Subgroups — C₁₂₇×#20559 #44584232

Order 32512.

1118 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#435×C₁₂₇	C₂	1	2	yes	maximal
2	C₄²×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1771	C₂	1	2	yes	maximal, complemented
4	C₄×C₄⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₂₇×#1753	C₂	1	2	yes	maximal, complemented
6	C₈×C₂²⋊C₄×C₁₂₇	C₂	1	2	yes	maximal, complemented
7	C₁₂₇×#1749	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₂²⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₂₇×#1745	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₂₇×#1728	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#336×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×M₁₆×C₁₂₇	C₂²	1	2	yes	—
13	C₂³×C₈×C₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₁₂₇	C₄	1	2	yes	—
15	C₂×C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
18	#435×C₁₂₇	C₄	1	16	yes	complemented
19	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
22	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
23	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₁₂₇	C₄	1	2	yes	—
27	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
28	C₄×C₂²⋊C₄×C₁₂₇	C₄	1	2	yes	—
29	C₄×C₂²⋊C₄×C₁₂₇	C₄	1	2	yes	—
30	C₄×C₂²⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
31	C₄×C₂²⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
32	C₄³×C₁₂₇	C₂²	1	1	yes	characteristic
33	#336×C₁₂₇	C₂²	1	1	yes	characteristic
34	#336×C₁₂₇	C₂²	1	1	yes	characteristic
35	#336×C₁₂₇	C₂²	1	2	yes	—
36	#336×C₁₂₇	C₂²	1	1	yes	characteristic
37	#336×C₁₂₇	C₂²	1	2	yes	—
38	#336×C₁₂₇	C₂²	1	1	yes	characteristic
39	#336×C₁₂₇	C₂²	1	2	yes	—
40	C₂²×D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
42	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
43	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₁₂₇	—	4	2	no	—
46	C₂×M₁₆×C₁₂₇	—	4	2	no	—
47	C₂×M₁₆×C₁₂₇	C₂×C₄	1	8	yes	complemented
48	C₂²×C₈×C₁₂₇	—	4	2	no	—
49	C₂²×C₈×C₁₂₇	—	4	2	no	—
50	C₂²×C₈×C₁₂₇	C₂×C₄	1	8	yes	complemented
51	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
52	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
54	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
55	C₄×D₄×C₁₂₇	—	2	8	no	—
56	C₄×D₄×C₁₂₇	C₂×C₄	1	8	yes	—
57	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
60	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₁₂₇	—	4	2	no	—
64	C₂×C₄²×C₁₂₇	—	4	2	no	—
65	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
66	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
67	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
68	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
69	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	complemented
75	C₂²⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	complemented
