Subgroups — C₂×C₁₂₇×#1764 #44584091

Order 32512.

1558 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#131×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#104×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#416×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#405×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₂₇×#1764	C₂	1	8	yes	maximal, direct product factor
7	C₂×#341×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#339×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×#131×C₁₂₇	—	2	4	no	—
11	C₂×#131×C₁₂₇	—	2	2	no	—
12	C₂×#131×C₁₂₇	C₄	1	8	yes	complemented
13	C₂×#131×C₁₂₇	C₂²	1	4	yes	—
14	C₂²×M₁₆×C₁₂₇	—	2	1	no	—
15	C₂²×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₁₂₇	—	2	1	no	—
18	C₂³×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂×#117×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂×#104×C₁₂₇	C₄	1	8	yes	complemented
23	C₂×#104×C₁₂₇	C₂²	1	4	yes	—
24	#416×C₁₂₇	C₄	1	4	yes	complemented
25	#416×C₁₂₇	C₂²	1	4	yes	—
26	C₂×#103×C₁₂₇	C₄	1	2	yes	complemented
27	C₂×#102×C₁₂₇	C₄	1	2	yes	complemented
28	#405×C₁₂₇	—	2	2	no	—
29	#405×C₁₂₇	C₂²	1	4	yes	complemented
30	C₂×#97×C₁₂₇	—	2	1	no	—
31	C₂×C₄×C₈×C₁₂₇	—	2	4	no	—
32	C₂×C₄×C₈×C₁₂₇	—	2	1	no	—
33	C₂×C₄×C₈×C₁₂₇	C₂²	1	4	yes	complemented
34	#341×C₁₂₇	C₂²	1	4	yes	complemented
35	#339×C₁₂₇	C₂²	1	4	yes	complemented
36	C₂×#41×C₁₂₇	—	2	2	no	—
37	C₂×#41×C₁₂₇	—	2	1	no	—
38	C₂×#41×C₁₂₇	—	2	1	no	—
39	C₂×#41×C₁₂₇	—	2	2	no	—
40	C₂×#41×C₁₂₇	C₂×C₄	1	4	yes	—
41	C₂×#41×C₁₂₇	C₂×C₄	1	4	yes	—
42	C₂×#41×C₁₂₇	C₂×C₄	1	2	yes	—
43	C₂×#41×C₁₂₇	C₂³	1	2	yes	—
44	C₂²×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₄×C₁₂₇	—	2	1	no	—
46	C₂²×D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₁₂₇	—	2	1	no	—
48	C₂³×C₄×C₁₂₇	—	2	1	no	—
49	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
50	#131×C₁₂₇	—	2	16	no	—
51	#131×C₁₂₇	—	2	8	no	—
52	#131×C₁₂₇	C₂×C₄	1	16	yes	complemented
53	C₂×M₁₆×C₁₂₇	—	2	2	no	—
54	C₂×M₁₆×C₁₂₇	—	2	1	no	—
55	C₂×M₁₆×C₁₂₇	—	2	1	no	—
56	C₂×M₁₆×C₁₂₇	—	2	2	no	—
57	C₂×M₁₆×C₁₂₇	—	2	8	no	—
58	C₂×M₁₆×C₁₂₇	—	2	4	no	—
59	C₂×M₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
60	C₂×M₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
61	C₂×M₁₆×C₁₂₇	C₂³	1	2	yes	—
62	C₂×M₁₆×C₁₂₇	C₂³	1	2	yes	—
63	C₂²×C₈×C₁₂₇	—	2	1	no	—
64	C₂²×C₈×C₁₂₇	—	2	2	no	—
65	C₂²×C₈×C₁₂₇	—	2	1	no	—
66	C₂²×C₈×C₁₂₇	—	2	2	no	—
67	C₂²×C₈×C₁₂₇	—	2	8	no	—
68	C₂²×C₈×C₁₂₇	—	2	4	no	—
69	C₂²×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
70	C₂²×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
71	C₂²×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
72	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
73	C₂²×C₈×C₁₂₇	D₄	1	4	yes	—
74	C₂²×C₈×C₁₂₇	D₄	1	2	yes	—
75	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
