Subgroups — #30648518

Order 16256.

388 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550433	C₂	1	1	yes	maximal, characteristic, complemented
2	#22550183	C₂	1	2	yes	maximal, complemented
3	#22550176	C₂	1	2	yes	maximal, complemented
4	#22550171	C₂	1	1	yes	maximal, characteristic, complemented
5	#22550166	C₂	1	1	yes	maximal, characteristic, complemented
6	#22550161	C₂	1	1	yes	maximal, characteristic, complemented
7	#22550156	C₂	1	1	yes	maximal, characteristic, complemented
8	#22549912	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊(C₁₂₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	#22549872	C₂	1	2	yes	maximal
11	C₄×#41×C₁₂₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
13	#16861209	C₂²	1	1	yes	characteristic
14	#16861209	C₂²	1	1	yes	characteristic
15	#16861201	C₂²	1	2	yes	—
16	#16861188	—	2	1	no	—
17	#16861188	—	2	1	no	—
18	C₄×Dic₂₅₄	—	2	1	no	—
19	C₄×Dic₂₅₄	—	2	1	no	—
20	#16861168	C₂²	1	2	yes	—
21	#16861142	C₂²	1	2	yes	—
22	#16861142	C₂²	1	2	yes	—
23	#16861140	C₂²	1	2	yes	—
24	#16861140	C₂²	1	2	yes	—
25	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
27	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	2	yes	—
28	#16861133	C₂²	1	2	yes	—
29	#16861133	C₂²	1	2	yes	—
30	#16861130	C₂²	1	2	yes	—
31	#16861130	C₂²	1	2	yes	—
32	C₂×#41×C₁₂₇	C₂²	1	1	yes	characteristic
33	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
35	C₄×D₄×C₁₂₇	—	2	1	no	—
36	C₄×D₄×C₁₂₇	—	2	1	no	—
37	C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
38	C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
39	#117×C₁₂₇	—	2	1	no	—
40	#117×C₁₂₇	C₂²	1	1	yes	characteristic
41	C₂×C₄²×C₁₂₇	—	2	1	no	—
42	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
43	C₂×Dic₂₅₄	—	2	1	no	—
44	C₂×Dic₂₅₄	—	2	1	no	—
45	C₂×C₁₂₇⋊C₈	—	2	4	no	—
46	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
47	C₂×C₁₂₇⋊C₈	C₂³	1	2	yes	—
48	C₄⋊Dic₁₂₇	—	2	1	no	—
49	C₄⋊Dic₁₂₇	—	2	1	no	—
50	C₄⋊Dic₁₂₇	C₂³	1	2	yes	—
51	C₄⋊Dic₁₂₇	C₂³	1	2	yes	—
52	Dic₁₂₇⋊C₄	—	2	2	no	—
53	Dic₁₂₇⋊C₄	—	2	2	no	—
54	C₂²⋊Dic₁₂₇	—	2	2	no	—
55	C₄×Dic₁₂₇	—	2	2	no	—
56	#41×C₁₂₇	—	2	2	no	—
57	#41×C₁₂₇	Q₈	1	4	yes	complemented
58	C₂×Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
59	C₂×D₄×C₁₂₇	—	2	1	no	—
60	C₂×D₄×C₁₂₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₂₇	—	2	1	no	—
62	C₂²×C₄×C₁₂₇	—	2	1	no	—
63	C₂²×C₄×C₁₂₇	—	2	1	no	—
64	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₁₂₇	—	2	1	no	—
68	C₄⋊C₄×C₁₂₇	—	2	1	no	—
69	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
71	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
72	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
73	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
74	C₄²×C₁₂₇	—	2	1	no	—
75	C₄²×C₁₂₇	—	2	1	no	—
76	C₄²×C₁₂₇	—	2	1	no	—
77	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
78	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
79	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
80	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
