Subgroups — C₁₂₇×#18056 #44581729

Order 32512.

676 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2984	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₂₇×#2974	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×SD₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1590	C₂	1	2	yes	maximal, complemented
5	C₁₂₇×#1588	C₂	1	2	yes	maximal, complemented
6	C₁₂₇×#1585	C₂	1	2	yes	maximal, complemented
7	C₈×SD₁₆×C₁₂₇	C₂	1	2	yes	maximal, complemented
8	C₁₂₇×#1568	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₂₇×#1491	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₂₇×#1490	C₂	1	1	yes	maximal, characteristic
11	C₈×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₁₂₇	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
15	#446×C₁₂₇	—	2	2	no	—
16	#446×C₁₂₇	C₂²	1	2	yes	complemented
17	C₈×Q₈×C₁₂₇	C₂²	1	2	yes	complemented
18	C₄×SD₁₆×C₁₂₇	C₄	1	1	yes	characteristic
19	C₄×SD₁₆×C₁₂₇	C₄	1	1	yes	characteristic
20	C₄×SD₁₆×C₁₂₇	C₄	1	1	yes	characteristic
21	C₄×SD₁₆×C₁₂₇	C₄	1	1	yes	characteristic
22	C₄×SD₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₄×SD₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₄×SD₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
25	C₄×SD₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
26	#436×C₁₂₇	C₂²	1	2	yes	—
27	#435×C₁₂₇	—	2	2	no	—
28	C₈×D₄×C₁₂₇	C₂²	1	2	yes	—
29	#432×C₁₂₇	—	2	1	no	—
30	#432×C₁₂₇	C₂²	1	1	yes	characteristic
31	#432×C₁₂₇	C₂²	1	1	yes	characteristic
32	#431×C₁₂₇	—	2	1	no	—
33	#431×C₁₂₇	C₂²	1	1	yes	characteristic
34	C₂×#106×C₁₂₇	C₄	1	1	yes	characteristic
35	C₂×#103×C₁₂₇	C₄	1	1	yes	characteristic
36	C₂×#102×C₁₂₇	C₄	1	1	yes	characteristic
37	#407×C₁₂₇	—	2	1	no	—
38	C₂×C₂²⋊C₈×C₁₂₇	—	2	1	no	—
39	C₄×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
41	#335×C₁₂₇	C₂²	1	2	yes	complemented
42	#334×C₁₂₇	C₂²	1	2	yes	complemented
43	Q₈⋊C₈×C₁₂₇	C₂²	1	2	yes	complemented
44	Q₈⋊C₈×C₁₂₇	C₂²	1	2	yes	complemented
45	D₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
46	D₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
47	#322×C₁₂₇	C₂²	1	2	yes	complemented
48	C₈²×C₁₂₇	C₂²	1	2	yes	complemented
49	C₂²×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
51	C₂³×C₄×C₁₂₇	—	2	1	no	—
52	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
53	C₂×SD₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
54	C₂×SD₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
55	C₂×SD₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
56	C₂×SD₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
57	C₂×M₁₆×C₁₂₇	—	2	1	no	—
58	C₂×M₁₆×C₁₂₇	—	2	2	no	—
59	C₂²×C₈×C₁₂₇	—	2	2	no	—
60	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
62	C₄×Q₈×C₁₂₇	—	2	1	no	—
63	C₄×Q₈×C₁₂₇	—	2	1	no	—
64	C₄×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
