Subgroups — #31888647

Order 18176.

438 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588195	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985752	C₂	1	1	yes	maximal, characteristic
4	#22586174	C₂	1	2	yes	maximal
5	C₂×#335×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
7	C₂×#14556038	—	2	2	no	—
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₈×Dic₇₁	C₄	1	2	yes	—
10	C₂×C₈×Dic₇₁	C₄	1	2	yes	—
11	C₄²×Dic₇₁	C₂²	1	1	yes	characteristic
12	#17985753	—	2	2	no	—
13	#17985752	C₄	1	4	yes	complemented
14	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	#335×C₇₁	C₄	1	4	yes	complemented
17	C₂²×C₇₁⋊C₈	—	2	2	no	—
18	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₇₁	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
21	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
22	C₂×C₄×Dic₇₁	Q₈	1	2	yes	—
23	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
25	#14556038	—	4	4	no	—
26	C₄×C₇₁⋊C₈	—	2	1	no	—
27	C₄×C₇₁⋊C₈	—	2	1	no	—
28	C₈×Dic₇₁	—	2	4	no	—
29	C₈×Dic₇₁	—	2	4	no	—
30	C₂²×C₈×C₇₁	—	2	2	no	—
31	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₇₁	—	2	2	no	—
35	C₄⋊C₈×C₇₁	C₂×C₄	1	4	yes	—
36	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
38	C₂²×Dic₇₁	C₄⋊C₄	1	2	yes	—
39	C₂²×Dic₇₁	C₄⋊C₄	1	2	yes	—
40	C₂×C₇₁⋊C₈	—	2	2	no	—
41	C₂×C₇₁⋊C₈	—	2	2	no	—
42	C₂×C₇₁⋊C₈	—	2	4	no	—
43	C₂×C₇₁⋊C₈	—	4	2	no	—
44	C₂×C₇₁⋊C₈	—	4	2	no	—
45	C₂×C₇₁⋊C₈	M₁₆	1	4	yes	—
46	C₄×Dic₇₁	—	2	2	no	—
47	C₄×Dic₇₁	—	2	2	no	—
48	C₄×Dic₇₁	—	2	1	no	—
49	C₄×Dic₇₁	—	2	1	no	—
50	C₄×Dic₇₁	—	2	4	no	—
51	C₄×Dic₇₁	D₈	1	2	yes	—
52	C₄×Dic₇₁	Q₁₆	1	2	yes	—
53	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₈×C₇₁	—	2	2	no	—
57	C₂×C₈×C₇₁	—	2	4	no	—
58	C₂×C₈×C₇₁	—	4	2	no	—
59	C₂×C₈×C₇₁	—	4	2	no	—
60	C₂×C₈×C₇₁	C₄²	1	2	yes	—
61	C₂×C₈×C₇₁	C₄²	1	2	yes	—
62	C₂×C₈×C₇₁	M₁₆	1	4	yes	—
63	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₇₁	#41	1	1	yes	characteristic
66	C₄²×C₇₁	#41	1	1	yes	characteristic
67	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₇₁	—	2	2	no	—
69	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
76	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
80	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
81	C₂×Dic₇₁	—	2	2	no	—
82	C₂×Dic₇₁	—	2	2	no	—
83	C₂×Dic₇₁	—	2	2	no	—
84	C₂×Dic₇₁	—	2	2	no	—
85	C₂×Dic₇₁	#107	1	4	yes	—
86	C₂×Dic₇₁	#108	1	4	yes	—
87	C₈×C₇₁	—	2	2	no	—
88	C₈×C₇₁	—	4	4	no	—
89	C₈×C₇₁	#97	1	4	yes	complemented
90	C₇₁⋊C₈	—	2	4	no	—
91	C₇₁⋊C₈	—	4	4	no	—
92	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
93	C₂²×C₇₁	#423	1	2	yes	—
94	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
95	C₂²×C₇₁	#427	1	1	yes	characteristic
96	C₂²×C₇₁	C₂×#108	1	2	yes	—
97	C₄×C₇₁	—	2	1	no	—
98	C₄×C₇₁	—	2	1	no	—
99	C₄×C₇₁	C₂×#97	1	1	yes	characteristic
100	C₄×C₇₁	C₄×M₁₆	1	1	yes	characteristic
101	C₄×C₇₁	#432	1	2	yes	—
102	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
103	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
104	C₄×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
105	C₄×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
106	Dic₇₁	—	2	4	no	—
107	Dic₇₁	—	2	4	no	—
108	#8274	—	71	1	no	maximal, Sylow
109	C₂×C₇₁	#1567	1	2	yes	—
110	C₂×C₇₁	#1568	1	2	yes	—
111	C₂×C₇₁	#1767	1	1	yes	characteristic
112	C₂×C₇₁	C₄×#107	1	1	yes	characteristic
113	C₂×C₇₁	C₄×#108	1	1	yes	characteristic
114	#1767	—	71	2	no	—
115	C₄²×C₈	—	71	1	no	—
116	C₂×#335	—	71	1	no	—
117	C₂×#335	D₇₁	1	1	yes	characteristic
118	#1277	—	71	2	no	—
119	C₇₁	#8274	1	1	yes	characteristic, complemented, Sylow
120	C₂×C₄⋊C₈	—	71	2	no	—
121	C₂×C₄⋊C₈	C₂×D₇₁	1	2	yes	—
122	C₂×#97	—	142	2	no	—
123	C₂×C₄×C₈	—	71	2	no	—
124	C₂×C₄×C₈	—	71	2	no	—
125	C₂×C₄×C₈	—	71	1	no	—
126	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
127	C₄³	—	71	1	no	—
