Subgroups — #30647299

Order 16256.

484 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861217	C₂	1	2	yes	maximal
2	C₄×#14523248	C₂	1	1	yes	maximal, characteristic
3	#22549904	C₂	1	2	yes	maximal, complemented
4	#22549901	C₂	1	4	yes	maximal, complemented
5	#22549898	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₈×D₁₂₇	C₂	1	2	yes	maximal, complemented
8	#423×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14523248	C₂²	1	1	yes	characteristic
10	#16861217	—	2	8	no	—
11	C₂×#14523231	C₂²	1	4	yes	—
12	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
13	C₂×C₈×D₁₂₇	C₂²	1	4	yes	—
14	C₄×Dic₂₅₄	C₄	1	1	yes	characteristic
15	C₄×Dic₂₅₄	C₄	1	1	yes	characteristic
16	C₄×D₅₀₈	C₄	1	1	yes	characteristic
17	C₄×D₅₀₈	C₄	1	1	yes	characteristic
18	C₄×C₂²⋊D₁₂₇	C₄	1	2	yes	—
19	C₄×C₂²⋊D₁₂₇	C₄	1	2	yes	—
20	#16861168	C₂²	1	1	yes	characteristic
21	#16861166	C₂²	1	2	yes	—
22	C₄²×D₁₂₇	C₂²	1	2	yes	—
23	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
24	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
25	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
26	Dic₁₂₇⋊C₈	C₂²	1	4	yes	—
27	Dic₁₂₇⋊C₈	C₂²	1	4	yes	—
28	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
29	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
30	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
31	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
32	#14523248	D₄	1	4	yes	—
33	#14523248	Q₈	1	4	yes	—
34	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
35	C₂×D₅₀₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
40	#14523231	—	2	8	no	—
41	C₂×C₁₂₇⋊C₈	—	2	4	no	—
42	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
43	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
44	C₈×D₁₂₇	—	2	8	no	—
45	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
46	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
47	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
48	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
49	C₂²⋊Dic₁₂₇	C₂×C₄	1	2	yes	—
50	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
51	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
53	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₂₇	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
58	M₁₆×C₁₂₇	—	2	4	no	—
59	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
60	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
61	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
62	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
63	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
64	C₂²×D₁₂₇	C₂²×C₄	1	2	yes	—
65	Dic₂₅₄	C₄⋊C₄	1	4	yes	—
66	D₅₀₈	C₄⋊C₄	1	4	yes	—
67	C₂²⋊D₁₂₇	C₄⋊C₄	1	8	yes	—
68	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
69	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
70	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
71	C₄×D₁₂₇	—	2	4	no	—
72	C₄×D₁₂₇	—	4	2	no	—
73	C₄×D₁₂₇	C₂×D₄	1	4	yes	—
74	C₄×D₁₂₇	C₂×Q₈	1	4	yes	—
75	C₁₂₇⋊C₈	—	2	2	no	—
76	C₁₂₇⋊C₈	—	2	2	no	—
77	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₂₇	—	2	2	no	—
79	C₂×C₄×C₁₂₇	—	4	1	no	—
80	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
87	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
88	C₈×C₁₂₇	—	2	4	no	—
89	C₂²×C₁₂₇	—	2	1	no	—
90	C₂²×C₁₂₇	—	2	1	no	—
91	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
92	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
93	C₂×D₁₂₇	—	2	2	no	—
94	C₂×D₁₂₇	—	2	2	no	—
95	C₂×D₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
96	C₄×C₁₂₇	—	2	1	no	—
97	C₄×C₁₂₇	—	2	1	no	—
98	C₄×C₁₂₇	—	2	1	no	—
99	C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
100	C₄×C₁₂₇	#131	1	2	yes	complemented
101	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
102	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
103	Dic₁₂₇	—	2	2	no	—
104	Dic₁₂₇	—	2	2	no	—
105	Dic₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
106	C₂×C₁₂₇	—	2	2	no	—