76	C₂²⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	complemented
77	#97×C₁₂₇	—	4	2	no	—
78	#97×C₁₂₇	C₂×C₄	1	8	yes	complemented
79	C₄×C₈×C₁₂₇	—	4	2	no	—
80	C₄×C₈×C₁₂₇	—	4	2	no	—
81	#95×C₁₂₇	C₂×C₄	1	2	yes	—
82	#95×C₁₂₇	C₂×C₄	1	2	yes	—
83	#95×C₁₂₇	C₂×C₄	1	2	yes	—
84	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
85	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×D₄×C₁₂₇	—	2	4	no	—
87	C₂×D₄×C₁₂₇	C₄²	1	4	yes	—
88	C₂²×C₄×C₁₂₇	—	2	2	no	—
89	C₂²×C₄×C₁₂₇	—	2	2	no	—
90	C₂²×C₄×C₁₂₇	—	2	1	no	—
91	C₂²×C₄×C₁₂₇	—	2	1	no	—
92	C₂²×C₄×C₁₂₇	—	2	1	no	—
93	C₂²×C₄×C₁₂₇	—	2	1	no	—
94	C₂²×C₄×C₁₂₇	—	2	4	no	—
95	C₂²×C₄×C₁₂₇	—	2	4	no	—
96	C₂²×C₄×C₁₂₇	—	2	4	no	—
97	C₂²×C₄×C₁₂₇	—	2	4	no	—
98	C₂²×C₄×C₁₂₇	—	4	2	no	—
99	C₂²×C₄×C₁₂₇	—	4	2	no	—
100	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
106	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
107	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
111	M₁₆×C₁₂₇	—	4	8	no	—
112	C₂×C₈×C₁₂₇	—	2	2	no	—
113	C₂×C₈×C₁₂₇	—	2	2	no	—
114	C₂×C₈×C₁₂₇	—	2	2	no	—
115	C₂×C₈×C₁₂₇	—	2	2	no	—
116	C₂×C₈×C₁₂₇	—	2	2	no	—
117	C₂×C₈×C₁₂₇	—	2	2	no	—
118	C₂×C₈×C₁₂₇	—	2	2	no	—
119	C₂×C₈×C₁₂₇	—	2	2	no	—
120	C₂×C₈×C₁₂₇	—	4	8	no	—
121	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
122	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
123	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
124	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
125	C₄⋊C₄×C₁₂₇	—	2	4	no	—
126	C₄⋊C₄×C₁₂₇	C₄²	1	4	yes	—
127	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
128	C₂²⋊C₄×C₁₂₇	C₄²	1	4	yes	—
129	C₂²⋊C₄×C₁₂₇	C₄²	1	4	yes	—
130	C₂²⋊C₄×C₁₂₇	M₁₆	1	8	yes	complemented
131	C₄²×C₁₂₇	—	2	4	no	—
132	C₄²×C₁₂₇	—	2	4	no	—
133	C₄²×C₁₂₇	—	2	1	no	—
134	C₄²×C₁₂₇	—	2	2	no	—
135	C₄²×C₁₂₇	—	2	2	no	—
136	C₄²×C₁₂₇	—	2	2	no	—
137	C₄²×C₁₂₇	—	2	1	no	—
138	C₄²×C₁₂₇	—	4	2	no	—
139	C₄²×C₁₂₇	—	4	2	no	—
140	C₄²×C₁₂₇	C₂²×C₄	1	4	yes	—
141	C₂³×C₁₂₇	—	2	1	no	—
142	C₂³×C₁₂₇	—	2	1	no	—
143	C₂³×C₁₂₇	—	2	2	no	—
144	C₂³×C₁₂₇	—	2	1	no	—
145	C₂³×C₁₂₇	—	2	1	no	—
146	C₂³×C₁₂₇	—	2	2	no	—
147	C₂³×C₁₂₇	—	2	2	no	—
148	C₂³×C₁₂₇	C₂×C₄²	1	2	yes	—
149	C₂³×C₁₂₇	C₂×C₄²	1	2	yes	—
150	C₂³×C₁₂₇	C₂×M₁₆	1	4	yes	—
151	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
152	D₄×C₁₂₇	—	2	8	no	—
153	C₂×C₄×C₁₂₇	—	2	2	no	—
154	C₂×C₄×C₁₂₇	—	2	2	no	—
155	C₂×C₄×C₁₂₇	—	2	1	no	—
156	C₂×C₄×C₁₂₇	—	2	1	no	—
157	C₂×C₄×C₁₂₇	—	2	2	no	—
158	C₂×C₄×C₁₂₇	—	2	1	no	—
159	C₂×C₄×C₁₂₇	—	2	1	no	—
160	C₂×C₄×C₁₂₇	—	2	2	no	—
161	C₂×C₄×C₁₂₇	—	2	2	no	—
162	C₂×C₄×C₁₂₇	—	2	2	no	—
163	C₂×C₄×C₁₂₇	—	2	2	no	—
164	C₂×C₄×C₁₂₇	—	2	2	no	—
165	C₂×C₄×C₁₂₇	—	2	4	no	—
166	C₂×C₄×C₁₂₇	—	2	4	no	—
167	C₂×C₄×C₁₂₇	—	2	4	no	—
168	C₂×C₄×C₁₂₇	—	2	4	no	—
169	C₂×C₄×C₁₂₇	—	2	2	no	—
170	C₂×C₄×C₁₂₇	—	2	2	no	—
171	C₂×C₄×C₁₂₇	—	2	2	no	—
