Subgroups — #30652781

Order 16256.

1124 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550599	C₂	1	1	yes	maximal, characteristic, complemented
2	#22550583	C₂	1	2	yes	maximal, complemented
3	#22550564	C₂	1	2	yes	maximal, complemented
4	#22550549	C₂	1	4	yes	maximal, complemented
5	Q₈×D₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#22550534	C₂	1	2	yes	maximal, complemented
7	#22550531	C₂	1	1	yes	maximal, characteristic, complemented
8	#22550530	C₂	1	1	yes	maximal, characteristic, complemented
9	#22550522	C₂	1	2	yes	maximal, complemented
10	#22550497	C₂	1	2	yes	maximal, complemented
11	#22550467	C₂	1	4	yes	maximal, complemented
12	#22550433	C₂	1	1	yes	maximal, characteristic, complemented
13	#22550386	C₂	1	2	yes	maximal, complemented
14	C₂²⋊Q₈×D₁₂₇	C₂	1	2	yes	maximal, complemented
15	#22550356	C₂	1	2	yes	maximal, complemented
16	C₂×C₄×D₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
17	#529×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#14523248	C₂²	1	2	yes	—
19	C₂×#14523247	C₂²	1	2	yes	complemented
20	C₂×Q₈×D₁₂₇	C₂²	1	2	yes	complemented
21	C₂²×D₅₀₈	C₂²	1	1	yes	characteristic
22	C₂²×C₄×D₁₂₇	C₂²	1	2	yes	—
23	#16861209	C₂²	1	1	yes	characteristic, complemented
24	#16861209	C₂²	1	1	yes	characteristic, complemented
25	#16861209	C₂²	1	2	yes	complemented
26	#16861204	C₂²	1	4	yes	complemented
27	#16861204	C₂²	1	4	yes	complemented
28	#16861201	C₂²	1	2	yes	complemented
29	#16861200	C₂²	1	4	yes	complemented
30	#16861197	C₂²	1	2	yes	—
31	#16861195	C₂²	1	4	yes	complemented
32	#16861194	C₂²	1	4	yes	—
33	#16861193	C₂²	1	4	yes	complemented
34	#16861191	C₂²	1	4	yes	complemented
35	#16861191	C₂²	1	4	yes	complemented
36	#16861189	C₂²	1	2	yes	complemented
37	#16861189	C₂²	1	4	yes	complemented
38	#16861189	C₂²	1	2	yes	complemented
39	#16861188	C₂²	1	2	yes	—
40	#16861188	C₂²	1	2	yes	—
41	#16861188	C₂²	1	2	yes	complemented
42	#16861187	C₂²	1	4	yes	complemented
43	#16861186	C₂²	1	4	yes	—
44	#16861186	C₂²	1	4	yes	complemented
45	#16861185	C₂²	1	2	yes	complemented
46	#16861183	C₂²	1	2	yes	—
47	#16861182	C₂²	1	4	yes	—
48	#16861180	C₂²	1	2	yes	complemented
49	#16861177	C₂²	1	4	yes	—
50	C₄×Dic₂₅₄	C₂²	1	2	yes	complemented
51	C₄×Dic₂₅₄	C₂²	1	2	yes	complemented
52	C₄×Dic₂₅₄	C₂²	1	2	yes	complemented
53	C₄×D₅₀₈	C₂²	1	1	yes	characteristic
54	C₄×D₅₀₈	C₂²	1	1	yes	characteristic
55	C₄×D₅₀₈	C₂²	1	2	yes	—
56	C₄×D₅₀₈	C₂²	1	2	yes	—
57	C₄×D₅₀₈	C₂²	1	1	yes	characteristic
58	C₄×D₅₀₈	C₂²	1	2	yes	—
59	C₄×D₅₀₈	C₂²	1	1	yes	characteristic
