	1	2	yes	—
90	C₄⋊C₄×C₇₁	C₄⋊C₄	1	4	yes	—
91	C₄⋊C₄×C₇₁	C₄⋊C₄	1	4	yes	—
92	C₄⋊C₄×C₇₁	C₄⋊C₄	1	2	yes	—
93	C₄⋊C₄×C₇₁	C₄⋊C₄	1	2	yes	—
94	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₇₁	C₂²×C₄	1	4	yes	—
97	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₇₁	C₂×D₄	1	4	yes	—
99	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
100	C₂³×C₇₁	C₂²×Q₈	1	1	yes	characteristic
101	Q₈×C₇₁	—	2	4	no	—
102	Q₈×C₇₁	—	4	2	no	—
103	C₂×C₄×C₇₁	—	2	2	no	—
104	C₂×C₄×C₇₁	—	4	1	no	—
105	C₂×C₄×C₇₁	—	4	2	no	—
106	C₂×C₄×C₇₁	—	4	2	no	—
107	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	4	yes	—
110	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	4	yes	—
112	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
119	C₈×C₇₁	—	2	2	no	—
120	C₈×C₇₁	—	2	4	no	—
121	C₈×C₇₁	C₄×Q₈	1	4	yes	—
122	C₇₁⋊C₈	—	2	4	no	—
123	C₇₁⋊C₈	—	4	2	no	—
124	C₇₁⋊C₈	—	4	4	no	—
125	C₂²×C₇₁	—	2	1	no	—
126	C₂²×C₇₁	—	2	1	no	—
127	C₂²×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₇₁	#557	1	1	yes	characteristic
129	C₂²×C₇₁	Q₈²	1	1	yes	characteristic
130	C₄×C₇₁	—	2	1	no	—
131	C₄×C₇₁	—	2	4	no	—
132	C₄×C₇₁	—	2	1	no	—
133	C₄×C₇₁	—	2	1	no	—
134	C₄×C₇₁	—	2	4	no	—
135	C₄×C₇₁	#430	1	2	yes	—
136	C₄×C₇₁	C₂×C₄×Q₈	1	1	yes	characteristic
137	C₄×C₇₁	#520	1	1	yes	characteristic
138	C₄×C₇₁	D₄×Q₈	1	1	yes	characteristic
139	C₄×C₇₁	#550	1	1	yes	characteristic
140	#22733	—	71	1	no	maximal, Sylow
141	C₂×C₇₁	—	2	1	no	—
142	C₂×C₇₁	—	2	1	no	—
143	C₂×C₇₁	Q₈×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₇₁	C₂×#430	1	1	yes	characteristic
145	C₂×C₇₁	#2916	1	1	yes	characteristic
146	Q₈×M₁₆	—	71	2	no	—
147	C₂×#446	D₇₁	1	1	yes	characteristic
148	#1950	—	71	4	no	—
149	#1950	—	71	2	no	—
150	#1849	—	71	2	no	—
151	#1848	—	71	1	no	—
152	#1778	—	71	2	no	—
153	C₄×#108	—	71	1	no	—
154	C₇₁	#22733	1	1	yes	characteristic, complemented, Sylow
155	C₂×C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
156	#446	—	142	2	no	—
157	#446	—	142	4	no	—
158	#446	Dic₇₁	1	4	yes	—
159	#446	Dic₇₁	1	4	yes	—
160	C₈×Q₈	—	142	2	no	—
161	C₂×#108	—	71	2	no	—
162	C₂×#108	—	71	4	no	—
163	C₂×#108	—	71	1	no	—
164	C₂×#108	—	71	2	no	—
165	C₂×#108	—	71	1	no	—
166	#423	—	71	2	no	—
167	#423	—	71	4	no	—
168	C₂×C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
169	C₂×C₄⋊C₈	C₂×D₇₁	1	2	yes	—
170	C₄×M₁₆	—	71	2	no	—
171	C₄×M₁₆	—	71	4	no	—
172	C₂×#97	C₂×D₇₁	1	2	yes	—
173	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
174	#341	—	71	2	no	—
175	#341	—	71	4	no	—
176	C₂²×Q₈	Dic₁₄₂	1	1	yes	characteristic
177	C₂×M₁₆	—	71	2	no	—
178	C₂×M₁₆	—	71	2	no	—
179	C₂×M₁₆	—	71	4	no	—
180	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₇₁	1	2	yes	—
182	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
183	C₄×Q₈	D₂₈₄	1	2	yes	—
184	C₄×Q₈	D₂₈₄	1	2	yes	—
185	C₄×Q₈	C₂×Dic₇₁	1	2	yes	—
186	C₄×Q₈	C₂×Dic₇₁	1	2	yes	—
187	C₂×C₄⋊C₄	Dic₁₄₂	1	1	yes	characteristic
188	C₂×C₄⋊C₄	Dic₁₄₂	1	2	yes	—
189	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
190	C₂×C₄²	C₂²×D₇₁	1	2	yes	—
191	#108	—	284	2	no	—
192	#108	—	284	4	no	—
193	#108	—	71	4	no	—
194	#108	—	142	4	no	—
195	#108	—	142	2	no	—
196	C₄⋊C₈	—	142	2	no	—
197	C₄⋊C₈	—	142	4	no	—
198	C₄⋊C₈	—	142	4	no	—
199	C₄⋊C₈	—	142	2	no	—
200	C₄⋊C₈	C₂×Dic₇₁	1	2	yes	—
201	C₄⋊C₈	C₂×Dic₇₁	1	2	yes	—
202	C₄⋊C₈	C₂×Dic₇₁	1	8	yes	—
