Subgroups — C₁₂₇×#21834 #44585507

Order 32512.

1302 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#431×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#2285	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1990	C₂	1	4	yes	maximal, complemented
5	C₁₂₇×#1987	C₂	1	4	yes	maximal, complemented
6	C₁₂₇×#1824	C₂	1	4	yes	maximal
7	C₂³×Q₈×C₁₂₇	—	2	1	no	—
8	C₂×C₄⋊Q₈×C₁₂₇	—	2	1	no	—
9	C₂×C₄⋊Q₈×C₁₂₇	—	2	1	no	—
10	C₂×C₄⋊Q₈×C₁₂₇	—	2	4	no	—
11	C₂×C₄⋊Q₈×C₁₂₇	C₄	1	8	yes	complemented
12	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
14	C₂×#117×C₁₂₇	—	2	1	no	—
15	C₂²×C₄⋊C₄×C₁₂₇	—	2	1	no	—
16	C₂²×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
17	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
18	#431×C₁₂₇	—	2	4	no	—
19	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
22	#384×C₁₂₇	C₂²	1	2	yes	—
23	#382×C₁₂₇	—	2	1	no	—
24	#380×C₁₂₇	—	2	1	no	—
25	#380×C₁₂₇	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₁₂₇	—	2	1	no	—
27	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
28	#340×C₁₂₇	C₂²	1	16	yes	—
29	C₂²×Q₈×C₁₂₇	—	2	8	no	—
30	C₂²×Q₈×C₁₂₇	—	2	2	no	—
31	C₂²×Q₈×C₁₂₇	—	2	2	no	—
32	C₂²×Q₈×C₁₂₇	—	4	8	no	—
33	C₂³×C₄×C₁₂₇	—	2	2	no	—
34	C₂³×C₄×C₁₂₇	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₁₂₇	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₂₇	C₂³	1	4	yes	—
38	C₄⋊Q₈×C₁₂₇	—	2	8	no	—
39	C₄⋊Q₈×C₁₂₇	—	2	8	no	—
40	C₄⋊Q₈×C₁₂₇	—	2	4	no	—
41	C₄⋊Q₈×C₁₂₇	—	2	4	no	—
42	C₄⋊Q₈×C₁₂₇	C₂×C₄	1	8	yes	—
43	C₄⋊Q₈×C₁₂₇	C₂×C₄	1	8	yes	—
44	#117×C₁₂₇	—	4	4	no	—
45	C₂×C₄⋊C₄×C₁₂₇	—	2	8	no	—
46	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₁₂₇	—	2	4	no	—
50	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	8	yes	—
53	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
54	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
55	C₂×C₂²⋊C₄×C₁₂₇	—	2	2	no	—
56	C₂×C₄²×C₁₂₇	—	2	2	no	—
57	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
58	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₁₂₇	D₄	1	4	yes	—
61	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₁₂₇	—	2	4	no	—
64	C₂²⋊C₈×C₁₂₇	—	2	4	no	—
65	C₄×C₈×C₁₂₇	—	2	4	no	—
66	#95×C₁₂₇	—	2	2	no	—
67	#95×C₁₂₇	—	2	2	no	—
68	C₂⁴×C₁₂₇	C₂×D₄	1	1	yes	characteristic
69	C₂×Q₈×C₁₂₇	—	2	8	no	—
70	C₂×Q₈×C₁₂₇	—	2	8	no	—
71	C₂×Q₈×C₁₂₇	—	4	8	no	—
72	C₂×Q₈×C₁₂₇	—	4	32	no	—
73	C₂×Q₈×C₁₂₇	—	4	8	no	—
74	C₂²×C₄×C₁₂₇	—	2	4	no	—
75	C₂²×C₄×C₁₂₇	—	2	4	no	—
76	C₂²×C₄×C₁₂₇	—	2	8	no	—
77	C₂²×C₄×C₁₂₇	—	2	4	no	—
78	C₂²×C₄×C₁₂₇	—	2	4	no	—
79	C₂²×C₄×C₁₂₇	—	4	8	no	—
80	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
81	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
82	C₂²×C₄×C₁₂₇	SD₁₆	1	4	yes	—
83	C₂²×C₄×C₁₂₇	Q₁₆	1	4	yes	—
84	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
92	C₂×C₈×C₁₂₇	—	2	4	no	—
93	C₂×C₈×C₁₂₇	—	2	8	no	—
94	C₄⋊C₄×C₁₂₇	—	2	8	no	—
95	C₄⋊C₄×C₁₂₇	—	2	8	no	—
96	C₄⋊C₄×C₁₂₇	—	2	8	no	—
97	C₄⋊C₄×C₁₂₇	—	2	4	no	—
98	C₄⋊C₄×C₁₂₇	—	2	8	no	—
99	C₄⋊C₄×C₁₂₇	—	4	4	no	—
100	C₄⋊C₄×C₁₂₇	—	4	4	no	—
