Subgroups — C₂²×#17986728 #32904498

Order 18176.

7414 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986728	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#14556128	C₂	1	3	yes	maximal
3	C₂²×#14556119	C₂	1	6	yes	maximal
4	C₂²×Q₈×Dic₇₁	C₂	1	2	yes	maximal
5	C₂²×C₄×Dic₁₄₂	C₂	1	3	yes	maximal
6	C₂²×C₄×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₄₂	C₂²	1	3	yes	—
8	#17986728	C₂²	1	256	yes	direct product factor
9	C₂×#14556128	C₂²	1	72	yes	—
10	C₂×#14556119	C₂²	1	144	yes	—
11	C₂×Q₈×Dic₇₁	C₂²	1	48	yes	—
12	C₂×C₄×Dic₁₄₂	C₂²	1	72	yes	—
13	C₂²×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊Dic₇₁	C₂²	1	6	yes	—
15	C₂²×C₄⋊Dic₇₁	C₂²	1	6	yes	—
16	C₂²×Dic₇₁⋊C₄	C₂²	1	6	yes	—
17	C₂²×C₄×Dic₇₁	C₂²	1	6	yes	—
18	C₂³×Q₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₇₁	C₂²	1	24	yes	—
20	C₂²×C₄⋊C₄×C₇₁	C₂²	1	3	yes	—
21	C₂²×C₄²×C₇₁	C₂²	1	3	yes	—
22	C₂²×Dic₁₄₂	—	2	24	no	—
23	C₂²×Dic₁₄₂	C₂³	1	36	yes	—
24	C₂³×Dic₇₁	C₂³	1	6	yes	—
25	C₂³×Dic₇₁	C₂³	1	2	yes	—
26	#14556128	C₂³	1	192	yes	—
27	#14556119	C₂³	1	384	yes	—
28	Q₈×Dic₇₁	C₂³	1	128	yes	—
29	C₄×Dic₁₄₂	C₂³	1	192	yes	—
30	C₂×C₄⋊Dic₇₁	C₂³	1	12	yes	—
31	C₂×C₄⋊Dic₇₁	C₂³	1	72	yes	—
32	C₂×C₄⋊Dic₇₁	C₂³	1	72	yes	—
33	C₂×Dic₇₁⋊C₄	C₂³	1	72	yes	—
34	C₂×C₄×Dic₇₁	C₂³	1	72	yes	—
35	C₂²×Q₈×C₇₁	Q₈	1	16	yes	complemented
36	C₂²×Q₈×C₇₁	C₂³	1	12	yes	—
37	C₂³×C₄×C₇₁	C₂³	1	3	yes	—
38	C₂³×C₄×C₇₁	C₂³	1	3	yes	—
39	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
40	C₄×Q₈×C₇₁	C₂³	1	64	yes	—
41	C₂×C₄⋊C₄×C₇₁	C₂³	1	36	yes	—
42	C₂×C₄²×C₇₁	C₂³	1	36	yes	—
43	C₂×Dic₁₄₂	—	2	144	no	—
44	C₂×Dic₁₄₂	C₂⁴	1	48	yes	—
45	C₂²×Dic₇₁	—	2	24	no	—
46	C₂²×Dic₇₁	—	2	8	no	—
47	C₂²×Dic₇₁	C₂⁴	1	36	yes	—
48	C₂²×Dic₇₁	C₂⁴	1	12	yes	—
49	C₄⋊Dic₇₁	C₂⁴	1	16	yes	—
50	C₄⋊Dic₇₁	C₂⁴	1	96	yes	—
51	C₄⋊Dic₇₁	C₂⁴	1	96	yes	—
52	Dic₇₁⋊C₄	C₂⁴	1	96	yes	—
53	C₄×Dic₇₁	C₂⁴	1	96	yes	—
54	C₂⁴×C₇₁	C₂⁴	1	1	yes	characteristic
55	C₂×Q₈×C₇₁	C₂×Q₈	1	96	yes	complemented
56	C₂×Q₈×C₇₁	C₂⁴	1	16	yes	—
57	C₂²×C₄×C₇₁	—	2	12	no	—
58	C₂²×C₄×C₇₁	C₂×Q₈	1	24	yes	—
59	C₂²×C₄×C₇₁	#41	1	8	yes	—
60	C₂²×C₄×C₇₁	C₂⁴	1	18	yes	—
61	C₂²×C₄×C₇₁	C₂⁴	1	18	yes	—
62	C₂²×C₄×C₇₁	C₂⁴	1	6	yes	—
63	C₄⋊C₄×C₇₁	C₂⁴	1	48	yes	—
64	C₄²×C₇₁	C₂⁴	1	48	yes	—
65	C₂³×C₇₁	C₂²×Q₈	1	4	yes	—
66	C₂³×C₇₁	C₂×#41	1	4	yes	—
67	C₂³×C₇₁	2₋¹⁺⁴	1	4	yes	—
68	C₂³×C₇₁	C₂⁵	1	3	yes	—