76	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
77	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
78	#117×C₁₂₇	—	2	1	no	—
79	#117×C₁₂₇	—	2	1	no	—
80	#117×C₁₂₇	C₂×C₄	1	2	yes	—
81	#117×C₁₂₇	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
83	C₂×C₂²⋊C₄×C₁₂₇	—	2	1	no	—
84	C₂×C₄²×C₁₂₇	—	2	2	no	—
85	C₂×C₄²×C₁₂₇	—	2	1	no	—
86	C₂×C₄²×C₁₂₇	—	2	1	no	—
87	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
88	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
89	#104×C₁₂₇	C₂×C₄	1	16	yes	complemented
90	#103×C₁₂₇	C₂×C₄	1	8	yes	complemented
91	#102×C₁₂₇	C₂×C₄	1	8	yes	complemented
92	#97×C₁₂₇	—	2	4	no	—
93	C₄×C₈×C₁₂₇	—	2	8	no	—
94	C₄×C₈×C₁₂₇	—	2	4	no	—
95	C₂⁴×C₁₂₇	—	2	1	no	—
96	C₂⁴×C₁₂₇	#41	1	1	yes	characteristic
97	#41×C₁₂₇	—	2	4	no	—
98	#41×C₁₂₇	—	2	4	no	—
99	#41×C₁₂₇	—	2	4	no	—
100	#41×C₁₂₇	—	2	4	no	—
101	#41×C₁₂₇	—	2	4	no	—
102	#41×C₁₂₇	—	2	2	no	—
103	#41×C₁₂₇	—	2	2	no	—
104	#41×C₁₂₇	C₄²	1	8	yes	—
105	#41×C₁₂₇	C₂²×C₄	1	4	yes	—
106	#41×C₁₂₇	C₂²×C₄	1	4	yes	—
107	C₂×Q₈×C₁₂₇	—	2	1	no	—
108	C₂×Q₈×C₁₂₇	—	2	2	no	—
109	C₂×Q₈×C₁₂₇	C₄²	1	4	yes	—
110	C₂×Q₈×C₁₂₇	C₂²×C₄	1	2	yes	—
111	C₂×D₄×C₁₂₇	—	2	4	no	—
112	C₂×D₄×C₁₂₇	—	2	2	no	—
113	C₂×D₄×C₁₂₇	—	2	4	no	—
114	C₂×D₄×C₁₂₇	—	2	2	no	—
115	C₂×D₄×C₁₂₇	—	2	2	no	—
116	C₂×D₄×C₁₂₇	—	2	1	no	—
117	C₂×D₄×C₁₂₇	C₄²	1	4	yes	—
118	C₂×D₄×C₁₂₇	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₁₂₇	—	2	8	no	—
120	C₂²×C₄×C₁₂₇	—	2	4	no	—
121	C₂²×C₄×C₁₂₇	—	2	1	no	—
122	C₂²×C₄×C₁₂₇	—	2	2	no	—
123	C₂²×C₄×C₁₂₇	—	2	4	no	—
124	C₂²×C₄×C₁₂₇	—	2	2	no	—
125	C₂²×C₄×C₁₂₇	—	2	4	no	—
126	C₂²×C₄×C₁₂₇	—	2	2	no	—
127	C₂²×C₄×C₁₂₇	—	2	2	no	—
128	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
129	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
130	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₁₂₇	#41	1	1	yes	characteristic
136	C₂²×C₄×C₁₂₇	#41	1	2	yes	—
137	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
138	M₁₆×C₁₂₇	—	2	4	no	—
139	M₁₆×C₁₂₇	—	2	4	no	—
140	M₁₆×C₁₂₇	—	2	2	no	—
141	M₁₆×C₁₂₇	—	2	2	no	—
142	M₁₆×C₁₂₇	—	2	16	no	—
143	M₁₆×C₁₂₇	C₂²×C₄	1	4	yes	—
144	M₁₆×C₁₂₇	C₂²×C₄	1	4	yes	—
145	C₂×C₈×C₁₂₇	—	2	4	no	—
146	C₂×C₈×C₁₂₇	—	2	4	no	—
147	C₂×C₈×C₁₂₇	—	2	4	no	—
148	C₂×C₈×C₁₂₇	—	2	4	no	—
149	C₂×C₈×C₁₂₇	—	2	2	no	—
150	C₂×C₈×C₁₂₇	—	2	2	no	—
151	C₂×C₈×C₁₂₇	—	2	2	no	—
152	C₂×C₈×C₁₂₇	—	2	2	no	—
153	C₂×C₈×C₁₂₇	—	2	16	no	—
154	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	4	yes	—
155	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	4	yes	—
156	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	8	yes	—
157	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
158	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
159	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
160	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
161	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