81	Dic₂₅₄	—	4	2	no	—
82	Dic₂₅₄	—	4	2	no	—
83	C₂×Dic₁₂₇	—	2	2	no	—
84	C₂×Dic₁₂₇	—	2	2	no	—
85	C₁₂₇⋊C₈	—	2	2	no	—
86	C₁₂₇⋊C₈	—	2	2	no	—
87	C₂³×C₁₂₇	—	2	1	no	—
88	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
89	Q₈×C₁₂₇	—	2	1	no	—
90	Q₈×C₁₂₇	C₂×Q₈	1	2	yes	—
91	D₄×C₁₂₇	—	2	4	no	—
92	D₄×C₁₂₇	—	2	1	no	—
93	D₄×C₁₂₇	C₂×Q₈	1	2	yes	—
94	C₂×C₄×C₁₂₇	—	2	1	no	—
95	C₂×C₄×C₁₂₇	—	2	4	no	—
96	C₂×C₄×C₁₂₇	—	2	2	no	—
97	C₂×C₄×C₁₂₇	—	2	1	no	—
98	C₂×C₄×C₁₂₇	—	2	1	no	—
99	C₂×C₄×C₁₂₇	—	4	1	no	—
100	C₂×C₄×C₁₂₇	—	4	2	no	—
101	C₂×C₄×C₁₂₇	—	4	2	no	—
102	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
108	C₂×C₄×C₁₂₇	#41	1	2	yes	—
109	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
110	C₂²×C₁₂₇	—	2	1	no	—
111	C₂²×C₁₂₇	—	2	1	no	—
112	C₂²×C₁₂₇	—	2	2	no	—
113	C₂²×C₁₂₇	—	4	1	no	—
114	C₂²×C₁₂₇	—	4	1	no	—
115	C₂²×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
116	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
117	C₄×C₁₂₇	—	2	2	no	—
118	C₄×C₁₂₇	—	2	1	no	—
119	C₄×C₁₂₇	—	2	1	no	—
120	C₄×C₁₂₇	—	2	1	no	—
121	C₄×C₁₂₇	—	4	1	no	—
122	C₄×C₁₂₇	—	4	1	no	—
123	C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
124	C₄×C₁₂₇	#135	1	2	yes	—
125	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
126	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
127	Dic₁₂₇	—	4	2	no	—
128	Dic₁₂₇	—	4	2	no	—
129	C₂×C₁₂₇	—	2	2	no	—
130	C₂×C₁₂₇	—	4	2	no	—
131	C₂×C₁₂₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₁₂₇	C₂×#135	1	1	yes	characteristic
133	C₂×C₁₂₇	#575	1	1	yes	characteristic
134	#3076	—	127	1	no	maximal, Sylow
135	C₁₂₇	#3076	1	1	yes	characteristic, complemented, Sylow
136	#533	—	127	1	no	—
137	C₄×#41	D₁₂₇	1	1	yes	characteristic
138	#479	—	127	2	no	—
139	#478	—	127	2	no	—
140	#477	—	127	1	no	—
141	#476	—	127	1	no	—
142	#475	—	127	1	no	—
143	#474	—	127	1	no	—
144	#427	—	127	2	no	—
145	C₂×C₄⋊C₈	—	127	1	no	—
146	#416	—	127	2	no	—
147	C₂×#41	C₂×D₁₂₇	1	1	yes	characteristic
148	C₂²×C₈	—	127	2	no	—
149	C₄⋊Q₈	—	127	1	no	—
150	C₄⋊Q₈	—	127	1	no	—
151	#125	—	127	2	no	—
152	C₂²⋊Q₈	—	254	1	no	—
153	C₂²⋊Q₈	—	254	1	no	—
154	C₄×Q₈	—	254	1	no	—
155	C₄×Q₈	—	254	1	no	—
156	C₄×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
157	C₄×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
158	C₄×D₄	—	2	1	no	—
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	C₂×D₁₂₇	1	1	yes	characteristic
161	C₄×D₄	C₂×D₁₂₇	1	1	yes	characteristic
162	#117	—	2	1	no	—
163	#117	—	127	2	no	—
164	#117	C₂×D₁₂₇	1	1	yes	characteristic
165	C₂×C₄²	—	2	1	no	—
166	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
167	#107	—	127	2	no	—
168	#107	—	127	2	no	—
169	#106	—	127	2	no	—
170	#106	—	127	2	no	—
171	C₄⋊C₈	—	127	1	no	—
172	C₄⋊C₈	—	127	1	no	—
173	C₄⋊C₈	—	127	2	no	—
174	#103	—	127	2	no	—
175	#103	—	127	2	no	—
176	#102	—	127	2	no	—
177	#102	—	127	2	no	—
178	#41	—	2	2	no	—
179	#41	Dic₂₅₄	1	4	yes	complemented
180	C₂×Q₈	—	254	1	no	—
181	C₂×Q₈	—	254	1	no	—
182	C₂×Q₈	C₂²×D₁₂₇	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	C₂²×D₁₂₇	1	1	yes	characteristic
