Subgroups — #3283×C₂₂₇ #44397239

Order 29056.

484 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#135×C₂₂₇	C₂	1	2	yes	maximal, complemented
2	#547×C₂₂₇	C₂	1	2	yes	maximal, complemented
3	#494×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#467×C₂₂₇	C₂	1	2	yes	maximal, complemented
5	#461×C₂₂₇	C₂	1	2	yes	maximal, complemented
6	#452×C₂₂₇	C₂	1	4	yes	maximal, complemented
7	C₄×D₈×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₄×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₂₂₇	C₂²	1	4	yes	—
10	#135×C₂₂₇	—	2	8	no	—
11	C₂×Q₁₆×C₂₂₇	C₂²	1	2	yes	—
12	C₂×Q₁₆×C₂₂₇	C₂²	1	2	yes	—
13	C₂×SD₁₆×C₂₂₇	C₂²	1	4	yes	—
14	C₂×D₈×C₂₂₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₂₂₇	C₂²	1	2	yes	—
16	C₄⋊Q₈×C₂₂₇	C₂²	1	2	yes	—
17	#123×C₂₂₇	—	2	4	no	—
18	C₂²⋊Q₈×C₂₂₇	C₂²	1	4	yes	—
19	#121×C₂₂₇	—	2	2	no	—
20	C₄×D₄×C₂₂₇	C₂²	1	2	yes	—
21	C₄×D₄×C₂₂₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₂₂₇	—	2	2	no	—
23	#107×C₂₂₇	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₂₂₇	C₂²	1	1	yes	characteristic
25	#103×C₂₂₇	C₂²	1	4	yes	—
26	#102×C₂₂₇	C₂²	1	2	yes	—
27	C₂²⋊C₈×C₂₂₇	C₂²	1	2	yes	—
28	C₄×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
29	#41×C₂₂₇	—	2	8	no	—
30	#41×C₂₂₇	—	4	4	no	—
31	C₂×Q₈×C₂₂₇	C₂³	1	4	yes	—
32	C₂×D₄×C₂₂₇	—	2	4	no	—
33	C₂×D₄×C₂₂₇	D₄	1	1	yes	characteristic
34	C₂×D₄×C₂₂₇	C₂³	1	2	yes	—
35	C₂²×C₄×C₂₂₇	—	2	2	no	—
36	C₂²×C₄×C₂₂₇	—	2	4	no	—
37	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
38	Q₁₆×C₂₂₇	—	2	4	no	—
39	Q₁₆×C₂₂₇	—	4	2	no	—
40	SD₁₆×C₂₂₇	—	2	8	no	—
41	D₈×C₂₂₇	D₄	1	4	yes	complemented
42	C₂×C₈×C₂₂₇	—	2	4	no	—
43	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₂₂₇	C₂³	1	2	yes	—
45	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₂₂₇	—	2	2	no	—
47	C₄⋊C₄×C₂₂₇	—	2	4	no	—
48	C₄⋊C₄×C₂₂₇	—	2	2	no	—
49	C₄⋊C₄×C₂₂₇	D₄	1	1	yes	characteristic
50	C₄⋊C₄×C₂₂₇	C₂³	1	2	yes	—
51	C₂²⋊C₄×C₂₂₇	—	2	4	no	—
52	C₂²⋊C₄×C₂₂₇	—	2	2	no	—
53	C₂²⋊C₄×C₂₂₇	D₄	1	2	yes	—
54	C₄²×C₂₂₇	C₂³	1	1	yes	characteristic
55	C₂³×C₂₂₇	—	2	2	no	—
56	C₂³×C₂₂₇	C₂×D₄	1	2	yes	—
57	Q₈×C₂₂₇	—	2	4	no	—
58	Q₈×C₂₂₇	—	2	4	no	—
59	D₄×C₂₂₇	—	2	2	no	—
60	D₄×C₂₂₇	—	2	2	no	—
61	D₄×C₂₂₇	—	4	8	no	—
62	D₄×C₂₂₇	C₂×D₄	1	4	yes	—
63	C₂×C₄×C₂₂₇	—	2	2	no	—
64	C₂×C₄×C₂₂₇	—	2	2	no	—
65	C₂×C₄×C₂₂₇	—	2	2	no	—
66	C₂×C₄×C₂₂₇	—	2	4	no	—
67	C₂×C₄×C₂₂₇	—	4	8	no	—
68	C₂×C₄×C₂₂₇	—	4	4	no	—
69	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
