Subgroups — #30647371

Order 16256.

400 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550344	C₂	1	1	yes	maximal, characteristic
2	#22549938	C₂	1	1	yes	maximal, characteristic, complemented
3	#22549937	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549935	C₂	1	1	yes	maximal, characteristic, complemented
5	#22549928	C₂	1	2	yes	maximal, complemented
6	#22549926	C₂	1	2	yes	maximal, complemented
7	#22549901	C₂	1	2	yes	maximal, complemented
8	#22549841	C₂	1	2	yes	maximal
9	#22549828	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₈×D₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#433×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14523248	C₂²	1	1	yes	characteristic
13	C₂×#14523231	C₂²	1	2	yes	—
14	C₂×C₈×D₁₂₇	C₂²	1	2	yes	—
15	#16861174	C₄	1	2	yes	—
16	#16861171	C₄	1	2	yes	—
17	(C₂²⋊D₁₂₇)⋊C₄	C₄	1	2	yes	—
18	(C₂²⋊D₁₂₇)⋊C₄	C₄	1	2	yes	—
19	#16861166	C₂²	1	1	yes	characteristic
20	#16861166	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₁₂₇	C₂²	1	1	yes	characteristic
22	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
23	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
24	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
25	Dic₁₂₇⋊C₈	C₂²	1	2	yes	—
26	Dic₁₂₇⋊C₈	C₂²	1	2	yes	—
27	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
28	(C₂×D₁₂₇)⋊C₈	C₂²	1	2	yes	—
29	(C₂×D₁₂₇)⋊C₈	C₂²	1	2	yes	—
30	#16861120	C₂²	1	1	yes	characteristic
31	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
32	C₈×Dic₁₂₇	C₂²	1	2	yes	—
33	C₈×Dic₁₂₇	C₂²	1	2	yes	—
34	#117×C₁₂₇	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
36	C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
37	#97×C₁₂₇	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
39	#14523248	—	2	4	no	—
40	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
41	C₂×D₅₀₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₁₂₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×Dic₁₂₇	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
48	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
49	#14523231	—	2	4	no	—
50	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
51	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
52	C₈×D₁₂₇	—	2	4	no	—
53	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
54	Dic₁₂₇⋊C₄	C₂×C₄	1	2	yes	—
55	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
56	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
59	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
61	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
64	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
65	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
66	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
67	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
68	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
69	C₂²×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
70	C₂²×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
71	Dic₂₅₄	—	2	2	no	—
72	D₅₀₈	—	2	2	no	—
73	C₂²⋊D₁₂₇	—	2	2	no	—
74	C₂²⋊D₁₂₇	—	2	2	no	—
75	C₂×Dic₁₂₇	—	2	4	no	—
76	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
78	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₁₂₇	C₂²×C₄	1	2	yes	—
80	C₄×D₁₂₇	—	2	2	no	—
81	C₄×D₁₂₇	—	2	4	no	—
82	C₄×D₁₂₇	—	2	4	no	—
83	C₄×D₁₂₇	#41	1	4	yes	—
84	C₁₂₇⋊C₈	—	2	2	no	—
85	C₁₂₇⋊C₈	—	2	2	no	—
86	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₂₇	—	2	2	no	—
88	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
92	C₈×C₁₂₇	—	2	2	no	—
93	C₈×C₁₂₇	—	2	2	no	—
94	C₂²×C₁₂₇	—	2	1	no	—
95	C₂²×C₁₂₇	—	2	2	no	—
96	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
97	C₂×D₁₂₇	—	2	2	no	—
98	C₂×D₁₂₇	—	2	1	no	—
99	C₂×D₁₂₇	—	2	2	no	—
100	C₂×D₁₂₇	#117	1	2	yes	—
101	C₄×C₁₂₇	—	2	1	no	—
102	C₄×C₁₂₇	—	2	2	no	—
103	C₄×C₁₂₇	—	2	2	no	—
104	C₄×C₁₂₇	C₂×#41	1	2	yes	—
105	Dic₁₂₇	—	2	1	no	—
106	Dic₁₂₇	—	2	2	no	—
107	Dic₁₂₇	#117	1	2	yes	—
