Subgroups — #107×C₇₁⋊C₈ #31888683

Order 18176.

446 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₄×C₇₁⋊C₈	C₂	1	2	yes	maximal
2	C₂×#17985752	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#22586181	C₂	1	2	yes	maximal
5	C₄×#107×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₇₁⋊C₈	—	2	2	no	—
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
10	#17985754	—	2	2	no	—
11	#17985752	C₄	1	4	yes	complemented
12	C₈×C₇₁⋊C₈	C₄	1	4	yes	complemented
13	C₂×#107×C₇₁	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
16	#340×C₇₁	C₄	1	2	yes	—
17	C₂²×C₇₁⋊C₈	—	2	2	no	—
18	C₂²×C₇₁⋊C₈	—	2	1	no	—
19	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
20	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
21	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
22	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
23	C₄×C₇₁⋊C₈	—	2	1	no	—
24	C₄×C₇₁⋊C₈	—	4	4	no	—
25	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
27	C₄×C₇₁⋊C₈	Q₈	1	2	yes	—
28	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₇₁	—	2	2	no	—
32	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
33	#107×C₇₁	C₈	1	8	yes	complemented
34	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
36	#95×C₇₁	—	2	2	no	—
37	C₂×C₇₁⋊C₈	—	2	4	no	—
38	C₂×C₇₁⋊C₈	—	2	2	no	—
39	C₂×C₇₁⋊C₈	—	2	1	no	—
40	C₂×C₇₁⋊C₈	—	2	1	no	—
41	C₂×C₇₁⋊C₈	—	2	4	no	—
42	C₂×C₇₁⋊C₈	—	4	2	no	—
43	C₂×C₇₁⋊C₈	—	4	2	no	—
44	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
45	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
46	C₂×C₇₁⋊C₈	D₈	1	2	yes	—
47	C₂×C₇₁⋊C₈	Q₁₆	1	2	yes	—
48	C₂²×C₄×C₇₁	—	2	4	no	—
49	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
52	C₂×C₈×C₇₁	—	2	2	no	—
53	C₂×C₈×C₇₁	C₄²	1	2	yes	—
54	C₂×C₈×C₇₁	C₄²	1	2	yes	—
55	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	—
56	C₄⋊C₄×C₇₁	—	2	4	no	—
57	C₄⋊C₄×C₇₁	C₂×C₈	1	8	yes	—
58	C₄²×C₇₁	—	4	2	no	—
59	C₄²×C₇₁	—	4	2	no	—
60	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₇₁	#41	1	1	yes	characteristic
64	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
65	C₂×C₄×C₇₁	—	2	2	no	—
66	C₂×C₄×C₇₁	—	2	8	no	—
67	C₂×C₄×C₇₁	—	4	4	no	—
68	C₂×C₄×C₇₁	—	4	4	no	—
69	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₇₁	#117	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	C₂²×C₈	1	2	yes	—
76	C₂×C₄×C₇₁	#131	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	C₈×C₇₁	—	2	2	no	—
82	C₈×C₇₁	C₄×C₈	1	4	yes	complemented
83	C₇₁⋊C₈	—	2	4	no	—
84	C₇₁⋊C₈	—	2	2	no	—
85	C₇₁⋊C₈	—	4	4	no	—
86	C₇₁⋊C₈	#107	1	4	yes	direct product factor
87	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₇₁	C₈×D₄	1	2	yes	—
89	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
90	C₂²×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
91	C₂²×C₇₁	#443	1	2	yes	—
92	C₄×C₇₁	—	2	1	no	—
93	C₄×C₇₁	—	2	1	no	—
94	C₄×C₇₁	—	4	8	no	—
95	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
96	C₄×C₇₁	#405	1	1	yes	characteristic
97	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
98	C₄×C₇₁	#427	1	1	yes	characteristic
99	C₄×C₇₁	C₄×D₈	1	1	yes	characteristic
100	C₄×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
101	C₄×C₇₁	C₈×Q₈	1	2	yes	—
102	C₈×#107	—	71	1	no	maximal, Sylow
103	C₂×C₇₁	C₈×D₈	1	2	yes	—
104	C₂×C₇₁	C₈×Q₁₆	1	2	yes	—
105	C₂×C₇₁	C₈×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₇₁	#1772	1	1	yes	characteristic
107	C₂×C₇₁	C₄×#107	1	1	yes	characteristic
108	C₄×#107	D₇₁	1	1	yes	characteristic
109	C₈×C₄⋊C₄	—	71	2	no	—
110	C₂×#335	—	71	1	no	—
111	C₂×C₈²	—	71	1	no	—
112	#1279	—	71	2	no	—
113	C₇₁	C₈×#107	1	1	yes	characteristic, complemented, Sylow
114	C₂×#107	Dic₇₁	1	2	yes	—
115	C₂×C₄⋊C₈	—	71	2	no	—
116	C₂×C₄×C₈	—	71	1	no	—
117	C₂×C₄×C₈	—	71	1	no	—
118	C₂×C₄×C₈	—	142	2	no	—
119	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
120	C₄×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
121	#340	Dic₇₁	1	2	yes	—
