Subgroups — C₂³×#14556108 #32904096

Order 18176.

14078 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₄×D₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×#14556108	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₄⋊Dic₇₁	C₂	1	2	yes	maximal, complemented
5	C₂³×(C₂×D₇₁)⋊C₄	C₂	1	2	yes	maximal
6	C₂³×C₄⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₇₁	C₂²	1	1	yes	characteristic
8	C₂³×Dic₁₄₂	—	2	16	no	—
9	C₂³×Dic₁₄₂	C₂²	1	28	yes	complemented
10	C₂⁴×Dic₇₁	C₂²	1	2	yes	—
11	C₂⁴×Dic₇₁	C₂²	1	1	yes	characteristic
12	C₂³×C₄×D₇₁	—	2	16	no	—
13	C₂³×C₄×D₇₁	C₂²	1	28	yes	—
14	C₂×#14556108	C₂²	1	448	yes	direct product factor
15	C₂²×C₄⋊Dic₇₁	C₂²	1	56	yes	complemented
16	C₂²×(C₂×D₇₁)⋊C₄	C₂²	1	56	yes	—
17	C₂⁴×C₄×C₇₁	C₂²	1	2	yes	—
18	C₂⁴×C₄×C₇₁	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₇₁	C₂²	1	28	yes	complemented
20	C₂⁴×D₇₁	—	2	8	no	—
21	C₂⁴×D₇₁	—	2	8	no	—
22	C₂⁴×D₇₁	Q₈	1	16	yes	complemented
23	C₂⁴×D₇₁	C₂³	1	14	yes	—
24	C₂²×Dic₁₄₂	—	2	224	no	—
25	C₂²×Dic₁₄₂	C₂³	1	112	yes	complemented
26	C₂³×Dic₇₁	—	2	8	no	—
27	C₂³×Dic₇₁	—	2	16	no	—
28	C₂³×Dic₇₁	C₂³	1	14	yes	—
29	C₂³×Dic₇₁	C₂³	1	28	yes	—
30	C₂²×C₄×D₇₁	—	2	224	no	—
31	C₂²×C₄×D₇₁	C₂³	1	112	yes	—
32	#14556108	C₂³	1	512	yes	direct product factor
33	C₂×C₄⋊Dic₇₁	C₂³	1	224	yes	complemented
34	C₂×(C₂×D₇₁)⋊C₄	C₂³	1	224	yes	—
35	C₂⁵×C₇₁	C₂³	1	1	yes	characteristic
36	C₂³×C₄×C₇₁	—	2	16	no	—
37	C₂³×C₄×C₇₁	D₄	1	16	yes	—
38	C₂³×C₄×C₇₁	C₂³	1	28	yes	—
39	C₂³×C₄×C₇₁	C₂³	1	14	yes	—
40	C₂×C₄⋊C₄×C₇₁	C₂³	1	112	yes	complemented
41	C₂³×D₇₁	—	2	128	no	—
42	C₂³×D₇₁	—	2	56	no	—
43	C₂³×D₇₁	—	2	56	no	—
44	C₂³×D₇₁	C₂×Q₈	1	112	yes	complemented
45	C₂³×D₇₁	C₂⁴	1	28	yes	—
46	C₂×Dic₁₄₂	—	2	448	no	—
47	C₂×Dic₁₄₂	C₂⁴	1	64	yes	complemented
48	C₂²×Dic₇₁	—	2	56	no	—
49	C₂²×Dic₇₁	—	2	112	no	—
50	C₂²×Dic₇₁	C₂⁴	1	28	yes	—
51	C₂²×Dic₇₁	C₂⁴	1	56	yes	—
52	C₂×C₄×D₇₁	—	2	448	no	—
53	C₂×C₄×D₇₁	C₂⁴	1	64	yes	—
