Subgroups — #31915709

Order 18176.

470 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985815	C₂	1	4	yes	maximal
2	C₄×#14556068	C₂	1	1	yes	maximal, characteristic
3	#22588512	C₂	1	1	yes	maximal, characteristic
4	C₂×#494×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556068	C₂²	1	2	yes	—
6	C₂×#14556064	—	2	1	no	—
7	C₂×#14556064	C₂²	1	2	yes	—
8	#17986023	—	2	1	no	—
9	#17985815	—	2	8	no	—
10	#17985775	—	2	1	no	—
11	C₄×C₇₁⋊C₁₆	—	2	1	no	—
12	C₂²×Q₁₆×C₇₁	—	2	1	no	—
13	C₂²×Q₁₆×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄⋊Q₈×C₇₁	—	2	1	no	—
15	#494×C₇₁	—	2	1	no	—
16	#494×C₇₁	—	2	1	no	—
17	#494×C₇₁	C₄	1	2	yes	—
18	#494×C₇₁	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×#11765849	—	2	2	no	—
21	#14556068	—	2	2	no	—
22	#14556068	D₄	1	4	yes	—
23	C₂×C₇₁⋊C₁₆	—	2	2	no	—
24	#14556064	—	2	4	no	—
25	#14556064	—	4	2	no	—
26	C₄⋊(C₇₁⋊C₈)	—	4	2	no	—
27	C₂²×Q₈×C₇₁	—	2	2	no	—
28	C₂×Q₁₆×C₇₁	—	2	2	no	—
29	C₂×Q₁₆×C₇₁	—	2	2	no	—
30	C₂×Q₁₆×C₇₁	—	2	4	no	—
31	C₂×Q₁₆×C₇₁	—	2	2	no	—
32	C₂×Q₁₆×C₇₁	—	2	4	no	—
33	C₂×Q₁₆×C₇₁	—	4	2	no	—
34	C₂×Q₁₆×C₇₁	—	4	2	no	—
35	C₂×Q₁₆×C₇₁	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
38	C₄⋊Q₈×C₇₁	—	2	2	no	—
39	C₄⋊Q₈×C₇₁	—	2	2	no	—
40	C₄⋊Q₈×C₇₁	—	4	2	no	—
41	C₂×C₄⋊C₄×C₇₁	—	4	2	no	—
42	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
43	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
47	#11765849	—	4	2	no	—
48	#11765849	—	4	2	no	—
49	C₂×C₇₁⋊C₈	—	4	2	no	—
50	C₇₁⋊C₁₆	—	2	4	no	—
51	C₂×Q₈×C₇₁	—	2	4	no	—
52	C₂×Q₈×C₇₁	—	4	2	no	—
53	C₂×Q₈×C₇₁	—	4	2	no	—
54	C₂×Q₈×C₇₁	—	4	2	no	—
55	C₂×Q₈×C₇₁	—	4	2	no	—
56	C₂×Q₈×C₇₁	—	4	2	no	—
57	C₂²×C₄×C₇₁	—	4	2	no	—
58	C₂²×C₄×C₇₁	SD₁₆	1	2	yes	—
59	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
60	Q₁₆×C₇₁	—	2	8	no	—
61	Q₁₆×C₇₁	—	4	4	no	—
62	Q₁₆×C₇₁	—	4	4	no	—
63	Q₁₆×C₇₁	—	4	4	no	—
64	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₇₁	D₈	1	2	yes	—
67	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
68	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₇₁	#41	1	2	yes	—
72	C₄⋊C₄×C₇₁	—	4	4	no	—
73	C₄⋊C₄×C₇₁	—	4	4	no	—
74	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₇₁	D₈	1	2	yes	—
77	C₂³×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
78	Q₈×C₇₁	—	4	4	no	—
79	Q₈×C₇₁	—	4	4	no	—
80	Q₈×C₇₁	—	8	2	no	—
81	Q₈×C₇₁	—	8	2	no	—
82	C₂×C₄×C₇₁	—	2	4	no	—
83	C₂×C₄×C₇₁	—	4	4	no	—
84	C₂×C₄×C₇₁	—	8	2	no	—
85	C₂×C₄×C₇₁	—	8	2	no	—
86	C₂×C₄×C₇₁	#102	1	2	yes	—
87	C₂×C₄×C₇₁	#102	1	2	yes	—
88	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
90	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
91	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
94	C₈×C₇₁	#102	1	2	yes	—
95	C₈×C₇₁	#102	1	2	yes	—
96	C₈×C₇₁	C₄×D₄	1	2	yes	—
97	C₈×C₇₁	#121	1	2	yes	—
98	C₇₁⋊C₈	—	8	2	no	—
99	C₂²×C₇₁	—	2	2	no	—
100	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
101	C₂²×C₇₁	#486	1	1	yes	characteristic
102	C₂²×C₇₁	#492	1	1	yes	characteristic
103	C₄×C₇₁	—	2	2	no	—
104	C₄×C₇₁	—	8	4	no	—
105	C₄×C₇₁	#362	1	4	yes	complemented
106	C₄×C₇₁	#390	1	1	yes	characteristic
107	C₄×C₇₁	C₂×#102	1	1	yes	characteristic
108	C₄×C₇₁	#486	1	1	yes	characteristic
109	C₄×C₇₁	#493	1	1	yes	characteristic
110	#13212	—	71	1	no	maximal, Sylow
111	C₂×C₇₁	—	2	1	no	—
112	C₂×C₇₁	—	2	1	no	—
113	C₂×C₇₁	#1957	1	1	yes	characteristic
114	C₂×C₇₁	C₂×#362	1	2	yes	—
115	C₂×#494	D₇₁	1	1	yes	characteristic
116	C₂×#362	—	71	4	no	—
117	C₄×M₃₂	—	71	1	no	—
118	#1848	—	71	1	no	—
119	C₇₁	#13212	1	1	yes	characteristic, complemented, Sylow
120	C₂²×Q₁₆	—	2	1	no	—
121	C₂²×Q₁₆	C₂×D₇₁	1	2	yes	—
122	C₂×C₄⋊Q₈	—	2	1	no	—
123	C₂×M₃₂	—	71	2	no	—
124	#494	—	2	1	no	—
125	#494	—	2	1	no	—
126	#494	Dic₇₁	1	2	yes	—
