Subgroups — #30647666

Order 16256.

348 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550430	C₂	1	1	yes	maximal, characteristic
2	#22550007	C₂	1	1	yes	maximal, characteristic, complemented
3	#22550006	C₂	1	1	yes	maximal, characteristic, complemented
4	#22550005	C₂	1	1	yes	maximal, characteristic, complemented
5	#22550005	C₂	1	1	yes	maximal, characteristic, complemented
6	#22550004	C₂	1	1	yes	maximal, characteristic, complemented
7	#22550004	C₂	1	1	yes	maximal, characteristic, complemented
8	#22550003	C₂	1	1	yes	maximal, characteristic, complemented
9	#22549901	C₂	1	1	yes	maximal, characteristic
10	#22549901	C₂	1	1	yes	maximal, characteristic
11	#22549898	C₂	1	1	yes	maximal, characteristic
12	#22549833	C₂	1	1	yes	maximal, characteristic
13	#22549830	C₂	1	1	yes	maximal, characteristic
14	#97×D₁₂₇	C₂	1	1	yes	maximal, characteristic
15	#446×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14523231	C₂²	1	1	yes	characteristic
17	C₂×#14523231	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
20	#16861208	C₄	1	1	yes	characteristic
21	#16861207	C₄	1	1	yes	characteristic
22	#16861199	C₄	1	1	yes	characteristic
23	#16861199	C₄	1	1	yes	characteristic
24	#16861189	C₄	1	1	yes	characteristic
25	#16861187	C₄	1	1	yes	characteristic
26	#16861187	C₄	1	1	yes	characteristic
27	#16861185	C₄	1	1	yes	characteristic
28	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
29	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
30	C₄×Dic₂₅₄	C₂²	1	1	yes	characteristic
31	#16861166	C₂²	1	1	yes	characteristic
32	#16861166	C₂²	1	1	yes	characteristic
33	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
34	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
35	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
36	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	1	yes	characteristic
37	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
38	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
39	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
41	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
43	#16861120	C₂²	1	1	yes	characteristic
44	#16861120	C₂²	1	1	yes	characteristic
45	#16861119	C₂²	1	1	yes	characteristic
46	#16861119	C₂²	1	1	yes	characteristic
47	#16861119	C₂²	1	1	yes	characteristic
48	#16861119	C₂²	1	1	yes	characteristic
49	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
52	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
56	#97×C₁₂₇	C₂²	1	1	yes	characteristic
57	#97×C₁₂₇	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
60	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
61	C₂×Dic₂₅₄	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₁₂₇	C₂³	1	1	yes	characteristic
65	#14523231	—	2	2	no	—
66	#14523231	—	2	2	no	—
67	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₁₂₇⋊C₈	C₂³	1	1	yes	characteristic
71	C₈×D₁₂₇	—	2	2	no	—
72	C₈×D₁₂₇	Q₈	1	4	yes	complemented
73	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
74	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
75	C₄⋊Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
76	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
77	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
78	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
79	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
80	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
81	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
82	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
83	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
