Subgroups — #31888646

Order 18176.

438 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588193	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985751	C₂	1	2	yes	maximal
4	#22586174	C₂	1	2	yes	maximal
5	C₇₁×#1767	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
7	C₂×#14556039	—	2	1	no	—
8	C₂×C₄×C₇₁⋊C₈	C₄	1	2	yes	—
9	C₂×C₄×C₇₁⋊C₈	C₄	1	2	yes	—
10	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
12	#17985754	—	2	1	no	—
13	#17985751	C₄	1	8	yes	complemented
14	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
15	C₂×#97×C₇₁	—	2	1	no	—
16	C₄³×C₇₁	C₂²	1	1	yes	characteristic
17	#336×C₇₁	—	2	1	no	—
18	C₂²×C₇₁⋊C₈	—	2	2	no	—
19	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
24	#14556039	—	4	2	no	—
25	C₄×C₇₁⋊C₈	—	2	4	no	—
26	C₄×C₇₁⋊C₈	—	2	4	no	—
27	C₄×C₇₁⋊C₈	—	2	2	no	—
28	C₄×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₇₁	—	2	2	no	—
30	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₇₁	Q₈	1	2	yes	—
34	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₇₁	—	2	2	no	—
36	C₄⋊C₈×C₇₁	C₂×C₄	1	4	yes	—
37	#97×C₇₁	—	4	2	no	—
38	C₂×C₇₁⋊C₈	—	2	2	no	—
39	C₂×C₇₁⋊C₈	—	2	2	no	—
40	C₂×C₇₁⋊C₈	—	2	2	no	—
41	C₂×C₇₁⋊C₈	—	2	4	no	—
42	C₂×C₇₁⋊C₈	—	4	2	no	—
43	C₂×C₇₁⋊C₈	—	4	2	no	—
44	C₂×C₇₁⋊C₈	C₄²	1	2	yes	—
45	C₂×C₇₁⋊C₈	C₄²	1	2	yes	—
46	C₂×C₇₁⋊C₈	M₁₆	1	4	yes	—
47	C₂×C₇₁⋊C₈	M₁₆	1	4	yes	—
48	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₇₁	—	2	4	no	—
54	C₂×C₈×C₇₁	—	4	2	no	—
55	C₂×C₈×C₇₁	—	4	2	no	—
56	C₄²×C₇₁	—	2	4	no	—
57	C₄²×C₇₁	—	2	2	no	—
58	C₄²×C₇₁	—	2	2	no	—
59	C₄²×C₇₁	—	2	2	no	—
60	C₄²×C₇₁	D₈	1	2	yes	—
61	C₄²×C₇₁	Q₁₆	1	2	yes	—
62	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
63	C₄²×C₇₁	#41	1	2	yes	—
64	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₁	—	2	2	no	—
66	C₂×C₄×C₇₁	—	2	2	no	—
67	C₂×C₄×C₇₁	—	2	2	no	—
68	C₂×C₄×C₇₁	—	2	2	no	—
69	C₂×C₄×C₇₁	—	2	2	no	—
70	C₂×C₄×C₇₁	#107	1	4	yes	—
71	C₂×C₄×C₇₁	#108	1	4	yes	—
72	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
74	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
75	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
76	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
78	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×C₄×C₇₁	#135	1	2	yes	—
81	C₈×C₇₁	—	4	4	no	—
82	C₇₁⋊C₈	—	2	2	no	—
83	C₇₁⋊C₈	—	2	4	no	—
84	C₇₁⋊C₈	—	4	4	no	—
85	C₇₁⋊C₈	#97	1	4	yes	complemented
86	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₇₁	#423	1	2	yes	—
88	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
89	C₂²×C₇₁	#427	1	1	yes	characteristic
90	C₂²×C₇₁	C₂×#108	1	2	yes	—
91	C₄×C₇₁	—	2	4	no	—
92	C₄×C₇₁	—	2	4	no	—
93	C₄×C₇₁	—	2	2	no	—
94	C₄×C₇₁	C₂×#97	1	1	yes	characteristic
95	C₄×C₇₁	C₄×M₁₆	1	1	yes	characteristic
96	C₄×C₇₁	#432	1	2	yes	—
97	C₄×C₇₁	C₄×D₈	1	2	yes	—
98	C₄×C₇₁	C₄×Q₁₆	1	2	yes	—
99	#8274	—	71	1	no	maximal, Sylow
100	C₂×C₇₁	#1567	1	2	yes	—
101	C₂×C₇₁	#1568	1	2	yes	—
102	C₂×C₇₁	#1767	1	1	yes	characteristic
103	C₂×C₇₁	C₄×#107	1	1	yes	characteristic
104	C₂×C₇₁	C₄×#108	1	1	yes	characteristic
105	#1767	—	71	1	no	—
106	#1767	D₇₁	1	1	yes	characteristic
107	C₄²×C₈	—	71	1	no	—
108	C₂×#335	—	71	2	no	—
109	#1277	—	71	2	no	—