203	#97	—	142	4	no	—
204	#97	—	142	2	no	—
205	#97	C₂×Dic₇₁	1	8	yes	—
206	C₄×C₈	—	142	2	no	—
207	C₄×C₈	—	142	4	no	—
208	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
209	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
210	C₂×Q₈	—	4	2	no	—
211	C₂×Q₈	C₄⋊Dic₇₁	1	2	yes	—
212	C₂×Q₈	C₄⋊Dic₇₁	1	2	yes	—
213	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
214	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
215	C₂²×C₄	C₂×Dic₁₄₂	1	2	yes	—
216	C₂²×C₄	C₂×Dic₁₄₂	1	2	yes	—
217	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
218	M₁₆	—	71	4	no	—
219	M₁₆	—	142	4	no	—
220	M₁₆	—	142	2	no	—
221	M₁₆	—	142	4	no	—
222	M₁₆	—	142	2	no	—
223	M₁₆	—	142	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	142	2	no	—
229	C₂×C₈	—	142	4	no	—
230	C₂×C₈	—	142	2	no	—
231	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
232	C₂×C₈	C₂²×Dic₇₁	1	4	yes	—
233	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
234	C₂×C₈	Q₈×D₇₁	1	2	yes	—
235	C₂×C₈	#11765866	1	2	yes	—
236	C₄⋊C₄	C₄⋊Dic₇₁	1	4	yes	—
237	C₄⋊C₄	C₄⋊Dic₇₁	1	4	yes	—
238	C₄⋊C₄	C₄⋊Dic₇₁	1	2	yes	—
239	C₄⋊C₄	C₄⋊Dic₇₁	1	2	yes	—
240	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
241	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
242	C₄²	C₂²×Dic₇₁	1	4	yes	—
243	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
244	C₄²	C₂×D₂₈₄	1	4	yes	—
245	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
246	C₂³	C₂²×Dic₁₄₂	1	1	yes	characteristic
247	Q₈	—	2	4	no	—
248	Q₈	—	4	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
254	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄⋊Dic₇₁	1	4	yes	—
256	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄⋊Dic₇₁	1	4	yes	—
258	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
259	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₂₈₄	1	1	yes	characteristic
261	C₂×C₄	C₂×Q₈×D₇₁	1	1	yes	characteristic
262	C₂×C₄	C₂×#11765866	1	1	yes	characteristic
263	C₂×C₄	#14556174	1	1	yes	characteristic
264	C₂×C₄	#14556175	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	2	4	no	—
267	C₈	—	284	2	no	—
268	C₈	—	284	4	no	—
269	C₈	—	142	4	no	—
270	C₈	Q₈×Dic₇₁	1	4	yes	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	C₂²×C₄⋊Dic₇₁	1	1	yes	characteristic
274	C₂²	#17986728	1	1	yes	characteristic
275	C₂²	Q₈×Dic₁₄₂	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	4	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	4	no	—
281	C₄	#17986048	1	2	yes	—
282	C₄	C₂×Q₈×Dic₇₁	1	1	yes	characteristic
283	C₄	#17986482	1	1	yes	characteristic
284	C₄	Q₈×D₂₈₄	1	1	yes	characteristic
285	C₄	#17986671	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	Q₈×C₄⋊Dic₇₁	1	1	yes	characteristic
289	C₂	C₂×#17986048	1	1	yes	characteristic
290	C₂	#22596951	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₁
Fitting	C₂×#446×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×C₄⋊C₄
prime radical	C₂³×C₇₁	C₂²×Q₈
cosocle	C₂²×C₄×C₇₁	C₂⁴
hypercenter	C₂×#446	D₇₁
hypocenter	C₇₁	#22733
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
37. #14556064	Q₈	semidirect
154. C₇₁	#22733	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11765849	2
#22589116	2
#22588515	2
#22588512	2
#22588246	2
C₄×#14556064	2
C₂×#446×C₇₁	2
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