71	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
73	C₈×C₂₂₇	—	2	2	no	—
74	C₈×C₂₂₇	—	2	1	no	—
75	C₈×C₂₂₇	C₂×D₄	1	2	yes	—
76	C₂²×C₂₂₇	—	2	2	no	—
77	C₂²×C₂₂₇	—	2	4	no	—
78	C₂²×C₂₂₇	—	2	2	no	—
79	C₂²×C₂₂₇	—	2	2	no	—
80	C₂²×C₂₂₇	—	4	2	no	—
81	C₂²×C₂₂₇	—	4	2	no	—
82	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
83	C₄×C₂₂₇	—	2	2	no	—
84	C₄×C₂₂₇	—	2	1	no	—
85	C₄×C₂₂₇	—	4	4	no	—
86	C₄×C₂₂₇	—	4	2	no	—
87	C₄×C₂₂₇	#135	1	2	yes	complemented
88	C₄×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
89	C₄×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
90	C₂×C₂₂₇	—	4	2	no	—
91	C₂×C₂₂₇	—	4	4	no	—
92	C₂×C₂₂₇	D₄²	1	1	yes	characteristic
93	C₂×C₂₂₇	C₂×#135	1	1	yes	characteristic
94	C₂×C₂₂₇	#578	1	1	yes	characteristic
95	C₂₂₇	#3283	1	1	yes	characteristic, direct product factor, Sylow
96	#3283	C₂₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂×#135	C₂×C₂₂₇	1	2	yes	complemented
98	#547	C₂×C₂₂₇	1	2	yes	complemented
99	#494	C₂×C₂₂₇	1	1	yes	characteristic, complemented
100	#467	C₂×C₂₂₇	1	2	yes	complemented
101	#461	C₂×C₂₂₇	1	2	yes	complemented
102	#452	C₂×C₂₂₇	1	4	yes	complemented
103	C₄×D₈	C₂×C₂₂₇	1	1	yes	characteristic, complemented
104	C₈×D₄	C₂×C₂₂₇	1	1	yes	characteristic, complemented
105	C₂×#41	C₂²×C₂₂₇	1	4	yes	—
106	#135	—	2	8	no	—
107	C₂×Q₁₆	C₂²×C₂₂₇	1	2	yes	—
108	C₂×Q₁₆	C₂²×C₂₂₇	1	2	yes	—
109	C₂×SD₁₆	C₂²×C₂₂₇	1	4	yes	—
110	C₂×D₈	C₂²×C₂₂₇	1	1	yes	characteristic
111	C₂²×C₈	C₂²×C₂₂₇	1	2	yes	—
112	C₄⋊Q₈	C₂²×C₂₂₇	1	2	yes	—
113	#123	—	2	4	no	—
114	C₂²⋊Q₈	C₂²×C₂₂₇	1	4	yes	—
115	#121	—	2	2	no	—
116	C₄×D₄	C₂²×C₂₂₇	1	2	yes	—
117	C₄×D₄	C₂²×C₂₂₇	1	1	yes	characteristic
118	C₂×C₄⋊C₄	—	2	2	no	—
119	#107	C₂²×C₂₂₇	1	1	yes	characteristic
120	C₄⋊C₈	C₂²×C₂₂₇	1	1	yes	characteristic
121	#103	C₂²×C₂₂₇	1	4	yes	—
122	#102	C₂²×C₂₂₇	1	2	yes	—
123	C₂²⋊C₈	C₂²×C₂₂₇	1	2	yes	—
124	C₄×C₈	C₂²×C₂₂₇	1	1	yes	characteristic
125	#41	—	2	8	no	—
126	#41	—	4	4	no	—
127	C₂×Q₈	C₂³×C₂₂₇	1	4	yes	—
128	C₂×D₄	—	2	4	no	—
129	C₂×D₄	D₄×C₂₂₇	1	1	yes	characteristic
130	C₂×D₄	C₂³×C₂₂₇	1	2	yes	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	2	4	no	—
133	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
134	Q₁₆	—	2	4	no	—
135	Q₁₆	—	4	2	no	—
136	SD₁₆	—	2	8	no	—
137	D₈	D₄×C₂₂₇	1	4	yes	complemented
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	C₂³×C₂₂₇	1	1	yes	characteristic
140	C₂×C₈	C₂³×C₂₂₇	1	2	yes	—
