Subgroups — #30647717

Order 16256.

362 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	#22550007	C₂	1	2	yes	maximal, complemented
2	#22549954	C₂	1	2	yes	maximal, complemented
3	#22549948	C₂	1	2	yes	maximal, complemented
4	#22549948	C₂	1	2	yes	maximal, complemented
5	#22549937	C₂	1	2	yes	maximal, complemented
6	#22549933	C₂	1	1	yes	maximal, characteristic, complemented
7	#22549832	C₂	1	2	yes	maximal, complemented
8	C₄×#14523232	C₂	1	1	yes	maximal, characteristic, complemented
9	#529×C₁₂₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#14523232	C₂²	1	2	yes	—
11	C₂×#14523232	C₂²	1	2	yes	—
12	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
13	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
14	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
15	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
16	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
18	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
19	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
20	#16861120	C₂²	1	2	yes	—
21	#16861120	C₂²	1	2	yes	—
22	#16861120	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	#126×C₁₂₇	C₄	1	2	yes	—
27	#125×C₁₂₇	C₄	1	1	yes	characteristic
28	#124×C₁₂₇	C₄	1	1	yes	characteristic
29	#123×C₁₂₇	C₄	1	2	yes	—
30	C₂²⋊Q₈×C₁₂₇	C₄	1	1	yes	characteristic
31	#121×C₁₂₇	C₄	1	1	yes	characteristic
32	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
33	C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
34	C₄×D₄×C₁₂₇	C₂²	1	2	yes	—
35	#117×C₁₂₇	C₂²	1	2	yes	—
36	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
37	#14523232	—	2	4	no	—
38	#14523232	—	2	4	no	—
39	C₂×C₁₂₇⋊C₈	—	2	4	no	—
40	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
41	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
42	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
43	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
44	C₂×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
45	C₂×D₄×C₁₂₇	C₂×C₄	1	2	yes	—
46	C₂×D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
50	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
52	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
53	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
54	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
55	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
56	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
57	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
58	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
59	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
60	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
61	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
62	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
63	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
64	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
65	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
66	C₄²×C₁₂₇	C₂³	1	2	yes	—
67	C₁₂₇⋊C₈	—	2	2	no	—
68	C₁₂₇⋊C₈	—	2	2	no	—
69	C₁₂₇⋊C₈	—	2	2	no	—
70	C₁₂₇⋊C₈	#41	1	4	yes	—
71	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
72	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
73	Q₈×C₁₂₇	—	4	1	no	—
74	D₄×C₁₂₇	—	4	1	no	—
75	D₄×C₁₂₇	—	4	2	no	—
76	C₂×C₄×C₁₂₇	—	2	2	no	—
77	C₂×C₄×C₁₂₇	—	2	2	no	—
78	C₂×C₄×C₁₂₇	—	2	2	no	—
79	C₂×C₄×C₁₂₇	—	4	1	no	—
80	C₂×C₄×C₁₂₇	—	4	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
90	C₂²×C₁₂₇	—	2	2	no	—
91	C₂²×C₁₂₇	—	2	1	no	—
92	C₂²×C₁₂₇	—	2	1	no	—
93	C₂²×C₁₂₇	—	2	1	no	—
94	C₂²×C₁₂₇	—	2	2	no	—
95	C₂²×C₁₂₇	—	2	2	no	—
96	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
97	C₄×C₁₂₇	—	2	2	no	—
98	C₄×C₁₂₇	—	2	1	no	—
99	C₄×C₁₂₇	—	2	1	no	—
100	C₄×C₁₂₇	—	2	1	no	—
101	C₄×C₁₂₇	—	2	1	no	—
102	C₄×C₁₂₇	—	2	1	no	—
103	C₄×C₁₂₇	—	2	2	no	—
104	C₄×C₁₂₇	—	2	2	no	—
105	C₄×C₁₂₇	C₂×#41	1	2	yes	—
106	C₂×C₁₂₇	—	4	1	no	—
107	C₂×C₁₂₇	—	4	2	no	—
108	C₂×C₁₂₇	C₄×#41	1	1	yes	characteristic
109	C₂×C₁₂₇	#568	1	1	yes	characteristic
110	C₂×C₁₂₇	#568	1	1	yes	characteristic
111	#2967	—	127	1	no	maximal, Sylow
112	C₁₂₇	#2967	1	1	yes	characteristic, complemented, Sylow
113	#529	D₁₂₇	1	1	yes	characteristic
114	#446	—	127	2	no	—
115	#436	—	127	2	no	—