172	C₂×C₄×C₁₂₇	—	4	8	no	—
173	C₂×C₄×C₁₂₇	—	4	8	no	—
174	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
175	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
176	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
177	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
178	C₂×C₄×C₁₂₇	C₂×M₁₆	1	4	yes	—
179	C₂×C₄×C₁₂₇	C₂×M₁₆	1	4	yes	—
180	C₂×C₄×C₁₂₇	#131	1	4	yes	—
181	C₂×C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
182	C₂×C₄×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
183	C₂×C₄×C₁₂₇	H(5,𝔽₂)	1	2	yes	—
184	C₂×C₄×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
185	C₈×C₁₂₇	—	2	4	no	—
186	C₈×C₁₂₇	—	2	4	no	—
187	C₈×C₁₂₇	—	2	4	no	—
188	C₈×C₁₂₇	—	2	4	no	—
189	C₂²×C₁₂₇	—	2	2	no	—
190	C₂²×C₁₂₇	—	2	4	no	—
191	C₂²×C₁₂₇	—	2	4	no	—
192	C₂²×C₁₂₇	—	2	2	no	—
193	C₂²×C₁₂₇	—	2	2	no	—
194	C₂²×C₁₂₇	—	2	2	no	—
195	C₂²×C₁₂₇	—	2	2	no	—
196	C₂²×C₁₂₇	—	4	2	no	—
197	C₂²×C₁₂₇	—	4	2	no	—
198	C₂²×C₁₂₇	C₄×M₁₆	1	4	yes	—
199	C₂²×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
200	C₂²×C₁₂₇	#518	1	1	yes	characteristic
201	C₂²×C₁₂₇	#520	1	1	yes	characteristic
202	C₂²×C₁₂₇	C₂²×M₁₆	1	2	yes	—
203	C₂²×C₁₂₇	C₂×#131	1	2	yes	—
204	C₄×C₁₂₇	—	2	2	no	—
205	C₄×C₁₂₇	—	2	2	no	—
206	C₄×C₁₂₇	—	2	4	no	—
207	C₄×C₁₂₇	—	2	4	no	—
208	C₄×C₁₂₇	—	2	4	no	—
209	C₄×C₁₂₇	—	2	2	no	—
210	C₄×C₁₂₇	—	2	2	no	—
211	C₄×C₁₂₇	—	4	2	no	—
212	C₄×C₁₂₇	#518	1	2	yes	—
213	C₄×C₁₂₇	#520	1	2	yes	—
214	#20559	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
215	C₂×C₁₂₇	—	2	4	no	—
216	C₂×C₁₂₇	—	4	2	no	—
217	C₂×C₁₂₇	#2275	1	1	yes	characteristic
218	C₂×C₁₂₇	C₂×C₄×M₁₆	1	1	yes	characteristic
219	C₂×C₁₂₇	C₄×#131	1	1	yes	characteristic
220	C₂×C₁₂₇	#2974	1	2	yes	—
221	C₂×C₁₂₇	#2990	1	2	yes	—
222	C₂×#435	C₂×C₁₂₇	1	2	yes	—
223	C₄²×D₄	C₂×C₁₂₇	1	1	yes	characteristic
224	#1771	C₂×C₁₂₇	1	2	yes	complemented
225	C₄×C₄⋊C₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
226	#1753	C₂×C₁₂₇	1	2	yes	complemented
227	C₈×C₂²⋊C₄	C₂×C₁₂₇	1	2	yes	complemented
228	#1749	C₂×C₁₂₇	1	1	yes	characteristic, complemented
229	C₄×C₂²⋊C₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
230	#1745	C₂×C₁₂₇	1	1	yes	characteristic, complemented
231	#1728	C₂×C₁₂₇	1	1	yes	characteristic, complemented
232	C₂×#336	C₂×C₁₂₇	1	1	yes	characteristic, complemented
233	C₁₂₇	#20559	1	1	yes	characteristic, direct product factor, Sylow
234	C₂²×M₁₆	C₂²×C₁₂₇	1	2	yes	—
235	C₂³×C₈	C₂²×C₁₂₇	1	2	yes	—
236	C₂×C₄×D₄	C₄×C₁₂₇	1	2	yes	—
237	C₂×C₄×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
238	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
239	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
240	#435	C₄×C₁₂₇	1	16	yes	complemented
241	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
242	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