127	#494	Dic₇₁	1	2	yes	—
128	C₂×#108	—	71	2	no	—
129	C₂×#108	—	142	1	no	—
130	#423	—	142	1	no	—
131	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
132	#362	—	142	8	no	—
133	#346	—	142	1	no	—
134	C₄×C₁₆	—	142	1	no	—
135	C₂²×Q₈	—	2	2	no	—
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×Q₁₆	—	2	2	no	—
138	C₂×Q₁₆	—	2	4	no	—
139	C₂×Q₁₆	—	2	2	no	—
140	C₂×Q₁₆	—	2	4	no	—
141	C₂×Q₁₆	—	4	2	no	—
142	C₂×Q₁₆	—	4	2	no	—
143	C₂×Q₁₆	C₂×Dic₇₁	1	4	yes	—
144	C₂×M₁₆	—	142	2	no	—
145	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
146	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	4	2	no	—
150	C₂×C₄⋊C₄	—	4	2	no	—
151	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
152	M₃₂	—	71	4	no	—
153	M₃₂	—	142	2	no	—
154	C₂×C₁₆	—	142	2	no	—
155	#108	—	284	2	no	—
156	#108	—	142	4	no	—
157	C₄⋊C₈	—	284	2	no	—
158	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
159	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
162	C₂×Q₈	—	2	4	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	4	2	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂×Q₈	—	4	2	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	#11765853	1	2	yes	—
170	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
171	Q₁₆	—	2	8	no	—
172	Q₁₆	—	4	4	no	—
173	Q₁₆	—	4	4	no	—
174	Q₁₆	—	4	4	no	—
175	M₁₆	—	284	2	no	—
176	M₁₆	—	284	2	no	—
177	C₂×C₈	—	284	2	no	—
178	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
179	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
180	C₂×C₈	#11765850	1	2	yes	—
181	C₂×C₈	#11765853	1	2	yes	—
182	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
183	C₂×C₈	D₄×D₇₁	1	2	yes	—
184	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
185	C₂×C₈	#11765864	1	2	yes	—
186	C₄⋊C₄	—	4	4	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
189	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
190	C₄²	#11765850	1	2	yes	—
191	C₁₆	—	142	4	no	—
192	C₂³	C₂×#11765853	1	1	yes	characteristic
193	Q₈	—	4	4	no	—
194	Q₈	—	4	4	no	—
195	Q₈	—	8	2	no	—
196	Q₈	—	8	2	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	8	2	no	—
200	C₂×C₄	—	8	2	no	—
201	C₂×C₄	#14556049	1	2	yes	—
202	C₂×C₄	#14556049	1	2	yes	—
203	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
204	C₂×C₄	#14556115	1	1	yes	characteristic
205	C₂×C₄	#14556124	1	1	yes	characteristic
206	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
207	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
208	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
209	C₈	—	568	2	no	—
210	C₈	#14556049	1	2	yes	—
211	C₈	#14556049	1	2	yes	—
212	C₈	D₄×Dic₇₁	1	2	yes	—
213	C₈	#14556099	1	2	yes	—
214	C₂²	—	2	2	no	—
215	C₂²	C₂×#14556049	1	1	yes	characteristic
216	C₂²	#17986345	1	1	yes	characteristic
217	C₂²	#17986371	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	8	4	no	—
220	C₄	#17985815	1	4	yes	complemented
221	C₄	#17985902	1	1	yes	characteristic
222	C₄	C₂×#14556049	1	1	yes	characteristic
223	C₄	#17986345	1	1	yes	characteristic
224	C₄	#17986375	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	2	1	no	—
227	C₂	#22589147	1	1	yes	characteristic
228	C₂	C₂×#17985815	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14556049
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#494×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#102
prime radical	C₂³×C₇₁	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#494	D₇₁
hypocenter	C₇₁	#13212
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
105. C₄×C₇₁	#362	semidirect
119. C₇₁	#13212	semidirect
220. C₄	#17985815	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985815	2
C₄×#14556068	2
#22588512	2
C₂×#494×C₇₁	2
#13212	71