162	C₄⋊C₄×C₁₂₇	—	2	2	no	—
163	C₄⋊C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
164	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
165	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
166	C₄²×C₁₂₇	—	2	4	no	—
167	C₄²×C₁₂₇	—	2	2	no	—
168	C₄²×C₁₂₇	—	2	2	no	—
169	C₄²×C₁₂₇	—	2	2	no	—
170	C₄²×C₁₂₇	C₂²×C₄	1	4	yes	—
171	C₂³×C₁₂₇	—	2	4	no	—
172	C₂³×C₁₂₇	—	2	2	no	—
173	C₂³×C₁₂₇	—	2	4	no	—
174	C₂³×C₁₂₇	—	4	4	no	—
175	C₂³×C₁₂₇	#117	1	2	yes	—
176	C₂³×C₁₂₇	C₄×D₄	1	1	yes	characteristic
177	C₂³×C₁₂₇	C₄×D₄	1	2	yes	—
178	C₂³×C₁₂₇	C₄×D₄	1	1	yes	characteristic
179	C₂³×C₁₂₇	C₂×#41	1	1	yes	characteristic
180	Q₈×C₁₂₇	—	2	2	no	—
181	Q₈×C₁₂₇	—	2	4	no	—
182	Q₈×C₁₂₇	C₂×C₄²	1	4	yes	—
183	D₄×C₁₂₇	—	2	4	no	—
184	D₄×C₁₂₇	—	2	4	no	—
185	D₄×C₁₂₇	—	2	8	no	—
186	D₄×C₁₂₇	—	2	2	no	—
187	D₄×C₁₂₇	—	2	4	no	—
188	D₄×C₁₂₇	C₂×C₄²	1	4	yes	—
189	C₂×C₄×C₁₂₇	—	2	8	no	—
190	C₂×C₄×C₁₂₇	—	2	16	no	—
191	C₂×C₄×C₁₂₇	—	2	4	no	—
192	C₂×C₄×C₁₂₇	—	2	2	no	—
193	C₂×C₄×C₁₂₇	—	2	4	no	—
194	C₂×C₄×C₁₂₇	—	2	4	no	—
195	C₂×C₄×C₁₂₇	—	2	4	no	—
196	C₂×C₄×C₁₂₇	—	2	8	no	—
197	C₂×C₄×C₁₂₇	—	2	4	no	—
198	C₂×C₄×C₁₂₇	—	4	4	no	—
199	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
200	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
201	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
202	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
203	C₂×C₄×C₁₂₇	#117	1	2	yes	—
204	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
205	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
206	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
207	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
208	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
209	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
210	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
211	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
212	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
214	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
217	C₈×C₁₂₇	—	2	4	no	—
218	C₈×C₁₂₇	—	2	4	no	—
219	C₈×C₁₂₇	—	2	4	no	—
220	C₈×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
221	C₈×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
222	C₂²×C₁₂₇	—	2	2	no	—
223	C₂²×C₁₂₇	—	2	2	no	—
224	C₂²×C₁₂₇	—	2	4	no	—
225	C₂²×C₁₂₇	—	2	4	no	—
226	C₂²×C₁₂₇	—	2	4	no	—
227	C₂²×C₁₂₇	—	4	4	no	—
228	C₂²×C₁₂₇	—	4	8	no	—
229	C₂²×C₁₂₇	C₄×C₂²⋊C₄	1	2	yes	—
230	C₂²×C₁₂₇	C₄×C₂²⋊C₄	1	2	yes	—
231	C₂²×C₁₂₇	#443	1	4	yes	—
232	C₂²×C₁₂₇	C₂×#117	1	1	yes	characteristic
233	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
234	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
235	C₄×C₁₂₇	—	2	8	no	—
236	C₄×C₁₂₇	—	2	4	no	—
237	C₄×C₁₂₇	—	4	4	no	—
238	C₄×C₁₂₇	C₄×C₂²⋊C₄	1	2	yes	—
239	C₄×C₁₂₇	C₄×C₂²⋊C₄	1	2	yes	—
240	C₄×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
