Subgroups — C₂×#31773657 #46836070

Order 34688.

488 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#16864638	C₂	1	1	yes	maximal, characteristic
2	C₂×#22576119	C₂	1	1	yes	maximal, characteristic
3	#31773657	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₄²×C₂₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#16864638	—	2	2	no	—
6	C₂×#16864638	—	2	1	no	—
7	C₂×#16864638	C₄	1	4	yes	complemented
8	C₂×#16864638	C₂²	1	2	yes	—
9	C₂²×C₂₇₁⋊C₈	—	2	1	no	—
10	#22576119	—	2	1	no	—
11	#22576119	—	2	1	no	—
12	#22576119	C₄	1	2	yes	complemented
13	#22576119	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₂₇₁	C₄	1	2	yes	complemented
15	C₂×C₂²⋊Dic₂₇₁	—	2	1	no	—
16	C₂×C₄×Dic₂₇₁	—	2	1	no	—
17	C₂³×C₄×C₂₇₁	—	2	1	no	—
18	C₂³×C₄×C₂₇₁	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₂₇₁	—	2	1	no	—
20	C₂×C₄²×C₂₇₁	—	2	2	no	—
21	C₂×C₄²×C₂₇₁	C₄	1	4	yes	complemented
22	C₂×C₄²×C₂₇₁	C₂²	1	2	yes	—
23	C₂²×Dic₂₇₁	—	2	2	no	—
24	#16864638	—	2	4	no	—
25	#16864638	—	2	2	no	—
26	#16864638	C₂×C₄	1	4	yes	complemented
27	C₂×C₂₇₁⋊C₈	—	2	4	no	—
28	C₂×C₂₇₁⋊C₈	—	2	2	no	—
29	C₄⋊Dic₂₇₁	—	2	2	no	—
30	C₄⋊Dic₂₇₁	C₂×C₄	1	4	yes	complemented
31	C₂²⋊Dic₂₇₁	—	2	2	no	—
32	C₂²⋊Dic₂₇₁	—	2	2	no	—
33	C₄×Dic₂₇₁	—	2	2	no	—
34	C₄×Dic₂₇₁	—	2	2	no	—
35	C₂⁴×C₂₇₁	D₄	1	1	yes	characteristic
36	C₂²×C₄×C₂₇₁	—	2	8	no	—
37	C₂²×C₄×C₂₇₁	—	2	4	no	—
38	C₂²×C₄×C₂₇₁	—	2	2	no	—
39	C₂²×C₄×C₂₇₁	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₇₁	C₂×C₄	1	2	yes	—
43	C₂²×C₄×C₂₇₁	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₇₁	D₄	1	2	yes	—
45	C₂²×C₄×C₂₇₁	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₇₁	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₂₇₁	Q₈	1	2	yes	—
48	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
49	C₄²×C₂₇₁	—	2	4	no	—
50	C₄²×C₂₇₁	—	2	2	no	—
51	C₄²×C₂₇₁	C₂×C₄	1	4	yes	complemented
52	C₂×Dic₂₇₁	—	2	4	no	—
53	C₂×Dic₂₇₁	—	4	4	no	—
54	C₂₇₁⋊C₈	—	2	4	no	—
55	C₂³×C₂₇₁	—	2	4	no	—
56	C₂³×C₂₇₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂³×C₂₇₁	C₂²⋊C₄	1	1	yes	characteristic
58	C₂³×C₂₇₁	C₂²⋊C₄	1	2	yes	—
59	C₂³×C₂₇₁	C₄⋊C₄	1	2	yes	—
60	C₂³×C₂₇₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₂₇₁	—	2	16	no	—
62	C₂×C₄×C₂₇₁	—	2	8	no	—
63	C₂×C₄×C₂₇₁	—	2	4	no	—
64	C₂×C₄×C₂₇₁	C₄²	1	2	yes	—
65	C₂×C₄×C₂₇₁	C₄²	1	2	yes	—
66	C₂×C₄×C₂₇₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₇₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₂₇₁	C₂²⋊C₄	1	2	yes	—
69	C₂×C₄×C₂₇₁	C₂²⋊C₄	1	2	yes	—
70	C₂×C₄×C₂₇₁	C₂²⋊C₄	1	2	yes	—
71	C₂×C₄×C₂₇₁	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₂₇₁	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₂₇₁	C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₂₇₁	C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₂₇₁	C₄⋊C₄	1	2	yes	—