65	C₄×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
66	C₄×Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
67	C₄×Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
68	C₄×D₄×C₁₂₇	—	2	1	no	—
69	C₄×D₄×C₁₂₇	—	2	1	no	—
70	C₄×D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
71	C₄×D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
72	C₄×D₄×C₁₂₇	C₂³	1	1	yes	characteristic
73	C₄×D₄×C₁₂₇	C₂³	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
75	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
77	C₂×C₂²⋊C₄×C₁₂₇	—	2	1	no	—
78	C₂×C₄²×C₁₂₇	—	2	1	no	—
79	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
80	#106×C₁₂₇	C₂×C₄	1	1	yes	characteristic
81	#106×C₁₂₇	C₂×C₄	1	1	yes	characteristic
82	#106×C₁₂₇	C₂×C₄	1	1	yes	characteristic
83	#106×C₁₂₇	C₂×C₄	1	1	yes	characteristic
84	C₄⋊C₈×C₁₂₇	—	2	2	no	—
85	C₄⋊C₈×C₁₂₇	—	2	2	no	—
86	C₄⋊C₈×C₁₂₇	D₄	1	2	yes	—
87	C₄⋊C₈×C₁₂₇	C₂³	1	2	yes	—
88	C₄⋊C₈×C₁₂₇	C₂³	1	2	yes	—
89	#103×C₁₂₇	C₂×C₄	1	1	yes	characteristic
90	#103×C₁₂₇	C₂×C₄	1	1	yes	characteristic
91	#103×C₁₂₇	C₂×C₄	1	1	yes	characteristic
92	#103×C₁₂₇	C₂×C₄	1	1	yes	characteristic
93	#102×C₁₂₇	C₂×C₄	1	1	yes	characteristic
94	#102×C₁₂₇	C₂×C₄	1	1	yes	characteristic
95	#102×C₁₂₇	C₂×C₄	1	1	yes	characteristic
96	#102×C₁₂₇	C₂×C₄	1	1	yes	characteristic
97	C₂²⋊C₈×C₁₂₇	—	2	1	no	—
98	C₂²⋊C₈×C₁₂₇	—	2	1	no	—
99	C₂²⋊C₈×C₁₂₇	—	2	2	no	—
100	C₂²⋊C₈×C₁₂₇	—	2	2	no	—
101	C₂²⋊C₈×C₁₂₇	D₄	1	1	yes	characteristic
102	C₂²⋊C₈×C₁₂₇	D₄	1	1	yes	characteristic
103	#97×C₁₂₇	—	2	2	no	—
104	#97×C₁₂₇	—	2	1	no	—
105	#97×C₁₂₇	—	2	1	no	—
106	#97×C₁₂₇	—	2	1	no	—
107	C₄×C₈×C₁₂₇	—	2	2	no	—
108	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
109	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
110	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
111	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
112	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
113	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
114	C₂⁴×C₁₂₇	—	2	1	no	—
115	C₂×Q₈×C₁₂₇	—	2	1	no	—
116	C₂×Q₈×C₁₂₇	—	2	2	no	—
117	C₂×Q₈×C₁₂₇	—	4	1	no	—
118	C₂×Q₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
119	C₂×Q₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
120	C₂×D₄×C₁₂₇	—	2	2	no	—
121	C₂×D₄×C₁₂₇	—	2	1	no	—
122	C₂×D₄×C₁₂₇	—	4	1	no	—
123	C₂×D₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
124	C₂×D₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₄×C₁₂₇	—	2	1	no	—
126	C₂²×C₄×C₁₂₇	—	2	1	no	—
127	C₂²×C₄×C₁₂₇	—	2	1	no	—
128	C₂²×C₄×C₁₂₇	—	2	1	no	—
129	C₂²×C₄×C₁₂₇	—	2	1	no	—
130	C₂²×C₄×C₁₂₇	—	2	1	no	—
131	C₂²×C₄×C₁₂₇	—	2	1	no	—
132	C₂²×C₄×C₁₂₇	—	8	2	no	—
133	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₁₂₇	#41	1	1	yes	characteristic
136	SD₁₆×C₁₂₇	—	2	2	no	—
