Subgroups — C₂³×#14556118 #32904118

Order 18176.

8414 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×#14556118	C₂	1	56	yes	maximal, direct product factor
2	C₂³×C₄⋊Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂³×Dic₇₁⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₇₁⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₇₁⋊C₄	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₇₁⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×Dic₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₄⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic
9	C₂⁴×Dic₇₁	C₂²	1	1	yes	characteristic
10	C₂⁴×Dic₇₁	C₂²	1	1	yes	characteristic
11	C₂⁴×Dic₇₁	C₂²	1	1	yes	characteristic
12	C₂⁴×Dic₇₁	C₂²	1	1	yes	characteristic
13	C₂×#14556118	C₂²	1	448	yes	direct product factor
14	C₂²×C₄⋊Dic₇₁	C₂²	1	28	yes	—
15	C₂²×Dic₇₁⋊C₄	C₂²	1	28	yes	—
16	C₂²×Dic₇₁⋊C₄	C₂²	1	28	yes	—
17	C₂²×Dic₇₁⋊C₄	C₂²	1	28	yes	—
18	C₂²×Dic₇₁⋊C₄	C₂²	1	28	yes	—
19	C₂²×C₄×Dic₇₁	C₂²	1	28	yes	—
20	C₂⁴×C₄×C₇₁	C₂²	1	1	yes	characteristic
21	C₂⁴×C₄×C₇₁	C₂²	1	1	yes	characteristic
22	C₂⁴×C₄×C₇₁	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₇₁	C₂²	1	28	yes	—
24	C₂³×Dic₇₁	—	2	8	no	—
25	C₂³×Dic₇₁	—	2	8	no	—
26	C₂³×Dic₇₁	—	2	8	no	—
27	C₂³×Dic₇₁	Q₈	1	16	yes	—
28	C₂³×Dic₇₁	C₂³	1	14	yes	—
29	C₂³×Dic₇₁	C₂³	1	14	yes	—
30	C₂³×Dic₇₁	C₂³	1	14	yes	—
31	C₂³×Dic₇₁	C₂³	1	14	yes	—
32	#14556118	C₂³	1	512	yes	direct product factor
33	C₂×C₄⋊Dic₇₁	C₂³	1	112	yes	—
34	C₂×Dic₇₁⋊C₄	C₂³	1	112	yes	—
35	C₂×Dic₇₁⋊C₄	C₂³	1	112	yes	—
36	C₂×Dic₇₁⋊C₄	C₂³	1	112	yes	—
37	C₂×Dic₇₁⋊C₄	C₂³	1	112	yes	—
38	C₂×C₄×Dic₇₁	C₂³	1	112	yes	—
39	C₂⁵×C₇₁	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₇₁	—	2	8	no	—
41	C₂³×C₄×C₇₁	—	2	8	no	—
42	C₂³×C₄×C₇₁	—	2	8	no	—
43	C₂³×C₄×C₇₁	C₂³	1	14	yes	—
44	C₂³×C₄×C₇₁	C₂³	1	14	yes	—
45	C₂³×C₄×C₇₁	C₂³	1	14	yes	—
46	C₂×C₄⋊C₄×C₇₁	C₂³	1	112	yes	—
47	C₂²×Dic₇₁	—	2	56	no	—
48	C₂²×Dic₇₁	—	2	56	no	—
49	C₂²×Dic₇₁	—	2	56	no	—
50	C₂²×Dic₇₁	C₂×Q₈	1	112	yes	—
51	C₂²×Dic₇₁	C₂⁴	1	28	yes	—
52	C₂²×Dic₇₁	C₂⁴	1	28	yes	—
53	C₂²×Dic₇₁	C₂⁴	1	28	yes	—
54	C₂²×Dic₇₁	C₂⁴	1	28	yes	—
55	C₄⋊Dic₇₁	C₂⁴	1	64	yes	—
56	Dic₇₁⋊C₄	C₂⁴	1	64	yes	—
57	Dic₇₁⋊C₄	C₂⁴	1	64	yes	—
58	Dic₇₁⋊C₄	C₂⁴	1	64	yes	—
59	Dic₇₁⋊C₄	C₂⁴	1	64	yes	—
60	C₄×Dic₇₁	C₂⁴	1	64	yes	—
61	C₂⁴×C₇₁	C₂×Q₈	1	8	yes	—
62	C₂⁴×C₇₁	#41	1	8	yes	—
63	C₂⁴×C₇₁	#41	1	8	yes	—
64	C₂⁴×C₇₁	C₂⁴	1	7	yes	—
65	C₂²×C₄×C₇₁	—	2	56	no	—
66	C₂²×C₄×C₇₁	—	2	56	no	—
67	C₂²×C₄×C₇₁	—	2	56	no	—
68	C₂²×C₄×C₇₁	C₂⁴	1	28	yes	—