110	C₇₁	#8274	1	1	yes	characteristic, complemented, Sylow
111	C₂×C₄⋊C₈	—	71	2	no	—
112	C₂×C₄⋊C₈	C₂×D₇₁	1	2	yes	—
113	C₂×#97	—	2	1	no	—
114	C₂×#97	—	142	1	no	—
115	C₂×C₄×C₈	—	71	1	no	—
116	C₂×C₄×C₈	—	71	2	no	—
117	C₂×C₄×C₈	—	71	2	no	—
118	C₂×C₄×C₈	—	71	1	no	—
119	C₄³	C₂×D₇₁	1	1	yes	characteristic
120	#336	—	2	1	no	—
121	#336	—	142	1	no	—
122	#335	—	71	8	no	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	—	71	1	no	—
125	C₂²×C₈	—	71	1	no	—
126	C₂²×C₈	—	71	2	no	—
127	C₂²×C₈	—	142	2	no	—
128	C₂×C₄²	Dic₁₄₂	1	2	yes	—
129	C₂×C₄²	C₄×D₇₁	1	2	yes	—
130	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
131	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
132	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	71	4	no	—
135	C₄⋊C₈	—	142	2	no	—
136	C₄⋊C₈	C₄×D₇₁	1	4	yes	—
137	#97	—	4	2	no	—
138	#97	—	284	2	no	—
139	C₄×C₈	—	71	4	no	—
140	C₄×C₈	—	142	4	no	—
141	C₄×C₈	—	142	4	no	—
142	C₄×C₈	—	142	2	no	—
143	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
144	C₂²×C₄	Dic₇₁⋊C₄	1	2	yes	—
145	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
147	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	284	2	no	—
152	C₂×C₈	—	284	2	no	—
153	C₂×C₈	—	71	2	no	—
154	C₂×C₈	—	71	4	no	—
155	C₂×C₈	—	71	2	no	—
156	C₂×C₈	—	71	4	no	—
157	C₂×C₈	—	142	4	no	—
158	C₂×C₈	—	142	2	no	—
159	C₂×C₈	—	142	2	no	—
160	C₂×C₈	—	142	2	no	—
161	C₄²	—	2	4	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	#11765850	1	2	yes	—
166	C₄²	#11765855	1	2	yes	—
167	C₄²	C₂×C₄×D₇₁	1	2	yes	—
168	C₄²	#11765865	1	2	yes	—
169	C₂³	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	#14556061	1	4	yes	—
176	C₂×C₄	#14556063	1	4	yes	—
177	C₂×C₄	C₄²×D₇₁	1	1	yes	characteristic
178	C₂×C₄	#14556085	1	1	yes	characteristic
179	C₂×C₄	C₄×C₂²⋊D₇₁	1	2	yes	—
180	C₂×C₄	C₄×Dic₁₄₂	1	2	yes	—
181	C₂×C₄	M₁₆×D₇₁	1	2	yes	—
182	C₂×C₄	M₁₆×D₇₁	1	2	yes	—
183	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
184	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
185	C₂×C₄	#14556148	1	2	yes	—
186	C₈	—	4	4	no	—
187	C₈	—	284	4	no	—
188	C₈	—	71	4	no	—
189	C₈	—	142	4	no	—
190	C₈	—	142	2	no	—
191	C₂²	C₄×Dic₇₁⋊C₄	1	1	yes	characteristic
192	C₂²	#17986022	1	2	yes	—
193	C₂²	C₂×#14556061	1	1	yes	characteristic
194	C₂²	#17986035	1	1	yes	characteristic
195	C₂²	C₂×#14556063	1	2	yes	—
196	C₄	—	2	2	no	—
197	C₄	—	2	4	no	—
198	C₄	—	2	4	no	—
199	C₄	#97×D₇₁	1	1	yes	characteristic
200	C₄	#17985951	1	1	yes	characteristic
201	C₄	#17986054	1	2	yes	—
202	C₄	C₄×#11765850	1	2	yes	—
203	C₄	C₄×#11765855	1	2	yes	—
204	C₂	#22587098	1	2	yes	—
205	C₂	#22587101	1	2	yes	—
206	C₂	#22588194	1	1	yes	characteristic
207	C₂	C₄×#14556061	1	1	yes	characteristic
208	C₂	C₄×#14556063	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1767	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1767	D₇₁
hypocenter	C₇₁	#8274
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
13. #17985751	C₄	semidirect
85. C₇₁⋊C₈	#97	semidirect
110. C₇₁	#8274	semidirect

Maximal subgroups

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

subgroup	index
#22588193	2
C₄²×C₇₁⋊C₈	2
C₂×#17985751	2
#22586174	2
C₇₁×#1767	2
#8274	71