Subgroups — C₈×#14556063 #31888745

Order 18176.

374 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556063	C₂	1	1	yes	maximal, characteristic
2	#22587086	C₂	1	1	yes	maximal, characteristic
3	C₈×#11765849	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#22586178	C₂	1	1	yes	maximal, characteristic
6	#22586176	C₂	1	1	yes	maximal, characteristic
7	C₈×M₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
8	#17986052	C₂²	1	1	yes	characteristic
9	C₂×#14556063	C₄	1	2	yes	—
10	C₄×#11765849	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
13	#17985763	C₄	1	2	yes	—
14	#17985751	C₄	1	2	yes	—
15	#17985749	C₄	1	2	yes	—
16	#17985721	—	2	1	no	—
17	C₈×C₇₁⋊C₈	—	2	1	no	—
18	C₈×C₇₁⋊C₈	C₄	1	2	yes	—
19	C₈×C₇₁⋊C₈	C₄	1	2	yes	—
20	#431×C₇₁	C₂²	1	1	yes	characteristic
21	C₄×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
23	#322×C₇₁	—	2	1	no	—
24	C₈²×C₇₁	—	2	1	no	—
25	C₂×#11765849	C₂×C₄	1	2	yes	—
26	C₂²×C₇₁⋊C₈	—	2	1	no	—
27	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
29	#14556063	C₈	1	8	yes	direct product factor
30	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
31	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
32	#14556039	—	2	1	no	—
33	C₄×C₇₁⋊C₈	—	2	1	no	—
34	C₄×C₇₁⋊C₈	—	2	1	no	—
35	C₄×C₇₁⋊C₈	—	2	1	no	—
36	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
37	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₄×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×M₁₆×C₇₁	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₇₁	Q₈	1	2	yes	—
43	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
45	C₂²⋊C₈×C₇₁	—	2	1	no	—
46	#97×C₇₁	—	2	1	no	—
47	C₄×C₈×C₇₁	—	2	1	no	—
48	C₄×C₈×C₇₁	—	2	1	no	—
49	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
52	#11765849	—	2	2	no	—
53	#11765849	C₂×C₈	1	4	yes	—
54	C₂×C₇₁⋊C₈	—	2	4	no	—
55	C₂×C₇₁⋊C₈	—	2	2	no	—
56	C₂×C₇₁⋊C₈	—	2	2	no	—
57	C₂×C₇₁⋊C₈	—	2	2	no	—
58	C₂×C₇₁⋊C₈	—	2	2	no	—
59	C₂×C₇₁⋊C₈	C₄²	1	1	yes	characteristic
60	C₂×C₇₁⋊C₈	C₄²	1	1	yes	characteristic
61	C₂×C₇₁⋊C₈	C₄²	1	1	yes	characteristic
62	C₂×C₇₁⋊C₈	C₄²	1	1	yes	characteristic
63	C₂×C₇₁⋊C₈	C₂×C₈	1	4	yes	—
64	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
67	M₁₆×C₇₁	—	2	2	no	—
68	M₁₆×C₇₁	C₂×C₈	1	4	yes	—
69	C₂×C₈×C₇₁	—	2	2	no	—
70	C₂×C₈×C₇₁	—	2	4	no	—
71	C₂×C₈×C₇₁	—	2	2	no	—
72	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
73	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
74	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₇₁	#41	1	1	yes	characteristic
78	C₂³×C₇₁	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₇₁	—	2	2	no	—
80	C₂×C₄×C₇₁	—	2	2	no	—
81	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
84	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
87	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₇₁	#131	1	2	yes	—
89	C₈×C₇₁	—	2	2	no	—
90	C₈×C₇₁	—	2	4	no	—
91	C₈×C₇₁	—	4	2	no	—
92	C₈×C₇₁	#108	1	4	yes	complemented
93	C₇₁⋊C₈	—	2	4	no	—
94	C₇₁⋊C₈	—	2	1	no	—
95	C₇₁⋊C₈	—	2	1	no	—
96	C₇₁⋊C₈	—	2	4	no	—
97	C₇₁⋊C₈	—	4	2	no	—
98	C₇₁⋊C₈	C₄×C₈	1	2	yes	—
99	C₇₁⋊C₈	C₄×C₈	1	2	yes	—
100	C₂²×C₇₁	—	2	1	no	—
101	C₂²×C₇₁	—	2	1	no	—
102	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₇₁	C₈×Q₈	1	2	yes	—
104	C₄×C₇₁	—	2	1	no	—
105	C₄×C₇₁	—	2	1	no	—
106	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
107	C₄×C₇₁	#405	1	1	yes	characteristic
108	C₄×C₇₁	C₂×#108	1	2	yes	—
109	C₄×C₇₁	C₈×D₄	1	2	yes	—
110	C₄×C₇₁	#443	1	2	yes	—
111	C₈×#108	—	71	1	no	maximal, Sylow
112	C₂×C₇₁	—	2	2	no	—
113	C₂×C₇₁	C₈×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₇₁	#1772	1	1	yes	characteristic
115	C₂×C₇₁	C₄×#108	1	1	yes	characteristic
116	C₄×#108	—	71	1	no	—
117	#1564	—	71	1	no	—
118	C₈×M₁₆	—	71	1	no	—
119	C₈×M₁₆	D₇₁	1	1	yes	characteristic
120	C₂×C₈²	—	71	1	no	—
121	#1278	—	71	1	no	—
122	#1278	—	71	1	no	—
123	C₇₁	C₈×#108	1	1	yes	characteristic, complemented, Sylow
124	#431	—	71	1	no	—
