Subgroups — #32016349

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	M₁₆×Dic₁₄₂	C₂	1	2	yes	maximal
2	C₂×#17986129	C₂	1	1	yes	maximal, characteristic
3	#22589115	C₂	1	2	yes	maximal
4	#22589114	C₂	1	4	yes	maximal
5	#22588513	C₂	1	2	yes	maximal
6	#22588512	C₂	1	1	yes	maximal, characteristic
7	#22588248	C₂	1	2	yes	maximal
8	C₄×#108×C₇₁	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
10	#17986129	—	2	2	no	—
11	#17986129	C₄	1	4	yes	—
12	#17986129	C₄	1	4	yes	—
13	#17986127	—	2	4	no	—
14	C₈×Dic₁₄₂	—	2	2	no	—
15	C₂×#14556064	C₂²	1	2	yes	—
16	C₂×#14556064	C₂²	1	2	yes	—
17	C₂×#14556063	C₂²	1	4	yes	—
18	#17986023	C₂²	1	2	yes	—
19	#17986022	C₂²	1	4	yes	—
20	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
21	C₂×Dic₇₁⋊C₈	C₂²	1	2	yes	—
22	M₁₆×Dic₇₁	C₂²	1	4	yes	—
23	C₂×#14556038	C₂²	1	2	yes	—
24	#17985763	C₂²	1	4	yes	—
25	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
26	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₇₁	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
29	#341×C₇₁	C₂²	1	2	yes	—
30	C₂²×Dic₁₄₂	Q₈	1	1	yes	characteristic
31	C₂×#11765849	C₂³	1	4	yes	—
32	C₂²×C₇₁⋊C₈	C₂³	1	2	yes	—
33	C₄×Dic₁₄₂	C₂×C₄	1	2	yes	—
34	C₄×Dic₁₄₂	C₂×C₄	1	2	yes	—
35	C₄×Dic₁₄₂	D₄	1	2	yes	—
36	C₄×Dic₁₄₂	D₄	1	2	yes	—
37	C₂×C₄⋊Dic₇₁	Q₈	1	1	yes	characteristic
38	C₂×Dic₇₁⋊C₄	Q₈	1	2	yes	—
39	C₂×C₄×Dic₇₁	C₂³	1	2	yes	—
40	#14556064	—	2	4	no	—
41	#14556064	—	4	2	no	—
42	#14556063	—	4	4	no	—
43	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
44	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
45	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
46	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
47	Dic₇₁⋊C₈	—	2	4	no	—
48	Dic₇₁⋊C₈	—	2	4	no	—
49	Dic₇₁⋊C₈	C₂×C₄	1	8	yes	—
50	#14556038	—	2	4	no	—
51	#14556038	C₂×C₄	1	8	yes	—
52	C₈×Dic₇₁	—	2	4	no	—
53	C₂×M₁₆×C₇₁	C₂³	1	2	yes	—
54	C₂×M₁₆×C₇₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
58	#108×C₇₁	—	2	2	no	—
59	#108×C₇₁	Q₈	1	4	yes	complemented
60	#97×C₇₁	—	2	2	no	—
61	C₄×C₈×C₇₁	—	2	2	no	—
62	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
64	C₂×Dic₁₄₂	—	4	2	no	—
65	C₂×Dic₁₄₂	C₄⋊C₄	1	2	yes	—
66	C₂×Dic₁₄₂	C₄⋊C₄	1	2	yes	—
67	C₂²×Dic₇₁	C₂×Q₈	1	2	yes	—
68	C₂²×Dic₇₁	C₂×Q₈	1	2	yes	—
69	#11765849	—	2	4	no	—
70	#11765849	—	2	4	no	—
71	C₂×C₇₁⋊C₈	—	2	2	no	—
72	C₂×C₇₁⋊C₈	—	2	2	no	—
73	C₂×C₇₁⋊C₈	—	2	4	no	—
74	C₂×C₇₁⋊C₈	C₂²×C₄	1	4	yes	—
75	C₄⋊Dic₇₁	C₄⋊C₄	1	2	yes	—
76	C₄⋊Dic₇₁	C₄⋊C₄	1	2	yes	—
77	Dic₇₁⋊C₄	C₄⋊C₄	1	4	yes	—
78	Dic₇₁⋊C₄	C₄⋊C₄	1	4	yes	—
79	C₄×Dic₇₁	C₂²×C₄	1	4	yes	—
80	C₄×Dic₇₁	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
84	M₁₆×C₇₁	—	2	2	no	—
85	M₁₆×C₇₁	—	2	2	no	—
86	M₁₆×C₇₁	—	2	2	no	—
87	M₁₆×C₇₁	C₂×Q₈	1	4	yes	—
88	C₂×C₈×C₇₁	—	2	2	no	—
89	C₂×C₈×C₇₁	—	2	1	no	—
90	C₂×C₈×C₇₁	—	2	1	no	—
91	C₂×C₈×C₇₁	—	2	2	no	—
