Subgroups — C₂×#31773854 #46844778

Order 34688.

592 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#16864637	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×D₂₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#22576117	C₂	1	1	yes	maximal, characteristic
4	#31773854	C₂	1	8	yes	maximal, direct product factor
5	C₂×#22576070	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₂₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×Dic₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#97×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₄×D₂₇₁	C₂²	1	1	yes	characteristic
10	C₂×#16864637	C₄	1	8	yes	complemented
11	C₂×#16864637	C₂²	1	4	yes	—
12	C₂²×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₂₇₁	C₄	1	8	yes	complemented
15	C₂×C₈×D₂₇₁	C₂²	1	4	yes	—
16	#22576117	C₄	1	4	yes	—
17	#22576117	C₂²	1	4	yes	—
18	C₂×Dic₂₇₁⋊C₄	C₄	1	2	yes	—
19	C₂×(C₂×D₂₇₁)⋊C₄	C₄	1	2	yes	—
20	C₂×C₄×Dic₂₇₁	C₂²	1	1	yes	characteristic
21	#22576070	C₂²	1	4	yes	complemented
22	C₄×C₂₇₁⋊C₈	C₂²	1	4	yes	complemented
23	C₈×Dic₂₇₁	C₂²	1	4	yes	complemented
24	C₂²×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
25	C₂²×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₂₇₁	C₂²	1	1	yes	characteristic
27	#97×C₂₇₁	C₂²	1	4	yes	complemented
28	C₂³×D₂₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×Dic₂₇₁	C₂×C₄	1	2	yes	—
30	C₂²×Dic₂₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₂₇₁	C₂×C₄	1	2	yes	—
32	C₂×C₄×D₂₇₁	C₂×C₄	1	4	yes	—
33	C₂×C₄×D₂₇₁	C₂×C₄	1	4	yes	—
34	C₂×C₄×D₂₇₁	C₂³	1	2	yes	—
35	#16864637	C₂×C₄	1	16	yes	complemented
36	C₂×C₂₇₁⋊C₈	C₂×C₄	1	4	yes	—
37	C₂×C₂₇₁⋊C₈	C₂×C₄	1	4	yes	—
38	C₂×C₂₇₁⋊C₈	C₂³	1	2	yes	—
39	C₂×C₂₇₁⋊C₈	C₂³	1	2	yes	—
40	C₈×D₂₇₁	C₂×C₄	1	16	yes	complemented
41	Dic₂₇₁⋊C₄	C₂×C₄	1	8	yes	—
42	(C₂×D₂₇₁)⋊C₄	C₂×C₄	1	8	yes	—
43	C₄×Dic₂₇₁	C₂×C₄	1	2	yes	—
44	C₄×Dic₂₇₁	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₇₁	C₂×C₄	1	2	yes	—
46	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₂₇₁	C₂×C₄	1	4	yes	—
48	C₂×C₈×C₂₇₁	C₂×C₄	1	4	yes	—
49	C₂×C₈×C₂₇₁	C₂³	1	2	yes	—
50	C₂×C₈×C₂₇₁	C₂³	1	2	yes	—
51	C₄²×C₂₇₁	C₂×C₄	1	2	yes	—
52	C₄²×C₂₇₁	C₂³	1	2	yes	—
53	C₂²×D₂₇₁	—	2	4	no	—
54	C₂²×D₂₇₁	C₄²	1	4	yes	—
55	C₂²×D₂₇₁	C₂²×C₄	1	2	yes	—
56	C₂×Dic₂₇₁	—	2	4	no	—
57	C₂×Dic₂₇₁	C₄²	1	4	yes	—
58	C₂×Dic₂₇₁	C₂²×C₄	1	4	yes	—
59	C₂×Dic₂₇₁	C₂²×C₄	1	2	yes	—
60	C₄×D₂₇₁	C₄²	1	8	yes	—