125	#431	C₂×D₇₁	1	1	yes	characteristic
126	C₂×#108	—	71	2	no	—
127	C₄×M₁₆	—	71	1	no	—
128	C₄×M₁₆	C₂×D₇₁	1	1	yes	characteristic
129	C₂×C₄×C₈	—	71	1	no	—
130	C₂×C₄×C₈	—	71	1	no	—
131	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
132	#341	—	71	2	no	—
133	#335	—	71	2	no	—
134	#334	—	71	2	no	—
135	#322	—	2	1	no	—
136	#322	—	142	1	no	—
137	C₈²	—	2	1	no	—
138	C₈²	—	71	2	no	—
139	C₈²	—	71	2	no	—
140	C₈²	—	142	1	no	—
141	C₂×M₁₆	—	71	2	no	—
142	C₂×M₁₆	C₄×D₇₁	1	2	yes	—
143	C₂²×C₈	—	71	1	no	—
144	C₂²×C₈	—	71	1	no	—
145	C₂²×C₈	—	142	1	no	—
146	C₂²×C₈	Dic₁₄₂	1	2	yes	—
147	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
148	#108	—	71	8	no	—
149	C₄⋊C₈	—	71	2	no	—
150	C₄⋊C₈	C₄×D₇₁	1	2	yes	—
151	C₂²⋊C₈	—	2	1	no	—
152	C₂²⋊C₈	—	142	1	no	—
153	#97	—	2	1	no	—
154	#97	—	142	1	no	—
155	C₄×C₈	—	2	1	no	—
156	C₄×C₈	—	2	1	no	—
157	C₄×C₈	—	71	1	no	—
158	C₄×C₈	—	71	1	no	—
159	C₄×C₈	—	71	1	no	—
160	C₄×C₈	—	71	1	no	—
161	C₄×C₈	—	71	2	no	—
162	C₄×C₈	—	142	1	no	—
163	C₄×C₈	—	142	1	no	—
164	C₄×C₈	—	142	1	no	—
165	C₄×C₈	C₄×D₇₁	1	2	yes	—
166	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
167	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
169	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
170	C₂²×C₄	#11765865	1	1	yes	characteristic
171	M₁₆	—	2	2	no	—
172	M₁₆	—	71	4	no	—
173	M₁₆	—	142	2	no	—
174	M₁₆	C₈×D₇₁	1	4	yes	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	71	1	no	—
179	C₂×C₈	—	71	1	no	—
180	C₂×C₈	—	71	4	no	—
181	C₂×C₈	—	71	1	no	—
182	C₂×C₈	—	71	1	no	—
183	C₂×C₈	—	142	2	no	—
184	C₂×C₈	—	142	4	no	—
185	C₂×C₈	—	142	2	no	—
186	C₂×C₈	—	142	2	no	—
187	C₂×C₈	—	142	2	no	—
188	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
189	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
190	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
191	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
192	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
193	C₄²	#11765865	1	1	yes	characteristic
194	C₂³	C₄×Dic₁₄₂	1	1	yes	characteristic
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	C₄²×D₇₁	1	1	yes	characteristic
198	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
199	C₂×C₄	#14556085	1	1	yes	characteristic
200	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
201	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
202	C₂×C₄	C₄×Dic₁₄₂	1	1	yes	characteristic
203	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
204	C₂×C₄	#14556134	1	2	yes	—
205	C₈	—	2	4	no	—
206	C₈	—	2	2	no	—
207	C₈	—	4	2	no	—
208	C₈	—	284	2	no	—
209	C₈	—	71	2	no	—
210	C₈	—	71	2	no	—
211	C₈	—	142	4	no	—
212	C₈	—	142	4	no	—
213	C₈	—	142	1	no	—
214	C₈	—	142	1	no	—
215	C₈	#14556063	1	4	yes	direct product factor
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	C₄×Dic₇₁⋊C₄	1	1	yes	characteristic
219	C₂²	C₈×Dic₁₄₂	1	2	yes	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	C₄×C₈×D₇₁	1	1	yes	characteristic
223	C₄	#17985953	1	1	yes	characteristic
224	C₄	C₂×#14556063	1	2	yes	—
225	C₄	C₈×C₂²⋊D₇₁	1	2	yes	—
226	C₄	#17986114	1	2	yes	—
227	C₂	—	2	2	no	—
228	C₂	C₈×Dic₇₁⋊C₄	1	1	yes	characteristic
229	C₂	#22588220	1	1	yes	characteristic
230	C₂	C₄×#14556063	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²⋊D₇₁
commutator	C₄×C₇₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₈×M₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₄×C₄⋊C₄
prime radical	C₂³×C₇₁	C₄×Q₈
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₈×M₁₆	D₇₁
hypocenter	C₇₁	C₈×#108
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
29. #14556063	C₈	direct (N × H)
92. C₈×C₇₁	#108	semidirect
123. C₇₁	C₈×#108	semidirect
215. C₈	#14556063	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#14556063	2
#22587086	2
C₈×#11765849	2
C₂×C₈×C₇₁⋊C₈	2
#22586178	2
#22586176	2
C₈×M₁₆×C₇₁	2
C₈×#108	71

Subgroups — C8×#14556063 #31888745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#14556063 #31888745