Subgroups — #31915697

Order 18176.

482 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985806	C₂	1	4	yes	maximal, complemented
2	C₂×#17985775	C₂	1	1	yes	maximal, characteristic, complemented
3	#22588512	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₁×#3147	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14556064	—	2	1	no	—
7	C₂×#14556064	C₂²	1	2	yes	—
8	#17986023	—	2	1	no	—
9	#17985806	—	2	8	no	—
10	#17985775	—	2	1	no	—
11	#17985775	—	2	1	no	—
12	C₂×#135×C₇₁	—	2	1	no	—
13	C₂×#135×C₇₁	C₂²	1	2	yes	—
14	#532×C₇₁	—	2	1	no	—
15	#495×C₇₁	—	2	1	no	—
16	#495×C₇₁	C₄	1	2	yes	—
17	#494×C₇₁	C₄	1	1	yes	characteristic
18	#493×C₇₁	C₄	1	1	yes	characteristic
19	#492×C₇₁	—	2	1	no	—
20	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×#11765849	—	2	2	no	—
22	C₂×C₇₁⋊C₁₆	—	2	2	no	—
23	C₂×C₇₁⋊C₁₆	—	2	2	no	—
24	C₂×C₇₁⋊C₁₆	D₄	1	4	yes	complemented
25	#14556064	—	2	4	no	—
26	#14556064	—	4	2	no	—
27	C₄⋊(C₇₁⋊C₈)	—	4	2	no	—
28	C₂×#41×C₇₁	—	2	2	no	—
29	#135×C₇₁	—	2	4	no	—
30	#135×C₇₁	—	2	4	no	—
31	#135×C₇₁	—	4	2	no	—
32	#135×C₇₁	—	4	2	no	—
33	C₂×Q₁₆×C₇₁	—	2	1	no	—
34	C₂×Q₁₆×C₇₁	C₂×C₄	1	2	yes	—
35	C₂×SD₁₆×C₇₁	—	2	2	no	—
36	C₂×SD₁₆×C₇₁	—	2	2	no	—
37	C₂×D₈×C₇₁	—	2	1	no	—
38	C₂×D₈×C₇₁	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₇₁	—	2	1	no	—
42	#127×C₇₁	—	2	1	no	—
43	#124×C₇₁	—	2	2	no	—
44	#121×C₇₁	—	4	2	no	—
45	C₄×D₄×C₇₁	—	4	2	no	—
46	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
50	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
51	#11765849	—	4	2	no	—
52	#11765849	—	4	2	no	—
53	C₂×C₇₁⋊C₈	—	4	2	no	—
54	C₇₁⋊C₁₆	—	2	4	no	—
55	#41×C₇₁	—	4	2	no	—
56	#41×C₇₁	—	4	2	no	—
57	#41×C₇₁	—	4	2	no	—
58	#41×C₇₁	—	4	2	no	—
59	C₂×Q₈×C₇₁	—	2	2	no	—
60	C₂×D₄×C₇₁	—	2	2	no	—
61	C₂×D₄×C₇₁	—	4	2	no	—
62	C₂×D₄×C₇₁	—	4	2	no	—
63	C₂²×C₄×C₇₁	—	4	2	no	—
64	C₂²×C₄×C₇₁	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
66	Q₁₆×C₇₁	—	2	4	no	—
67	Q₁₆×C₇₁	—	4	2	no	—
68	SD₁₆×C₇₁	—	4	4	no	—
69	SD₁₆×C₇₁	—	4	4	no	—
70	D₈×C₇₁	—	2	4	no	—
71	D₈×C₇₁	—	4	2	no	—
72	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₇₁	D₈	1	2	yes	—
75	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
76	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₇₁	#41	1	2	yes	—
80	C₄⋊C₄×C₇₁	—	4	2	no	—
81	C₂²⋊C₄×C₇₁	—	4	4	no	—
82	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₇₁	D₈	1	2	yes	—
85	C₂³×C₇₁	—	4	2	no	—
86	C₂³×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
87	Q₈×C₇₁	—	4	2	no	—
88	Q₈×C₇₁	—	4	2	no	—
89	D₄×C₇₁	—	4	2	no	—
90	D₄×C₇₁	—	4	2	no	—
91	D₄×C₇₁	—	8	4	no	—
92	D₄×C₇₁	—	8	2	no	—
93	D₄×C₇₁	—	8	2	no	—
94	C₂×C₄×C₇₁	—	4	2	no	—