101	C₄⋊C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
102	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
103	C₄²×C₁₂₇	—	4	4	no	—
104	C₄²×C₁₂₇	C₂²⋊C₄	1	4	yes	—
105	C₄²×C₁₂₇	C₂²⋊C₄	1	4	yes	—
106	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
107	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
108	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
109	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
110	C₂³×C₁₂₇	—	2	2	no	—
111	C₂³×C₁₂₇	—	2	2	no	—
112	C₂³×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
113	C₂³×C₁₂₇	C₂×SD₁₆	1	2	yes	—
114	C₂³×C₁₂₇	C₂×Q₁₆	1	2	yes	—
115	C₂³×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
116	Q₈×C₁₂₇	—	4	16	no	—
117	Q₈×C₁₂₇	—	4	16	no	—
118	C₂×C₄×C₁₂₇	—	2	2	no	—
119	C₂×C₄×C₁₂₇	—	2	2	no	—
120	C₂×C₄×C₁₂₇	—	2	4	no	—
121	C₂×C₄×C₁₂₇	—	2	4	no	—
122	C₂×C₄×C₁₂₇	—	2	8	no	—
123	C₂×C₄×C₁₂₇	—	2	8	no	—
124	C₂×C₄×C₁₂₇	—	4	16	no	—
125	C₂×C₄×C₁₂₇	—	4	8	no	—
126	C₂×C₄×C₁₂₇	—	4	4	no	—
127	C₂×C₄×C₁₂₇	—	4	4	no	—
128	C₂×C₄×C₁₂₇	—	4	4	no	—
129	C₂×C₄×C₁₂₇	#103	1	16	yes	—
130	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
131	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
132	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
133	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
134	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
135	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	2	yes	—
136	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	2	yes	—
137	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	2	yes	—
138	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	2	yes	—
139	C₂×C₄×C₁₂₇	AGL(1,ℤ/8)	1	4	yes	—
140	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₂₇	2₋¹⁺⁴	1	4	yes	—
143	C₈×C₁₂₇	—	4	4	no	—
144	C₂²×C₁₂₇	—	2	2	no	—
145	C₂²×C₁₂₇	—	2	2	no	—
146	C₂²×C₁₂₇	—	2	4	no	—
147	C₂²×C₁₂₇	—	2	4	no	—
148	C₂²×C₁₂₇	C₂×#103	1	4	yes	—
149	C₂²×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₁₂₇	#536	1	2	yes	—
151	C₂²×C₁₂₇	C₂²×SD₁₆	1	1	yes	characteristic
152	C₂²×C₁₂₇	C₂²×Q₁₆	1	1	yes	characteristic
153	C₂²×C₁₂₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
154	C₄×C₁₂₇	—	2	2	no	—
155	C₄×C₁₂₇	—	2	2	no	—
156	C₄×C₁₂₇	—	4	8	no	—
157	C₄×C₁₂₇	—	8	4	no	—
158	C₄×C₁₂₇	C₂×#103	1	4	yes	—
159	C₄×C₁₂₇	C₂×#103	1	4	yes	—
160	C₄×C₁₂₇	#418	1	4	yes	—
161	C₄×C₁₂₇	#519	1	2	yes	—
162	C₄×C₁₂₇	#536	1	2	yes	—
163	#21834	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂×C₁₂₇	—	2	4	no	—
165	C₂×C₁₂₇	#2310	1	1	yes	characteristic
166	C₂×C₁₂₇	C₂²×#103	1	1	yes	characteristic
167	C₂×C₁₂₇	C₂×#418	1	1	yes	characteristic
168	C₂×C₁₂₇	#3227	1	2	yes	—
169	C₂×C₁₂₇	#3230	1	2	yes	—
170	C₂²×C₄⋊Q₈	C₂×C₁₂₇	1	1	yes	characteristic
171	C₂×#431	C₂×C₁₂₇	1	1	yes	characteristic
172	#2285	C₂×C₁₂₇	1	1	yes	characteristic
173	#1990	C₂×C₁₂₇	1	4	yes	complemented
174	#1987	C₂×C₁₂₇	1	4	yes	complemented
175	#1824	C₂×C₁₂₇	1	4	yes	—
176	C₁₂₇	#21834	1	1	yes	characteristic, direct product factor, Sylow