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊D₁₂₇	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
191	C₂×C₈	—	127	2	no	—
192	C₂×C₈	—	127	2	no	—
193	C₂×C₈	—	254	4	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	127	2	no	—
197	C₄⋊C₄	—	127	2	no	—
198	C₄⋊C₄	—	254	2	no	—
199	C₄⋊C₄	—	254	2	no	—
200	C₄⋊C₄	—	254	1	no	—
201	C₄⋊C₄	—	254	1	no	—
202	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
203	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	254	2	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	254	2	no	—
212	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
213	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
214	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
215	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
216	C₂³	—	2	1	no	—
217	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
218	Q₈	—	2	1	no	—
219	Q₈	—	508	2	no	—
220	Q₈	—	508	2	no	—
221	Q₈	C₂×Dic₂₅₄	1	2	yes	—
222	D₄	—	2	4	no	—
223	D₄	—	2	1	no	—
224	D₄	C₂×Dic₂₅₄	1	2	yes	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	254	2	no	—
234	C₂×C₄	—	254	2	no	—
235	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
237	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
238	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
239	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
240	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
241	C₂×C₄	#14523248	1	2	yes	—
242	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
243	C₈	—	254	2	no	—
244	C₈	—	254	2	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	—	4	1	no	—
250	C₂²	#16861188	1	2	yes	—
251	C₂²	C₂²×C₂²⋊D₁₂₇	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	1	no	—
258	C₄	—	508	2	no	—
259	C₄	—	508	2	no	—
260	C₄	#16861188	1	2	yes	—
261	C₄	#16861229	1	2	yes	—
262	C₄	C₂²×Dic₂₅₄	1	1	yes	characteristic
263	C₄	C₂×#14523248	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	C₂×#16861188	1	1	yes	characteristic
267	C₂	C₂×#16861229	1	1	yes	characteristic
268	C₂	#22550674	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₂₇
commutator, cosocle	C₂×C₄×C₁₂₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₄×#41×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²×D₄
prime radical	C₂×#41×C₁₂₇	C₂²
hypercenter	C₄×#41	D₁₂₇
hypocenter	C₁₂₇	#3076
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #22550433	C₂	semidirect
2. #22550183	C₂	semidirect
3. #22550176	C₂	semidirect
4. #22550171	C₂	semidirect
5. #22550166	C₂	semidirect
6. #22550161	C₂	semidirect
7. #22550156	C₂	semidirect
8. #22549912	C₂	semidirect
9. C₂×C₄⋊(C₁₂₇⋊C₈)	C₂	semidirect
57. #41×C₁₂₇	Q₈	semidirect
135. C₁₂₇	#3076	semidirect
179. #41	Dic₂₅₄	semidirect

Maximal subgroups

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

subgroup	index
#22550433	2
#22550183	2
#22550176	2
#22550171	2
#22550166	2
#22550161	2
#22550156	2
#22549912	2
C₂×C₄⋊(C₁₂₇⋊C₈)	2
#22549872	2
C₄×#41×C₁₂₇	2
#3076	127