61	C₄×D₂₇₁	C₂²×C₄	1	4	yes	—
62	C₄×D₂₇₁	C₂²×C₄	1	4	yes	—
63	C₂₇₁⋊C₈	C₂²×C₄	1	4	yes	—
64	C₂₇₁⋊C₈	C₂²×C₄	1	4	yes	—
65	C₂³×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₂₇₁	—	2	4	no	—
67	C₂×C₄×C₂₇₁	C₂²×C₄	1	4	yes	—
68	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₂₇₁	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₂₇₁	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
72	C₈×C₂₇₁	C₂²×C₄	1	4	yes	—
73	C₈×C₂₇₁	C₂²×C₄	1	4	yes	—
74	C₂²×C₂₇₁	C₂×C₄²	1	2	yes	—
75	C₂²×C₂₇₁	#131	1	4	yes	—
76	C₂²×C₂₇₁	C₂³×C₄	1	1	yes	characteristic
77	C₂×D₂₇₁	—	2	4	no	—
78	C₂×D₂₇₁	—	2	8	no	—
79	C₂×D₂₇₁	C₂×C₄²	1	4	yes	—
80	C₄×C₂₇₁	—	2	4	no	—
81	C₄×C₂₇₁	C₂×C₄²	1	2	yes	—
82	C₄×C₂₇₁	C₂³×C₄	1	1	yes	characteristic
83	C₄×C₂₇₁	C₂³×C₄	1	1	yes	characteristic
84	Dic₂₇₁	—	2	4	no	—
85	Dic₂₇₁	C₂×C₄²	1	4	yes	—
86	C₂×C₂₇₁	#405	1	4	yes	complemented
87	C₂×C₂₇₁	C₂²×C₄²	1	1	yes	characteristic
88	C₂×C₂₇₁	C₂×#131	1	2	yes	—
89	D₂₇₁	—	2	4	no	—
90	C₂₇₁	C₂×#405	1	1	yes	characteristic, complemented, Sylow
91	C₂×#405	—	271	1	no	maximal, Sylow
92	C₂²×M₁₆	—	271	1	no	—
93	C₂³×C₈	—	271	1	no	—
94	C₂×#117	—	271	1	no	—
95	#405	—	271	8	no	—
96	C₂×#97	—	271	1	no	—
97	C₂×#97	D₂₇₁	1	1	yes	characteristic, complemented
98	C₂×C₄×C₈	—	271	1	no	—
99	C₂×C₄×C₈	—	271	1	no	—
100	C₂³×C₄	—	271	1	no	—
101	C₂×M₁₆	—	271	8	no	—
102	C₂×M₁₆	—	271	4	no	—
103	C₂²×C₈	—	271	1	no	—
104	C₂²×C₈	—	271	1	no	—
105	C₂²×C₈	—	271	8	no	—
106	C₂²×C₈	—	271	4	no	—
107	C₂²×C₈	C₂×D₂₇₁	1	1	yes	characteristic
108	C₂²×C₈	C₂×D₂₇₁	1	1	yes	characteristic
109	#117	—	271	4	no	—
110	#117	—	271	4	no	—
111	C₂×C₄⋊C₄	—	271	2	no	—
112	C₂×C₂²⋊C₄	—	271	2	no	—
113	C₂×C₄²	—	271	1	no	—
114	C₂×C₄²	C₂×D₂₇₁	1	1	yes	characteristic
115	#97	—	271	4	no	—
116	#97	C₂×D₂₇₁	1	4	yes	complemented
117	C₄×C₈	—	271	4	no	—
118	C₄×C₈	—	271	4	no	—
119	C₂⁴	—	271	1	no	—
120	C₂²×C₄	—	271	1	no	—
121	C₂²×C₄	—	271	2	no	—
122	C₂²×C₄	—	271	4	no	—
123	C₂²×C₄	—	271	4	no	—
124	C₂²×C₄	—	271	2	no	—
125	C₂²×C₄	—	271	2	no	—
126	C₂²×C₄	C₄×D₂₇₁	1	2	yes	—
127	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
128	M₁₆	—	271	16	no	—
129	C₂×C₈	—	271	4	no	—
130	C₂×C₈	—	271	4	no	—
131	C₂×C₈	—	271	2	no	—
132	C₂×C₈	—	271	2	no	—
133	C₂×C₈	—	271	16	no	—
134	C₂×C₈	C₄×D₂₇₁	1	4	yes	—
135	C₂×C₈	C₄×D₂₇₁	1	4	yes	—
136	C₂×C₈	C₂²×D₂₇₁	1	2	yes	—
137	C₂×C₈	C₂²×D₂₇₁	1	2	yes	—
138	C₄⋊C₄	—	271	8	no	—
139	C₂²⋊C₄	—	271	8	no	—