92	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
95	C₂×C₈×C₇₁	#41	1	2	yes	—
96	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
100	C₂³×C₇₁	C₂²×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₇₁	—	2	2	no	—
102	C₂×C₄×C₇₁	—	4	1	no	—
103	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
111	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
112	C₂×C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
113	C₂×Dic₇₁	—	4	2	no	—
114	C₂×Dic₇₁	—	4	2	no	—
115	C₂×Dic₇₁	C₂×C₄⋊C₄	1	4	yes	—
116	C₂×Dic₇₁	C₂×C₄⋊C₄	1	4	yes	—
117	Dic₁₄₂	—	2	4	no	—
118	Dic₁₄₂	—	4	2	no	—
119	C₈×C₇₁	—	2	2	no	—
120	C₈×C₇₁	—	2	4	no	—
121	C₈×C₇₁	—	4	2	no	—
122	C₈×C₇₁	C₄×Q₈	1	4	yes	—
123	C₇₁⋊C₈	—	2	4	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	1	no	—
132	C₄×C₇₁	—	2	1	no	—
133	C₄×C₇₁	#430	1	2	yes	—
134	C₄×C₇₁	C₂×C₄×Q₈	1	1	yes	characteristic
135	C₄×C₇₁	#520	1	1	yes	characteristic
136	C₄×C₇₁	D₄×Q₈	1	1	yes	characteristic
137	C₄×C₇₁	#550	1	1	yes	characteristic
138	Dic₇₁	—	2	4	no	—
139	Dic₇₁	—	2	4	no	—
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	—	71	1	no	—
148	#1950	—	71	2	no	—
149	#1950	—	71	4	no	—
150	#1849	—	71	2	no	—
151	#1848	—	71	1	no	—
152	#1778	—	71	2	no	—
153	C₄×#108	D₇₁	1	1	yes	characteristic
154	C₇₁	#22733	1	1	yes	characteristic, complemented, Sylow
155	C₂×C₄×Q₈	—	71	1	no	—
156	#446	—	71	4	no	—
157	#446	—	71	4	no	—
158	#446	—	142	2	no	—
159	#446	—	142	4	no	—
160	C₈×Q₈	—	142	2	no	—
161	C₂×#108	—	71	4	no	—
162	C₂×#108	—	71	2	no	—
163	C₂×#108	—	71	2	no	—
164	C₂×#108	C₂×D₇₁	1	1	yes	characteristic
165	C₂×#108	C₂×D₇₁	1	1	yes	characteristic
166	#423	—	71	4	no	—
167	#423	—	71	2	no	—
168	C₂×C₄⋊C₈	—	71	2	no	—
169	C₂×C₄⋊C₈	—	71	1	no	—
170	C₄×M₁₆	—	71	4	no	—
171	C₄×M₁₆	C₂×D₇₁	1	2	yes	—
172	C₂×#97	—	71	2	no	—
173	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
174	#341	—	71	4	no	—
175	#341	C₂×D₇₁	1	2	yes	—
176	C₂²×Q₈	—	71	1	no	—
177	C₂×M₁₆	—	71	4	no	—
178	C₂×M₁₆	C₂²×D₇₁	1	2	yes	—
179	C₂×M₁₆	C₂²×D₇₁	1	2	yes	—
180	C₂²×C₈	—	71	2	no	—
181	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
183	C₄×Q₈	—	71	2	no	—
184	C₄×Q₈	—	71	2	no	—
185	C₄×Q₈	—	71	2	no	—
186	C₄×Q₈	—	71	2	no	—
187	C₂×C₄⋊C₄	—	71	2	no	—
188	C₂×C₄⋊C₄	—	71	1	no	—
189	C₂×C₄²	—	71	2	no	—
190	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
191	#108	—	2	2	no	—
192	#108	—	284	2	no	—
193	#108	—	284	4	no	—
194	#108	—	142	4	no	—
195	#108	Dic₁₄₂	1	4	yes	complemented
196	C₄⋊C₈	—	71	8	no	—
197	C₄⋊C₈	—	71	2	no	—
198	C₄⋊C₈	—	71	2	no	—
199	C₄⋊C₈	—	142	4	no	—
200	C₄⋊C₈	—	142	4	no	—
201	C₄⋊C₈	—	142	2	no	—
202	C₄⋊C₈	—	142	2	no	—
203	#97	—	2	2	no	—
204	#97	—	71	8	no	—
205	#97	—	142	4	no	—
206	C₄×C₈	—	2	2	no	—
207	C₄×C₈	—	142	4	no	—
208	C₄×C₈	C₄×D₇₁	1	2	yes	—
209	C₄×C₈	C₄×D₇₁	1	2	yes	—
210	C₂×Q₈	—	284	2	no	—
211	C₂×Q₈	—	71	2	no	—
212	C₂×Q₈	—	71	2	no	—
213	C₂²×C₄	—	71	2	no	—