69	Q₈×C₇₁	C₂²×Q₈	1	64	yes	complemented
70	C₂×C₄×C₇₁	—	2	36	no	—
71	C₂×C₄×C₇₁	C₂²×Q₈	1	72	yes	—
72	C₂×C₄×C₇₁	C₂×#41	1	24	yes	—
73	C₂×C₄×C₇₁	C₂⁵	1	12	yes	—
74	C₂×C₄×C₇₁	C₂⁵	1	12	yes	—
75	C₂×C₄×C₇₁	C₂⁵	1	4	yes	—
76	C₂×Dic₇₁	—	2	72	no	—
77	C₂×Dic₇₁	—	2	24	no	—
78	C₂×Dic₇₁	C₂⁵	1	8	yes	—
79	C₂×Dic₇₁	C₂⁵	1	24	yes	—
80	Dic₁₄₂	—	2	96	no	—
81	C₂²×C₇₁	#557	1	16	yes	complemented
82	C₂²×C₇₁	C₂³×Q₈	1	6	yes	—
83	C₂²×C₇₁	C₂²×#41	1	6	yes	—
84	C₂²×C₇₁	C₂×2₋¹⁺⁴	1	6	yes	—
85	C₂²×C₇₁	C₂⁶	1	1	yes	characteristic
86	C₄×C₇₁	—	2	12	no	—
87	C₄×C₇₁	C₂³×Q₈	1	24	yes	—
88	C₄×C₇₁	C₂²×#41	1	8	yes	—
89	Dic₇₁	—	2	24	no	—
90	Dic₇₁	—	2	8	no	—
91	C₂²×#557	—	71	1	no	maximal, Sylow
92	C₂×C₇₁	C₂×#557	1	12	yes	complemented
93	C₂×C₇₁	C₂⁴×Q₈	1	1	yes	characteristic
94	C₂×C₇₁	C₂³×#41	1	1	yes	characteristic
95	C₂×C₇₁	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
96	C₂×#557	—	71	48	no	—
97	C₂²×C₄⋊Q₈	—	71	3	no	—
98	C₂²×#125	—	71	6	no	—
99	C₂²×C₄×Q₈	—	71	3	no	—
100	C₂²×C₄×Q₈	—	71	2	no	—
101	C₂²×C₄×Q₈	D₇₁	1	1	yes	characteristic
102	C₇₁	C₂²×#557	1	1	yes	characteristic, complemented, Sylow
103	C₂³×Q₈	—	71	3	no	—
104	C₂³×Q₈	C₂×D₇₁	1	1	yes	characteristic
105	#557	—	71	256	no	—
106	C₂×C₄⋊Q₈	—	71	72	no	—
107	C₂×#125	—	71	144	no	—
108	C₂×C₄×Q₈	—	71	48	no	—
109	C₂×C₄×Q₈	—	71	72	no	—
110	C₂×C₄×Q₈	C₂×D₇₁	1	24	yes	—
111	C₂²×C₄⋊C₄	—	71	1	no	—
112	C₂²×C₄⋊C₄	—	71	6	no	—
113	C₂²×C₄⋊C₄	—	71	6	no	—
114	C₂²×C₄⋊C₄	—	71	6	no	—
115	C₂²×C₄⋊C₄	C₂×D₇₁	1	3	yes	—
116	C₂²×C₄²	—	71	6	no	—
117	C₂²×C₄²	C₂×D₇₁	1	3	yes	—
118	C₂²×Q₈	—	71	36	no	—
119	C₂²×Q₈	—	142	24	no	—
120	C₂²×Q₈	Dic₁₄₂	1	16	yes	complemented
121	C₂²×Q₈	C₂²×D₇₁	1	12	yes	—
122	C₂³×C₄	—	71	6	no	—
123	C₂³×C₄	—	71	2	no	—
124	C₂³×C₄	C₂²×D₇₁	1	3	yes	—
125	C₂³×C₄	C₂²×D₇₁	1	3	yes	—
126	C₂³×C₄	C₂²×D₇₁	1	1	yes	characteristic
127	C₄⋊Q₈	—	71	192	no	—
128	#125	—	71	384	no	—
129	C₄×Q₈	—	71	128	no	—
130	C₄×Q₈	—	71	192	no	—
131	C₄×Q₈	C₂²×D₇₁	1	64	yes	—
132	C₂×C₄⋊C₄	—	71	12	no	—
133	C₂×C₄⋊C₄	—	71	72	no	—
134	C₂×C₄⋊C₄	—	71	72	no	—
135	C₂×C₄⋊C₄	—	71	72	no	—
136	C₂×C₄⋊C₄	C₂²×D₇₁	1	36	yes	—
137	C₂×C₄²	—	71	72	no	—
138	C₂×C₄²	C₂²×D₇₁	1	36	yes	—
139	C₂⁴	C₂³×D₇₁	1	1	yes	characteristic
140	C₂×Q₈	—	71	48	no	—
141	C₂×Q₈	—	142	144	no	—
142	C₂×Q₈	C₂×Dic₁₄₂	1	96	yes	complemented
143	C₂×Q₈	C₂³×D₇₁	1	16	yes	—
144	C₂²×C₄	—	2	12	no	—