241	C₄×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
242	C₄×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
243	C₄×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
244	C₂×#1764	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
245	C₂×C₁₂₇	—	2	1	no	—
246	C₂×C₁₂₇	—	2	1	no	—
247	C₂×C₁₂₇	—	2	2	no	—
248	C₂×C₁₂₇	—	4	4	no	—
249	C₂×C₁₂₇	#1764	1	4	yes	direct product factor
250	C₂×C₁₂₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
251	C₂×C₁₂₇	C₂×#443	1	2	yes	—
252	C₂²×#131	C₂×C₁₂₇	1	1	yes	characteristic
253	C₂²×#104	C₂×C₁₂₇	1	1	yes	characteristic
254	C₂×#416	C₂×C₁₂₇	1	1	yes	characteristic
255	C₂×#405	C₂×C₁₂₇	1	1	yes	characteristic, complemented
256	C₂²×C₄×C₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
257	#1764	C₂×C₁₂₇	1	8	yes	direct product factor
258	C₂×#341	C₂×C₁₂₇	1	1	yes	characteristic, complemented
259	C₂×#339	C₂×C₁₂₇	1	1	yes	characteristic, complemented
260	C₁₂₇	C₂×#1764	1	1	yes	characteristic, direct product factor, Sylow
261	C₂²×#41	C₂²×C₁₂₇	1	1	yes	characteristic
262	C₂×#131	—	2	4	no	—
263	C₂×#131	—	2	2	no	—
264	C₂×#131	C₄×C₁₂₇	1	8	yes	complemented
265	C₂×#131	C₂²×C₁₂₇	1	4	yes	—
266	C₂²×M₁₆	—	2	1	no	—
267	C₂²×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
268	C₂²×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
269	C₂³×C₈	—	2	1	no	—
270	C₂³×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
271	C₂³×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
272	C₂×#117	C₂²×C₁₂₇	1	1	yes	characteristic
273	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
274	C₂×#104	C₄×C₁₂₇	1	8	yes	complemented
275	C₂×#104	C₂²×C₁₂₇	1	4	yes	—
276	#416	C₄×C₁₂₇	1	4	yes	complemented
277	#416	C₂²×C₁₂₇	1	4	yes	—
278	C₂×#103	C₄×C₁₂₇	1	2	yes	complemented
279	C₂×#102	C₄×C₁₂₇	1	2	yes	complemented
280	#405	—	2	2	no	—
281	#405	C₂²×C₁₂₇	1	4	yes	complemented
282	C₂×#97	—	2	1	no	—
283	C₂×C₄×C₈	—	2	4	no	—
284	C₂×C₄×C₈	—	2	1	no	—
285	C₂×C₄×C₈	C₂²×C₁₂₇	1	4	yes	complemented
286	#341	C₂²×C₁₂₇	1	4	yes	complemented
287	#339	C₂²×C₁₂₇	1	4	yes	complemented
288	C₂×#41	—	2	2	no	—
289	C₂×#41	—	2	1	no	—
290	C₂×#41	—	2	1	no	—
291	C₂×#41	—	2	2	no	—
292	C₂×#41	C₂×C₄×C₁₂₇	1	4	yes	—
293	C₂×#41	C₂×C₄×C₁₂₇	1	4	yes	—
294	C₂×#41	C₂×C₄×C₁₂₇	1	2	yes	—
295	C₂×#41	C₂³×C₁₂₇	1	2	yes	—
296	C₂²×Q₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
297	C₂²×D₄	—	2	1	no	—
298	C₂²×D₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
299	C₂³×C₄	—	2	1	no	—
300	C₂³×C₄	—	2	1	no	—
301	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
302	#131	—	2	16	no	—
303	#131	—	2	8	no	—
304	#131	C₂×C₄×C₁₂₇	1	16	yes	complemented
305	C₂×M₁₆	—	2	2	no	—
306	C₂×M₁₆	—	2	1	no	—
307	C₂×M₁₆	—	2	1	no	—
308	C₂×M₁₆	—	2	2	no	—
309	C₂×M₁₆	—	2	8	no	—
310	C₂×M₁₆	—	2	4	no	—