60	D₄×Dic₁₂₇	C₂²	1	4	yes	—
61	(C₂²⋊D₁₂₇)⋊C₄	C₂²	1	4	yes	complemented
62	(C₂²⋊D₁₂₇)⋊C₄	C₂²	1	4	yes	complemented
63	#16861168	C₂²	1	2	yes	—
64	#16861167	C₂²	1	4	yes	complemented
65	#16861167	C₂²	1	2	yes	complemented
66	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
67	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	complemented
68	C₄⋊C₄×D₁₂₇	C₂²	1	2	yes	complemented
69	C₄⋊C₄×D₁₂₇	C₂²	1	4	yes	complemented
70	C₂×(C₂×D₁₂₇)⋊C₄	C₂²	1	2	yes	—
71	C₂²⋊C₄×D₁₂₇	C₂²	1	4	yes	—
72	#126×C₁₂₇	C₂²	1	2	yes	complemented
73	#125×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
74	#124×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
75	#123×C₁₂₇	C₂²	1	2	yes	—
76	C₂²⋊Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
77	#121×C₁₂₇	C₂²	1	1	yes	characteristic
78	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
79	C₄×D₄×C₁₂₇	C₂²	1	2	yes	—
80	C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
81	#117×C₁₂₇	C₂²	1	2	yes	complemented
82	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
83	C₂³×D₁₂₇	C₂³	1	2	yes	—
84	#14523248	—	2	8	no	—
85	#14523247	—	4	4	no	—
86	C₂×Dic₂₅₄	C₂³	1	2	yes	—
87	C₂×Dic₂₅₄	C₂³	1	2	yes	—
88	C₂×Dic₂₅₄	C₂³	1	4	yes	—
89	Q₈×D₁₂₇	—	4	4	no	—
90	C₂×D₅₀₈	—	2	4	no	—
91	C₂×D₅₀₈	C₂³	1	2	yes	—
92	C₂×D₅₀₈	C₂³	1	1	yes	characteristic
93	C₂×D₅₀₈	C₂³	1	1	yes	characteristic
94	C₂×C₂²⋊D₁₂₇	C₂³	1	4	yes	—
95	C₂×C₂²⋊D₁₂₇	C₂³	1	4	yes	—
96	C₂²×Dic₁₂₇	C₂³	1	2	yes	—
97	C₂²×Dic₁₂₇	C₂³	1	4	yes	—
98	C₂×C₄×D₁₂₇	—	4	4	no	—
99	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
100	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
101	C₂×C₄×D₁₂₇	C₂³	1	4	yes	—
102	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
103	C₂×C₄×D₁₂₇	C₂³	1	4	yes	—
104	C₂×C₄×D₁₂₇	C₂³	1	2	yes	—
105	C₄⋊Dic₁₂₇	C₂³	1	2	yes	—
106	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
107	C₄⋊Dic₁₂₇	C₂³	1	2	yes	—
108	C₄⋊Dic₁₂₇	C₂³	1	2	yes	—
109	C₄⋊Dic₁₂₇	C₂³	1	1	yes	characteristic
110	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
111	C₄⋊Dic₁₂₇	C₂³	1	2	yes	—
112	C₄⋊Dic₁₂₇	C₂³	1	2	yes	—
113	C₄⋊Dic₁₂₇	C₂³	1	1	yes	characteristic
114	Dic₁₂₇⋊C₄	C₂³	1	4	yes	—
115	Dic₁₂₇⋊C₄	C₂³	1	4	yes	—
116	Dic₁₂₇⋊C₄	C₂³	1	4	yes	—
117	Dic₁₂₇⋊C₄	C₂³	1	4	yes	—
118	C₂²⋊Dic₁₂₇	C₂³	1	4	yes	—
119	C₂²⋊Dic₁₂₇	C₂³	1	4	yes	—
120	(C₂×D₁₂₇)⋊C₄	C₂³	1	4	yes	—
121	(C₂×D₁₂₇)⋊C₄	C₂³	1	4	yes	—
122	(C₂×D₁₂₇)⋊C₄	C₂³	1	2	yes	—
123	(C₂×D₁₂₇)⋊C₄	C₂³	1	4	yes	—
124	(C₂×D₁₂₇)⋊C₄	C₂³	1	4	yes	—
125	(C₂×D₁₂₇)⋊C₄	C₂³	1	2	yes	—
126	(C₂×D₁₂₇)⋊C₄	C₂³	1	2	yes	—