177	C₂³×Q₈	—	2	1	no	—
178	C₂×C₄⋊Q₈	—	2	1	no	—
179	C₂×C₄⋊Q₈	—	2	1	no	—
180	C₂×C₄⋊Q₈	—	2	4	no	—
181	C₂×C₄⋊Q₈	C₄×C₁₂₇	1	8	yes	complemented
182	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
183	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
184	C₂×#117	—	2	1	no	—
185	C₂²×C₄⋊C₄	—	2	1	no	—
186	C₂²×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
187	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
188	#431	—	2	4	no	—
189	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
190	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
191	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
192	#384	C₂²×C₁₂₇	1	2	yes	—
193	#382	—	2	1	no	—
194	#380	—	2	1	no	—
195	#380	C₂²×C₁₂₇	1	2	yes	—
196	C₄×C₄⋊C₄	—	2	1	no	—
197	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
198	#340	C₂²×C₁₂₇	1	16	yes	—
199	C₂²×Q₈	—	2	8	no	—
200	C₂²×Q₈	—	2	2	no	—
201	C₂²×Q₈	—	2	2	no	—
202	C₂²×Q₈	—	4	8	no	—
203	C₂³×C₄	—	2	2	no	—
204	C₂³×C₄	D₄×C₁₂₇	1	1	yes	characteristic
205	C₂³×C₄	D₄×C₁₂₇	1	1	yes	characteristic
206	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
207	C₂²×C₈	C₂³×C₁₂₇	1	4	yes	—
208	C₄⋊Q₈	—	2	8	no	—
209	C₄⋊Q₈	—	2	8	no	—
210	C₄⋊Q₈	—	2	4	no	—
211	C₄⋊Q₈	—	2	4	no	—
212	C₄⋊Q₈	C₂×C₄×C₁₂₇	1	8	yes	—
213	C₄⋊Q₈	C₂×C₄×C₁₂₇	1	8	yes	—
214	#117	—	4	4	no	—
215	C₂×C₄⋊C₄	—	2	8	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	4	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	8	yes	—
223	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	4	yes	—
224	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	4	yes	—
225	C₂×C₂²⋊C₄	—	2	2	no	—
226	C₂×C₄²	—	2	2	no	—
227	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
228	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
229	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
230	C₂×C₄²	D₄×C₁₂₇	1	4	yes	—
231	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
232	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
233	C₄⋊C₈	—	2	4	no	—
234	C₂²⋊C₈	—	2	4	no	—
235	C₄×C₈	—	2	4	no	—
236	#95	—	2	2	no	—
237	#95	—	2	2	no	—
238	C₂⁴	C₂×D₄×C₁₂₇	1	1	yes	characteristic
239	C₂×Q₈	—	2	8	no	—
240	C₂×Q₈	—	2	8	no	—
241	C₂×Q₈	—	4	8	no	—
242	C₂×Q₈	—	4	32	no	—
243	C₂×Q₈	—	4	8	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	8	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	4	8	no	—
250	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
251	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
252	C₂²×C₄	SD₁₆×C₁₂₇	1	4	yes	—
253	C₂²×C₄	Q₁₆×C₁₂₇	1	4	yes	—
254	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
255	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
258	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
259	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
260	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
261	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	2	8	no	—
264	C₄⋊C₄	—	2	8	no	—
265	C₄⋊C₄	—	2	8	no	—
266	C₄⋊C₄	—	2	8	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	8	no	—