243	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
244	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
245	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
246	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
247	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
248	C₄×C₄⋊C₄	C₄×C₁₂₇	1	2	yes	—
249	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
250	C₄×C₂²⋊C₄	C₄×C₁₂₇	1	2	yes	—
251	C₄×C₂²⋊C₄	C₄×C₁₂₇	1	2	yes	—
252	C₄×C₂²⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
253	C₄×C₂²⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
254	C₄³	C₂²×C₁₂₇	1	1	yes	characteristic
255	#336	C₂²×C₁₂₇	1	1	yes	characteristic
256	#336	C₂²×C₁₂₇	1	1	yes	characteristic
257	#336	C₂²×C₁₂₇	1	2	yes	—
258	#336	C₂²×C₁₂₇	1	1	yes	characteristic
259	#336	C₂²×C₁₂₇	1	2	yes	—
260	#336	C₂²×C₁₂₇	1	1	yes	characteristic
261	#336	C₂²×C₁₂₇	1	2	yes	—
262	C₂²×D₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
263	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
264	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
265	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
266	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
267	C₂×M₁₆	—	4	2	no	—
268	C₂×M₁₆	—	4	2	no	—
269	C₂×M₁₆	C₂×C₄×C₁₂₇	1	8	yes	complemented
270	C₂²×C₈	—	4	2	no	—
271	C₂²×C₈	—	4	2	no	—
272	C₂²×C₈	C₂×C₄×C₁₂₇	1	8	yes	complemented
273	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
274	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
275	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
276	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
277	C₄×D₄	—	2	8	no	—
278	C₄×D₄	C₂×C₄×C₁₂₇	1	8	yes	—
279	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
280	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
281	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
282	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
283	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
284	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
285	C₂×C₄²	—	4	2	no	—
286	C₂×C₄²	—	4	2	no	—
287	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
288	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
289	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
290	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
291	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
292	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
293	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
294	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
295	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
296	C₄⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	complemented
297	C₂²⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	complemented