76	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₂₇₁	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₂₇₁	—	2	4	no	—
83	C₂²×C₂₇₁	—	2	4	no	—
84	C₂²×C₂₇₁	—	2	2	no	—
85	C₂²×C₂₇₁	—	2	2	no	—
86	C₂²×C₂₇₁	#95	1	2	yes	—
87	C₂²×C₂₇₁	#95	1	2	yes	—
88	C₂²×C₂₇₁	#104	1	4	yes	—
89	C₂²×C₂₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₂₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₂₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₂₇₁	—	2	8	no	—
93	C₄×C₂₇₁	#95	1	2	yes	—
94	C₄×C₂₇₁	#95	1	2	yes	—
95	C₄×C₂₇₁	C₂×C₄²	1	1	yes	characteristic
96	C₄×C₂₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₄×C₂₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₄×C₂₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
99	Dic₂₇₁	—	4	4	no	—
100	C₂×C₂₇₁	—	2	1	no	—
101	C₂×C₂₇₁	—	2	2	no	—
102	C₂×C₂₇₁	—	2	1	no	—
103	C₂×C₂₇₁	#339	1	4	yes	complemented
104	C₂×C₂₇₁	C₂×#95	1	1	yes	characteristic
105	C₂×C₂₇₁	C₂×#104	1	2	yes	—
106	C₂₇₁	C₂×#339	1	1	yes	characteristic, complemented, Sylow
107	C₂×#339	—	271	1	no	maximal, Sylow
108	C₂²×M₁₆	—	271	1	no	—
109	C₂×#117	—	271	1	no	—
110	C₂²×C₄²	D₂₇₁	1	1	yes	characteristic
111	#339	—	271	4	no	—
112	C₂³×C₄	—	2	1	no	—
113	C₂³×C₄	C₂×D₂₇₁	1	1	yes	characteristic
114	C₂×M₁₆	—	271	4	no	—
115	C₂×M₁₆	—	271	2	no	—
116	C₂×M₁₆	—	542	2	no	—
117	C₂×M₁₆	—	542	1	no	—
118	C₂²×C₈	—	542	1	no	—
119	#117	—	271	2	no	—
120	#117	—	271	2	no	—
121	#117	—	542	1	no	—
122	#117	—	542	1	no	—
123	C₂×C₄⋊C₄	—	271	2	no	—
124	C₂×C₂²⋊C₄	—	542	1	no	—
125	C₂×C₄²	—	2	1	no	—
126	C₂×C₄²	—	2	2	no	—
127	C₂×C₄²	—	542	1	no	—
128	C₂×C₄²	Dic₂₇₁	1	4	yes	complemented
129	C₂×C₄²	C₂×D₂₇₁	1	2	yes	—
130	C₂⁴	C₂²⋊D₂₇₁	1	1	yes	characteristic
131	C₂²×C₄	—	2	8	no	—
132	C₂²×C₄	—	2	4	no	—
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	542	2	no	—
135	C₂²×C₄	C₄×D₂₇₁	1	2	yes	—
136	C₂²×C₄	C₄×D₂₇₁	1	1	yes	characteristic
137	C₂²×C₄	C₄×D₂₇₁	1	1	yes	characteristic
138	C₂²×C₄	C₂×Dic₂₇₁	1	2	yes	—
139	C₂²×C₄	C₂²⋊D₂₇₁	1	1	yes	characteristic
140	C₂²×C₄	C₂²⋊D₂₇₁	1	2	yes	—
141	C₂²×C₄	D₁₀₈₄	1	1	yes	characteristic
142	C₂²×C₄	D₁₀₈₄	1	1	yes	characteristic
143	C₂²×C₄	Dic₅₄₂	1	2	yes	—
144	C₂²×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
145	M₁₆	—	271	4	no	—
146	M₁₆	—	542	4	no	—
147	M₁₆	—	542	2	no	—
148	C₂×C₈	—	542	4	no	—
149	C₂×C₈	—	542	2	no	—
150	C₄⋊C₄	—	271	4	no	—
151	C₄⋊C₄	—	542	2	no	—
152	C₂²⋊C₄	—	542	2	no	—
153	C₂²⋊C₄	—	542	2	no	—
154	C₄²	—	2	4	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	542	2	no	—
157	C₄²	—	542	2	no	—
158	C₄²	C₂×Dic₂₇₁	1	4	yes	complemented
159	C₂³	—	2	4	no	—
160	C₂³	(C₂×D₂₇₁)⋊C₄	1	1	yes	characteristic