311	C₂×M₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
312	C₂×M₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
313	C₂×M₁₆	C₂³×C₁₂₇	1	2	yes	—
314	C₂×M₁₆	C₂³×C₁₂₇	1	2	yes	—
315	C₂²×C₈	—	2	1	no	—
316	C₂²×C₈	—	2	2	no	—
317	C₂²×C₈	—	2	1	no	—
318	C₂²×C₈	—	2	2	no	—
319	C₂²×C₈	—	2	8	no	—
320	C₂²×C₈	—	2	4	no	—
321	C₂²×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
322	C₂²×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
323	C₂²×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
324	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
325	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
326	C₂²×C₈	D₄×C₁₂₇	1	4	yes	—
327	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
328	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
329	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
330	#117	—	2	1	no	—
331	#117	—	2	1	no	—
332	#117	C₂×C₄×C₁₂₇	1	2	yes	—
333	#117	C₂³×C₁₂₇	1	2	yes	—
334	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
335	C₂×C₂²⋊C₄	—	2	1	no	—
336	C₂×C₄²	—	2	1	no	—
337	C₂×C₄²	—	2	2	no	—
338	C₂×C₄²	—	2	1	no	—
339	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
340	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
341	#104	C₂×C₄×C₁₂₇	1	16	yes	complemented
342	#103	C₂×C₄×C₁₂₇	1	8	yes	complemented
343	#102	C₂×C₄×C₁₂₇	1	8	yes	complemented
344	#97	—	2	4	no	—
345	C₄×C₈	—	2	8	no	—
346	C₄×C₈	—	2	4	no	—
347	C₂⁴	—	2	1	no	—
348	C₂⁴	#41×C₁₂₇	1	1	yes	characteristic
349	#41	—	2	4	no	—
350	#41	—	2	4	no	—
351	#41	—	2	4	no	—
352	#41	—	2	4	no	—
353	#41	—	2	4	no	—
354	#41	—	2	2	no	—
355	#41	—	2	2	no	—
356	#41	C₄²×C₁₂₇	1	8	yes	—
357	#41	C₂²×C₄×C₁₂₇	1	4	yes	—
358	#41	C₂²×C₄×C₁₂₇	1	4	yes	—
359	C₂×Q₈	—	2	1	no	—
360	C₂×Q₈	—	2	2	no	—
361	C₂×Q₈	C₄²×C₁₂₇	1	4	yes	—
362	C₂×Q₈	C₂²×C₄×C₁₂₇	1	2	yes	—
363	C₂×D₄	—	2	4	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	—	2	4	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	2	1	no	—
369	C₂×D₄	C₄²×C₁₂₇	1	4	yes	—
370	C₂×D₄	C₂²×C₄×C₁₂₇	1	2	yes	—
371	C₂²×C₄	—	2	8	no	—
372	C₂²×C₄	—	2	4	no	—
373	C₂²×C₄	—	2	1	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	4	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	4	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
381	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
382	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
383	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
384	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
385	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
386	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
387	C₂²×C₄	#41×C₁₂₇	1	1	yes	characteristic
388	C₂²×C₄	#41×C₁₂₇	1	2	yes	—
389	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
390	M₁₆	—	2	4	no	—
391	M₁₆	—	2	4	no	—
392	M₁₆	—	2	2	no	—
393	M₁₆	—	2	2	no	—
394	M₁₆	—	2	16	no	—
395	M₁₆	C₂²×C₄×C₁₂₇	1	4	yes	—