298	C₂²⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	complemented
299	#97	—	4	2	no	—
300	#97	C₂×C₄×C₁₂₇	1	8	yes	complemented
301	C₄×C₈	—	4	2	no	—
302	C₄×C₈	—	4	2	no	—
303	#95	C₂×C₄×C₁₂₇	1	2	yes	—
304	#95	C₂×C₄×C₁₂₇	1	2	yes	—
305	#95	C₂×C₄×C₁₂₇	1	2	yes	—
306	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
307	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
308	C₂×D₄	—	2	4	no	—
309	C₂×D₄	C₄²×C₁₂₇	1	4	yes	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	1	no	—
313	C₂²×C₄	—	2	1	no	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	4	no	—
317	C₂²×C₄	—	2	4	no	—
318	C₂²×C₄	—	2	4	no	—
319	C₂²×C₄	—	2	4	no	—
320	C₂²×C₄	—	4	2	no	—
321	C₂²×C₄	—	4	2	no	—
322	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
323	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
324	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
325	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
326	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
327	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
328	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
329	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
330	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
331	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
332	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
333	M₁₆	—	4	8	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	4	8	no	—
343	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
344	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
345	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
346	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
347	C₄⋊C₄	—	2	4	no	—
348	C₄⋊C₄	C₄²×C₁₂₇	1	4	yes	—
349	C₂²⋊C₄	—	2	4	no	—
350	C₂²⋊C₄	C₄²×C₁₂₇	1	4	yes	—
351	C₂²⋊C₄	C₄²×C₁₂₇	1	4	yes	—
352	C₂²⋊C₄	M₁₆×C₁₂₇	1	8	yes	complemented
353	C₄²	—	2	4	no	—
354	C₄²	—	2	4	no	—
355	C₄²	—	2	1	no	—
356	C₄²	—	2	2	no	—
357	C₄²	—	2	2	no	—
358	C₄²	—	2	2	no	—
359	C₄²	—	2	1	no	—
360	C₄²	—	4	2	no	—
361	C₄²	—	4	2	no	—
362	C₄²	C₂²×C₄×C₁₂₇	1	4	yes	—
363	C₂³	—	2	1	no	—
364	C₂³	—	2	1	no	—
365	C₂³	—	2	2	no	—
366	C₂³	—	2	1	no	—
367	C₂³	—	2	1	no	—
368	C₂³	—	2	2	no	—
369	C₂³	—	2	2	no	—
370	C₂³	C₂×C₄²×C₁₂₇	1	2	yes	—
371	C₂³	C₂×C₄²×C₁₂₇	1	2	yes	—
372	C₂³	C₂×M₁₆×C₁₂₇	1	4	yes	—
373	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
374	D₄	—	2	8	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	4	no	—
388	C₂×C₄	—	2	4	no	—
389	C₂×C₄	—	2	4	no	—
390	C₂×C₄	—	2	4	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	4	8	no	—
395	C₂×C₄	—	4	8	no	—
396	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
397	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
398	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
399	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
