Subgroups — C₂²×#17986403 #32867049

Order 18176.

2398 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	C₂²×#14556127	C₂	1	2	yes	maximal
2	C₂×#17986403	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#14556062	C₂	1	1	yes	maximal, characteristic
4	C₂²×#14556061	C₂	1	2	yes	maximal
5	C₂²×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#107×C₇₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₄₂	—	2	2	no	—
8	C₂³×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×#14556127	—	2	12	no	—
10	C₂×#14556127	C₂²	1	24	yes	—
11	C₂²×C₄⋊Dic₇₁	C₂²	1	2	yes	—
12	C₂²×Dic₇₁⋊C₄	—	2	2	no	—
13	C₂²×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
14	#17986403	C₂²	1	64	yes	direct product factor
15	C₂×#14556062	C₂²	1	12	yes	—
16	C₂×#14556061	C₂²	1	24	yes	—
17	C₂×C₈×Dic₇₁	C₂²	1	12	yes	—
18	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
20	C₂×#107×C₇₁	C₂²	1	12	yes	—
21	C₂²×Dic₁₄₂	—	2	24	no	—
22	C₂²×Dic₁₄₂	—	4	16	no	—
23	C₂³×Dic₇₁	—	2	2	no	—
24	C₂³×Dic₇₁	D₄	1	1	yes	characteristic
25	C₂³×Dic₇₁	D₄	1	1	yes	characteristic
26	C₂²×C₇₁⋊C₈	Q₈	1	8	yes	complemented
27	C₂²×C₇₁⋊C₈	C₂³	1	6	yes	—
28	#14556127	—	2	48	no	—
29	#14556127	C₂³	1	32	yes	—
30	C₂×C₄⋊Dic₇₁	—	2	6	no	—
31	C₂×C₄⋊Dic₇₁	C₂³	1	12	yes	—
32	C₂×Dic₇₁⋊C₄	—	2	24	no	—
33	C₂×C₄×Dic₇₁	D₄	1	6	yes	—
34	C₂×C₄×Dic₇₁	C₂³	1	6	yes	—
35	#14556062	C₂³	1	16	yes	—
36	#14556061	C₂³	1	32	yes	—
37	C₈×Dic₇₁	C₂³	1	16	yes	—
38	C₂³×C₄×C₇₁	—	2	2	no	—
39	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₇₁	Q₈	1	8	yes	complemented
41	C₂²×C₈×C₇₁	C₂³	1	6	yes	—
42	C₂×C₄⋊C₄×C₇₁	—	2	6	no	—
43	C₂×C₄⋊C₄×C₇₁	C₂³	1	12	yes	—
44	#107×C₇₁	C₂³	1	16	yes	—
45	C₂×Dic₁₄₂	—	2	32	no	—
46	C₂×Dic₁₄₂	—	4	96	no	—
47	C₂²×Dic₇₁	—	2	12	no	—
48	C₂²×Dic₇₁	—	4	8	no	—
49	C₂²×Dic₇₁	D₈	1	8	yes	—
50	C₂²×Dic₇₁	Q₁₆	1	8	yes	—
51	C₂²×Dic₇₁	C₂×D₄	1	6	yes	—
52	C₂²×Dic₇₁	C₂×D₄	1	6	yes	—
53	C₂×C₇₁⋊C₈	C₂×Q₈	1	24	yes	complemented
54	C₂×C₇₁⋊C₈	C₂⁴	1	4	yes	—
55	C₄⋊Dic₇₁	—	2	12	no	—