145	C₂²×C₄	—	71	12	no	—
146	C₂²×C₄	—	71	36	no	—
147	C₂²×C₄	—	142	24	no	—
148	C₂²×C₄	—	142	8	no	—
149	C₂²×C₄	C₂×Dic₁₄₂	1	24	yes	—
150	C₂²×C₄	#11765866	1	8	yes	—
151	C₂²×C₄	C₂³×D₇₁	1	18	yes	—
152	C₂²×C₄	C₂³×D₇₁	1	18	yes	—
153	C₂²×C₄	C₂³×D₇₁	1	6	yes	—
154	C₄⋊C₄	—	71	96	no	—
155	C₄⋊C₄	—	71	16	no	—
156	C₄⋊C₄	—	71	96	no	—
157	C₄⋊C₄	—	71	96	no	—
158	C₄⋊C₄	C₂³×D₇₁	1	48	yes	—
159	C₄²	—	71	96	no	—
160	C₄²	C₂³×D₇₁	1	48	yes	—
161	C₂³	C₂²×Dic₁₄₂	1	4	yes	—
162	C₂³	C₂×#11765866	1	4	yes	—
163	C₂³	#14556175	1	4	yes	—
164	C₂³	C₂⁴×D₇₁	1	3	yes	—
165	Q₈	—	142	96	no	—
166	Q₈	C₂²×Dic₁₄₂	1	64	yes	complemented
167	C₂×C₄	—	2	36	no	—
168	C₂×C₄	—	71	8	no	—
169	C₂×C₄	—	71	24	no	—
170	C₂×C₄	—	142	72	no	—
171	C₂×C₄	—	142	24	no	—
172	C₂×C₄	C₂²×Dic₁₄₂	1	72	yes	—
173	C₂×C₄	C₂×#11765866	1	24	yes	—
174	C₂×C₄	C₂⁴×D₇₁	1	12	yes	—
175	C₂×C₄	C₂⁴×D₇₁	1	12	yes	—
176	C₂×C₄	C₂⁴×D₇₁	1	4	yes	—
177	C₂²	#17986728	1	16	yes	direct product factor
178	C₂²	C₂³×Dic₁₄₂	1	6	yes	—
179	C₂²	C₂²×#11765866	1	6	yes	—
180	C₂²	C₂×#14556175	1	6	yes	—
181	C₂²	C₂⁵×D₇₁	1	1	yes	characteristic
182	C₄	—	2	12	no	—
183	C₄	—	142	24	no	—
184	C₄	—	142	8	no	—
185	C₄	C₂³×Dic₁₄₂	1	24	yes	—
186	C₄	C₂²×#11765866	1	8	yes	—
187	C₂	C₂×#17986728	1	12	yes	direct product factor
188	C₂	C₂⁴×Dic₁₄₂	1	1	yes	characteristic
189	C₂	C₂³×#11765866	1	1	yes	characteristic
190	C₂	C₂²×#14556175	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₇₁
commutator, cosocle	C₂²×C₇₁	C₂⁶
Frattini	C₂²	C₂⁵×D₇₁
Fitting	C₂²×C₄×Q₈×C₇₁	C₂
socle, prime radical	C₂⁴×C₇₁	C₂⁴
hypercenter	C₂²×C₄×Q₈	D₇₁
hypocenter	C₇₁	C₂²×#557
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#17986728	C₂	direct (N × H)
8. #17986728	C₂²	direct (N × H)
35. C₂²×Q₈×C₇₁	Q₈	semidirect
55. C₂×Q₈×C₇₁	C₂×Q₈	semidirect
69. Q₈×C₇₁	C₂²×Q₈	semidirect
81. C₂²×C₇₁	#557	semidirect
92. C₂×C₇₁	C₂×#557	semidirect
102. C₇₁	C₂²×#557	semidirect
120. C₂²×Q₈	Dic₁₄₂	semidirect
142. C₂×Q₈	C₂×Dic₁₄₂	semidirect
166. Q₈	C₂²×Dic₁₄₂	semidirect
177. C₂²	#17986728	direct (N × H)
187. C₂	C₂×#17986728	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17986728	2
C₂²×#14556128	2
C₂²×#14556119	2
C₂²×Q₈×Dic₇₁	2
C₂²×C₄×Dic₁₄₂	2
C₂²×C₄×Q₈×C₇₁	2
C₂²×#557	71

Subgroups — C22×#17986728 #32904498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17986728 #32904498