127	(C₂×D₁₂₇)⋊C₄	C₂³	1	2	yes	—
128	C₄×Dic₁₂₇	C₂³	1	4	yes	—
129	C₄×Dic₁₂₇	C₂³	1	4	yes	—
130	C₂×Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
131	C₂×D₄×C₁₂₇	C₂³	1	1	yes	characteristic
132	C₂×D₄×C₁₂₇	C₂³	1	2	yes	—
133	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
134	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
135	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
136	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
137	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
138	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
139	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
140	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
141	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
142	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
143	C₂²⋊C₄×C₁₂₇	C₂³	1	2	yes	—
144	C₂²⋊C₄×C₁₂₇	C₂³	1	2	yes	—
145	C₂²⋊C₄×C₁₂₇	C₂³	1	2	yes	—
146	C₂²⋊C₄×C₁₂₇	C₂³	1	2	yes	—
147	C₂²⋊C₄×C₁₂₇	C₂³	1	2	yes	—
148	C₄²×C₁₂₇	C₂³	1	2	yes	—
149	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
150	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
151	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
152	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
153	C₂²×D₁₂₇	—	2	4	no	—
154	C₂²×D₁₂₇	—	2	4	no	—
155	C₂²×D₁₂₇	—	4	2	no	—
156	C₂²×D₁₂₇	C₂⁴	1	2	yes	—
157	C₂²×D₁₂₇	C₂⁴	1	2	yes	—
158	Dic₂₅₄	—	2	4	no	—
159	Dic₂₅₄	—	2	4	no	—
160	Dic₂₅₄	—	4	4	no	—
161	D₅₀₈	—	2	4	no	—
162	D₅₀₈	—	2	2	no	—
163	D₅₀₈	#41	1	4	yes	complemented
164	C₂²⋊D₁₂₇	—	2	8	no	—
165	C₂²⋊D₁₂₇	—	4	4	no	—
166	C₂×Dic₁₂₇	—	4	2	no	—
167	C₂×Dic₁₂₇	—	4	4	no	—
168	C₂×Dic₁₂₇	C₂⁴	1	4	yes	—
169	C₂×Dic₁₂₇	C₂⁴	1	2	yes	—
170	C₂×Dic₁₂₇	C₂⁴	1	4	yes	—
171	C₂×Dic₁₂₇	C₂⁴	1	2	yes	—
172	C₄×D₁₂₇	—	2	8	no	—
173	C₄×D₁₂₇	—	2	4	no	—
174	C₄×D₁₂₇	—	2	4	no	—
175	C₄×D₁₂₇	—	4	2	no	—
176	C₄×D₁₂₇	—	4	2	no	—
177	C₄×D₁₂₇	—	4	4	no	—
178	C₂³×C₁₂₇	C₂⁴	1	2	yes	—
179	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
180	Q₈×C₁₂₇	—	2	2	no	—
181	D₄×C₁₂₇	—	2	4	no	—
182	D₄×C₁₂₇	—	2	2	no	—
183	C₂×C₄×C₁₂₇	—	2	2	no	—
184	C₂×C₄×C₁₂₇	—	2	2	no	—
185	C₂×C₄×C₁₂₇	—	2	4	no	—
186	C₂×C₄×C₁₂₇	—	2	2	no	—
187	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
188	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
189	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
190	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
191	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
192	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