56	C₄⋊Dic₇₁	C₂⁴	1	8	yes	—
57	Dic₇₁⋊C₄	—	2	32	no	—
58	C₄×Dic₇₁	C₂×D₄	1	12	yes	—
59	C₄×Dic₇₁	C₂⁴	1	4	yes	—
60	C₂⁴×C₇₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₁	—	2	12	no	—
62	C₂²×C₄×C₇₁	—	4	8	no	—
63	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
64	C₂²×C₄×C₇₁	C₂×Q₈	1	4	yes	—
65	C₂²×C₄×C₇₁	C₂×Q₈	1	4	yes	—
66	C₂²×C₄×C₇₁	C₂⁴	1	3	yes	—
67	C₂×C₈×C₇₁	C₂×Q₈	1	24	yes	complemented
68	C₂×C₈×C₇₁	C₂⁴	1	4	yes	—
69	C₄⋊C₄×C₇₁	—	2	12	no	—
70	C₄⋊C₄×C₇₁	C₂⁴	1	8	yes	—
71	C₂³×C₇₁	C₄⋊Q₈	1	4	yes	—
72	C₂³×C₇₁	C₂×D₈	1	4	yes	—
73	C₂³×C₇₁	C₂×Q₁₆	1	4	yes	—
74	C₂³×C₇₁	C₂²×D₄	1	3	yes	—
75	C₂×C₄×C₇₁	—	2	8	no	—
76	C₂×C₄×C₇₁	—	4	24	no	—
77	C₂×C₄×C₇₁	C₄⋊Q₈	1	12	yes	—
78	C₂×C₄×C₇₁	C₂²×D₄	1	3	yes	—
79	C₂×C₄×C₇₁	C₂²×Q₈	1	6	yes	—
80	C₂×C₄×C₇₁	C₂²×Q₈	1	6	yes	—
81	C₂×C₄×C₇₁	C₂⁵	1	1	yes	characteristic
82	C₂×Dic₇₁	—	2	8	no	—
83	C₂×Dic₇₁	—	4	24	no	—
84	C₂×Dic₇₁	C₂×D₈	1	24	yes	—
85	C₂×Dic₇₁	C₂×Q₁₆	1	24	yes	—
86	C₂×Dic₇₁	C₂²×D₄	1	4	yes	—
87	C₂×Dic₇₁	C₂²×D₄	1	4	yes	—
88	Dic₁₄₂	—	4	64	no	—
89	C₈×C₇₁	C₂²×Q₈	1	8	yes	complemented
90	C₇₁⋊C₈	C₂²×Q₈	1	8	yes	complemented
91	C₂²×C₇₁	#500	1	16	yes	complemented
92	C₂²×C₇₁	C₂×C₄⋊Q₈	1	6	yes	—
93	C₂²×C₇₁	C₂²×D₈	1	6	yes	—
94	C₂²×C₇₁	C₂²×Q₁₆	1	6	yes	—
95	C₂²×C₇₁	C₂³×D₄	1	1	yes	characteristic
96	C₄×C₇₁	—	4	8	no	—
97	C₄×C₇₁	C₂×C₄⋊Q₈	1	6	yes	—
98	C₄×C₇₁	C₂³×Q₈	1	1	yes	characteristic
99	C₄×C₇₁	C₂³×Q₈	1	1	yes	characteristic
100	Dic₇₁	—	4	8	no	—
101	Dic₇₁	C₂²×D₈	1	8	yes	—
102	Dic₇₁	C₂²×Q₁₆	1	8	yes	—
103	C₂²×#500	—	71	1	no	maximal, Sylow
104	C₂×C₇₁	C₂×#500	1	12	yes	complemented
105	C₂×C₇₁	C₂²×C₄⋊Q₈	1	1	yes	characteristic
106	C₂×C₇₁	C₂³×D₈	1	1	yes	characteristic
107	C₂×C₇₁	C₂³×Q₁₆	1	1	yes	characteristic
108	C₂²×C₄⋊Q₈	—	71	2	no	—
109	C₂×#500	—	71	24	no	—
110	C₂²×#107	—	71	1	no	—
111	C₂²×#107	—	71	2	no	—
112	C₂²×#107	D₇₁	1	1	yes	characteristic
113	C₂²×C₄×C₈	—	71	1	no	—
114	C₇₁	C₂²×#500	1	1	yes	characteristic, complemented, Sylow
115	C₂³×Q₈	—	142	2	no	—
