Subgroups — C₂×#28409016 #44408209

Order 29056.

756 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#16813868	C₂	1	2	yes	maximal, complemented
2	#28409016	C₂	1	8	yes	maximal, direct product factor
3	C₂×#20300207	C₂	1	2	yes	maximal, complemented
4	C₂×#20300205	C₂	1	2	yes	maximal, complemented
5	C₂²×#41×C₂₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#16813868	—	2	2	no	—
7	C₂×#16813868	—	2	4	no	—
8	C₂×#16813868	C₂²	1	4	yes	—
9	C₂×#16813868	C₂²	1	8	yes	complemented
10	C₂²×C₂₂₇⋊C₈	—	2	2	no	—
11	#20300207	C₂²	1	8	yes	complemented
12	#20300205	C₂²	1	8	yes	complemented
13	C₂×#41×C₂₂₇	—	2	2	no	—
14	C₂×#41×C₂₂₇	—	2	2	no	—
15	C₂×#41×C₂₂₇	—	2	4	no	—
16	C₂×#41×C₂₂₇	C₄	1	4	yes	—
17	C₂×#41×C₂₂₇	C₂²	1	4	yes	—
18	C₂²×Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
19	C₂²×D₄×C₂₂₇	C₄	1	2	yes	—
20	C₂²×D₄×C₂₂₇	C₂²	1	1	yes	characteristic
21	C₂³×C₄×C₂₂₇	C₄	1	2	yes	—
22	C₂³×C₄×C₂₂₇	C₂²	1	1	yes	characteristic
23	#16813868	—	2	4	no	—
24	#16813868	—	2	8	no	—
25	#16813868	C₂³	1	8	yes	—
26	C₂×C₂₂₇⋊C₈	—	2	4	no	—
27	C₂×C₂₂₇⋊C₈	—	2	8	no	—
28	C₂⁴×C₂₂₇	C₂×C₄	1	2	yes	—
29	C₂⁴×C₂₂₇	C₂×C₄	1	1	yes	characteristic
30	#41×C₂₂₇	—	2	8	no	—
31	#41×C₂₂₇	—	2	8	no	—
32	#41×C₂₂₇	—	2	8	no	—
33	#41×C₂₂₇	—	2	8	no	—
34	C₂×Q₈×C₂₂₇	—	2	2	no	—
35	C₂×Q₈×C₂₂₇	—	2	2	no	—
36	C₂×Q₈×C₂₂₇	C₂×C₄	1	2	yes	—
37	C₂×Q₈×C₂₂₇	C₂³	1	2	yes	—
38	C₂×D₄×C₂₂₇	—	2	4	no	—
39	C₂×D₄×C₂₂₇	—	2	4	no	—
40	C₂×D₄×C₂₂₇	—	2	2	no	—
41	C₂×D₄×C₂₂₇	—	2	2	no	—
42	C₂×D₄×C₂₂₇	C₂×C₄	1	8	yes	—
43	C₂×D₄×C₂₂₇	C₂×C₄	1	2	yes	—
44	C₂×D₄×C₂₂₇	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₂₇	—	2	4	no	—
46	C₂²×C₄×C₂₂₇	—	2	2	no	—
47	C₂²×C₄×C₂₂₇	—	2	2	no	—
48	C₂²×C₄×C₂₂₇	C₂×C₄	1	2	yes	—
49	C₂²×C₄×C₂₂₇	C₂×C₄	1	8	yes	—
50	C₂²×C₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₂₇	C₂×C₄	1	2	yes	—
52	C₂²×C₄×C₂₂₇	D₄	1	2	yes	—
53	C₂²×C₄×C₂₂₇	D₄	1	2	yes	—
54	C₂²×C₄×C₂₂₇	D₄	1	4	yes	—
55	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
57	C₂₂₇⋊C₈	—	2	8	no	—
58	C₂³×C₂₂₇	—	2	2	no	—
59	C₂³×C₂₂₇	—	2	2	no	—
60	C₂³×C₂₂₇	—	4	2	no	—
61	C₂³×C₂₂₇	—	4	4	no	—
62	C₂³×C₂₂₇	C₂²⋊C₄	1	2	yes	—
63	C₂³×C₂₂₇	C₂²⋊C₄	1	2	yes	—
64	C₂³×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
65	C₂³×C₂₂₇	C₂²×C₄	1	4	yes	—
66	C₂³×C₂₂₇	C₂²×C₄	1	2	yes	—
67	Q₈×C₂₂₇	—	2	4	no	—
68	Q₈×C₂₂₇	—	2	4	no	—
69	D₄×C₂₂₇	—	2	16	no	—
70	D₄×C₂₂₇	—	2	4	no	—
71	D₄×C₂₂₇	—	2	4	no	—
72	C₂×C₄×C₂₂₇	—	2	2	no	—
73	C₂×C₄×C₂₂₇	—	2	2	no	—
74	C₂×C₄×C₂₂₇	—	2	8	no	—
75	C₂×C₄×C₂₂₇	—	2	8	no	—
76	C₂×C₄×C₂₂₇	C₂²⋊C₄	1	2	yes	—
77	C₂×C₄×C₂₂₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₂₂₇	C₂²⋊C₄	1	4	yes	—
79	C₂×C₄×C₂₂₇	C₂²⋊C₄	1	4	yes	—