54	C₄⋊Dic₇₁	C₂⁴	1	128	yes	complemented
55	(C₂×D₇₁)⋊C₄	C₂⁴	1	128	yes	—
56	C₂⁴×C₇₁	C₂×D₄	1	8	yes	—
57	C₂⁴×C₇₁	C₂×Q₈	1	8	yes	—
58	C₂⁴×C₇₁	#41	1	8	yes	—
59	C₂⁴×C₇₁	C₂⁴	1	7	yes	—
60	C₂²×C₄×C₇₁	—	2	112	no	—
61	C₂²×C₄×C₇₁	C₂×D₄	1	112	yes	—
62	C₂²×C₄×C₇₁	C₂⁴	1	56	yes	—
63	C₂²×C₄×C₇₁	C₂⁴	1	28	yes	—
64	C₄⋊C₄×C₇₁	C₂⁴	1	64	yes	complemented
65	C₂³×C₇₁	C₂²⋊Q₈	1	64	yes	complemented
66	C₂³×C₇₁	C₂²×D₄	1	28	yes	—
67	C₂³×C₇₁	C₂²×Q₈	1	28	yes	—
68	C₂³×C₇₁	C₂×#41	1	28	yes	—
69	C₂³×C₇₁	C₂⁵	1	7	yes	—
70	C₂²×D₇₁	—	2	448	no	—
71	C₂²×D₇₁	—	2	56	no	—
72	C₂²×D₇₁	—	2	56	no	—
73	C₂²×D₇₁	C₂²×Q₈	1	112	yes	complemented
74	C₂²×D₇₁	C₂⁵	1	8	yes	—
75	C₂×C₄×C₇₁	—	2	112	no	—
76	C₂×C₄×C₇₁	C₂²×D₄	1	112	yes	—
77	C₂×C₄×C₇₁	C₂⁵	1	16	yes	—
78	C₂×C₄×C₇₁	C₂⁵	1	8	yes	—
79	C₂×Dic₇₁	—	2	56	no	—
80	C₂×Dic₇₁	—	2	112	no	—
81	C₂×Dic₇₁	C₂⁵	1	16	yes	—
82	C₂×Dic₇₁	C₂⁵	1	8	yes	—
83	C₄×D₇₁	—	2	128	no	—
84	Dic₁₄₂	—	2	128	no	—
85	C₂²×C₇₁	C₂×C₂²⋊Q₈	1	112	yes	complemented
86	C₂²×C₇₁	C₂³×D₄	1	14	yes	—
87	C₂²×C₇₁	C₂³×Q₈	1	14	yes	—
88	C₂²×C₇₁	C₂²×#41	1	14	yes	—
89	C₂²×C₇₁	C₂⁶	1	1	yes	characteristic
90	C₂×D₇₁	—	2	8	no	—
91	C₂×D₇₁	—	2	8	no	—
92	C₂×D₇₁	—	2	224	no	—
93	C₂×D₇₁	C₂³×Q₈	1	16	yes	complemented
94	C₄×C₇₁	—	2	16	no	—
95	C₄×C₇₁	C₂³×D₄	1	16	yes	—
96	Dic₇₁	—	2	16	no	—
97	Dic₇₁	—	2	8	no	—
98	C₂³×C₂²⋊Q₈	—	71	1	no	maximal, Sylow
99	C₂×C₇₁	C₂²×C₂²⋊Q₈	1	28	yes	complemented
100	C₂×C₇₁	C₂⁴×D₄	1	1	yes	characteristic
101	C₂×C₇₁	C₂⁴×Q₈	1	1	yes	characteristic
102	C₂×C₇₁	C₂³×#41	1	1	yes	characteristic
103	D₇₁	—	2	16	no	—
104	C₂⁴×Q₈	—	71	1	no	—
105	C₂⁵×C₄	—	71	1	no	—
106	C₂²×C₂²⋊Q₈	—	71	56	no	—
107	C₂³×C₄⋊C₄	—	71	2	no	—
108	C₂³×C₄⋊C₄	D₇₁	1	1	yes	characteristic, complemented
109	C₂³×C₂²⋊C₄	—	71	2	no	—
110	C₇₁	C₂³×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
111	C₂⁶	—	71	1	no	—
112	C₂³×Q₈	—	71	28	no	—
113	C₂³×Q₈	—	142	16	no	—
114	C₂⁴×C₄	—	71	1	no	—
115	C₂⁴×C₄	—	71	2	no	—