137	SD₁₆×C₁₂₇	—	2	2	no	—
138	SD₁₆×C₁₂₇	—	2	2	no	—
139	SD₁₆×C₁₂₇	M₁₆	1	4	yes	complemented
140	M₁₆×C₁₂₇	—	2	4	no	—
141	M₁₆×C₁₂₇	—	8	2	no	—
142	C₂×C₈×C₁₂₇	—	2	2	no	—
143	C₂×C₈×C₁₂₇	—	2	2	no	—
144	C₂×C₈×C₁₂₇	—	2	2	no	—
145	C₂×C₈×C₁₂₇	—	2	2	no	—
146	C₂×C₈×C₁₂₇	—	4	1	no	—
147	C₂×C₈×C₁₂₇	—	8	2	no	—
148	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
149	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
150	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
151	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
152	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
153	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
154	C₄⋊C₄×C₁₂₇	—	2	1	no	—
155	C₄⋊C₄×C₁₂₇	—	2	1	no	—
156	C₄⋊C₄×C₁₂₇	—	2	1	no	—
157	C₄⋊C₄×C₁₂₇	—	2	1	no	—
158	C₄⋊C₄×C₁₂₇	—	2	1	no	—
159	C₄⋊C₄×C₁₂₇	—	2	1	no	—
160	C₄⋊C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
161	C₄⋊C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
162	C₄⋊C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
163	C₄⋊C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
164	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
165	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
166	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
167	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
168	C₄²×C₁₂₇	—	2	1	no	—
169	C₄²×C₁₂₇	—	2	1	no	—
170	C₄²×C₁₂₇	—	2	1	no	—
171	C₄²×C₁₂₇	—	2	1	no	—
172	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
173	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₁₂₇	#41	1	1	yes	characteristic
175	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
176	C₂³×C₁₂₇	—	2	1	no	—
177	C₂³×C₁₂₇	—	2	1	no	—
178	C₂³×C₁₂₇	—	4	2	no	—
179	C₂³×C₁₂₇	—	4	2	no	—
180	C₂³×C₁₂₇	—	8	1	no	—
181	C₂³×C₁₂₇	C₄×D₄	1	1	yes	characteristic
182	Q₈×C₁₂₇	—	2	1	no	—
183	Q₈×C₁₂₇	—	2	1	no	—
184	Q₈×C₁₂₇	—	2	1	no	—
185	Q₈×C₁₂₇	—	4	2	no	—
186	Q₈×C₁₂₇	C₂×M₁₆	1	2	yes	—
187	D₄×C₁₂₇	—	2	1	no	—
188	D₄×C₁₂₇	—	2	1	no	—
189	D₄×C₁₂₇	—	2	1	no	—
190	D₄×C₁₂₇	—	4	2	no	—
191	D₄×C₁₂₇	C₂×M₁₆	1	2	yes	—
192	C₂×C₄×C₁₂₇	—	2	1	no	—
193	C₂×C₄×C₁₂₇	—	2	2	no	—
194	C₂×C₄×C₁₂₇	—	2	1	no	—
195	C₂×C₄×C₁₂₇	—	2	1	no	—
196	C₂×C₄×C₁₂₇	—	2	2	no	—
197	C₂×C₄×C₁₂₇	—	2	1	no	—
198	C₂×C₄×C₁₂₇	—	2	1	no	—
199	C₂×C₄×C₁₂₇	—	2	1	no	—
200	C₂×C₄×C₁₂₇	—	2	1	no	—
201	C₂×C₄×C₁₂₇	—	2	1	no	—
202	C₂×C₄×C₁₂₇	—	4	2	no	—
203	C₂×C₄×C₁₂₇	—	4	2	no	—
204	C₂×C₄×C₁₂₇	—	4	2	no	—
205	C₂×C₄×C₁₂₇	—	4	2	no	—
206	C₂×C₄×C₁₂₇	—	8	1	no	—
207	C₂×C₄×C₁₂₇	—	8	1	no	—
208	C₂×C₄×C₁₂₇	—	8	1	no	—
209	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
210	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
211	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
212	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
215	C₈×C₁₂₇	—	2	4	no	—
216	C₈×C₁₂₇	—	2	1	no	—
217	C₈×C₁₂₇	—	2	1	no	—