69	C₂²×C₄×C₇₁	C₂⁴	1	28	yes	—
70	C₂²×C₄×C₇₁	C₂⁴	1	28	yes	—
71	C₄⋊C₄×C₇₁	C₂⁴	1	64	yes	—
72	C₂³×C₇₁	#125	1	64	yes	complemented
73	C₂³×C₇₁	C₂²×Q₈	1	28	yes	—
74	C₂³×C₇₁	C₂×#41	1	28	yes	—
75	C₂³×C₇₁	C₂×#41	1	28	yes	—
76	C₂³×C₇₁	C₂⁵	1	7	yes	—
77	C₂×C₄×C₇₁	—	2	56	no	—
78	C₂×C₄×C₇₁	—	2	56	no	—
79	C₂×C₄×C₇₁	—	2	56	no	—
80	C₂×C₄×C₇₁	C₂⁵	1	8	yes	—
81	C₂×C₄×C₇₁	C₂⁵	1	8	yes	—
82	C₂×C₄×C₇₁	C₂⁵	1	8	yes	—
83	C₂×Dic₇₁	—	2	56	no	—
84	C₂×Dic₇₁	—	2	56	no	—
85	C₂×Dic₇₁	—	2	56	no	—
86	C₂×Dic₇₁	C₂²×Q₈	1	112	yes	—
87	C₂×Dic₇₁	C₂⁵	1	8	yes	—
88	C₂×Dic₇₁	C₂⁵	1	8	yes	—
89	C₂×Dic₇₁	C₂⁵	1	8	yes	—
90	C₂×Dic₇₁	C₂⁵	1	8	yes	—
91	C₂²×C₇₁	C₂×#125	1	112	yes	complemented
92	C₂²×C₇₁	C₂³×Q₈	1	14	yes	—
93	C₂²×C₇₁	C₂²×#41	1	14	yes	—
94	C₂²×C₇₁	C₂²×#41	1	14	yes	—
95	C₂²×C₇₁	C₂⁶	1	1	yes	characteristic
96	C₄×C₇₁	—	2	8	no	—
97	C₄×C₇₁	—	2	8	no	—
98	C₄×C₇₁	—	2	8	no	—
99	Dic₇₁	—	2	8	no	—
100	Dic₇₁	—	2	8	no	—
101	Dic₇₁	—	2	8	no	—
102	Dic₇₁	C₂³×Q₈	1	16	yes	—
103	C₂³×#125	—	71	1	no	maximal, Sylow
104	C₂×C₇₁	C₂²×#125	1	28	yes	complemented
105	C₂×C₇₁	C₂⁴×Q₈	1	1	yes	characteristic
106	C₂×C₇₁	C₂³×#41	1	1	yes	characteristic
107	C₂×C₇₁	C₂³×#41	1	1	yes	characteristic
108	C₂²×#125	—	71	56	no	—
109	C₂³×C₄⋊C₄	—	71	1	no	—
110	C₂³×C₄⋊C₄	—	71	1	no	—
111	C₂³×C₄⋊C₄	—	71	1	no	—
112	C₂³×C₄⋊C₄	—	71	1	no	—
113	C₂³×C₄⋊C₄	—	71	1	no	—
114	C₂³×C₄⋊C₄	D₇₁	1	1	yes	characteristic
115	C₂³×C₄²	—	71	1	no	—
116	C₇₁	C₂³×#125	1	1	yes	characteristic, complemented, Sylow
117	C₂⁴×C₄	—	71	1	no	—
118	C₂⁴×C₄	—	71	1	no	—
119	C₂⁴×C₄	—	71	1	no	—
120	C₂⁴×C₄	—	71	1	no	—
121	C₂⁴×C₄	C₂×D₇₁	1	1	yes	characteristic
122	C₂⁴×C₄	C₂×D₇₁	1	1	yes	characteristic
123	C₂⁴×C₄	C₂×D₇₁	1	1	yes	characteristic
124	C₂×#125	—	71	448	no	—
125	C₂²×C₄⋊C₄	—	71	28	no	—
126	C₂²×C₄⋊C₄	—	71	28	no	—
127	C₂²×C₄⋊C₄	—	71	28	no	—
128	C₂²×C₄⋊C₄	—	71	28	no	—
129	C₂²×C₄⋊C₄	—	71	28	no	—
130	C₂²×C₄⋊C₄	C₂×D₇₁	1	28	yes	—
131	C₂²×C₄²	—	71	28	no	—
132	C₂⁵	C₂²×D₇₁	1	1	yes	characteristic
133	C₂³×C₄	—	2	8	no	—
134	C₂³×C₄	—	2	8	no	—
135	C₂³×C₄	—	2	8	no	—
136	C₂³×C₄	—	71	16	no	—
137	C₂³×C₄	—	71	14	no	—
138	C₂³×C₄	—	71	14	no	—
139	C₂³×C₄	—	71	14	no	—
140	C₂³×C₄	—	71	14	no	—
141	C₂³×C₄	—	142	8	no	—
142	C₂³×C₄	—	142	8	no	—
143	C₂³×C₄	—	142	8	no	—
144	C₂³×C₄	C₂²×D₇₁	1	14	yes	—
145	C₂³×C₄	C₂²×D₇₁	1	14	yes	—
146	C₂³×C₄	C₂²×D₇₁	1	14	yes	—
147	#125	—	71	512	no	—
148	C₂×C₄⋊C₄	—	71	112	no	—
149	C₂×C₄⋊C₄	—	71	112	no	—
150	C₂×C₄⋊C₄	—	71	112	no	—
151	C₂×C₄⋊C₄	—	71	112	no	—
152	C₂×C₄⋊C₄	—	71	112	no	—