116	C₂³×C₈	—	71	1	no	—
117	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
118	C₂×C₄⋊Q₈	—	71	24	no	—
119	C₂×C₄⋊Q₈	—	142	12	no	—
120	C₂²×C₄⋊C₄	—	71	2	no	—
121	C₂²×C₄⋊C₄	—	142	2	no	—
122	C₂²×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
123	C₂²×C₄²	—	71	1	no	—
124	#500	—	71	64	no	—
125	C₂×#107	—	71	12	no	—
126	C₂×#107	—	71	24	no	—
127	C₂×#107	C₂×D₇₁	1	12	yes	—
128	C₂×C₄×C₈	—	71	12	no	—
129	C₂²×Q₈	—	284	16	no	—
130	C₂²×Q₈	—	142	24	no	—
131	C₂³×C₄	—	2	2	no	—
132	C₂³×C₄	—	71	1	no	—
133	C₂³×C₄	—	71	1	no	—
134	C₂³×C₄	—	142	2	no	—
135	C₂³×C₄	C₂²×D₇₁	1	1	yes	characteristic
136	C₂²×C₈	—	71	8	no	—
137	C₂²×C₈	—	71	6	no	—
138	C₂²×C₈	Dic₁₄₂	1	8	yes	complemented
139	C₂²×C₈	C₂²×D₇₁	1	6	yes	—
140	C₄⋊Q₈	—	71	32	no	—
141	C₄⋊Q₈	—	142	48	no	—
142	C₂×C₄⋊C₄	—	2	6	no	—
143	C₂×C₄⋊C₄	—	71	12	no	—
144	C₂×C₄⋊C₄	—	142	24	no	—
145	C₂×C₄⋊C₄	—	142	6	no	—
146	C₂×C₄⋊C₄	C₂²×D₇₁	1	12	yes	—
147	C₂×C₄²	—	71	6	no	—
148	C₂×C₄²	—	71	6	no	—
149	#107	—	71	16	no	—
150	#107	—	71	32	no	—
151	#107	C₂²×D₇₁	1	16	yes	—
152	C₄×C₈	—	71	16	no	—
153	C₂⁴	D₄×D₇₁	1	1	yes	characteristic
154	C₂×Q₈	—	284	96	no	—
155	C₂×Q₈	—	142	32	no	—
156	C₂²×C₄	—	2	12	no	—
157	C₂²×C₄	—	4	8	no	—
158	C₂²×C₄	—	284	8	no	—
159	C₂²×C₄	—	71	6	no	—
160	C₂²×C₄	—	71	6	no	—
161	C₂²×C₄	—	71	8	no	—
162	C₂²×C₄	—	71	8	no	—
163	C₂²×C₄	—	142	12	no	—
164	C₂²×C₄	D₄×D₇₁	1	3	yes	—
165	C₂²×C₄	Q₈×D₇₁	1	4	yes	—
166	C₂²×C₄	C₂×Dic₁₄₂	1	4	yes	—
167	C₂²×C₄	C₂³×D₇₁	1	3	yes	—
168	C₂×C₈	—	71	24	no	—
169	C₂×C₈	—	71	4	no	—
170	C₂×C₈	C₂×Dic₁₄₂	1	24	yes	complemented
171	C₂×C₈	C₂³×D₇₁	1	4	yes	—
172	C₄⋊C₄	—	2	12	no	—
173	C₄⋊C₄	—	71	8	no	—
174	C₄⋊C₄	—	142	32	no	—
175	C₄⋊C₄	—	142	12	no	—
176	C₄⋊C₄	C₂³×D₇₁	1	8	yes	—
177	C₄²	—	71	12	no	—
178	C₄²	—	71	4	no	—
179	C₂³	#14556127	1	4	yes	—
180	C₂³	D₈×D₇₁	1	4	yes	—
181	C₂³	Q₁₆×D₇₁	1	4	yes	—
182	C₂³	C₂×D₄×D₇₁	1	3	yes	—
183	Q₈	—	284	64	no	—
184	C₂×C₄	—	2	8	no	—
185	C₂×C₄	—	4	24	no	—
186	C₂×C₄	—	284	24	no	—
