Subgroups — #31913601

Order 18176.

396 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556068	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985821	C₂	1	2	yes	maximal, complemented
3	#22588378	C₂	1	1	yes	maximal, characteristic
4	#22586438	C₂	1	2	yes	maximal, complemented
5	C₇₁×#1807	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	—	2	4	no	—
7	C₂×#14556068	C₄	1	2	yes	complemented
8	C₂×#14556068	C₂²	1	1	yes	characteristic
9	C₂×#14556068	C₂²	1	1	yes	characteristic
10	C₂²×C₇₁⋊C₁₆	C₄	1	2	yes	complemented
11	C₂×#14556064	C₂²	1	2	yes	—
12	#17985963	—	2	1	no	—
13	#17985821	—	2	4	no	—
14	#17985764	—	2	1	no	—
15	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×#106×C₇₁	C₂²	1	2	yes	—
17	#380×C₇₁	—	2	1	no	—
18	#340×C₇₁	—	2	1	no	—
19	C₂×#11765849	—	2	2	no	—
20	#14556068	—	2	4	no	—
21	#14556068	—	2	2	no	—
22	#14556068	D₄	1	4	yes	—
23	C₂×C₇₁⋊C₁₆	—	2	4	no	—
24	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
26	#14556064	—	2	2	no	—
27	#14556064	—	2	2	no	—
28	C₂²⋊(C₇₁⋊C₈)	—	4	1	no	—
29	C₂²⋊(C₇₁⋊C₈)	—	4	1	no	—
30	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₁	—	2	2	no	—
32	C₂²×C₈×C₇₁	—	2	2	no	—
33	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
37	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
38	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
40	C₂×C₂²⋊C₄×C₇₁	—	4	1	no	—
41	#106×C₇₁	—	2	2	no	—
42	#106×C₇₁	—	2	2	no	—
43	#95×C₇₁	—	4	1	no	—
44	#11765849	—	4	2	no	—
45	#11765849	—	4	2	no	—
46	C₂×C₇₁⋊C₈	—	4	2	no	—
47	C₇₁⋊C₁₆	—	2	2	no	—
48	C₇₁⋊C₁₆	C₂²⋊C₄	1	4	yes	complemented
49	C₂⁴×C₇₁	SD₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₇₁	—	2	2	no	—
51	C₂²×C₄×C₇₁	—	2	2	no	—
52	C₂²×C₄×C₇₁	—	4	2	no	—
53	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
58	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₇₁	—	2	2	no	—
60	C₂×C₈×C₇₁	—	2	1	no	—
61	C₂×C₈×C₇₁	—	2	1	no	—
62	C₂×C₈×C₇₁	—	2	1	no	—
63	C₂×C₈×C₇₁	—	2	1	no	—
64	C₂×C₈×C₇₁	—	2	2	no	—
65	C₂×C₈×C₇₁	—	2	2	no	—
66	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₈×C₇₁	#41	1	2	yes	—
72	C₄⋊C₄×C₇₁	—	4	2	no	—
73	C₄⋊C₄×C₇₁	—	4	2	no	—
74	C₂²⋊C₄×C₇₁	—	8	2	no	—
75	C₂³×C₇₁	—	2	2	no	—
76	C₂³×C₇₁	—	2	2	no	—
77	C₂³×C₇₁	—	4	1	no	—
78	C₂³×C₇₁	#106	1	1	yes	characteristic
79	C₂³×C₇₁	#106	1	1	yes	characteristic
80	C₂³×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₇₁	—	2	2	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	2	no	—
87	C₂×C₄×C₇₁	—	2	2	no	—
88	C₂×C₄×C₇₁	—	8	2	no	—
89	C₂×C₄×C₇₁	—	8	2	no	—
90	C₂×C₄×C₇₁	—	8	2	no	—
91	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
92	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
93	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₇₁	#121	1	2	yes	—
95	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₇₁	#135	1	2	yes	—
97	C₈×C₇₁	—	2	1	no	—
98	C₈×C₇₁	—	2	1	no	—
99	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₈×C₇₁	#117	1	1	yes	characteristic
101	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
102	C₇₁⋊C₈	—	8	2	no	—
103	C₂²×C₇₁	—	2	1	no	—
104	C₂²×C₇₁	—	2	1	no	—
105	C₂²×C₇₁	—	2	1	no	—
106	C₂²×C₇₁	—	2	1	no	—
107	C₂²×C₇₁	—	2	1	no	—
108	C₂²×C₇₁	—	2	1	no	—
109	C₂²×C₇₁	—	2	1	no	—
110	C₂²×C₇₁	—	4	4	no	—
111	C₂²×C₇₁	#365	1	2	yes	complemented
112	C₂²×C₇₁	C₂×#106	1	1	yes	characteristic
113	C₂²×C₇₁	#465	1	2	yes	—
114	C₄×C₇₁	—	2	1	no	—
115	C₄×C₇₁	—	2	1	no	—
116	C₄×C₇₁	—	16	2	no	—
117	C₄×C₇₁	#380	1	1	yes	characteristic
118	C₄×C₇₁	#427	1	1	yes	characteristic
119	C₄×C₇₁	#465	1	2	yes	—
120	#12831	—	71	1	no	maximal, Sylow
121	C₂×C₇₁	—	2	2	no	—
122	C₂×C₇₁	—	2	2	no	—
123	C₂×C₇₁	—	4	1	no	—
124	C₂×C₇₁	#1807	1	1	yes	characteristic
125	C₂×C₇₁	C₂×#365	1	1	yes	characteristic
126	C₂×C₇₁	#2155	1	1	yes	characteristic
127	C₂²×M₃₂	—	71	1	no	—
128	C₂×#365	—	71	2	no	—
129	#1810	—	71	1	no	—
130	#1807	D₇₁	1	1	yes	characteristic
131	#1383	—	71	2	no	—
132	C₇₁	#12831	1	1	yes	characteristic, complemented, Sylow