193	C₂×C₄×C₁₂₇	C₂⁴	1	2	yes	—
194	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
195	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
196	C₂²×C₁₂₇	—	2	1	no	—
197	C₂²×C₁₂₇	—	2	1	no	—
198	C₂²×C₁₂₇	—	2	2	no	—
199	C₂²×C₁₂₇	—	2	2	no	—
200	C₂²×C₁₂₇	—	2	2	no	—
201	C₂²×C₁₂₇	2₋¹⁺⁴	1	2	yes	—
202	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
203	C₂×D₁₂₇	—	2	2	no	—
204	C₂×D₁₂₇	—	2	2	no	—
205	C₂×D₁₂₇	—	2	2	no	—
206	C₂×D₁₂₇	—	2	2	no	—
207	C₂×D₁₂₇	—	2	2	no	—
208	C₂×D₁₂₇	—	4	8	no	—
209	C₂×D₁₂₇	C₂×#41	1	4	yes	—
210	C₄×C₁₂₇	—	2	2	no	—
211	C₄×C₁₂₇	—	2	1	no	—
212	C₄×C₁₂₇	—	2	1	no	—
213	C₄×C₁₂₇	—	2	2	no	—
214	C₄×C₁₂₇	—	2	1	no	—
215	C₄×C₁₂₇	—	2	1	no	—
216	C₄×C₁₂₇	—	2	2	no	—
217	C₄×C₁₂₇	C₂×#41	1	2	yes	—
218	C₄×C₁₂₇	H(5,𝔽₂)	1	2	yes	—
219	Dic₁₂₇	—	2	4	no	—
220	Dic₁₂₇	—	2	2	no	—
221	Dic₁₂₇	—	2	2	no	—
222	Dic₁₂₇	—	2	4	no	—
223	C₂×C₁₂₇	—	2	2	no	—
224	C₂×C₁₂₇	—	4	2	no	—
225	C₂×C₁₂₇	C₂²×#41	1	1	yes	characteristic
226	C₂×C₁₂₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
227	C₂×C₁₂₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
228	D₁₂₇	—	2	4	no	—
229	D₁₂₇	—	4	2	no	—
230	#3520	—	127	1	no	maximal, Sylow
231	C₁₂₇	#3520	1	1	yes	characteristic, complemented, Sylow
232	#556	—	127	1	no	—
233	#554	—	127	2	no	—
234	#552	—	127	2	no	—
235	#551	—	127	4	no	—
236	D₄×Q₈	—	127	1	no	—
237	#548	—	127	2	no	—
238	#547	—	127	1	no	—
239	#547	—	127	1	no	—
240	#546	—	127	2	no	—
241	#542	—	127	2	no	—
242	#539	—	127	4	no	—
243	#533	—	127	1	no	—
244	#529	D₁₂₇	1	1	yes	characteristic, complemented
245	#525	—	127	2	no	—
246	C₂×C₂²⋊Q₈	—	127	2	no	—
247	#520	—	127	2	no	—
248	C₂×C₄×D₄	—	127	1	no	—
249	C₂×#41	—	127	2	no	—
250	C₂×#41	—	127	2	no	—
251	C₂²×Q₈	—	127	2	no	—
252	C₂²×D₄	—	127	1	no	—
253	C₂³×C₄	—	127	2	no	—
254	C₄⋊Q₈	—	127	1	no	—
255	C₄⋊Q₈	—	127	1	no	—
256	C₄⋊Q₈	—	127	2	no	—
257	#126	—	127	4	no	—
258	#126	—	127	4	no	—
259	#126	C₂×D₁₂₇	1	2	yes	complemented
260	#125	—	127	4	no	—
261	#125	—	127	2	no	—
262	#125	C₂×D₁₂₇	1	1	yes	characteristic, complemented
263	#124	—	127	4	no	—
264	#124	—	127	2	no	—
265	#124	C₂×D₁₂₇	1	1	yes	characteristic, complemented
266	#123	—	127	4	no	—
267	#123	—	127	4	no	—
268	#123	—	127	4	no	—
269	#123	—	127	4	no	—
270	#123	C₂×D₁₂₇	1	2	yes	—
271	C₂²⋊Q₈	—	127	2	no	—
272	C₂²⋊Q₈	—	127	2	no	—
273	C₂²⋊Q₈	—	127	2	no	—
274	C₂²⋊Q₈	—	127	2	no	—
275	C₂²⋊Q₈	—	127	4	no	—
276	C₂²⋊Q₈	—	127	2	no	—
277	C₂²⋊Q₈	—	127	2	no	—
278	C₂²⋊Q₈	—	127	4	no	—
279	C₂²⋊Q₈	—	127	4	no	—
280	C₂²⋊Q₈	—	127	4	no	—
281	C₂²⋊Q₈	C₂×D₁₂₇	1	1	yes	characteristic