218	C₈×C₁₂₇	—	2	1	no	—
219	C₈×C₁₂₇	—	2	2	no	—
220	C₈×C₁₂₇	—	2	2	no	—
221	C₈×C₁₂₇	—	4	2	no	—
222	C₈×C₁₂₇	—	4	2	no	—
223	C₈×C₁₂₇	C₂×M₁₆	1	2	yes	—
224	C₂²×C₁₂₇	—	2	1	no	—
225	C₂²×C₁₂₇	—	2	1	no	—
226	C₂²×C₁₂₇	—	2	2	no	—
227	C₂²×C₁₂₇	—	4	1	no	—
228	C₂²×C₁₂₇	—	4	1	no	—
229	C₂²×C₁₂₇	—	4	1	no	—
230	C₂²×C₁₂₇	—	4	1	no	—
231	C₂²×C₁₂₇	—	4	1	no	—
232	C₂²×C₁₂₇	—	8	4	no	—
233	C₂²×C₁₂₇	#443	1	2	yes	—
234	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
235	C₄×C₁₂₇	—	2	1	no	—
236	C₄×C₁₂₇	—	2	2	no	—
237	C₄×C₁₂₇	—	2	1	no	—
238	C₄×C₁₂₇	—	2	1	no	—
239	C₄×C₁₂₇	—	2	1	no	—
240	C₄×C₁₂₇	—	4	1	no	—
241	C₄×C₁₂₇	—	4	1	no	—
242	C₄×C₁₂₇	—	4	1	no	—
243	C₄×C₁₂₇	—	4	1	no	—
244	C₄×C₁₂₇	—	4	1	no	—
245	C₄×C₁₂₇	C₂²×M₁₆	1	1	yes	characteristic
246	C₄×C₁₂₇	#568	1	1	yes	characteristic
247	C₄×C₁₂₇	#577	1	2	yes	—
248	#18056	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
249	C₂×C₁₂₇	—	2	2	no	—
250	C₂×C₁₂₇	—	4	2	no	—
251	C₂×C₁₂₇	—	8	1	no	—
252	C₂×C₁₂₇	D₄×M₁₆	1	1	yes	characteristic
253	C₂×C₁₂₇	#2949	1	1	yes	characteristic
254	C₂×C₁₂₇	C₂×#443	1	1	yes	characteristic
255	#2984	C₂×C₁₂₇	1	1	yes	characteristic, complemented
256	#2974	C₂×C₁₂₇	1	1	yes	characteristic
257	C₂×C₄×SD₁₆	C₂×C₁₂₇	1	1	yes	characteristic
258	#1590	C₂×C₁₂₇	1	2	yes	complemented
259	#1588	C₂×C₁₂₇	1	2	yes	complemented
260	#1585	C₂×C₁₂₇	1	2	yes	complemented
261	C₈×SD₁₆	C₂×C₁₂₇	1	2	yes	complemented
262	#1568	C₂×C₁₂₇	1	1	yes	characteristic, complemented
263	#1491	C₂×C₁₂₇	1	1	yes	characteristic, complemented
264	#1490	C₂×C₁₂₇	1	1	yes	characteristic
265	C₈×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
266	C₁₂₇	#18056	1	1	yes	characteristic, direct product factor, Sylow
267	C₂²×SD₁₆	C₄×C₁₂₇	1	1	yes	characteristic
268	C₂×C₄×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
269	C₂×C₄×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
270	#446	—	2	2	no	—
271	#446	C₂²×C₁₂₇	1	2	yes	complemented
272	C₈×Q₈	C₂²×C₁₂₇	1	2	yes	complemented
273	C₄×SD₁₆	C₄×C₁₂₇	1	1	yes	characteristic
274	C₄×SD₁₆	C₄×C₁₂₇	1	1	yes	characteristic
275	C₄×SD₁₆	C₄×C₁₂₇	1	1	yes	characteristic
276	C₄×SD₁₆	C₄×C₁₂₇	1	1	yes	characteristic
277	C₄×SD₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
278	C₄×SD₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
279	C₄×SD₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
280	C₄×SD₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
281	#436	C₂²×C₁₂₇	1	2	yes	—
282	#435	—	2	2	no	—
283	C₈×D₄	C₂²×C₁₂₇	1	2	yes	—
284	#432	—	2	1	no	—
285	#432	C₂²×C₁₂₇	1	1	yes	characteristic
286	#432	C₂²×C₁₂₇	1	1	yes	characteristic
287	#431	—	2	1	no	—
288	#431	C₂²×C₁₂₇	1	1	yes	characteristic
289	C₂×#106	C₄×C₁₂₇	1	1	yes	characteristic
290	C₂×#103	C₄×C₁₂₇	1	1	yes	characteristic
291	C₂×#102	C₄×C₁₂₇	1	1	yes	characteristic