128	#336	—	142	2	no	—
129	#335	—	71	4	no	—
130	#335	Dic₇₁	1	4	yes	complemented
131	C₂²×C₈	—	2	2	no	—
132	C₂²×C₈	—	71	2	no	—
133	C₂²×C₈	—	142	2	no	—
134	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
135	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
136	C₂×C₄²	—	71	1	no	—
137	C₂×C₄²	—	71	2	no	—
138	C₂×C₄²	—	71	1	no	—
139	C₂×C₄²	—	71	2	no	—
140	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	71	4	no	—
143	C₄⋊C₈	—	142	2	no	—
144	C₄⋊C₈	C₂×Dic₇₁	1	4	yes	—
145	#97	—	284	4	no	—
146	C₄×C₈	—	142	4	no	—
147	C₄×C₈	—	142	4	no	—
148	C₄×C₈	—	142	1	no	—
149	C₄×C₈	—	142	1	no	—
150	C₄×C₈	C₄×D₇₁	1	2	yes	—
151	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
152	C₂²×C₄	—	71	2	no	—
153	C₂²×C₄	—	71	2	no	—
154	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
155	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
156	C₂²×C₄	Q₈×D₇₁	1	1	yes	characteristic
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	4	2	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	284	2	no	—
162	C₂×C₈	—	284	2	no	—
163	C₂×C₈	—	71	4	no	—
164	C₂×C₈	—	142	4	no	—
165	C₂×C₈	—	142	2	no	—
166	C₂×C₈	—	142	2	no	—
167	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
168	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
169	C₂×C₈	#11765848	1	4	yes	—
170	C₄²	—	71	2	no	—
171	C₄²	—	71	2	no	—
172	C₄²	—	142	4	no	—
173	C₄²	—	142	2	no	—
174	C₄²	—	142	2	no	—
175	C₄²	—	142	1	no	—
176	C₄²	—	142	1	no	—
177	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
178	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
179	C₄²	#11765864	1	1	yes	characteristic
180	C₄²	#11765866	1	1	yes	characteristic
181	C₂³	C₄⋊C₄×D₇₁	1	1	yes	characteristic
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	71	4	no	—
184	C₂×C₄	—	71	4	no	—
185	C₂×C₄	—	142	2	no	—
186	C₂×C₄	—	142	2	no	—
187	C₂×C₄	—	142	2	no	—
188	C₂×C₄	—	142	2	no	—
189	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
190	C₂×C₄	#14556086	1	1	yes	characteristic
191	C₂×C₄	#14556090	1	1	yes	characteristic
192	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
193	C₂×C₄	#14556093	1	1	yes	characteristic
194	C₂×C₄	Q₈×Dic₇₁	1	1	yes	characteristic
195	C₂×C₄	M₁₆×D₇₁	1	2	yes	—
196	C₂×C₄	C₂×#11765848	1	2	yes	—
197	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
198	C₂×C₄	Q₁₆×D₇₁	1	1	yes	characteristic
199	C₂×C₄	#14556147	1	1	yes	characteristic
200	C₂×C₄	#14556151	1	1	yes	characteristic
201	C₈	—	2	2	no	—
202	C₈	—	4	4	no	—
203	C₈	—	284	4	no	—
204	C₈	—	142	4	no	—
205	C₈	#14556038	1	4	yes	complemented
206	C₂²	C₄⋊C₄×Dic₇₁	1	1	yes	characteristic
207	C₂²	#17986021	1	2	yes	—
208	C₂²	#107×D₇₁	1	1	yes	characteristic
209	C₂²	#17986036	1	1	yes	characteristic
210	C₂²	#108×D₇₁	1	2	yes	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	142	4	no	—
214	C₄	—	142	4	no	—
215	C₄	C₂×#14556038	1	1	yes	characteristic
216	C₄	M₁₆×Dic₇₁	1	1	yes	characteristic
217	C₄	#17986055	1	2	yes	—
218	C₄	#17986081	1	1	yes	characteristic
219	C₄	D₈×Dic₇₁	1	1	yes	characteristic
220	C₄	#17986093	1	1	yes	characteristic
221	C₄	Q₁₆×Dic₇₁	1	1	yes	characteristic
222	C₂	#22587097	1	2	yes	—
223	C₂	#22587102	1	2	yes	—
224	C₂	#22588195	1	1	yes	characteristic
225	C₂	#107×Dic₇₁	1	1	yes	characteristic
226	C₂	#108×Dic₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×#335×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×#335	D₇₁
hypocenter	C₇₁	#8274
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
13. #17985752	C₄	semidirect
16. #335×C₇₁	C₄	semidirect
89. C₈×C₇₁	#97	semidirect
119. C₇₁	#8274	semidirect
130. #335	Dic₇₁	semidirect
205. C₈	#14556038	semidirect

Maximal subgroups

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

subgroup	index
#22588195	2
C₄×C₈×Dic₇₁	2
C₂×#17985752	2
#22586174	2
C₂×#335×C₇₁	2
#8274	71