107	C₂×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
109	C₂×C₁₂₇	#568	1	1	yes	characteristic
110	D₁₂₇	—	2	4	no	—
111	#2906	—	127	1	no	maximal, Sylow
112	C₁₂₇	#2906	1	1	yes	characteristic, complemented, Sylow
113	C₂×#131	—	127	2	no	—
114	C₄×#41	—	127	1	no	—
115	#424	—	127	2	no	—
116	#424	—	127	4	no	—
117	#423	—	127	2	no	—
118	#423	D₁₂₇	1	1	yes	characteristic, complemented
119	C₂×C₄⋊C₈	—	127	1	no	—
120	C₂×C₄⋊C₈	—	127	2	no	—
121	C₂×#41	—	127	1	no	—
122	#131	—	254	8	no	—
123	C₂×M₁₆	—	127	4	no	—
124	C₂×M₁₆	C₂×D₁₂₇	1	2	yes	—
125	C₂²×C₈	—	127	2	no	—
126	C₂²×C₈	—	127	4	no	—
127	C₄×Q₈	—	127	1	no	—
128	C₄×Q₈	—	127	1	no	—
129	C₄×D₄	—	127	2	no	—
130	C₄×D₄	—	127	2	no	—
131	C₄×D₄	—	127	1	no	—
132	C₄×D₄	—	127	1	no	—
133	#117	—	127	1	no	—
134	#117	—	127	2	no	—
135	C₂×C₄²	—	127	2	no	—
136	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
137	C₄⋊C₈	—	127	4	no	—
138	C₄⋊C₈	—	127	4	no	—
139	C₄⋊C₈	—	127	1	no	—
140	C₄⋊C₈	—	127	1	no	—
141	C₄⋊C₈	—	127	2	no	—
142	C₄⋊C₈	C₂×D₁₂₇	1	2	yes	—
143	C₄⋊C₈	C₂×D₁₂₇	1	2	yes	—
144	#41	—	127	4	no	—
145	#41	—	127	4	no	—
146	C₂×Q₈	—	127	1	no	—
147	C₂×D₄	—	127	2	no	—
148	C₂×D₄	—	127	1	no	—
149	C₂²×C₄	—	127	2	no	—
150	C₂²×C₄	—	127	2	no	—
151	C₂²×C₄	—	127	2	no	—
152	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
153	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
155	M₁₆	—	2	4	no	—
156	M₁₆	—	254	8	no	—
157	C₂×C₈	—	127	2	no	—
158	C₂×C₈	—	127	2	no	—
159	C₂×C₈	—	254	8	no	—
160	C₂×C₈	—	254	4	no	—
161	C₂×C₈	C₂²×D₁₂₇	1	4	yes	—
162	C₄⋊C₄	—	127	2	no	—
163	C₄⋊C₄	—	127	2	no	—
164	C₄⋊C₄	—	127	1	no	—
165	C₄⋊C₄	—	127	1	no	—
166	C₂²⋊C₄	—	127	2	no	—
167	C₂²⋊C₄	—	127	2	no	—
168	C₂²⋊C₄	—	127	2	no	—
169	C₄²	—	127	2	no	—
170	C₄²	—	127	2	no	—
171	C₄²	—	127	2	no	—
172	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
173	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
174	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
175	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
176	C₂³	—	127	2	no	—
177	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
178	Q₈	—	127	4	no	—
179	D₄	—	127	8	no	—
180	D₄	—	127	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	—	127	2	no	—
184	C₂×C₄	—	127	4	no	—
185	C₂×C₄	—	127	4	no	—
186	C₂×C₄	—	127	2	no	—
187	C₂×C₄	—	127	2	no	—
188	C₂×C₄	—	508	2	no	—
189	C₂×C₄	—	254	4	no	—
190	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
191	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
192	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
195	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
196	C₂×C₄	Q₈×D₁₂₇	1	2	yes	—
197	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
198	C₈	—	2	4	no	—
199	C₈	—	254	2	no	—
200	C₈	—	254	2	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	127	4	no	—
204	C₂²	—	254	2	no	—
205	C₂²	—	254	2	no	—
206	C₂²	C₄⋊C₄×D₁₂₇	1	2	yes	—
207	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	127	4	no	—
212	C₄	—	254	2	no	—
213	C₄	—	254	2	no	—
214	C₄	C₄⋊C₄×D₁₂₇	1	2	yes	—
215	C₄	#16861217	1	2	yes	complemented
216	C₄	C₂×D₄×D₁₂₇	1	1	yes	characteristic
217	C₄	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
218	C₂	—	2	2	no	—
219	C₂	—	254	4	no	—
220	C₂	C₂×C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
221	C₂	C₂×#16861217	1	1	yes	characteristic
222	C₂	#22550642	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	#423×C₁₂₇	C₂
prime radical	C₂×#14523248	C₂²
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#423	D₁₂₇
hypocenter	C₁₂₇	#2906
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
3. #22549904	C₂	semidirect
4. #22549901	C₂	semidirect
5. #22549898	C₂	semidirect
6. C₂×C₄⋊(C₁₂₇⋊C₈)	C₂	semidirect
7. C₄⋊C₈×D₁₂₇	C₂	semidirect
8. #423×C₁₂₇	C₂	semidirect
100. C₄×C₁₂₇	#131	semidirect
112. C₁₂₇	#2906	semidirect
118. #423	D₁₂₇	semidirect
215. C₄	#16861217	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861217	2
C₄×#14523248	2
#22549904	2
#22549901	2
#22549898	2
C₂×C₄⋊(C₁₂₇⋊C₈)	2
C₄⋊C₈×D₁₂₇	2
#423×C₁₂₇	2
#2906	127