133	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
134	C₂×M₃₂	—	71	1	no	—
135	C₂×M₃₂	—	71	2	no	—
136	C₂×M₃₂	—	71	1	no	—
137	C₂×M₃₂	—	142	4	no	—
138	C₂²×C₁₆	—	71	2	no	—
139	C₂×#108	—	71	2	no	—
140	C₂×#106	C₂×D₇₁	1	2	yes	—
141	#407	—	142	1	no	—
142	#380	—	2	1	no	—
143	#365	—	142	4	no	—
144	#341	—	142	1	no	—
145	#340	—	2	1	no	—
146	C₂³×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
147	C₂×M₁₆	—	142	2	no	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
151	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
152	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
153	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
154	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₂²⋊C₄	—	4	1	no	—
158	M₃₂	—	71	4	no	—
159	M₃₂	—	142	4	no	—
160	M₃₂	—	142	2	no	—
161	C₂×C₁₆	—	71	2	no	—
162	C₂×C₁₆	—	71	2	no	—
163	C₂×C₁₆	—	142	4	no	—
164	#108	—	142	2	no	—
165	#108	—	142	2	no	—
166	#106	—	2	2	no	—
167	#106	—	2	2	no	—
168	C₂²⋊C₈	—	284	1	no	—
169	C₂²⋊C₈	—	284	1	no	—
170	#95	—	4	1	no	—
171	C₂⁴	#11765853	1	1	yes	characteristic
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
176	C₂²×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
177	C₂²×C₄	#11765852	1	1	yes	characteristic
178	C₂²×C₄	#11765852	1	1	yes	characteristic
179	C₂²×C₄	#11765853	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
181	M₁₆	—	284	2	no	—
182	M₁₆	—	284	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	284	2	no	—
191	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
192	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
193	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
194	C₂×C₈	D₄×D₇₁	1	2	yes	—
195	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
196	C₂×C₈	#11765864	1	2	yes	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₂²⋊C₄	—	8	2	no	—
200	C₁₆	—	71	4	no	—
201	C₁₆	—	142	2	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	4	1	no	—
205	C₂³	#14556059	1	1	yes	characteristic
206	C₂³	#14556059	1	1	yes	characteristic
207	C₂³	C₂×#11765853	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	8	2	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	#14556059	1	1	yes	characteristic
219	C₂×C₄	#14556059	1	1	yes	characteristic
220	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
221	C₂×C₄	#14556100	1	2	yes	—
222	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
223	C₂×C₄	#14556148	1	2	yes	—
224	C₈	—	2	1	no	—
225	C₈	—	2	1	no	—
226	C₈	—	568	2	no	—
227	C₈	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
228	C₈	#14556084	1	1	yes	characteristic
229	C₈	#14556105	1	2	yes	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	4	4	no	—
238	C₂²	#17985821	1	2	yes	complemented
239	C₂²	C₂×#14556059	1	1	yes	characteristic
240	C₂²	#17986226	1	2	yes	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	16	2	no	—
244	C₄	#17985857	1	1	yes	characteristic
245	C₄	#17986035	1	1	yes	characteristic
246	C₄	#17986230	1	2	yes	—
247	C₂	—	2	2	no	—
248	C₂	—	2	2	no	—
249	C₂	—	4	1	no	—
250	C₂	#22588363	1	1	yes	characteristic
251	C₂	C₂×#17985821	1	1	yes	characteristic
252	C₂	#22590112	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556059
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1807	C₂
socle	C₂²×C₇₁	C₂×#106
prime radical	C₂⁴×C₇₁	SD₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1807	D₇₁
hypocenter	C₇₁	#12831
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1807

Complements

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

N	H ≅ G/N	type
2. C₂×#17985821	C₂	semidirect
4. #22586438	C₂	semidirect
7. C₂×#14556068	C₄	semidirect
10. C₂²×C₇₁⋊C₁₆	C₄	semidirect
48. C₇₁⋊C₁₆	C₂²⋊C₄	semidirect
111. C₂²×C₇₁	#365	semidirect
132. C₇₁	#12831	semidirect
238. C₂²	#17985821	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14556068	2
C₂×#17985821	2
#22588378	2
#22586438	2
C₇₁×#1807	2
#12831	71