214	C₂²×C₄	—	71	2	no	—
215	C₂²×C₄	Q₈×D₇₁	1	1	yes	characteristic
216	C₂²×C₄	Q₈×D₇₁	1	1	yes	characteristic
217	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	142	4	no	—
222	M₁₆	—	142	4	no	—
223	M₁₆	C₂×Dic₁₄₂	1	4	yes	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	71	4	no	—
229	C₂×C₈	—	142	4	no	—
230	C₂×C₈	—	142	2	no	—
231	C₂×C₈	—	142	2	no	—
232	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
233	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
234	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
235	C₂×C₈	#11765865	1	2	yes	—
236	C₄⋊C₄	—	71	4	no	—
237	C₄⋊C₄	—	71	4	no	—
238	C₄⋊C₄	—	71	2	no	—
239	C₄⋊C₄	—	71	2	no	—
240	C₄²	—	71	4	no	—
241	C₄²	—	71	4	no	—
242	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
243	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
244	C₄²	D₄×D₇₁	1	1	yes	characteristic
245	C₄²	D₄×D₇₁	1	1	yes	characteristic
246	C₂³	C₂×Q₈×D₇₁	1	1	yes	characteristic
247	Q₈	—	284	2	no	—
248	Q₈	—	142	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	284	2	no	—
252	C₂×C₄	—	284	2	no	—
253	C₂×C₄	—	71	4	no	—
254	C₂×C₄	—	71	4	no	—
255	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
256	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
257	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
258	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
260	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
261	C₂×C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
262	C₂×C₄	C₂×#11765865	1	1	yes	characteristic
263	C₂×C₄	#14556174	1	1	yes	characteristic
264	C₂×C₄	#14556175	1	1	yes	characteristic
265	C₈	—	2	4	no	—
266	C₈	—	2	2	no	—
267	C₈	—	4	2	no	—
268	C₈	—	284	4	no	—
269	C₈	—	142	4	no	—
270	C₈	C₄×Dic₁₄₂	1	4	yes	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	C₂×C₄⋊C₄×D₇₁	1	1	yes	characteristic
274	C₂²	#17986727	1	1	yes	characteristic
275	C₂²	Q₈×Dic₁₄₂	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	142	4	no	—
280	C₄	—	142	4	no	—
281	C₄	#17986046	1	2	yes	—
282	C₄	C₂×C₄×Dic₁₄₂	1	1	yes	characteristic
283	C₄	#17986479	1	1	yes	characteristic
284	C₄	D₄×Dic₁₄₂	1	1	yes	characteristic
285	C₄	#17986670	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	C₄⋊C₄×Dic₁₄₂	1	1	yes	characteristic
289	C₂	C₂×#17986046	1	1	yes	characteristic
290	C₂	#22596948	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₁
Fitting	C₄×#108×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×C₄⋊C₄
prime radical	C₂³×C₇₁	C₂²×Q₈
cosocle	C₂²×C₄×C₇₁	C₂⁴
hypercenter	C₄×#108	D₇₁
hypocenter	C₇₁	#22733
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
59. #108×C₇₁	Q₈	semidirect
154. C₇₁	#22733	semidirect
195. #108	Dic₁₄₂	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₁₄₂	2
C₂×#17986129	2
#22589115	2
#22589114	2
#22588513	2
#22588512	2
#22588248	2
C₄×#108×C₇₁	2
#22733	71