292	#407	—	2	1	no	—
293	C₂×C₂²⋊C₈	—	2	1	no	—
294	C₄×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
295	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
296	#335	C₂²×C₁₂₇	1	2	yes	complemented
297	#334	C₂²×C₁₂₇	1	2	yes	complemented
298	Q₈⋊C₈	C₂²×C₁₂₇	1	2	yes	complemented
299	Q₈⋊C₈	C₂²×C₁₂₇	1	2	yes	complemented
300	D₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
301	D₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
302	#322	C₂²×C₁₂₇	1	2	yes	complemented
303	C₈²	C₂²×C₁₂₇	1	2	yes	complemented
304	C₂²×Q₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
305	C₂²×D₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
306	C₂³×C₄	—	2	1	no	—
307	C₂×SD₁₆	—	2	4	no	—
308	C₂×SD₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
309	C₂×SD₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
310	C₂×SD₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
311	C₂×SD₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
312	C₂×M₁₆	—	2	1	no	—
313	C₂×M₁₆	—	2	2	no	—
314	C₂²×C₈	—	2	2	no	—
315	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
316	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
317	C₄×Q₈	—	2	1	no	—
318	C₄×Q₈	—	2	1	no	—
319	C₄×Q₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
320	C₄×Q₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
321	C₄×Q₈	C₂³×C₁₂₇	1	1	yes	characteristic
322	C₄×Q₈	C₂³×C₁₂₇	1	1	yes	characteristic
323	C₄×D₄	—	2	1	no	—
324	C₄×D₄	—	2	1	no	—
325	C₄×D₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
326	C₄×D₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
327	C₄×D₄	C₂³×C₁₂₇	1	1	yes	characteristic
328	C₄×D₄	C₂³×C₁₂₇	1	1	yes	characteristic
329	C₂×C₄⋊C₄	—	2	1	no	—
330	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
331	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
332	C₂×C₂²⋊C₄	—	2	1	no	—
333	C₂×C₄²	—	2	1	no	—
334	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
335	#106	C₂×C₄×C₁₂₇	1	1	yes	characteristic
336	#106	C₂×C₄×C₁₂₇	1	1	yes	characteristic
337	#106	C₂×C₄×C₁₂₇	1	1	yes	characteristic
338	#106	C₂×C₄×C₁₂₇	1	1	yes	characteristic
339	C₄⋊C₈	—	2	2	no	—
340	C₄⋊C₈	—	2	2	no	—
341	C₄⋊C₈	D₄×C₁₂₇	1	2	yes	—
342	C₄⋊C₈	C₂³×C₁₂₇	1	2	yes	—
343	C₄⋊C₈	C₂³×C₁₂₇	1	2	yes	—
344	#103	C₂×C₄×C₁₂₇	1	1	yes	characteristic
345	#103	C₂×C₄×C₁₂₇	1	1	yes	characteristic
346	#103	C₂×C₄×C₁₂₇	1	1	yes	characteristic
347	#103	C₂×C₄×C₁₂₇	1	1	yes	characteristic
348	#102	C₂×C₄×C₁₂₇	1	1	yes	characteristic
349	#102	C₂×C₄×C₁₂₇	1	1	yes	characteristic
350	#102	C₂×C₄×C₁₂₇	1	1	yes	characteristic
351	#102	C₂×C₄×C₁₂₇	1	1	yes	characteristic
352	C₂²⋊C₈	—	2	1	no	—
353	C₂²⋊C₈	—	2	1	no	—
354	C₂²⋊C₈	—	2	2	no	—
355	C₂²⋊C₈	—	2	2	no	—
356	C₂²⋊C₈	D₄×C₁₂₇	1	1	yes	characteristic
357	C₂²⋊C₈	D₄×C₁₂₇	1	1	yes	characteristic
358	#97	—	2	2	no	—