282	#121	—	127	2	no	—
283	#121	—	127	2	no	—
284	#121	—	127	4	no	—
285	#121	C₂×D₁₂₇	1	1	yes	characteristic
286	#120	—	127	4	no	—
287	C₄×Q₈	—	127	2	no	—
288	C₄×Q₈	—	127	2	no	—
289	C₄×Q₈	—	127	2	no	—
290	C₄×Q₈	C₂×D₁₂₇	1	1	yes	characteristic, complemented
291	C₄×D₄	—	127	4	no	—
292	C₄×D₄	—	127	4	no	—
293	C₄×D₄	—	127	4	no	—
294	C₄×D₄	—	127	1	no	—
295	C₄×D₄	—	127	2	no	—
296	C₄×D₄	—	127	1	no	—
297	C₄×D₄	—	127	1	no	—
298	C₄×D₄	—	127	1	no	—
299	C₄×D₄	—	127	2	no	—
300	C₄×D₄	—	127	2	no	—
301	C₄×D₄	C₂×D₁₂₇	1	2	yes	—
302	C₄×D₄	C₂×D₁₂₇	1	1	yes	characteristic, complemented
303	#117	—	127	2	no	—
304	#117	—	127	4	no	—
305	#117	—	127	2	no	—
306	#117	C₂×D₁₂₇	1	2	yes	complemented
307	C₂×C₄⋊C₄	—	127	1	no	—
308	C₂×C₄⋊C₄	—	127	2	no	—
309	C₂×C₄⋊C₄	—	127	4	no	—
310	C₂×C₄⋊C₄	—	127	2	no	—
311	C₂×C₂²⋊C₄	—	127	2	no	—
312	C₂×C₂²⋊C₄	—	127	4	no	—
313	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
314	C₂⁴	—	127	2	no	—
315	#41	—	508	4	no	—
316	#41	—	254	8	no	—
317	C₂×Q₈	—	127	4	no	—
318	C₂×Q₈	—	127	2	no	—
319	C₂×Q₈	—	127	2	no	—
320	C₂×Q₈	—	508	4	no	—
321	C₂×Q₈	C₂²×D₁₂₇	1	1	yes	characteristic
322	C₂×D₄	—	127	4	no	—
323	C₂×D₄	—	127	4	no	—
324	C₂×D₄	—	127	1	no	—
325	C₂×D₄	—	127	2	no	—
326	C₂×D₄	—	127	1	no	—
327	C₂×D₄	—	254	4	no	—
328	C₂×D₄	C₂²×D₁₂₇	1	2	yes	—
329	C₂×D₄	C₂²×D₁₂₇	1	1	yes	characteristic
330	C₂²×C₄	—	127	2	no	—
331	C₂²×C₄	—	127	4	no	—
332	C₂²×C₄	—	127	4	no	—
333	C₂²×C₄	—	127	2	no	—
334	C₂²×C₄	—	127	2	no	—
335	C₂²×C₄	—	127	4	no	—
336	C₂²×C₄	—	127	2	no	—
337	C₂²×C₄	—	127	2	no	—
338	C₂²×C₄	—	508	4	no	—
339	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
340	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
341	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
342	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
343	C₄⋊C₄	—	127	4	no	—
344	C₄⋊C₄	—	127	4	no	—
345	C₄⋊C₄	—	127	4	no	—
346	C₄⋊C₄	—	127	4	no	—
347	C₄⋊C₄	—	127	1	no	—
348	C₄⋊C₄	—	127	4	no	—
349	C₄⋊C₄	—	127	2	no	—
350	C₄⋊C₄	—	127	2	no	—
351	C₄⋊C₄	—	127	1	no	—
352	C₄⋊C₄	—	127	2	no	—
353	C₄⋊C₄	—	127	2	no	—
354	C₄⋊C₄	—	127	2	no	—
355	C₄⋊C₄	—	127	4	no	—
356	C₄⋊C₄	C₂²×D₁₂₇	1	2	yes	—
357	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
358	C₄⋊C₄	C₂²×D₁₂₇	1	2	yes	—
359	C₄⋊C₄	C₂²×D₁₂₇	1	1	yes	characteristic
360	C₄⋊C₄	C₂²×D₁₂₇	1	2	yes	—
361	C₄⋊C₄	C₂²×D₁₂₇	1	2	yes	—
362	C₂²⋊C₄	—	127	4	no	—
363	C₂²⋊C₄	—	127	4	no	—
364	C₂²⋊C₄	—	127	2	no	—
365	C₂²⋊C₄	—	127	2	no	—
366	C₂²⋊C₄	—	127	2	no	—
367	C₂²⋊C₄	—	127	2	no	—
368	C₂²⋊C₄	—	127	4	no	—
369	C₂²⋊C₄	—	127	4	no	—