400	C₂×C₄	C₂×M₁₆×C₁₂₇	1	4	yes	—
401	C₂×C₄	C₂×M₁₆×C₁₂₇	1	4	yes	—
402	C₂×C₄	#131×C₁₂₇	1	4	yes	—
403	C₂×C₄	C₂³×C₄×C₁₂₇	1	2	yes	—
404	C₂×C₄	H(5,𝔽₂)×C₁₂₇	1	1	yes	characteristic
405	C₂×C₄	H(5,𝔽₂)×C₁₂₇	1	2	yes	—
406	C₂×C₄	2₋¹⁺⁴×C₁₂₇	1	1	yes	characteristic
407	C₈	—	2	4	no	—
408	C₈	—	2	4	no	—
409	C₈	—	2	4	no	—
410	C₈	—	2	4	no	—
411	C₂²	—	2	2	no	—
412	C₂²	—	2	4	no	—
413	C₂²	—	2	4	no	—
414	C₂²	—	2	2	no	—
415	C₂²	—	2	2	no	—
416	C₂²	—	2	2	no	—
417	C₂²	—	2	2	no	—
418	C₂²	—	4	2	no	—
419	C₂²	—	4	2	no	—
420	C₂²	C₄×M₁₆×C₁₂₇	1	4	yes	—
421	C₂²	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
422	C₂²	#518×C₁₂₇	1	1	yes	characteristic
423	C₂²	#520×C₁₂₇	1	1	yes	characteristic
424	C₂²	C₂²×M₁₆×C₁₂₇	1	2	yes	—
425	C₂²	C₂×#131×C₁₂₇	1	2	yes	—
426	C₄	—	2	2	no	—
427	C₄	—	2	2	no	—
428	C₄	—	2	4	no	—
429	C₄	—	2	4	no	—
430	C₄	—	2	4	no	—
431	C₄	—	2	2	no	—
432	C₄	—	2	2	no	—
433	C₄	—	4	2	no	—
434	C₄	#518×C₁₂₇	1	2	yes	—
435	C₄	#520×C₁₂₇	1	2	yes	—
436	C₂	—	2	4	no	—
437	C₂	—	4	2	no	—
438	C₂	C₁₂₇×#2275	1	1	yes	characteristic
439	C₂	C₂×C₄×M₁₆×C₁₂₇	1	1	yes	characteristic
440	C₂	C₄×#131×C₁₂₇	1	1	yes	characteristic
441	C₂	C₁₂₇×#2974	1	2	yes	—
442	C₂	C₁₂₇×#2990	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂³×C₁₂₇	C₂³×C₄
prime radical	C₂²×D₄×C₁₂₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₂₇×#1771	C₂	semidirect
4. C₄×C₄⋊C₈×C₁₂₇	C₂	semidirect
5. C₁₂₇×#1753	C₂	semidirect
6. C₈×C₂²⋊C₄×C₁₂₇	C₂	semidirect
7. C₁₂₇×#1749	C₂	semidirect
8. C₄×C₂²⋊C₈×C₁₂₇	C₂	semidirect
9. C₁₂₇×#1745	C₂	semidirect
10. C₁₂₇×#1728	C₂	semidirect
11. C₂×#336×C₁₂₇	C₂	semidirect
18. #435×C₁₂₇	C₄	semidirect
47. C₂×M₁₆×C₁₂₇	C₂×C₄	semidirect
50. C₂²×C₈×C₁₂₇	C₂×C₄	semidirect
74. C₄⋊C₈×C₁₂₇	C₂×C₄	semidirect
75. C₂²⋊C₈×C₁₂₇	C₂×C₄	semidirect
76. C₂²⋊C₈×C₁₂₇	C₂×C₄	semidirect
78. #97×C₁₂₇	C₂×C₄	semidirect
130. C₂²⋊C₄×C₁₂₇	M₁₆	semidirect
214. #20559	C₁₂₇	direct (N × H)
224. #1771	C₂×C₁₂₇	semidirect
225. C₄×C₄⋊C₈	C₂×C₁₂₇	semidirect
226. #1753	C₂×C₁₂₇	semidirect
227. C₈×C₂²⋊C₄	C₂×C₁₂₇	semidirect
228. #1749	C₂×C₁₂₇	semidirect
229. C₄×C₂²⋊C₈	C₂×C₁₂₇	semidirect
230. #1745	C₂×C₁₂₇	semidirect
231. #1728	C₂×C₁₂₇	semidirect
232. C₂×#336	C₂×C₁₂₇	semidirect
233. C₁₂₇	#20559	direct (N × H)
240. #435	C₄×C₁₂₇	semidirect
269. C₂×M₁₆	C₂×C₄×C₁₂₇	semidirect
272. C₂²×C₈	C₂×C₄×C₁₂₇	semidirect
296. C₄⋊C₈	C₂×C₄×C₁₂₇	semidirect
297. C₂²⋊C₈	C₂×C₄×C₁₂₇	semidirect
298. C₂²⋊C₈	C₂×C₄×C₁₂₇	semidirect
300. #97	C₂×C₄×C₁₂₇	semidirect
352. C₂²⋊C₄	M₁₆×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#435×C₁₂₇	2
C₄²×D₄×C₁₂₇	2
C₁₂₇×#1771	2
C₄×C₄⋊C₈×C₁₂₇	2
C₁₂₇×#1753	2
C₈×C₂²⋊C₄×C₁₂₇	2
C₁₂₇×#1749	2
C₄×C₂²⋊C₈×C₁₂₇	2
C₁₂₇×#1745	2
C₁₂₇×#1728	2
C₂×#336×C₁₂₇	2
#20559	127

Subgroups — C127×#20559 #44584232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#20559 #44584232