116	#435	—	127	2	no	—
117	#435	—	127	2	no	—
118	#433	—	127	2	no	—
119	#432	—	127	1	no	—
120	#405	—	127	2	no	—
121	C₄×M₁₆	—	127	1	no	—
122	C₂×M₁₆	—	127	2	no	—
123	C₂×M₁₆	—	127	2	no	—
124	C₂²×C₈	—	127	2	no	—
125	#126	Dic₁₂₇	1	2	yes	—
126	#125	Dic₁₂₇	1	1	yes	characteristic
127	#124	Dic₁₂₇	1	1	yes	characteristic
128	#123	Dic₁₂₇	1	2	yes	—
129	C₂²⋊Q₈	Dic₁₂₇	1	1	yes	characteristic
130	#121	Dic₁₂₇	1	1	yes	characteristic
131	C₄×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
132	C₄×D₄	C₂×D₁₂₇	1	1	yes	characteristic
133	C₄×D₄	C₂×D₁₂₇	1	2	yes	—
134	#117	C₂×D₁₂₇	1	2	yes	—
135	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
136	C₄⋊C₈	—	127	2	no	—
137	C₄⋊C₈	—	127	2	no	—
138	C₄⋊C₈	—	127	2	no	—
139	C₂²⋊C₈	—	127	1	no	—
140	C₂²⋊C₈	—	127	1	no	—
141	C₂²⋊C₈	—	127	2	no	—
142	C₂²⋊C₈	—	127	2	no	—
143	#97	—	127	2	no	—
144	#97	—	127	2	no	—
145	#97	—	127	2	no	—
146	C₄×C₈	—	127	1	no	—
147	C₄×C₈	—	127	1	no	—
148	C₄×C₈	—	127	2	no	—
149	C₂×Q₈	C₂×Dic₁₂₇	1	1	yes	characteristic
150	C₂×D₄	C₂×Dic₁₂₇	1	2	yes	—
151	C₂×D₄	C₂×Dic₁₂₇	1	1	yes	characteristic
152	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
153	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
155	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
156	M₁₆	—	254	4	no	—
157	M₁₆	—	254	4	no	—
158	C₂×C₈	—	127	2	no	—
159	C₂×C₈	—	127	2	no	—
160	C₂×C₈	—	127	2	no	—
161	C₂×C₈	—	127	2	no	—
162	C₂×C₈	—	254	4	no	—
163	C₄⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
164	C₄⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
165	C₄⋊C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
166	C₄⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
167	C₄⋊C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
168	C₄⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
169	C₂²⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
170	C₂²⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
171	C₂²⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
172	C₂²⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
173	C₂²⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
174	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
175	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
176	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
177	C₄²	C₂²×D₁₂₇	1	2	yes	—
178	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
179	C₂³	C₂²×Dic₁₂₇	1	2	yes	—
180	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
181	Q₈	—	4	1	no	—
182	D₄	—	4	1	no	—
183	D₄	—	4	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	4	1	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
190	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
191	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
192	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
193	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
194	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
195	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
196	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
197	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
198	C₈	—	127	4	no	—
199	C₈	—	254	2	no	—
200	C₈	—	254	2	no	—
201	C₈	—	254	2	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	C₂³×Dic₁₂₇	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	2	no	—
216	C₄	—	2	2	no	—
217	C₄	#41×D₁₂₇	1	2	yes	—
218	C₂	—	4	1	no	—
219	C₂	—	4	2	no	—
220	C₂	#41×Dic₁₂₇	1	1	yes	characteristic
221	C₂	#22550643	1	1	yes	characteristic
222	C₂	#22550643	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	#529×C₁₂₇	C₂
prime radical	#121×C₁₂₇	C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#529	D₁₂₇
hypocenter	C₁₂₇	#2967
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
1. #22550007	C₂	semidirect
2. #22549954	C₂	semidirect
3. #22549948	C₂	semidirect
4. #22549948	C₂	semidirect
5. #22549937	C₂	semidirect
6. #22549933	C₂	semidirect
7. #22549832	C₂	semidirect
8. C₄×#14523232	C₂	semidirect
112. C₁₂₇	#2967	semidirect

Maximal subgroups

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

subgroup	index
#22550007	2
#22549954	2
#22549948	2
#22549937	2
#22549933	2
#22549832	2
C₄×#14523232	2
#529×C₁₂₇	2
#2967	127