370	C₂²⋊C₄	—	127	4	no	—
371	C₂²⋊C₄	—	127	4	no	—
372	C₂²⋊C₄	C₂²×D₁₂₇	1	2	yes	—
373	C₂²⋊C₄	C₂²×D₁₂₇	1	2	yes	—
374	C₂²⋊C₄	C₂²×D₁₂₇	1	2	yes	—
375	C₂²⋊C₄	C₂²×D₁₂₇	1	2	yes	—
376	C₂²⋊C₄	C₂²×D₁₂₇	1	2	yes	—
377	C₄²	—	127	4	no	—
378	C₄²	—	127	4	no	—
379	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
380	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
381	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
382	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
383	C₄²	C₂²×D₁₂₇	1	2	yes	—
384	C₂³	—	127	2	no	—
385	C₂³	—	127	2	no	—
386	C₂³	—	508	2	no	—
387	C₂³	—	254	4	no	—
388	C₂³	—	254	4	no	—
389	C₂³	C₂³×D₁₂₇	1	2	yes	—
390	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
391	Q₈	—	2	2	no	—
392	Q₈	—	508	4	no	—
393	Q₈	—	254	4	no	—
394	Q₈	—	254	4	no	—
395	D₄	—	2	4	no	—
396	D₄	—	2	2	no	—
397	D₄	—	127	4	no	—
398	D₄	—	508	4	no	—
399	D₄	—	254	4	no	—
400	D₄	—	254	8	no	—
401	D₄	—	254	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	4	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	127	4	no	—
407	C₂×C₄	—	127	2	no	—
408	C₂×C₄	—	127	4	no	—
409	C₂×C₄	—	127	2	no	—
410	C₂×C₄	—	508	2	no	—
411	C₂×C₄	—	508	2	no	—
412	C₂×C₄	—	508	4	no	—
413	C₂×C₄	—	508	2	no	—
414	C₂×C₄	—	508	4	no	—
415	C₂×C₄	—	254	8	no	—
416	C₂×C₄	—	254	4	no	—
417	C₂×C₄	—	254	4	no	—
418	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
419	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
420	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
421	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
422	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
423	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
424	C₂×C₄	C₂³×D₁₂₇	1	2	yes	—
425	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
426	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
427	C₂²	—	2	1	no	—
428	C₂²	—	2	1	no	—
429	C₂²	—	2	2	no	—
430	C₂²	—	2	2	no	—
431	C₂²	—	2	2	no	—
432	C₂²	—	127	4	no	—
433	C₂²	—	508	8	no	—
434	C₂²	—	254	2	no	—
435	C₂²	—	254	2	no	—
436	C₂²	—	254	2	no	—
437	C₂²	—	254	2	no	—
438	C₂²	—	254	2	no	—
439	C₂²	#16861256	1	2	yes	—
440	C₂²	C₂⁴×D₁₂₇	1	1	yes	characteristic
441	C₄	—	2	2	no	—
442	C₄	—	2	1	no	—
443	C₄	—	2	1	no	—
444	C₄	—	2	2	no	—
445	C₄	—	2	1	no	—
446	C₄	—	2	1	no	—
447	C₄	—	2	2	no	—
448	C₄	—	254	4	no	—
449	C₄	—	254	2	no	—
450	C₄	—	254	2	no	—
451	C₄	—	254	4	no	—
452	C₄	#41×D₁₂₇	1	2	yes	—
453	C₄	#16861255	1	2	yes	—
454	C₂	—	2	2	no	—
455	C₂	—	4	2	no	—
456	C₂	—	508	2	no	—
457	C₂	—	254	4	no	—
458	C₂	C₂×#41×D₁₂₇	1	1	yes	characteristic