116	C₂⁴×C₄	—	71	28	no	—
117	C₂⁴×C₄	—	142	16	no	—
118	C₂⁴×C₄	C₂×D₇₁	1	2	yes	—
119	C₂⁴×C₄	C₂×D₇₁	1	1	yes	characteristic
120	C₂×C₂²⋊Q₈	—	71	448	no	—
121	C₂²×C₄⋊C₄	—	71	56	no	—
122	C₂²×C₄⋊C₄	C₂×D₇₁	1	28	yes	complemented
123	C₂²×C₂²⋊C₄	—	71	56	no	—
124	C₂⁵	—	71	16	no	—
125	C₂⁵	—	71	14	no	—
126	C₂⁵	—	142	8	no	—
127	C₂⁵	—	142	8	no	—
128	C₂⁵	C₂²×D₇₁	1	1	yes	characteristic
129	C₂²×Q₈	—	71	112	no	—
130	C₂²×Q₈	—	142	224	no	—
131	C₂³×C₄	—	2	16	no	—
132	C₂³×C₄	—	71	28	no	—
133	C₂³×C₄	—	71	14	no	—
134	C₂³×C₄	—	71	112	no	—
135	C₂³×C₄	—	142	8	no	—
136	C₂³×C₄	—	142	16	no	—
137	C₂³×C₄	—	142	224	no	—
138	C₂³×C₄	D₂₈₄	1	16	yes	—
139	C₂³×C₄	C₂²×D₇₁	1	28	yes	—
140	C₂³×C₄	C₂²×D₇₁	1	14	yes	—
141	C₂²⋊Q₈	—	71	512	no	—
142	C₂×C₄⋊C₄	—	71	224	no	—
143	C₂×C₄⋊C₄	C₂²×D₇₁	1	112	yes	complemented
144	C₂×C₂²⋊C₄	—	71	224	no	—
145	C₂⁴	—	71	112	no	—
146	C₂⁴	—	71	28	no	—
147	C₂⁴	—	142	56	no	—
148	C₂⁴	—	142	56	no	—
149	C₂⁴	—	142	128	no	—
150	C₂⁴	C₂×D₂₈₄	1	8	yes	—
151	C₂⁴	Q₈×D₇₁	1	8	yes	—
152	C₂⁴	#11765864	1	8	yes	—
153	C₂⁴	C₂³×D₇₁	1	7	yes	—
154	C₂×Q₈	—	71	64	no	—
155	C₂×Q₈	—	142	448	no	—
156	C₂²×C₄	—	2	112	no	—
157	C₂²×C₄	—	71	56	no	—
158	C₂²×C₄	—	71	28	no	—
159	C₂²×C₄	—	71	64	no	—
160	C₂²×C₄	—	142	56	no	—
161	C₂²×C₄	—	142	112	no	—
162	C₂²×C₄	—	142	448	no	—
163	C₂²×C₄	C₂×D₂₈₄	1	112	yes	—
164	C₂²×C₄	C₂³×D₇₁	1	56	yes	—
165	C₂²×C₄	C₂³×D₇₁	1	28	yes	—
166	C₄⋊C₄	—	71	128	no	—
167	C₄⋊C₄	C₂³×D₇₁	1	64	yes	complemented
168	C₂²⋊C₄	—	71	128	no	—
169	C₂³	—	71	112	no	—
170	C₂³	—	71	8	no	—
171	C₂³	—	142	56	no	—
172	C₂³	—	142	56	no	—
173	C₂³	—	142	448	no	—
174	C₂³	#14556108	1	64	yes	direct product factor
175	C₂³	C₂²×D₂₈₄	1	28	yes	—
176	C₂³	C₂×Q₈×D₇₁	1	28	yes	—
177	C₂³	C₂×#11765864	1	28	yes	—
178	C₂³	C₂⁴×D₇₁	1	7	yes	—
179	Q₈	—	142	128	no	—
180	C₂×C₄	—	2	112	no	—
181	C₂×C₄	—	71	16	no	—
182	C₂×C₄	—	71	8	no	—
183	C₂×C₄	—	142	56	no	—
184	C₂×C₄	—	142	112	no	—
185	C₂×C₄	—	142	128	no	—