80	C₂×C₄×C₂₂₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₂₂₇	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₂₂₇	C₂×D₄	1	4	yes	—
87	C₂×C₄×C₂₂₇	C₂×D₄	1	4	yes	—
88	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
89	C₂²×C₂₂₇	—	2	2	no	—
90	C₂²×C₂₂₇	—	2	4	no	—
91	C₂²×C₂₂₇	—	2	4	no	—
92	C₂²×C₂₂₇	—	4	2	no	—
93	C₂²×C₂₂₇	—	4	2	no	—
94	C₂²×C₂₂₇	—	4	4	no	—
95	C₂²×C₂₂₇	—	4	4	no	—
96	C₂²×C₂₂₇	—	4	4	no	—
97	C₂²×C₂₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂²×C₂₂₇	C₂×C₂²⋊C₄	1	4	yes	—
99	C₂²×C₂₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
101	C₄×C₂₂₇	—	2	4	no	—
102	C₄×C₂₂₇	C₂×C₂²⋊C₄	1	2	yes	—
103	C₄×C₂₂₇	C₂×C₂²⋊C₄	1	2	yes	—
104	C₄×C₂₂₇	C₂×C₂²⋊C₄	1	2	yes	—
105	C₄×C₂₂₇	C₂²×D₄	1	2	yes	—
106	C₂×C₂₂₇	—	2	1	no	—
107	C₂×C₂₂₇	—	2	1	no	—
108	C₂×C₂₂₇	—	2	4	no	—
109	C₂×C₂₂₇	—	4	4	no	—
110	C₂×C₂₂₇	#413	1	2	yes	complemented
111	C₂×C₂₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
112	C₂₂₇	C₂×#413	1	1	yes	characteristic, complemented, Sylow
113	C₂×#413	—	227	1	no	maximal, Sylow
114	C₂²×#41	D₂₂₇	1	1	yes	characteristic
115	C₂²×M₁₆	—	227	2	no	—
116	#413	—	227	8	no	—
117	C₂×#101	—	227	2	no	—
118	C₂×#100	—	227	2	no	—
119	C₂×#41	—	2	2	no	—
120	C₂×#41	—	2	4	no	—
121	C₂×#41	—	2	2	no	—
122	C₂×#41	Dic₂₂₇	1	4	yes	—
123	C₂×#41	C₂×D₂₂₇	1	4	yes	—
124	C₂²×Q₈	C₂×D₂₂₇	1	1	yes	characteristic
125	C₂²×D₄	Dic₂₂₇	1	2	yes	—
126	C₂²×D₄	C₂×D₂₂₇	1	1	yes	characteristic
127	C₂³×C₄	Dic₂₂₇	1	2	yes	—
128	C₂³×C₄	C₂×D₂₂₇	1	1	yes	characteristic
129	C₂×M₁₆	—	454	2	no	—
130	C₂×M₁₆	—	454	4	no	—
131	C₂×M₁₆	—	227	4	no	—
132	C₂×M₁₆	—	227	8	no	—
133	C₂²×C₈	—	454	2	no	—
134	#101	—	227	8	no	—
135	#100	—	227	8	no	—
136	C₂⁴	C₂×Dic₂₂₇	1	2	yes	—
137	C₂⁴	C₂×Dic₂₂₇	1	1	yes	characteristic
138	#41	—	2	8	no	—
139	#41	—	2	8	no	—
140	#41	—	2	8	no	—
141	#41	—	2	8	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂×Q₈	—	2	2	no	—
144	C₂×Q₈	C₂×Dic₂₂₇	1	2	yes	—
145	C₂×Q₈	C₂²×D₂₂₇	1	2	yes	—
146	C₂×D₄	—	2	4	no	—
147	C₂×D₄	—	2	4	no	—
148	C₂×D₄	—	2	2	no	—
149	C₂×D₄	—	2	2	no	—
150	C₂×D₄	C₂×Dic₂₂₇	1	8	yes	—
151	C₂×D₄	C₂×Dic₂₂₇	1	2	yes	—
152	C₂×D₄	C₂²×D₂₂₇	1	2	yes	—
153	C₂²×C₄	—	2	4	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	C₂×Dic₂₂₇	1	2	yes	—
157	C₂²×C₄	C₂×Dic₂₂₇	1	8	yes	—
158	C₂²×C₄	C₂×Dic₂₂₇	1	2	yes	—
159	C₂²×C₄	C₂×Dic₂₂₇	1	1	yes	characteristic
160	C₂²×C₄	C₂²⋊D₂₂₇	1	2	yes	—
161	C₂²×C₄	C₂²⋊D₂₂₇	1	4	yes	—
162	C₂²×C₄	C₂²⋊D₂₂₇	1	2	yes	—
163	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
164	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
165	M₁₆	—	454	4	no	—
166	M₁₆	—	454	8	no	—
167	M₁₆	—	227	8	no	—