359	#97	—	2	1	no	—
360	#97	—	2	1	no	—
361	#97	—	2	1	no	—
362	C₄×C₈	—	2	2	no	—
363	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
364	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
365	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
366	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
367	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
368	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
369	C₂⁴	—	2	1	no	—
370	C₂×Q₈	—	2	1	no	—
371	C₂×Q₈	—	2	2	no	—
372	C₂×Q₈	—	4	1	no	—
373	C₂×Q₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
374	C₂×Q₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	4	1	no	—
378	C₂×D₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
379	C₂×D₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	—	2	1	no	—
387	C₂²×C₄	—	8	2	no	—
388	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
389	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
390	C₂²×C₄	#41×C₁₂₇	1	1	yes	characteristic
391	SD₁₆	—	2	2	no	—
392	SD₁₆	—	2	2	no	—
393	SD₁₆	—	2	2	no	—
394	SD₁₆	M₁₆×C₁₂₇	1	4	yes	complemented
395	M₁₆	—	2	4	no	—
396	M₁₆	—	8	2	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	4	1	no	—
402	C₂×C₈	—	8	2	no	—
403	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
404	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
405	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
406	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
407	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
408	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	1	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
416	C₄⋊C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
417	C₄⋊C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
418	C₄⋊C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
419	C₂²⋊C₄	—	2	1	no	—
420	C₂²⋊C₄	—	2	1	no	—
421	C₂²⋊C₄	—	2	1	no	—
422	C₂²⋊C₄	—	2	1	no	—
423	C₄²	—	2	1	no	—
424	C₄²	—	2	1	no	—
425	C₄²	—	2	1	no	—
426	C₄²	—	2	1	no	—
427	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
428	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
429	C₄²	#41×C₁₂₇	1	1	yes	characteristic
430	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
431	C₂³	—	2	1	no	—
432	C₂³	—	2	1	no	—
433	C₂³	—	4	2	no	—
434	C₂³	—	4	2	no	—
435	C₂³	—	8	1	no	—
436	C₂³	C₄×D₄×C₁₂₇	1	1	yes	characteristic
437	Q₈	—	2	1	no	—
438	Q₈	—	2	1	no	—
439	Q₈	—	2	1	no	—
440	Q₈	—	4	2	no	—
441	Q₈	C₂×M₁₆×C₁₂₇	1	2	yes	—
442	D₄	—	2	1	no	—
443	D₄	—	2	1	no	—
444	D₄	—	2	1	no	—
445	D₄	—	4	2	no	—
446	D₄	C₂×M₁₆×C₁₂₇	1	2	yes	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	8	1	no	—
462	C₂×C₄	—	8	1	no	—
463	C₂×C₄	—	8	1	no	—
464	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
465	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