186	C₂×C₄	C₂²×D₂₈₄	1	112	yes	—
187	C₂×C₄	C₂⁴×D₇₁	1	16	yes	—
188	C₂×C₄	C₂⁴×D₇₁	1	8	yes	—
189	C₂²	—	71	16	no	—
190	C₂²	—	142	224	no	—
191	C₂²	—	142	8	no	—
192	C₂²	—	142	8	no	—
193	C₂²	C₂×#14556108	1	112	yes	direct product factor
194	C₂²	C₂³×D₂₈₄	1	14	yes	—
195	C₂²	C₂²×Q₈×D₇₁	1	14	yes	—
196	C₂²	C₂²×#11765864	1	14	yes	—
197	C₂²	C₂⁵×D₇₁	1	1	yes	characteristic
198	C₄	—	2	16	no	—
199	C₄	—	142	8	no	—
200	C₄	—	142	16	no	—
201	C₄	C₂³×D₂₈₄	1	16	yes	—
202	C₂	—	142	16	no	—
203	C₂	C₂²×#14556108	1	28	yes	direct product factor
204	C₂	C₂⁴×D₂₈₄	1	1	yes	characteristic
205	C₂	C₂³×Q₈×D₇₁	1	1	yes	characteristic
206	C₂	C₂³×#11765864	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂⁴×Dic₁₄₂	C₂	semidirect
3. C₂²×#14556108	C₂	direct (N × H)
4. C₂³×C₄⋊Dic₇₁	C₂	semidirect
6. C₂³×C₄⋊C₄×C₇₁	C₂	semidirect
9. C₂³×Dic₁₄₂	C₂²	semidirect
14. C₂×#14556108	C₂²	direct (N × H)
15. C₂²×C₄⋊Dic₇₁	C₂²	semidirect
19. C₂²×C₄⋊C₄×C₇₁	C₂²	semidirect
22. C₂⁴×D₇₁	Q₈	semidirect
25. C₂²×Dic₁₄₂	C₂³	semidirect
32. #14556108	C₂³	direct (N × H)
33. C₂×C₄⋊Dic₇₁	C₂³	semidirect
40. C₂×C₄⋊C₄×C₇₁	C₂³	semidirect
44. C₂³×D₇₁	C₂×Q₈	semidirect
47. C₂×Dic₁₄₂	C₂⁴	semidirect
54. C₄⋊Dic₇₁	C₂⁴	semidirect
64. C₄⋊C₄×C₇₁	C₂⁴	semidirect
65. C₂³×C₇₁	C₂²⋊Q₈	semidirect
73. C₂²×D₇₁	C₂²×Q₈	semidirect
85. C₂²×C₇₁	C₂×C₂²⋊Q₈	semidirect
93. C₂×D₇₁	C₂³×Q₈	semidirect
99. C₂×C₇₁	C₂²×C₂²⋊Q₈	semidirect
108. C₂³×C₄⋊C₄	D₇₁	semidirect
110. C₇₁	C₂³×C₂²⋊Q₈	semidirect
122. C₂²×C₄⋊C₄	C₂×D₇₁	semidirect
143. C₂×C₄⋊C₄	C₂²×D₇₁	semidirect
167. C₄⋊C₄	C₂³×D₇₁	semidirect
174. C₂³	#14556108	direct (N × H)
193. C₂²	C₂×#14556108	direct (N × H)
203. C₂	C₂²×#14556108	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Dic₁₄₂	2
C₂⁴×C₄×D₇₁	2
C₂²×#14556108	2
C₂³×C₄⋊Dic₇₁	2
C₂³×(C₂×D₇₁)⋊C₄	2
C₂³×C₄⋊C₄×C₇₁	2
C₂³×C₂²⋊Q₈	71

Subgroups — C23×#14556108 #32904096

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14556108 #32904096