84	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
85	Dic₁₂₇⋊C₄	C₂×C₄	1	1	yes	characteristic
86	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
87	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
88	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
89	(C₂×D₁₂₇)⋊C₄	C₂×C₄	1	1	yes	characteristic
90	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
91	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
92	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
93	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
94	C₄×Dic₁₂₇	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
101	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
102	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
103	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
104	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
105	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
106	C₂²×D₁₂₇	C₂²×C₄	1	1	yes	characteristic
107	Dic₂₅₄	—	4	1	no	—
108	Dic₂₅₄	—	4	1	no	—
109	Dic₂₅₄	—	4	1	no	—
110	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
116	C₂×Dic₁₂₇	C₂²×C₄	1	1	yes	characteristic
117	C₄×D₁₂₇	—	4	1	no	—
118	C₄×D₁₂₇	—	4	1	no	—
119	C₄×D₁₂₇	C₂×Q₈	1	2	yes	—
120	C₄×D₁₂₇	#41	1	2	yes	—
121	C₁₂₇⋊C₈	—	2	1	no	—
122	C₁₂₇⋊C₈	—	2	1	no	—
123	C₁₂₇⋊C₈	—	2	1	no	—
124	C₁₂₇⋊C₈	C₂×Q₈	1	2	yes	—
125	Q₈×C₁₂₇	—	4	1	no	—
126	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
133	C₈×C₁₂₇	—	2	1	no	—
134	C₈×C₁₂₇	—	2	1	no	—
135	C₈×C₁₂₇	—	2	1	no	—
136	C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
137	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
138	C₂×D₁₂₇	—	2	1	no	—
139	C₂×D₁₂₇	—	2	1	no	—
140	C₂×D₁₂₇	C₄×Q₈	1	2	yes	—
141	C₄×C₁₂₇	—	2	1	no	—
142	C₄×C₁₂₇	—	2	1	no	—
143	C₄×C₁₂₇	—	2	1	no	—
144	C₄×C₁₂₇	—	2	1	no	—
145	C₄×C₁₂₇	—	2	1	no	—
146	C₄×C₁₂₇	—	2	1	no	—
147	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
148	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
149	Dic₁₂₇	—	2	1	no	—
150	Dic₁₂₇	—	2	1	no	—
151	Dic₁₂₇	—	2	1	no	—
152	Dic₁₂₇	—	2	1	no	—
153	Dic₁₂₇	—	2	1	no	—
154	Dic₁₂₇	—	2	1	no	—
155	Dic₁₂₇	C₄×Q₈	1	2	yes	—
156	C₂×C₁₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
157	C₂×C₁₂₇	#568	1	1	yes	characteristic
158	C₂×C₁₂₇	#568	1	1	yes	characteristic
159	D₁₂₇	—	2	2	no	—
160	#2961	—	127	1	no	maximal, Sylow
161	C₁₂₇	#2961	1	1	yes	characteristic, complemented, Sylow
162	#533	—	127	1	no	—
163	#446	—	127	1	no	—
164	#446	—	127	1	no	—
165	#446	—	127	1	no	—
166	#446	—	127	1	no	—
167	#446	—	127	1	no	—
168	#446	—	127	1	no	—
169	#446	—	127	1	no	—
170	#446	D₁₂₇	1	1	yes	characteristic, complemented
171	#424	—	127	1	no	—
172	#424	—	127	1	no	—
173	#423	—	127	1	no	—
174	#405	—	127	1	no	—
175	#405	—	127	1	no	—
176	C₂×#97	—	127	1	no	—
177	C₂×M₁₆	—	127	1	no	—
178	C₂×M₁₆	—	127	1	no	—
179	C₂²×C₈	—	127	1	no	—
180	C₂²×C₈	—	127	1	no	—
181	C₄⋊Q₈	—	127	1	no	—
182	C₄⋊Q₈	—	127	1	no	—
183	#125	—	127	1	no	—
184	#125	—	127	1	no	—
185	C₂²⋊Q₈	—	127	1	no	—
186	C₂²⋊Q₈	—	127	1	no	—
187	C₂²⋊Q₈	—	127	1	no	—
188	C₂²⋊Q₈	—	127	1	no	—
189	C₄×Q₈	—	127	1	no	—
190	C₄×Q₈	—	127	1	no	—
191	C₄×Q₈	—	127	1	no	—
192	C₄×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
193	#117	—	127	1	no	—
194	#117	—	127	1	no	—
195	C₂×C₄²	—	127	1	no	—
196	C₄⋊C₈	—	127	1	no	—
197	C₄⋊C₈	—	127	1	no	—
198	C₄⋊C₈	—	127	1	no	—
199	C₄⋊C₈	—	127	1	no	—
200	C₄⋊C₈	—	127	1	no	—
201	C₄⋊C₈	—	127	1	no	—
202	C₄⋊C₈	—	127	1	no	—
203	C₄⋊C₈	—	127	1	no	—
204	C₄⋊C₈	—	127	1	no	—
205	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
206	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
207	C₄⋊C₈	C₂×D₁₂₇	1	1	yes	characteristic