466	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
467	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
468	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
469	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
470	C₈	—	2	4	no	—
471	C₈	—	2	1	no	—
472	C₈	—	2	1	no	—
473	C₈	—	2	1	no	—
474	C₈	—	2	2	no	—
475	C₈	—	2	2	no	—
476	C₈	—	4	2	no	—
477	C₈	—	4	2	no	—
478	C₈	C₂×M₁₆×C₁₂₇	1	2	yes	—
479	C₂²	—	2	1	no	—
480	C₂²	—	2	1	no	—
481	C₂²	—	2	2	no	—
482	C₂²	—	4	1	no	—
483	C₂²	—	4	1	no	—
484	C₂²	—	4	1	no	—
485	C₂²	—	4	1	no	—
486	C₂²	—	4	1	no	—
487	C₂²	—	8	4	no	—
488	C₂²	#443×C₁₂₇	1	2	yes	—
489	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
490	C₄	—	2	1	no	—
491	C₄	—	2	2	no	—
492	C₄	—	2	1	no	—
493	C₄	—	2	1	no	—
494	C₄	—	2	1	no	—
495	C₄	—	4	1	no	—
496	C₄	—	4	1	no	—
497	C₄	—	4	1	no	—
498	C₄	—	4	1	no	—
499	C₄	—	4	1	no	—
500	C₄	C₂²×M₁₆×C₁₂₇	1	1	yes	characteristic
501	C₄	#568×C₁₂₇	1	1	yes	characteristic
502	C₄	#577×C₁₂₇	1	2	yes	—
503	C₂	—	2	2	no	—
504	C₂	—	4	2	no	—
505	C₂	—	8	1	no	—
506	C₂	D₄×M₁₆×C₁₂₇	1	1	yes	characteristic
507	C₂	C₁₂₇×#2949	1	1	yes	characteristic
508	C₂	C₂×#443×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₄²	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂×C₄×D₄
prime radical	C₂²×D₄×C₁₂₇	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ #431×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₂₇×#2984	C₂	semidirect
4. C₁₂₇×#1590	C₂	semidirect
5. C₁₂₇×#1588	C₂	semidirect
6. C₁₂₇×#1585	C₂	semidirect
7. C₈×SD₁₆×C₁₂₇	C₂	semidirect
8. C₁₂₇×#1568	C₂	semidirect
9. C₁₂₇×#1491	C₂	semidirect
11. C₈×M₁₆×C₁₂₇	C₂	semidirect
16. #446×C₁₂₇	C₂²	semidirect
17. C₈×Q₈×C₁₂₇	C₂²	semidirect
41. #335×C₁₂₇	C₂²	semidirect
42. #334×C₁₂₇	C₂²	semidirect
43. Q₈⋊C₈×C₁₂₇	C₂²	semidirect
44. Q₈⋊C₈×C₁₂₇	C₂²	semidirect
47. #322×C₁₂₇	C₂²	semidirect
48. C₈²×C₁₂₇	C₂²	semidirect
139. SD₁₆×C₁₂₇	M₁₆	semidirect
248. #18056	C₁₂₇	direct (N × H)
255. #2984	C₂×C₁₂₇	semidirect
258. #1590	C₂×C₁₂₇	semidirect
259. #1588	C₂×C₁₂₇	semidirect
260. #1585	C₂×C₁₂₇	semidirect
261. C₈×SD₁₆	C₂×C₁₂₇	semidirect
262. #1568	C₂×C₁₂₇	semidirect
263. #1491	C₂×C₁₂₇	semidirect
265. C₈×M₁₆	C₂×C₁₂₇	semidirect
266. C₁₂₇	#18056	direct (N × H)
271. #446	C₂²×C₁₂₇	semidirect
272. C₈×Q₈	C₂²×C₁₂₇	semidirect
296. #335	C₂²×C₁₂₇	semidirect
297. #334	C₂²×C₁₂₇	semidirect
298. Q₈⋊C₈	C₂²×C₁₂₇	semidirect
299. Q₈⋊C₈	C₂²×C₁₂₇	semidirect
302. #322	C₂²×C₁₂₇	semidirect
303. C₈²	C₂²×C₁₂₇	semidirect
394. SD₁₆	M₁₆×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#2984	2
C₁₂₇×#2974	2
C₂×C₄×SD₁₆×C₁₂₇	2
C₁₂₇×#1590	2
C₁₂₇×#1588	2
C₁₂₇×#1585	2
C₈×SD₁₆×C₁₂₇	2
C₁₂₇×#1568	2
C₁₂₇×#1491	2
C₁₂₇×#1490	2
C₈×M₁₆×C₁₂₇	2
#18056	127

Subgroups — C127×#18056 #44581729

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#18056 #44581729