161	C₂³	(C₂×D₂₇₁)⋊C₄	1	1	yes	characteristic
162	C₂³	C₂²⋊Dic₂₇₁	1	2	yes	—
163	C₂³	Dic₂₇₁⋊C₄	1	2	yes	—
164	C₂³	C₂×C₂²⋊D₂₇₁	1	1	yes	characteristic
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	16	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	542	4	no	—
169	C₂×C₄	—	1084	4	no	—
170	C₂×C₄	C₄×Dic₂₇₁	1	2	yes	—
171	C₂×C₄	C₄×Dic₂₇₁	1	2	yes	—
172	C₂×C₄	(C₂×D₂₇₁)⋊C₄	1	1	yes	characteristic
173	C₂×C₄	(C₂×D₂₇₁)⋊C₄	1	1	yes	characteristic
174	C₂×C₄	(C₂×D₂₇₁)⋊C₄	1	2	yes	—
175	C₂×C₄	(C₂×D₂₇₁)⋊C₄	1	2	yes	—
176	C₂×C₄	C₂²⋊Dic₂₇₁	1	2	yes	—
177	C₂×C₄	Dic₂₇₁⋊C₄	1	2	yes	—
178	C₂×C₄	Dic₂₇₁⋊C₄	1	2	yes	—
179	C₂×C₄	Dic₂₇₁⋊C₄	1	2	yes	—
180	C₂×C₄	C₄⋊Dic₂₇₁	1	2	yes	—
181	C₂×C₄	C₄⋊Dic₂₇₁	1	2	yes	—
182	C₂×C₄	C₂×C₄×D₂₇₁	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×D₂₇₁	1	1	yes	characteristic
184	C₂×C₄	C₂²×Dic₂₇₁	1	1	yes	characteristic
185	C₂×C₄	C₂×C₂²⋊D₂₇₁	1	1	yes	characteristic
186	C₂×C₄	C₂×D₁₀₈₄	1	1	yes	characteristic
187	C₂×C₄	C₂×Dic₅₄₂	1	1	yes	characteristic
188	C₈	—	542	4	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	4	no	—
193	C₂²	C₂₇₁⋊#95	1	2	yes	—
194	C₂²	C₂₇₁⋊#95	1	2	yes	—
195	C₂²	#22576088	1	4	yes	—
196	C₂²	C₂×(C₂×D₂₇₁)⋊C₄	1	1	yes	characteristic
197	C₂²	C₂×C₂²⋊Dic₂₇₁	1	1	yes	characteristic
198	C₂²	C₂×Dic₂₇₁⋊C₄	1	1	yes	characteristic
199	C₄	—	2	8	no	—
200	C₄	—	1084	4	no	—
201	C₄	C₂₇₁⋊#95	1	2	yes	—
202	C₄	C₂₇₁⋊#95	1	2	yes	—
203	C₄	C₂×C₄×Dic₂₇₁	1	1	yes	characteristic
204	C₄	C₂×(C₂×D₂₇₁)⋊C₄	1	1	yes	characteristic
205	C₄	C₂×Dic₂₇₁⋊C₄	1	1	yes	characteristic
206	C₄	C₂×C₄⋊Dic₂₇₁	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	2	no	—
209	C₂	—	2	1	no	—
210	C₂	#31773657	1	4	yes	direct product factor
211	C₂	C₂×C₂₇₁⋊#95	1	1	yes	characteristic
212	C₂	C₂×#22576088	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₁₀₈₄
commutator	C₄×C₂₇₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₇₁
Fitting	C₂²×C₄²×C₂₇₁	C₂
socle	C₂³×C₂₇₁	C₂²⋊C₄
prime radical	C₂⁴×C₂₇₁	D₄
cosocle	C₂²×C₄×C₂₇₁	C₂³
hypercenter	C₂²×C₄²	D₂₇₁
hypocenter	C₂₇₁	C₂×#339
LCS	G ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
3. #31773657	C₂	direct (N × H)
7. C₂×#16864638	C₄	semidirect
12. #22576119	C₄	semidirect
14. C₂×C₄⋊Dic₂₇₁	C₄	semidirect
21. C₂×C₄²×C₂₇₁	C₄	semidirect
26. #16864638	C₂×C₄	semidirect
30. C₄⋊Dic₂₇₁	C₂×C₄	semidirect
51. C₄²×C₂₇₁	C₂×C₄	semidirect
103. C₂×C₂₇₁	#339	semidirect
106. C₂₇₁	C₂×#339	semidirect
128. C₂×C₄²	Dic₂₇₁	semidirect
158. C₄²	C₂×Dic₂₇₁	semidirect
210. C₂	#31773657	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×#16864638	2
C₂×#22576119	2
#31773657	2
C₂²×C₄²×C₂₇₁	2
C₂×#339	271

Subgroups — C2×#31773657 #46836070

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#31773657 #46836070