187	C₂×C₄	—	71	4	no	—
188	C₂×C₄	—	71	4	no	—
189	C₂×C₄	—	71	24	no	—
190	C₂×C₄	—	71	24	no	—
191	C₂×C₄	—	142	8	no	—
192	C₂×C₄	#14556127	1	12	yes	—
193	C₂×C₄	C₂×D₄×D₇₁	1	3	yes	—
194	C₂×C₄	C₂×Q₈×D₇₁	1	6	yes	—
195	C₂×C₄	C₂²×Dic₁₄₂	1	6	yes	—
196	C₂×C₄	C₂⁴×D₇₁	1	1	yes	characteristic
197	C₈	—	71	8	no	—
198	C₈	C₂²×Dic₁₄₂	1	8	yes	complemented
199	C₂²	#17986403	1	16	yes	direct product factor
200	C₂²	C₂×#14556127	1	6	yes	—
201	C₂²	C₂×D₈×D₇₁	1	6	yes	—
202	C₂²	C₂×Q₁₆×D₇₁	1	6	yes	—
203	C₂²	C₂²×D₄×D₇₁	1	1	yes	characteristic
204	C₄	—	4	8	no	—
205	C₄	—	284	8	no	—
206	C₄	—	71	8	no	—
207	C₄	—	71	8	no	—
208	C₄	C₂×#14556127	1	6	yes	—
209	C₄	C₂²×Q₈×D₇₁	1	1	yes	characteristic
210	C₄	C₂³×Dic₁₄₂	1	1	yes	characteristic
211	C₂	C₂×#17986403	1	12	yes	direct product factor
212	C₂	C₂²×#14556127	1	1	yes	characteristic
213	C₂	C₂²×D₈×D₇₁	1	1	yes	characteristic
214	C₂	C₂²×Q₁₆×D₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₇₁
commutator, cosocle	C₂×C₄×C₇₁	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₁
Fitting	C₂²×#107×C₇₁	C₂
socle, prime radical	C₂⁴×C₇₁	C₂×D₄
hypercenter	C₂²×#107	D₇₁
hypocenter	C₇₁	C₂²×#500
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
2. C₂×#17986403	C₂	direct (N × H)
14. #17986403	C₂²	direct (N × H)
26. C₂²×C₇₁⋊C₈	Q₈	semidirect
40. C₂²×C₈×C₇₁	Q₈	semidirect
53. C₂×C₇₁⋊C₈	C₂×Q₈	semidirect
67. C₂×C₈×C₇₁	C₂×Q₈	semidirect
89. C₈×C₇₁	C₂²×Q₈	semidirect
90. C₇₁⋊C₈	C₂²×Q₈	semidirect
91. C₂²×C₇₁	#500	semidirect
104. C₂×C₇₁	C₂×#500	semidirect
114. C₇₁	C₂²×#500	semidirect
138. C₂²×C₈	Dic₁₄₂	semidirect
170. C₂×C₈	C₂×Dic₁₄₂	semidirect
198. C₈	C₂²×Dic₁₄₂	semidirect
199. C₂²	#17986403	direct (N × H)
211. C₂	C₂×#17986403	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14556127	2
C₂×#17986403	2
C₂²×#14556062	2
C₂²×#14556061	2
C₂²×C₈×Dic₇₁	2
C₂²×#107×C₇₁	2
C₂²×#500	71

Subgroups — C22×#17986403 #32867049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17986403 #32867049