Subgroups — #46844972

Order 34688.

440 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31774663	C₂	1	2	yes	maximal
2	C₂×#22576166	C₂	1	1	yes	maximal, characteristic
3	#31773945	C₂	1	2	yes	maximal, complemented
4	#31773944	C₂	1	4	yes	maximal, complemented
5	C₂×#22576096	C₂	1	1	yes	maximal, characteristic, complemented
6	#108×D₂₇₁	C₂	1	4	yes	maximal, complemented
7	C₂×#108×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16864654	C₂²	1	1	yes	characteristic
9	#22576168	—	2	4	no	—
10	#22576166	—	2	4	no	—
11	#22576166	C₄	1	4	yes	—
12	#22576166	C₄	1	4	yes	—
13	C₂×#16864638	C₂²	1	2	yes	—
14	C₂×#16864638	C₂²	1	1	yes	characteristic
15	C₂×#16864637	C₂²	1	2	yes	—
16	M₁₆×D₂₇₁	C₂²	1	8	yes	—
17	C₂×C₈×D₂₇₁	C₂²	1	2	yes	—
18	#22576096	C₂²	1	2	yes	—
19	#22576096	C₂²	1	2	yes	—
20	#22576095	C₂²	1	8	yes	—
21	C₂×M₁₆×C₂₇₁	C₂²	1	2	yes	—
22	C₂²×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
23	#108×C₂₇₁	C₂²	1	2	yes	—
24	#108×C₂₇₁	C₂²	1	2	yes	—
25	#16864654	C₂×C₄	1	2	yes	—
26	#16864654	C₂×C₄	1	2	yes	—
27	#16864654	D₄	1	2	yes	—
28	#16864654	D₄	1	2	yes	—
29	C₂×Dic₅₄₂	Q₈	1	1	yes	characteristic
30	C₂×D₁₀₈₄	Q₈	1	1	yes	characteristic
31	C₂×C₂²⋊D₂₇₁	Q₈	1	2	yes	—
32	C₂×C₄×D₂₇₁	C₂³	1	2	yes	—
33	#16864638	—	2	2	no	—
34	#16864638	C₂×C₄	1	2	yes	—
35	#16864638	C₂×C₄	1	2	yes	—
36	#16864638	C₂³	1	4	yes	—
37	#16864637	—	2	8	no	—
38	#16864637	C₂×C₄	1	8	yes	—
39	C₂×C₂₇₁⋊C₈	—	2	2	no	—
40	C₂×C₂₇₁⋊C₈	C₂³	1	2	yes	—
41	C₈×D₂₇₁	—	2	8	no	—
42	C₈×D₂₇₁	C₂×C₄	1	8	yes	—
43	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
44	M₁₆×C₂₇₁	—	2	2	no	—
45	M₁₆×C₂₇₁	C₂³	1	4	yes	—
46	C₂×C₈×C₂₇₁	—	2	2	no	—
47	C₂×C₈×C₂₇₁	C₂×C₄	1	2	yes	—
48	C₂×C₈×C₂₇₁	C₂×C₄	1	2	yes	—
49	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
51	C₂²×D₂₇₁	C₂×Q₈	1	2	yes	—
52	Dic₅₄₂	C₄⋊C₄	1	2	yes	—
53	Dic₅₄₂	C₄⋊C₄	1	2	yes	—
54	D₁₀₈₄	C₄⋊C₄	1	2	yes	—
55	D₁₀₈₄	C₄⋊C₄	1	2	yes	—
56	C₂²⋊D₂₇₁	C₄⋊C₄	1	4	yes	—
57	C₂²⋊D₂₇₁	C₄⋊C₄	1	4	yes	—
58	C₂×Dic₂₇₁	C₂×Q₈	1	2	yes	—
59	C₄×D₂₇₁	C₂²×C₄	1	4	yes	—
60	C₄×D₂₇₁	C₂×D₄	1	4	yes	—
61	C₂₇₁⋊C₈	—	2	4	no	—
62	C₂₇₁⋊C₈	C₂²×C₄	1	4	yes	—
63	C₂³×C₂₇₁	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₂₇₁	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
70	C₈×C₂₇₁	—	2	4	no	—
71	C₈×C₂₇₁	C₂²×C₄	1	2	yes	—
72	C₈×C₂₇₁	C₂²×C₄	1	2	yes	—
73	C₂²×C₂₇₁	—	4	1	no	—
74	C₂²×C₂₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₂₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₂₇₁	C₂²×Q₈	1	1	yes	characteristic
77	C₂×D₂₇₁	—	4	2	no	—
78	C₂×D₂₇₁	C₂×C₄⋊C₄	1	4	yes	—
79	C₄×C₂₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₄×C₂₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₄×C₂₇₁	C₂³×C₄	1	1	yes	characteristic
82	C₄×C₂₇₁	C₂²×D₄	1	1	yes	characteristic
83	Dic₂₇₁	C₂×C₄⋊C₄	1	4	yes	—
84	C₂×C₂₇₁	—	2	1	no	—
85	C₂×C₂₇₁	—	2	1	no	—
86	C₂×C₂₇₁	—	2	1	no	—
87	C₂×C₂₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
88	D₂₇₁	—	2	4	no	—
89	C₂₇₁	#2916	1	1	yes	characteristic, complemented, Sylow
90	#2916	—	271	1	no	maximal, Sylow
91	#568	—	271	2	no	—
92	C₂×#131	—	271	1	no	—
93	#430	—	271	2	no	—