459	C₂	C₂×#16861255	1	1	yes	characteristic
460	C₂	C₂×#16861256	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₂₇
commutator, socle, cosocle	C₂²×C₁₂₇	C₂⁵
Fitting	#529×C₁₂₇	C₂
hypercenter	#529	D₁₂₇
hypocenter	C₁₂₇	#3520
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ #529

Complements

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

N	H ≅ G/N	type
1. #22550599	C₂	semidirect
2. #22550583	C₂	semidirect
3. #22550564	C₂	semidirect
4. #22550549	C₂	semidirect
5. Q₈×D₅₀₈	C₂	semidirect
6. #22550534	C₂	semidirect
7. #22550531	C₂	semidirect
8. #22550530	C₂	semidirect
9. #22550522	C₂	semidirect
10. #22550497	C₂	semidirect
11. #22550467	C₂	semidirect
12. #22550433	C₂	semidirect
13. #22550386	C₂	semidirect
14. C₂²⋊Q₈×D₁₂₇	C₂	semidirect
15. #22550356	C₂	semidirect
16. C₂×C₄×D₅₀₈	C₂	semidirect
17. #529×C₁₂₇	C₂	semidirect
19. C₂×#14523247	C₂²	semidirect
20. C₂×Q₈×D₁₂₇	C₂²	semidirect
23. #16861209	C₂²	semidirect
24. #16861209	C₂²	semidirect
25. #16861209	C₂²	semidirect
26. #16861204	C₂²	semidirect
27. #16861204	C₂²	semidirect
28. #16861201	C₂²	semidirect
29. #16861200	C₂²	semidirect
31. #16861195	C₂²	semidirect
33. #16861193	C₂²	semidirect
34. #16861191	C₂²	semidirect
35. #16861191	C₂²	semidirect
36. #16861189	C₂²	semidirect
37. #16861189	C₂²	semidirect
38. #16861189	C₂²	semidirect
41. #16861188	C₂²	semidirect
42. #16861187	C₂²	semidirect
44. #16861186	C₂²	semidirect
45. #16861185	C₂²	semidirect
48. #16861180	C₂²	semidirect
50. C₄×Dic₂₅₄	C₂²	semidirect
51. C₄×Dic₂₅₄	C₂²	semidirect
52. C₄×Dic₂₅₄	C₂²	semidirect
61. (C₂²⋊D₁₂₇)⋊C₄	C₂²	semidirect
62. (C₂²⋊D₁₂₇)⋊C₄	C₂²	semidirect
64. #16861167	C₂²	semidirect
65. #16861167	C₂²	semidirect
67. C₂×C₄⋊Dic₁₂₇	C₂²	semidirect
68. C₄⋊C₄×D₁₂₇	C₂²	semidirect
69. C₄⋊C₄×D₁₂₇	C₂²	semidirect
72. #126×C₁₂₇	C₂²	semidirect
73. #125×C₁₂₇	C₂²	semidirect
74. #124×C₁₂₇	C₂²	semidirect
78. C₄×Q₈×C₁₂₇	C₂²	semidirect
80. C₄×D₄×C₁₂₇	C₂²	semidirect
81. #117×C₁₂₇	C₂²	semidirect
163. D₅₀₈	#41	semidirect
231. C₁₂₇	#3520	semidirect
244. #529	D₁₂₇	semidirect
259. #126	C₂×D₁₂₇	semidirect
262. #125	C₂×D₁₂₇	semidirect
265. #124	C₂×D₁₂₇	semidirect
290. C₄×Q₈	C₂×D₁₂₇	semidirect
302. C₄×D₄	C₂×D₁₂₇	semidirect
306. #117	C₂×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550599	2
#22550583	2
#22550564	2
#22550549	2
Q₈×D₅₀₈	2
#22550534	2
#22550531	2
#22550530	2
#22550522	2
#22550497	2
#22550467	2
#22550433	2
#22550386	2
C₂²⋊Q₈×D₁₂₇	2
#22550356	2
C₂×C₄×D₅₀₈	2
#529×C₁₂₇	2
#3520	127