269	C₄⋊C₄	—	4	4	no	—
270	C₄⋊C₄	—	4	4	no	—
271	C₄⋊C₄	C₂²×C₄×C₁₂₇	1	8	yes	—
272	C₂²⋊C₄	—	4	4	no	—
273	C₄²	—	4	4	no	—
274	C₄²	C₂²⋊C₄×C₁₂₇	1	4	yes	—
275	C₄²	C₂²⋊C₄×C₁₂₇	1	4	yes	—
276	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
277	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
278	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
279	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
280	C₂³	—	2	2	no	—
281	C₂³	—	2	2	no	—
282	C₂³	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
283	C₂³	C₂×SD₁₆×C₁₂₇	1	2	yes	—
284	C₂³	C₂×Q₁₆×C₁₂₇	1	2	yes	—
285	C₂³	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
286	Q₈	—	4	16	no	—
287	Q₈	—	4	16	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	8	no	—
293	C₂×C₄	—	2	8	no	—
294	C₂×C₄	—	4	16	no	—
295	C₂×C₄	—	4	8	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	#103×C₁₂₇	1	16	yes	—
300	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
301	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
302	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
303	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
304	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
305	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	2	yes	—
306	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	2	yes	—
307	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	2	yes	—
308	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	2	yes	—
309	C₂×C₄	AGL(1,ℤ/8)×C₁₂₇	1	4	yes	—
310	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
311	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
312	C₂×C₄	2₋¹⁺⁴×C₁₂₇	1	4	yes	—
313	C₈	—	4	4	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	4	no	—
316	C₂²	—	2	4	no	—
317	C₂²	—	2	2	no	—
318	C₂²	C₂×#103×C₁₂₇	1	4	yes	—
319	C₂²	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
320	C₂²	#536×C₁₂₇	1	2	yes	—
321	C₂²	C₂²×SD₁₆×C₁₂₇	1	1	yes	characteristic
322	C₂²	C₂²×Q₁₆×C₁₂₇	1	1	yes	characteristic
323	C₂²	C₂×AGL(1,ℤ/8)×C₁₂₇	1	2	yes	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	4	8	no	—
327	C₄	—	8	4	no	—
328	C₄	C₂×#103×C₁₂₇	1	4	yes	—
329	C₄	C₂×#103×C₁₂₇	1	4	yes	—
330	C₄	#418×C₁₂₇	1	4	yes	—
331	C₄	#519×C₁₂₇	1	2	yes	—
332	C₄	#536×C₁₂₇	1	2	yes	—
333	C₂	—	2	4	no	—
334	C₂	C₁₂₇×#2310	1	1	yes	characteristic
335	C₂	C₂²×#103×C₁₂₇	1	1	yes	characteristic
336	C₂	C₂×#418×C₁₂₇	1	1	yes	characteristic
337	C₂	C₁₂₇×#3227	1	2	yes	—
338	C₂	C₁₂₇×#3230	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₁₂₇×#1990	C₂	semidirect
5. C₁₂₇×#1987	C₂	semidirect
11. C₂×C₄⋊Q₈×C₁₂₇	C₄	semidirect
163. #21834	C₁₂₇	direct (N × H)
173. #1990	C₂×C₁₂₇	semidirect
174. #1987	C₂×C₁₂₇	semidirect
176. C₁₂₇	#21834	direct (N × H)
181. C₂×C₄⋊Q₈	C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₁₂₇	2
C₂×#431×C₁₂₇	2
C₁₂₇×#2285	2
C₁₂₇×#1990	2
C₁₂₇×#1987	2
C₁₂₇×#1824	2
#21834	127

Subgroups — C127×#21834 #44585507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21834 #44585507