168	C₂×C₈	—	454	4	no	—
169	C₂×C₈	—	454	8	no	—
170	C₂³	—	2	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	4	2	no	—
173	C₂³	—	4	4	no	—
174	C₂³	C₂²⋊Dic₂₂₇	1	2	yes	—
175	C₂³	C₂²⋊Dic₂₂₇	1	2	yes	—
176	C₂³	C₂²×Dic₂₂₇	1	1	yes	characteristic
177	C₂³	C₂²×Dic₂₂₇	1	4	yes	—
178	C₂³	C₂²×Dic₂₂₇	1	2	yes	—
179	Q₈	—	2	4	no	—
180	Q₈	—	2	4	no	—
181	D₄	—	2	16	no	—
182	D₄	—	2	4	no	—
183	D₄	—	2	4	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	8	no	—
187	C₂×C₄	—	2	8	no	—
188	C₂×C₄	C₂²⋊Dic₂₂₇	1	1	yes	characteristic
189	C₂×C₄	C₂²⋊Dic₂₂₇	1	2	yes	—
190	C₂×C₄	C₂²⋊Dic₂₂₇	1	4	yes	—
191	C₂×C₄	C₂²⋊Dic₂₂₇	1	4	yes	—
192	C₂×C₄	C₂²⋊Dic₂₂₇	1	1	yes	characteristic
193	C₂×C₄	C₂²×Dic₂₂₇	1	4	yes	—
194	C₂×C₄	C₂²×Dic₂₂₇	1	2	yes	—
195	C₂×C₄	C₂²×Dic₂₂₇	1	1	yes	characteristic
196	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	2	yes	—
197	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	4	yes	—
198	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	2	yes	—
199	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	4	yes	—
200	C₂×C₄	C₂³×D₂₂₇	1	1	yes	characteristic
201	C₈	—	454	8	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	4	2	no	—
206	C₂²	—	4	2	no	—
207	C₂²	—	4	4	no	—
208	C₂²	—	4	4	no	—
209	C₂²	—	4	4	no	—
210	C₂²	C₂×C₂²⋊Dic₂₂₇	1	1	yes	characteristic
211	C₂²	C₂×C₂²⋊Dic₂₂₇	1	4	yes	—
212	C₂²	C₂×C₂²⋊Dic₂₂₇	1	1	yes	characteristic
213	C₂²	C₂³×Dic₂₂₇	1	1	yes	characteristic
214	C₄	—	2	4	no	—
215	C₄	C₂×C₂²⋊Dic₂₂₇	1	2	yes	—
216	C₄	C₂×C₂²⋊Dic₂₂₇	1	2	yes	—
217	C₄	C₂×C₂²⋊Dic₂₂₇	1	2	yes	—
218	C₄	C₂²×C₂²⋊D₂₂₇	1	2	yes	—
219	C₂	—	2	1	no	—
220	C₂	—	2	1	no	—
221	C₂	—	2	4	no	—
222	C₂	—	4	4	no	—
223	C₂	#28409016	1	2	yes	direct product factor
224	C₂	C₂²×C₂²⋊Dic₂₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₂₂₇
commutator	C₂²×C₂₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₂₇
Fitting, prime radical	C₂²×#41×C₂₂₇	C₂
socle	C₂²×C₂₂₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄×C₂₂₇	C₂⁴
hypercenter	C₂²×#41	D₂₂₇
hypocenter	C₂₂₇	C₂×#413
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#16813868	C₂	semidirect
2. #28409016	C₂	direct (N × H)
3. C₂×#20300207	C₂	semidirect
4. C₂×#20300205	C₂	semidirect
9. C₂×#16813868	C₂²	semidirect
11. #20300207	C₂²	semidirect
12. #20300205	C₂²	semidirect
110. C₂×C₂₂₇	#413	semidirect
112. C₂₂₇	C₂×#413	semidirect
223. C₂	#28409016	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×#16813868	2
#28409016	2
C₂×#20300207	2
C₂×#20300205	2
C₂²×#41×C₂₂₇	2
C₂×#413	227

Subgroups — C2×#28409016 #44408209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#28409016 #44408209