95	C₂×C₄×C₇₁	—	4	2	no	—
96	C₂×C₄×C₇₁	—	8	2	no	—
97	C₂×C₄×C₇₁	—	8	2	no	—
98	C₂×C₄×C₇₁	#102	1	2	yes	—
99	C₂×C₄×C₇₁	#102	1	2	yes	—
100	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
102	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
103	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
106	C₈×C₇₁	#102	1	2	yes	—
107	C₈×C₇₁	#102	1	2	yes	—
108	C₈×C₇₁	C₄×D₄	1	2	yes	—
109	C₈×C₇₁	#121	1	2	yes	—
110	C₇₁⋊C₈	—	8	2	no	—
111	C₂²×C₇₁	—	2	2	no	—
112	C₂²×C₇₁	—	8	4	no	—
113	C₂²×C₇₁	—	8	2	no	—
114	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
115	C₂²×C₇₁	#486	1	1	yes	characteristic
116	C₂²×C₇₁	#492	1	1	yes	characteristic
117	C₄×C₇₁	—	2	2	no	—
118	C₄×C₇₁	—	2	2	no	—
119	C₄×C₇₁	—	8	2	no	—
120	C₄×C₇₁	#390	1	1	yes	characteristic
121	C₄×C₇₁	C₂×#102	1	1	yes	characteristic
122	C₄×C₇₁	#486	1	1	yes	characteristic
123	C₄×C₇₁	#493	1	1	yes	characteristic
124	#13209	—	71	1	no	maximal, Sylow
125	C₂×C₇₁	—	2	1	no	—
126	C₂×C₇₁	—	2	1	no	—
127	C₂×C₇₁	—	16	2	no	—
128	C₂×C₇₁	#1957	1	1	yes	characteristic
129	C₂×C₇₁	#2143	1	2	yes	—
130	#3147	D₇₁	1	1	yes	characteristic
131	C₂×#359	—	71	4	no	—
132	C₂×#346	—	71	1	no	—
133	#1848	—	71	1	no	—
134	C₇₁	#13209	1	1	yes	characteristic, complemented, Sylow
135	C₂×#135	—	2	1	no	—
136	C₂×#135	C₂×D₇₁	1	2	yes	—
137	#532	—	2	1	no	—
138	C₂²×C₁₆	—	71	2	no	—
139	#495	—	2	1	no	—
140	#495	Dic₇₁	1	2	yes	—
141	#494	Dic₇₁	1	1	yes	characteristic
142	#493	Dic₇₁	1	1	yes	characteristic
143	#492	—	2	1	no	—
144	C₂×#108	—	71	2	no	—
145	C₂×#108	—	142	1	no	—
146	#423	—	142	1	no	—
147	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
148	#359	—	142	8	no	—
149	#346	—	142	1	no	—
150	#346	—	142	1	no	—
151	C₂×#41	—	2	2	no	—
152	#135	—	2	4	no	—
153	#135	—	2	4	no	—
154	#135	—	4	2	no	—
155	#135	—	4	2	no	—
156	C₂×Q₁₆	—	2	1	no	—
157	C₂×Q₁₆	C₂×Dic₇₁	1	2	yes	—
158	C₂×SD₁₆	—	2	2	no	—
159	C₂×SD₁₆	—	2	2	no	—
160	C₂×D₈	—	2	1	no	—
161	C₂×D₈	C₂×Dic₇₁	1	2	yes	—
162	C₂×M₁₆	—	142	2	no	—
163	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
165	C₄⋊Q₈	—	2	1	no	—
166	#127	—	2	1	no	—
167	#124	—	2	2	no	—
168	#121	—	4	2	no	—
169	C₄×D₄	—	4	2	no	—
170	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
171	C₂×C₁₆	—	71	4	no	—
172	C₂×C₁₆	—	142	2	no	—
173	C₂×C₁₆	—	142	2	no	—
174	#108	—	284	2	no	—
175	#108	—	142	4	no	—
176	C₄⋊C₈	—	284	2	no	—
177	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
178	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
179	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
181	#41	—	4	2	no	—
182	#41	—	4	2	no	—
183	#41	—	4	2	no	—
184	#41	—	4	2	no	—
185	C₂×Q₈	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	#11765853	1	2	yes	—