140	C₄²	—	271	2	no	—
141	C₄²	—	271	2	no	—
142	C₄²	C₄×D₂₇₁	1	2	yes	—
143	C₄²	C₂²×D₂₇₁	1	2	yes	—
144	C₂³	—	271	2	no	—
145	C₂³	—	271	4	no	—
146	C₂³	—	542	4	no	—
147	C₂³	C₂×C₄×D₂₇₁	1	1	yes	characteristic
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	271	2	no	—
150	C₂×C₄	—	271	4	no	—
151	C₂×C₄	—	271	4	no	—
152	C₂×C₄	—	271	4	no	—
153	C₂×C₄	—	271	8	no	—
154	C₂×C₄	—	271	4	no	—
155	C₂×C₄	—	542	4	no	—
156	C₂×C₄	C₂×C₄×D₂₇₁	1	4	yes	—
157	C₂×C₄	C₂×C₄×D₂₇₁	1	2	yes	—
158	C₂×C₄	C₂×C₄×D₂₇₁	1	2	yes	—
159	C₂×C₄	C₂×C₄×D₂₇₁	1	1	yes	characteristic
160	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
161	C₈	—	271	4	no	—
162	C₈	—	271	4	no	—
163	C₈	C₂×C₄×D₂₇₁	1	4	yes	—
164	C₈	C₂×C₄×D₂₇₁	1	4	yes	—
165	C₂²	—	271	4	no	—
166	C₂²	—	542	8	no	—
167	C₂²	—	542	4	no	—
168	C₂²	C₄²×D₂₇₁	1	2	yes	—
169	C₂²	#22576168	1	4	yes	—
170	C₂²	C₂²×C₄×D₂₇₁	1	1	yes	characteristic
171	C₄	—	2	4	no	—
172	C₄	—	271	4	no	—
173	C₄	—	542	4	no	—
174	C₄	C₄²×D₂₇₁	1	2	yes	—
175	C₄	C₂²×C₄×D₂₇₁	1	1	yes	characteristic
176	C₄	C₂²×C₄×D₂₇₁	1	1	yes	characteristic
177	C₂	—	542	4	no	—
178	C₂	#31773854	1	4	yes	direct product factor
179	C₂	C₂×C₄²×D₂₇₁	1	1	yes	characteristic
180	C₂	C₂×#22576168	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₇₁
commutator	C₂×C₂₇₁	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₂₇₁
Fitting	C₂×#97×C₂₇₁	C₂
socle	C₂³×C₂₇₁	C₂²×C₄
prime radical	C₂³×D₂₇₁	C₂×C₄
cosocle	C₂×C₄×C₂₇₁	C₂⁴
hypercenter	C₂×#97	D₂₇₁
hypocenter	C₂₇₁	C₂×#405
LCS	G ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
4. #31773854	C₂	direct (N × H)
5. C₂×#22576070	C₂	semidirect
6. C₂×C₄×C₂₇₁⋊C₈	C₂	semidirect
7. C₂×C₈×Dic₂₇₁	C₂	semidirect
8. C₂×#97×C₂₇₁	C₂	semidirect
10. C₂×#16864637	C₄	semidirect
14. C₂×C₈×D₂₇₁	C₄	semidirect
21. #22576070	C₂²	semidirect
22. C₄×C₂₇₁⋊C₈	C₂²	semidirect
23. C₈×Dic₂₇₁	C₂²	semidirect
27. #97×C₂₇₁	C₂²	semidirect
35. #16864637	C₂×C₄	semidirect
40. C₈×D₂₇₁	C₂×C₄	semidirect
86. C₂×C₂₇₁	#405	semidirect
90. C₂₇₁	C₂×#405	semidirect
97. C₂×#97	D₂₇₁	semidirect
116. #97	C₂×D₂₇₁	semidirect
178. C₂	#31773854	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×#16864637	2
C₂²×C₈×D₂₇₁	2
C₂×#22576117	2
#31773854	2
C₂×#22576070	2
C₂×C₄×C₂₇₁⋊C₈	2
C₂×C₈×Dic₂₇₁	2
C₂×#97×C₂₇₁	2
C₂×#405	271

Subgroups — C2×#31773854 #46844778

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31773854 #46844778