141	C₂×C₈	C₂³×C₂₂₇	1	1	yes	characteristic
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	D₄×C₂₂₇	1	1	yes	characteristic
146	C₄⋊C₄	C₂³×C₂₂₇	1	2	yes	—
147	C₂²⋊C₄	—	2	4	no	—
148	C₂²⋊C₄	—	2	2	no	—
149	C₂²⋊C₄	D₄×C₂₂₇	1	2	yes	—
150	C₄²	C₂³×C₂₂₇	1	1	yes	characteristic
151	C₂³	—	2	2	no	—
152	C₂³	C₂×D₄×C₂₂₇	1	2	yes	—
153	Q₈	—	2	4	no	—
154	Q₈	—	2	4	no	—
155	D₄	—	2	2	no	—
156	D₄	—	2	2	no	—
157	D₄	—	4	8	no	—
158	D₄	C₂×D₄×C₂₂₇	1	4	yes	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	4	8	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	C₂×D₄×C₂₂₇	1	1	yes	characteristic
166	C₂×C₄	C₂×D₄×C₂₂₇	1	2	yes	—
167	C₂×C₄	C₂×D₄×C₂₂₇	1	1	yes	characteristic
168	C₂×C₄	C₂⁴×C₂₂₇	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	—	2	1	no	—
171	C₈	C₂×D₄×C₂₂₇	1	2	yes	—
172	C₂²	—	2	2	no	—
173	C₂²	—	2	4	no	—
174	C₂²	—	2	2	no	—
175	C₂²	—	2	2	no	—
176	C₂²	—	4	2	no	—
177	C₂²	—	4	2	no	—
178	C₂²	C₂²×D₄×C₂₂₇	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	2	1	no	—
181	C₄	—	4	4	no	—
182	C₄	—	4	2	no	—
183	C₄	#135×C₂₂₇	1	2	yes	complemented
184	C₄	C₂²×D₄×C₂₂₇	1	1	yes	characteristic
185	C₄	H(5,𝔽₂)×C₂₂₇	1	1	yes	characteristic
186	C₂	—	4	2	no	—
187	C₂	—	4	4	no	—
188	C₂	D₄²×C₂₂₇	1	1	yes	characteristic
189	C₂	C₂×#135×C₂₂₇	1	1	yes	characteristic
190	C₂	#578×C₂₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₂₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₂₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₇ ◁ C₄×D₄×C₂₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#135×C₂₂₇	C₂	semidirect
2. #547×C₂₂₇	C₂	semidirect
3. #494×C₂₂₇	C₂	semidirect
4. #467×C₂₂₇	C₂	semidirect
5. #461×C₂₂₇	C₂	semidirect
6. #452×C₂₂₇	C₂	semidirect
7. C₄×D₈×C₂₂₇	C₂	semidirect
8. C₈×D₄×C₂₂₇	C₂	semidirect
41. D₈×C₂₂₇	D₄	semidirect
87. C₄×C₂₂₇	#135	semidirect
95. C₂₂₇	#3283	direct (N × H)
96. #3283	C₂₂₇	direct (N × H)
97. C₂×#135	C₂×C₂₂₇	semidirect
98. #547	C₂×C₂₂₇	semidirect
99. #494	C₂×C₂₂₇	semidirect
100. #467	C₂×C₂₂₇	semidirect
101. #461	C₂×C₂₂₇	semidirect
102. #452	C₂×C₂₂₇	semidirect
103. C₄×D₈	C₂×C₂₂₇	semidirect
104. C₈×D₄	C₂×C₂₂₇	semidirect
137. D₈	D₄×C₂₂₇	semidirect
183. C₄	#135×C₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#135×C₂₂₇	2
#547×C₂₂₇	2
#494×C₂₂₇	2
#467×C₂₂₇	2
#461×C₂₂₇	2
#452×C₂₂₇	2
C₄×D₈×C₂₂₇	2
C₈×D₄×C₂₂₇	2
#3283	227

Subgroups — #3283×C227 #44397239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3283×C₂₂₇ #44397239