208	#97	—	127	1	no	—
209	#97	—	127	1	no	—
210	#97	—	127	1	no	—
211	#97	—	127	1	no	—
212	#97	—	127	1	no	—
213	#97	—	127	1	no	—
214	#97	C₂×D₁₂₇	1	1	yes	characteristic
215	#97	C₂×D₁₂₇	1	1	yes	characteristic
216	C₄×C₈	—	127	1	no	—
217	C₄×C₈	—	127	1	no	—
218	C₄×C₈	—	127	1	no	—
219	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
220	C₂×Q₈	—	127	1	no	—
221	C₂×Q₈	—	127	1	no	—
222	C₂×Q₈	—	127	1	no	—
223	C₂×Q₈	C₄×D₁₂₇	1	1	yes	characteristic
224	C₂²×C₄	—	127	1	no	—
225	C₂²×C₄	—	127	1	no	—
226	C₂²×C₄	—	127	1	no	—
227	M₁₆	—	254	2	no	—
228	M₁₆	—	254	2	no	—
229	C₂×C₈	—	127	1	no	—
230	C₂×C₈	—	127	4	no	—
231	C₂×C₈	—	127	1	no	—
232	C₂×C₈	—	127	1	no	—
233	C₂×C₈	—	127	1	no	—
234	C₂×C₈	—	254	2	no	—
235	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
236	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
237	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
239	C₄⋊C₄	—	127	1	no	—
240	C₄⋊C₄	—	127	1	no	—
241	C₄⋊C₄	—	127	1	no	—
242	C₄⋊C₄	—	127	1	no	—
243	C₄⋊C₄	—	127	1	no	—
244	C₄⋊C₄	—	127	1	no	—
245	C₄⋊C₄	—	127	1	no	—
246	C₄⋊C₄	—	127	1	no	—
247	C₄⋊C₄	—	127	1	no	—
248	C₄⋊C₄	—	127	1	no	—
249	C₄⋊C₄	—	127	1	no	—
250	C₄⋊C₄	—	127	1	no	—
251	C₄⋊C₄	—	127	1	no	—
252	C₄⋊C₄	C₄×D₁₂₇	1	1	yes	characteristic
253	C₄⋊C₄	C₄×D₁₂₇	1	1	yes	characteristic
254	C₄⋊C₄	C₄×D₁₂₇	1	1	yes	characteristic
255	C₂²⋊C₄	—	127	1	no	—
256	C₂²⋊C₄	—	127	1	no	—
257	C₂²⋊C₄	—	127	1	no	—
258	C₂²⋊C₄	—	127	1	no	—
259	C₄²	—	127	1	no	—
260	C₄²	—	127	1	no	—
261	C₄²	—	127	1	no	—
262	C₄²	—	127	1	no	—
263	C₄²	—	127	1	no	—
264	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
265	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
266	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
267	C₂³	—	127	1	no	—
268	Q₈	—	4	1	no	—
269	Q₈	—	508	1	no	—
270	Q₈	—	508	1	no	—
271	Q₈	—	508	1	no	—
272	C₂×C₄	—	127	1	no	—
273	C₂×C₄	—	127	2	no	—
274	C₂×C₄	—	127	2	no	—
275	C₂×C₄	—	127	1	no	—
276	C₂×C₄	—	127	1	no	—
277	C₂×C₄	—	127	1	no	—
278	C₂×C₄	—	127	1	no	—
279	C₂×C₄	—	127	1	no	—
280	C₂×C₄	—	127	1	no	—
281	C₂×C₄	—	508	1	no	—
282	C₂×C₄	—	508	1	no	—
283	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
285	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
286	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	127	2	no	—
294	C₈	—	254	1	no	—
295	C₈	—	254	1	no	—
296	C₈	—	254	1	no	—
297	C₈	Q₈×D₁₂₇	1	2	yes	—
298	C₂²	—	127	2	no	—
299	C₂²	—	254	1	no	—
300	C₂²	—	254	1	no	—
301	C₂²	C₂²×C₄×D₁₂₇	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	127	2	no	—
309	C₄	—	254	1	no	—
310	C₄	—	254	1	no	—
311	C₄	—	254	1	no	—
312	C₄	—	254	1	no	—
313	C₄	—	254	1	no	—
314	C₄	—	254	1	no	—
315	C₄	C₂×Q₈×D₁₂₇	1	1	yes	characteristic
316	C₄	#41×D₁₂₇	1	1	yes	characteristic
317	C₂	—	254	2	no	—
318	C₂	C₄×Q₈×D₁₂₇	1	1	yes	characteristic
319	C₂	#22550641	1	1	yes	characteristic
320	C₂	#22550642	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Fitting	#446×C₁₂₇	C₂
prime radical	C₂²×D₁₂₇	C₂²×C₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	#446	D₁₂₇
hypocenter	C₁₂₇	#2961
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #22550007	C₂	semidirect
3. #22550006	C₂	semidirect
4. #22550005	C₂	semidirect
5. #22550005	C₂	semidirect
6. #22550004	C₂	semidirect
7. #22550004	C₂	semidirect
8. #22550003	C₂	semidirect
15. #446×C₁₂₇	C₂	semidirect
72. C₈×D₁₂₇	Q₈	semidirect
161. C₁₂₇	#2961	semidirect
170. #446	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550430	2
#22550007	2
#22550006	2
#22550005	2
#22550004	2
#22550003	2
#22549901	2
#22549898	2
#22549833	2
#22549830	2
#97×D₁₂₇	2
#446×C₁₂₇	2
#2961	127