108	Dic₁₂₇	#131	1	4	yes	—
109	C₂×C₁₂₇	—	4	1	no	—
110	C₂×C₁₂₇	C₂×#117	1	1	yes	characteristic
111	C₂×C₁₂₇	C₂×#131	1	2	yes	—
112	D₁₂₇	—	2	2	no	—
113	D₁₂₇	—	4	1	no	—
114	#2922	—	127	1	no	maximal, Sylow
115	C₁₂₇	#2922	1	1	yes	characteristic, complemented, Sylow
116	C₄×#41	—	127	1	no	—
117	#433	—	127	1	no	—
118	#433	—	127	1	no	—
119	#433	—	127	1	no	—
120	#433	D₁₂₇	1	1	yes	characteristic, complemented
121	#431	—	127	2	no	—
122	#431	—	127	2	no	—
123	#424	—	127	2	no	—
124	#408	—	127	2	no	—
125	C₄×M₁₆	—	127	1	no	—
126	C₂×C₄×C₈	—	127	1	no	—
127	C₂×#41	—	127	1	no	—
128	C₂×M₁₆	—	127	2	no	—
129	C₂²×C₈	—	127	2	no	—
130	C₄×Q₈	—	127	2	no	—
131	C₄×D₄	—	127	2	no	—
132	C₄×D₄	—	127	2	no	—
133	C₄×D₄	—	127	2	no	—
134	#117	—	127	1	no	—
135	#117	—	127	1	no	—
136	#117	C₂×D₁₂₇	1	1	yes	characteristic
137	C₂×C₄²	—	127	1	no	—
138	C₂×C₄²	—	127	1	no	—
139	C₂×C₄²	—	127	1	no	—
140	C₄⋊C₈	—	127	2	no	—
141	C₄⋊C₈	—	127	2	no	—
142	C₄⋊C₈	—	127	2	no	—
143	C₄⋊C₈	C₂×D₁₂₇	1	2	yes	—
144	C₂²⋊C₈	—	127	2	no	—
145	C₂²⋊C₈	—	127	2	no	—
146	C₂²⋊C₈	—	127	2	no	—
147	C₂²⋊C₈	C₂×D₁₂₇	1	2	yes	—
148	#97	—	127	1	no	—
149	#97	C₂×D₁₂₇	1	1	yes	characteristic
150	C₄×C₈	—	127	2	no	—
151	C₄×C₈	—	127	2	no	—
152	C₄×C₈	—	127	1	no	—
153	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
154	#41	—	254	4	no	—
155	C₂×Q₈	—	127	1	no	—
156	C₂×D₄	—	127	1	no	—
157	C₂×D₄	—	127	1	no	—
158	C₂×D₄	—	127	1	no	—
159	C₂²×C₄	—	127	1	no	—
160	C₂²×C₄	—	127	2	no	—
161	C₂²×C₄	—	127	2	no	—
162	C₂²×C₄	—	127	2	no	—
163	C₂²×C₄	—	127	1	no	—
164	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
165	M₁₆	—	254	4	no	—
166	C₂×C₈	—	127	2	no	—
167	C₂×C₈	—	127	2	no	—
168	C₂×C₈	—	254	4	no	—
169	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
170	C₂×C₈	C₂²×D₁₂₇	1	2	yes	—
171	C₄⋊C₄	—	127	2	no	—
172	C₄⋊C₄	—	127	2	no	—
173	C₄⋊C₄	C₄×D₁₂₇	1	2	yes	—
174	C₂²⋊C₄	—	127	2	no	—
175	C₂²⋊C₄	—	127	2	no	—
176	C₂²⋊C₄	C₄×D₁₂₇	1	2	yes	—
177	C₄²	—	127	2	no	—
178	C₄²	—	127	2	no	—
179	C₄²	—	127	2	no	—
180	C₄²	—	127	1	no	—
181	C₄²	—	127	1	no	—
182	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
183	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
184	C₂³	—	127	1	no	—
185	C₂³	—	127	1	no	—
186	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
187	Q₈	—	254	2	no	—
188	D₄	—	254	2	no	—
189	D₄	—	254	2	no	—
190	D₄	—	254	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	127	2	no	—
193	C₂×C₄	—	127	1	no	—
194	C₂×C₄	—	127	4	no	—
195	C₂×C₄	—	127	2	no	—
196	C₂×C₄	—	127	1	no	—
197	C₂×C₄	—	254	4	no	—
198	C₂×C₄	—	254	4	no	—
199	C₂×C₄	—	254	4	no	—
200	C₂×C₄	—	254	2	no	—
201	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
202	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
204	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₈	—	254	2	no	—
208	C₈	—	254	2	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	127	2	no	—
212	C₂²	—	254	1	no	—
213	C₂²	—	254	2	no	—
214	C₂²	—	254	2	no	—
215	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	127	2	no	—
220	C₄	—	127	4	no	—
221	C₄	—	254	1	no	—
222	C₄	—	254	2	no	—
223	C₄	#41×D₁₂₇	1	2	yes	—
224	C₂	—	4	1	no	—
225	C₂	—	508	1	no	—
226	C₂	—	254	2	no	—
227	C₂	#117×D₁₂₇	1	1	yes	characteristic
228	C₂	#131×D₁₂₇	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	#433×C₁₂₇	C₂
prime radical	C₂×#14523248	C₂²
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#433	D₁₂₇
hypocenter	C₁₂₇	#2922
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. #22549938	C₂	semidirect
3. #22549937	C₂	semidirect
4. #22549935	C₂	semidirect
5. #22549928	C₂	semidirect
6. #22549926	C₂	semidirect
7. #22549901	C₂	semidirect
9. #22549828	C₂	semidirect
10. C₄×C₈×D₁₂₇	C₂	semidirect
11. #433×C₁₂₇	C₂	semidirect
115. C₁₂₇	#2922	semidirect
120. #433	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550344	2
#22549938	2
#22549937	2
#22549935	2
#22549928	2
#22549926	2
#22549901	2
#22549841	2
#22549828	2
C₄×C₈×D₁₂₇	2
#433×C₁₂₇	2
#2922	127