396	M₁₆	C₂²×C₄×C₁₂₇	1	4	yes	—
397	C₂×C₈	—	2	4	no	—
398	C₂×C₈	—	2	4	no	—
399	C₂×C₈	—	2	4	no	—
400	C₂×C₈	—	2	4	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	2	no	—
404	C₂×C₈	—	2	2	no	—
405	C₂×C₈	—	2	16	no	—
406	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	4	yes	—
407	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	4	yes	—
408	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	8	yes	—
409	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
410	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
411	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
412	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
413	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
416	C₂²⋊C₄	—	2	2	no	—
417	C₂²⋊C₄	—	2	2	no	—
418	C₄²	—	2	4	no	—
419	C₄²	—	2	2	no	—
420	C₄²	—	2	2	no	—
421	C₄²	—	2	2	no	—
422	C₄²	C₂²×C₄×C₁₂₇	1	4	yes	—
423	C₂³	—	2	4	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	2	4	no	—
426	C₂³	—	4	4	no	—
427	C₂³	#117×C₁₂₇	1	2	yes	—
428	C₂³	C₄×D₄×C₁₂₇	1	1	yes	characteristic
429	C₂³	C₄×D₄×C₁₂₇	1	2	yes	—
430	C₂³	C₄×D₄×C₁₂₇	1	1	yes	characteristic
431	C₂³	C₂×#41×C₁₂₇	1	1	yes	characteristic
432	Q₈	—	2	2	no	—
433	Q₈	—	2	4	no	—
434	Q₈	C₂×C₄²×C₁₂₇	1	4	yes	—
435	D₄	—	2	4	no	—
436	D₄	—	2	4	no	—
437	D₄	—	2	8	no	—
438	D₄	—	2	2	no	—
439	D₄	—	2	4	no	—
440	D₄	C₂×C₄²×C₁₂₇	1	4	yes	—
441	C₂×C₄	—	2	8	no	—
442	C₂×C₄	—	2	16	no	—
443	C₂×C₄	—	2	4	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	4	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	8	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	4	4	no	—
451	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
452	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
453	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
454	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
455	C₂×C₄	#117×C₁₂₇	1	2	yes	—
456	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
457	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
458	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
459	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
460	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
461	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
462	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
463	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
464	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
465	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
466	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
467	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
468	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
469	C₈	—	2	4	no	—
470	C₈	—	2	4	no	—
471	C₈	—	2	4	no	—
472	C₈	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
473	C₈	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	2	4	no	—