122	#336	—	142	2	no	—
123	#335	—	71	4	no	—
124	C₈²	—	71	4	no	—
125	C₂²×C₈	—	71	1	no	—
126	C₂²×C₈	—	71	1	no	—
127	C₂²×C₈	—	142	2	no	—
128	C₂²×C₈	—	142	1	no	—
129	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
130	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
131	C₂×C₄⋊C₄	C₂×Dic₇₁	1	4	yes	—
132	C₂×C₄²	—	2	2	no	—
133	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
134	#107	C₇₁⋊C₈	1	8	yes	direct product factor
135	C₄⋊C₈	—	71	4	no	—
136	C₄⋊C₈	—	142	2	no	—
137	C₄×C₈	—	284	4	no	—
138	C₄×C₈	—	71	2	no	—
139	C₄×C₈	—	71	2	no	—
140	C₄×C₈	—	71	2	no	—
141	C₄×C₈	—	142	1	no	—
142	C₄×C₈	C₄×D₇₁	1	2	yes	—
143	C₄×C₈	C₄×D₇₁	1	2	yes	—
144	#95	—	2	2	no	—
145	C₂²×C₄	—	2	4	no	—
146	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
147	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
148	C₂²×C₄	#11765864	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	284	2	no	—
151	C₂×C₈	—	284	2	no	—
152	C₂×C₈	—	71	2	no	—
153	C₂×C₈	—	71	2	no	—
154	C₂×C₈	—	71	2	no	—
155	C₂×C₈	—	71	2	no	—
156	C₂×C₈	—	142	4	no	—
157	C₂×C₈	—	142	4	no	—
158	C₂×C₈	—	142	2	no	—
159	C₂×C₈	—	142	1	no	—
160	C₂×C₈	—	142	1	no	—
161	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
162	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
163	C₂×C₈	C₂×C₇₁⋊C₈	1	4	yes	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	C₂×C₇₁⋊C₈	1	8	yes	—
166	C₄²	—	4	2	no	—
167	C₄²	—	4	2	no	—
168	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
169	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
170	C₄²	Q₈×D₇₁	1	1	yes	characteristic
171	C₄²	#11765866	1	1	yes	characteristic
172	C₂³	D₄×Dic₇₁	1	1	yes	characteristic
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	8	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
178	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
179	C₂×C₄	#14556086	1	1	yes	characteristic
180	C₂×C₄	#14556090	1	1	yes	characteristic
181	C₂×C₄	#14556093	1	1	yes	characteristic
182	C₂×C₄	Q₈×Dic₇₁	1	1	yes	characteristic
183	C₂×C₄	C₂²×C₇₁⋊C₈	1	2	yes	—
184	C₂×C₄	#14556135	1	2	yes	—
185	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
186	C₂×C₄	Q₁₆×D₇₁	1	1	yes	characteristic
187	C₂×C₄	#14556147	1	1	yes	characteristic
188	C₂×C₄	#14556151	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	—	284	4	no	—
191	C₈	—	71	4	no	—
192	C₈	—	142	4	no	—
193	C₈	—	142	2	no	—
194	C₈	C₄×C₇₁⋊C₈	1	4	yes	complemented
195	C₂²	C₄⋊C₄×Dic₇₁	1	1	yes	characteristic
196	C₂²	D₄×C₇₁⋊C₈	1	2	yes	—
197	C₂²	D₈×Dic₇₁	1	1	yes	characteristic
198	C₂²	Q₁₆×Dic₇₁	1	1	yes	characteristic
199	C₂²	#17986116	1	2	yes	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	4	8	no	—
203	C₄	C₂×C₄×C₇₁⋊C₈	1	1	yes	characteristic
204	C₄	#17985955	1	1	yes	characteristic
205	C₄	#107×D₇₁	1	1	yes	characteristic
206	C₄	#17986036	1	1	yes	characteristic
207	C₄	#17986081	1	1	yes	characteristic
208	C₄	#17986093	1	1	yes	characteristic
209	C₄	Q₈×C₇₁⋊C₈	1	2	yes	—
210	C₂	D₈×C₇₁⋊C₈	1	2	yes	—
211	C₂	Q₁₆×C₇₁⋊C₈	1	2	yes	—
212	C₂	C₄⋊C₄×C₇₁⋊C₈	1	1	yes	characteristic
213	C₂	#22588221	1	1	yes	characteristic
214	C₂	#107×Dic₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₁
commutator	C₄×C₇₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₄×#107×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₄×D₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₄×#107	D₇₁
hypocenter	C₇₁	C₈×#107
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
11. #17985752	C₄	semidirect
12. C₈×C₇₁⋊C₈	C₄	semidirect
33. #107×C₇₁	C₈	semidirect
82. C₈×C₇₁	C₄×C₈	semidirect
86. C₇₁⋊C₈	#107	direct (N × H)
113. C₇₁	C₈×#107	semidirect
134. #107	C₇₁⋊C₈	direct (N × H)
194. C₈	C₄×C₇₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₄×C₇₁⋊C₈	2
C₂×#17985752	2
C₂×C₈×C₇₁⋊C₈	2
#22586181	2
C₄×#107×C₇₁	2
C₈×#107	71

Subgroups — #107×C71⋊C8 #31888683

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #107×C₇₁⋊C₈ #31888683