191	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
192	Q₁₆	—	2	4	no	—
193	Q₁₆	—	4	2	no	—
194	SD₁₆	—	4	4	no	—
195	SD₁₆	—	4	4	no	—
196	D₈	—	2	4	no	—
197	D₈	—	4	2	no	—
198	M₁₆	—	284	2	no	—
199	M₁₆	—	284	2	no	—
200	C₂×C₈	—	284	2	no	—
201	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
203	C₂×C₈	#11765850	1	2	yes	—
204	C₂×C₈	#11765853	1	2	yes	—
205	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
206	C₂×C₈	D₄×D₇₁	1	2	yes	—
207	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
208	C₂×C₈	#11765864	1	2	yes	—
209	C₄⋊C₄	—	4	2	no	—
210	C₂²⋊C₄	—	4	4	no	—
211	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
212	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
213	C₄²	#11765850	1	2	yes	—
214	C₁₆	—	142	4	no	—
215	C₂³	—	4	2	no	—
216	C₂³	C₂×#11765853	1	1	yes	characteristic
217	Q₈	—	4	2	no	—
218	Q₈	—	4	2	no	—
219	D₄	—	4	2	no	—
220	D₄	—	4	2	no	—
221	D₄	—	8	4	no	—
222	D₄	—	8	2	no	—
223	D₄	—	8	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	8	2	no	—
227	C₂×C₄	—	8	2	no	—
228	C₂×C₄	#14556049	1	2	yes	—
229	C₂×C₄	#14556049	1	2	yes	—
230	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
231	C₂×C₄	#14556115	1	1	yes	characteristic
232	C₂×C₄	#14556124	1	1	yes	characteristic
233	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
234	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
235	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
236	C₈	—	568	2	no	—
237	C₈	#14556049	1	2	yes	—
238	C₈	#14556049	1	2	yes	—
239	C₈	D₄×Dic₇₁	1	2	yes	—
240	C₈	#14556099	1	2	yes	—
241	C₂²	—	2	2	no	—
242	C₂²	—	8	4	no	—
243	C₂²	—	8	2	no	—
244	C₂²	C₂×#14556049	1	1	yes	characteristic
245	C₂²	#17986345	1	1	yes	characteristic
246	C₂²	#17986371	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	8	2	no	—
250	C₄	#17985902	1	1	yes	characteristic
251	C₄	C₂×#14556049	1	1	yes	characteristic
252	C₄	#17986345	1	1	yes	characteristic
253	C₄	#17986375	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	16	2	no	—
257	C₂	#22589147	1	1	yes	characteristic
258	C₂	#22590068	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11765850
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting, prime radical	C₇₁×#3147	C₂
socle	C₂²×C₇₁	C₂×#102
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#3147	D₇₁
hypocenter	C₇₁	#13209
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#17985806	C₂	semidirect
2. C₂×#17985775	C₂	semidirect
3. #22588512	C₂	semidirect
24. C₂×C₇₁⋊C₁₆	D₄	semidirect
134. C₇₁	#13209	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985806	2
C₂×#17985775	2
#22588512	2
C₇₁×#3147	2
#13209	71