477	C₂²	—	2	4	no	—
478	C₂²	—	2	4	no	—
479	C₂²	—	4	4	no	—
480	C₂²	—	4	8	no	—
481	C₂²	C₄×C₂²⋊C₄×C₁₂₇	1	2	yes	—
482	C₂²	C₄×C₂²⋊C₄×C₁₂₇	1	2	yes	—
483	C₂²	#443×C₁₂₇	1	4	yes	—
484	C₂²	C₂×#117×C₁₂₇	1	1	yes	characteristic
485	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
486	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
487	C₄	—	2	8	no	—
488	C₄	—	2	4	no	—
489	C₄	—	4	4	no	—
490	C₄	C₄×C₂²⋊C₄×C₁₂₇	1	2	yes	—
491	C₄	C₄×C₂²⋊C₄×C₁₂₇	1	2	yes	—
492	C₄	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
493	C₄	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
494	C₄	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
495	C₄	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
496	C₂	—	2	1	no	—
497	C₂	—	2	1	no	—
498	C₂	—	2	2	no	—
499	C₂	—	4	4	no	—
500	C₂	C₁₂₇×#1764	1	4	yes	direct product factor
501	C₂	C₂×C₄×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
502	C₂	C₂×#443×C₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₁₂₇	D₄
commutator	C₄	C₂²×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂³×C₁₂₇	C₄×D₄
prime radical	C₂²×#41×C₁₂₇	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₁₂₇ ◁ C₂³×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×#405×C₁₂₇	C₂	semidirect
5. C₂²×C₄×C₈×C₁₂₇	C₂	semidirect
6. C₁₂₇×#1764	C₂	direct (N × H)
7. C₂×#341×C₁₂₇	C₂	semidirect
8. C₂×#339×C₁₂₇	C₂	semidirect
12. C₂×#131×C₁₂₇	C₄	semidirect
22. C₂×#104×C₁₂₇	C₄	semidirect
24. #416×C₁₂₇	C₄	semidirect
26. C₂×#103×C₁₂₇	C₄	semidirect
27. C₂×#102×C₁₂₇	C₄	semidirect
29. #405×C₁₂₇	C₂²	semidirect
33. C₂×C₄×C₈×C₁₂₇	C₂²	semidirect
34. #341×C₁₂₇	C₂²	semidirect
35. #339×C₁₂₇	C₂²	semidirect
52. #131×C₁₂₇	C₂×C₄	semidirect
89. #104×C₁₂₇	C₂×C₄	semidirect
90. #103×C₁₂₇	C₂×C₄	semidirect
91. #102×C₁₂₇	C₂×C₄	semidirect
244. C₂×#1764	C₁₂₇	direct (N × H)
249. C₂×C₁₂₇	#1764	direct (N × H)
255. C₂×#405	C₂×C₁₂₇	semidirect
256. C₂²×C₄×C₈	C₂×C₁₂₇	semidirect
257. #1764	C₂×C₁₂₇	direct (N × H)
258. C₂×#341	C₂×C₁₂₇	semidirect
259. C₂×#339	C₂×C₁₂₇	semidirect
260. C₁₂₇	C₂×#1764	direct (N × H)
264. C₂×#131	C₄×C₁₂₇	semidirect
274. C₂×#104	C₄×C₁₂₇	semidirect
276. #416	C₄×C₁₂₇	semidirect
278. C₂×#103	C₄×C₁₂₇	semidirect
279. C₂×#102	C₄×C₁₂₇	semidirect
281. #405	C₂²×C₁₂₇	semidirect
285. C₂×C₄×C₈	C₂²×C₁₂₇	semidirect
286. #341	C₂²×C₁₂₇	semidirect
287. #339	C₂²×C₁₂₇	semidirect
304. #131	C₂×C₄×C₁₂₇	semidirect
341. #104	C₂×C₄×C₁₂₇	semidirect
342. #103	C₂×C₄×C₁₂₇	semidirect
343. #102	C₂×C₄×C₁₂₇	semidirect
500. C₂	C₁₂₇×#1764	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#131×C₁₂₇	2
C₂²×#104×C₁₂₇	2
C₂×#416×C₁₂₇	2
C₂×#405×C₁₂₇	2
C₂²×C₄×C₈×C₁₂₇	2
C₁₂₇×#1764	2
C₂×#341×C₁₂₇	2
C₂×#339×C₁₂₇	2
C₂×#1764	127

Subgroups — C2×C127×#1764 #44584091

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×#1764 #44584091