153	C₂×C₄⋊C₄	C₂²×D₇₁	1	112	yes	—
154	C₂×C₄²	—	71	112	no	—
155	C₂⁴	Q₈×D₇₁	1	8	yes	—
156	C₂⁴	#11765864	1	8	yes	—
157	C₂⁴	#11765865	1	8	yes	—
158	C₂⁴	C₂³×D₇₁	1	7	yes	—
159	C₂²×C₄	—	2	56	no	—
160	C₂²×C₄	—	2	56	no	—
161	C₂²×C₄	—	2	56	no	—
162	C₂²×C₄	—	71	112	no	—
163	C₂²×C₄	—	71	28	no	—
164	C₂²×C₄	—	71	28	no	—
165	C₂²×C₄	—	71	28	no	—
166	C₂²×C₄	—	71	28	no	—
167	C₂²×C₄	—	142	56	no	—
168	C₂²×C₄	—	142	56	no	—
169	C₂²×C₄	—	142	56	no	—
170	C₂²×C₄	C₂³×D₇₁	1	28	yes	—
171	C₂²×C₄	C₂³×D₇₁	1	28	yes	—
172	C₂²×C₄	C₂³×D₇₁	1	28	yes	—
173	C₄⋊C₄	—	71	64	no	—
174	C₄⋊C₄	—	71	64	no	—
175	C₄⋊C₄	—	71	64	no	—
176	C₄⋊C₄	—	71	64	no	—
177	C₄⋊C₄	—	71	64	no	—
178	C₄⋊C₄	C₂³×D₇₁	1	64	yes	—
179	C₄²	—	71	64	no	—
180	C₂³	#14556118	1	64	yes	direct product factor
181	C₂³	C₂×Q₈×D₇₁	1	28	yes	—
182	C₂³	C₂×#11765864	1	28	yes	—
183	C₂³	C₂×#11765865	1	28	yes	—
184	C₂³	C₂⁴×D₇₁	1	7	yes	—
185	C₂×C₄	—	2	56	no	—
186	C₂×C₄	—	2	56	no	—
187	C₂×C₄	—	2	56	no	—
188	C₂×C₄	—	71	112	no	—
189	C₂×C₄	—	71	8	no	—
190	C₂×C₄	—	71	8	no	—
191	C₂×C₄	—	71	8	no	—
192	C₂×C₄	—	71	8	no	—
193	C₂×C₄	—	142	56	no	—
194	C₂×C₄	—	142	56	no	—
195	C₂×C₄	—	142	56	no	—
196	C₂×C₄	C₂⁴×D₇₁	1	8	yes	—
197	C₂×C₄	C₂⁴×D₇₁	1	8	yes	—
198	C₂×C₄	C₂⁴×D₇₁	1	8	yes	—
199	C₂²	C₂×#14556118	1	112	yes	direct product factor
200	C₂²	C₂²×Q₈×D₇₁	1	14	yes	—
201	C₂²	C₂²×#11765864	1	14	yes	—
202	C₂²	C₂²×#11765865	1	14	yes	—
203	C₂²	C₂⁵×D₇₁	1	1	yes	characteristic
204	C₄	—	2	8	no	—
205	C₄	—	2	8	no	—
206	C₄	—	2	8	no	—
207	C₄	—	71	16	no	—
208	C₄	—	142	8	no	—
209	C₄	—	142	8	no	—
210	C₄	—	142	8	no	—
211	C₂	C₂²×#14556118	1	28	yes	direct product factor
212	C₂	C₂³×Q₈×D₇₁	1	1	yes	characteristic
213	C₂	C₂³×#11765864	1	1	yes	characteristic
214	C₂	C₂³×#11765865	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, prime radical	C₂⁵×C₇₁	C₂³
hypercenter	C₂³×C₄⋊C₄	D₇₁
hypocenter	C₇₁	C₂³×#125
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. C₂²×#14556118	C₂	direct (N × H)
13. C₂×#14556118	C₂²	direct (N × H)
32. #14556118	C₂³	direct (N × H)
72. C₂³×C₇₁	#125	semidirect
91. C₂²×C₇₁	C₂×#125	semidirect
104. C₂×C₇₁	C₂²×#125	semidirect
116. C₇₁	C₂³×#125	semidirect
180. C₂³	#14556118	direct (N × H)
199. C₂²	C₂×#14556118	direct (N × H)
211. C₂	C₂²×#14556118	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×#14556118	2
C₂³×C₄⋊Dic₇₁	2
C₂³×Dic₇₁⋊C₄	2
C₂³×C₄×Dic₇₁	2
C₂³×C₄⋊C₄×C₇₁	2
C₂³×#125	71

Subgroups — C23×#14556118 #32904118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14556118 #32904118