94	#430	—	271	4	no	—
95	C₂×#108	—	271	1	no	—
96	C₂×#108	—	271	4	no	—
97	C₂×#108	D₂₇₁	1	1	yes	characteristic, complemented
98	C₂×#41	—	271	1	no	—
99	#131	—	271	4	no	—
100	#131	—	271	4	no	—
101	#131	—	542	4	no	—
102	#131	—	542	4	no	—
103	C₂×M₁₆	—	271	1	no	—
104	C₂×M₁₆	—	271	2	no	—
105	C₂×M₁₆	—	271	2	no	—
106	C₂×M₁₆	—	271	8	no	—
107	C₂×M₁₆	C₂×D₂₇₁	1	2	yes	—
108	C₂²×C₈	—	271	2	no	—
109	C₂²×C₈	C₂×D₂₇₁	1	1	yes	characteristic
110	#108	—	271	8	no	—
111	#108	—	271	2	no	—
112	#108	—	271	2	no	—
113	#108	C₂×D₂₇₁	1	2	yes	—
114	#108	C₂×D₂₇₁	1	2	yes	—
115	#41	—	271	2	no	—
116	#41	—	271	2	no	—
117	#41	—	271	2	no	—
118	#41	—	271	2	no	—
119	C₂×Q₈	—	271	1	no	—
120	C₂×D₄	—	271	2	no	—
121	C₂×D₄	—	271	1	no	—
122	C₂²×C₄	—	271	2	no	—
123	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
124	M₁₆	—	2	2	no	—
125	M₁₆	—	271	2	no	—
126	M₁₆	—	271	2	no	—
127	M₁₆	—	271	8	no	—
128	M₁₆	—	271	4	no	—
129	M₁₆	—	542	8	no	—
130	M₁₆	—	542	2	no	—
131	M₁₆	C₂²×D₂₇₁	1	4	yes	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	271	2	no	—
134	C₂×C₈	—	271	8	no	—
135	C₂×C₈	—	542	8	no	—
136	C₂×C₈	—	542	2	no	—
137	C₂×C₈	C₄×D₂₇₁	1	2	yes	—
138	C₂×C₈	C₄×D₂₇₁	1	2	yes	—
139	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
140	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
141	C₂³	—	271	2	no	—
142	C₂³	Q₈×D₂₇₁	1	1	yes	characteristic
143	Q₈	—	271	2	no	—
144	Q₈	—	271	2	no	—
145	D₄	—	271	4	no	—
146	D₄	—	271	4	no	—
147	D₄	—	271	2	no	—
148	D₄	—	271	2	no	—
149	C₂×C₄	—	271	2	no	—
150	C₂×C₄	—	271	4	no	—
151	C₂×C₄	—	271	4	no	—
152	C₂×C₄	C₂×C₄×D₂₇₁	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×D₂₇₁	1	1	yes	characteristic
154	C₂×C₄	D₄×D₂₇₁	1	1	yes	characteristic
155	C₂×C₄	D₄×D₂₇₁	1	1	yes	characteristic
156	C₂×C₄	Q₈×D₂₇₁	1	1	yes	characteristic
157	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	271	4	no	—
160	C₈	—	542	4	no	—
161	C₈	C₂×C₄×D₂₇₁	1	2	yes	—
162	C₈	C₂×C₄×D₂₇₁	1	2	yes	—
163	C₂²	—	4	1	no	—
164	C₂²	—	271	4	no	—
165	C₂²	—	1084	2	no	—
166	C₂²	C₄⋊C₄×D₂₇₁	1	1	yes	characteristic
167	C₂²	C₄⋊C₄×D₂₇₁	1	1	yes	characteristic
168	C₂²	C₂×Q₈×D₂₇₁	1	1	yes	characteristic
169	C₄	—	271	4	no	—
170	C₄	C₄⋊C₄×D₂₇₁	1	1	yes	characteristic
171	C₄	C₄⋊C₄×D₂₇₁	1	1	yes	characteristic
172	C₄	C₂²×C₄×D₂₇₁	1	1	yes	characteristic
173	C₄	C₂×D₄×D₂₇₁	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	542	4	no	—
178	C₂	C₂×C₄⋊C₄×D₂₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₂₇₁
commutator	C₄×C₂₇₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₇₁
Fitting	C₂×#108×C₂₇₁	C₂
socle	C₂×C₂₇₁	C₂²×C₄⋊C₄
prime radical	C₂×#16864654	C₂²
cosocle	C₂×C₄×C₂₇₁	C₂⁴
hypercenter	C₂×#108	D₂₇₁
hypocenter	C₂₇₁	#2916
LCS	G ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
3. #31773945	C₂	semidirect
4. #31773944	C₂	semidirect
5. C₂×#22576096	C₂	semidirect
6. #108×D₂₇₁	C₂	semidirect
7. C₂×#108×C₂₇₁	C₂	semidirect
89. C₂₇₁	#2916	semidirect
97. C₂×#108	D₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
#31774663	2
C₂×#22576166	2
#31773945	2
#31773944	2
C₂×#22576096	2
#108×D₂₇₁	2
C₂×#108×C₂₇₁	2
#2916	271