Subgroups — C₁₂₇×#20678 #44584351

Order 32512.

1330 conjugacy classes of subgroups.

The subgroup lattice has 486 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×#108×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#427×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₂₇×#2896	C₂	1	2	yes	maximal
5	C₁₂₇×#1816	C₂	1	4	yes	maximal, complemented
6	C₁₂₇×#1815	C₂	1	4	yes	maximal, complemented
7	C₁₂₇×#1742	C₂	1	2	yes	maximal, complemented
8	C₂²×#41×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂×#131×C₁₂₇	—	2	4	no	—
10	C₂×#131×C₁₂₇	—	2	1	no	—
11	C₂×#131×C₁₂₇	—	2	1	no	—
12	C₂×#131×C₁₂₇	C₄	1	8	yes	complemented
13	C₂×#131×C₁₂₇	C₂²	1	2	yes	—
14	C₂×#131×C₁₂₇	C₂²	1	2	yes	—
15	C₂²×M₁₆×C₁₂₇	—	2	1	no	—
16	C₂²×M₁₆×C₁₂₇	C₂²	1	2	yes	—
17	C₂²×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₁₂₇	—	2	1	no	—
19	C₂³×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂×#117×C₁₂₇	C₂²	1	2	yes	—
21	C₂×#108×C₁₂₇	—	2	4	no	—
22	C₂×#108×C₁₂₇	C₂²	1	2	yes	complemented
23	C₂×#108×C₁₂₇	C₂²	1	2	yes	complemented
24	#427×C₁₂₇	—	2	1	no	—
25	#427×C₁₂₇	—	2	1	no	—
26	#427×C₁₂₇	—	2	1	no	—
27	#427×C₁₂₇	—	2	1	no	—
28	C₂×#107×C₁₂₇	C₂²	1	2	yes	complemented
29	C₂×#106×C₁₂₇	C₂²	1	2	yes	complemented
30	#419×C₁₂₇	—	2	2	no	—
31	#418×C₁₂₇	—	2	2	no	—
32	#417×C₁₂₇	C₂²	1	8	yes	—
33	#416×C₁₂₇	—	2	4	no	—
34	#343×C₁₂₇	C₂²	1	8	yes	complemented
35	C₂×#41×C₁₂₇	—	2	2	no	—
36	C₂×#41×C₁₂₇	—	2	1	no	—
37	C₂×#41×C₁₂₇	—	2	1	no	—
38	C₂×#41×C₁₂₇	—	2	2	no	—
39	C₂×#41×C₁₂₇	C₂×C₄	1	4	yes	—
40	C₂×#41×C₁₂₇	D₄	1	2	yes	—
41	C₂×#41×C₁₂₇	Q₈	1	4	yes	—
42	C₂×#41×C₁₂₇	C₂³	1	2	yes	—
43	C₂²×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
44	C₂²×D₄×C₁₂₇	—	2	1	no	—
45	C₂²×D₄×C₁₂₇	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₁₂₇	—	2	1	no	—
47	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
48	#131×C₁₂₇	—	2	8	no	—
49	#131×C₁₂₇	—	2	8	no	—
50	#131×C₁₂₇	—	2	4	no	—
51	#131×C₁₂₇	—	2	4	no	—
52	#131×C₁₂₇	C₂×C₄	1	8	yes	—
53	#131×C₁₂₇	C₂×C₄	1	8	yes	—
54	C₂×M₁₆×C₁₂₇	—	2	2	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	2	no	—
59	C₂×M₁₆×C₁₂₇	—	2	2	no	—
60	C₂×M₁₆×C₁₂₇	—	2	4	no	—
61	C₂×M₁₆×C₁₂₇	—	2	4	no	—
62	C₂×M₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
63	C₂×M₁₆×C₁₂₇	C₂³	1	4	yes	—
64	C₂×M₁₆×C₁₂₇	C₂³	1	2	yes	—
65	C₂²×C₈×C₁₂₇	—	2	1	no	—
66	C₂²×C₈×C₁₂₇	—	2	8	no	—
67	C₂²×C₈×C₁₂₇	—	2	2	no	—
68	C₂²×C₈×C₁₂₇	—	2	2	no	—
69	C₂²×C₈×C₁₂₇	—	2	2	no	—
70	C₂²×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
71	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
72	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
73	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
74	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
75	C₂²×C₈×C₁₂₇	D₄	1	4	yes	—
76	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
78	#117×C₁₂₇	—	2	2	no	—
79	#117×C₁₂₇	—	2	2	no	—
80	#117×C₁₂₇	—	2	2	no	—
81	#117×C₁₂₇	—	2	2	no	—
82	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
83	C₂×C₂²⋊C₄×C₁₂₇	—	2	2	no	—
84	C₂×C₄²×C₁₂₇	—	2	2	no	—
85	#108×C₁₂₇	—	2	4	no	—
86	#108×C₁₂₇	—	2	4	no	—
87	#107×C₁₂₇	—	2	2	no	—
88	#107×C₁₂₇	—	2	2	no	—
89	#106×C₁₂₇	—	2	2	no	—
90	#106×C₁₂₇	—	2	2	no	—
91	#103×C₁₂₇	—	2	8	no	—
92	#102×C₁₂₇	—	2	8	no	—
93	C₂⁴×C₁₂₇	—	2	1	no	—
94	C₂⁴×C₁₂₇	#41	1	1	yes	characteristic
95	#41×C₁₂₇	—	2	4	no	—
96	#41×C₁₂₇	—	2	4	no	—
97	#41×C₁₂₇	—	2	4	no	—
98	#41×C₁₂₇	—	2	4	no	—
99	#41×C₁₂₇	—	2	4	no	—
100	#41×C₁₂₇	—	2	2	no	—
101	#41×C₁₂₇	—	2	2	no	—
102	#41×C₁₂₇	C₄⋊C₄	1	8	yes	—
103	#41×C₁₂₇	C₂²×C₄	1	4	yes	—
104	#41×C₁₂₇	C₂×Q₈	1	4	yes	—
105	C₂×Q₈×C₁₂₇	—	2	1	no	—
106	C₂×Q₈×C₁₂₇	—	2	2	no	—
107	C₂×Q₈×C₁₂₇	C₄⋊C₄	1	4	yes	—
108	C₂×Q₈×C₁₂₇	C₂×D₄	1	2	yes	—
109	C₂×D₄×C₁₂₇	—	2	4	no	—
110	C₂×D₄×C₁₂₇	—	2	2	no	—
111	C₂×D₄×C₁₂₇	—	2	4	no	—
112	C₂×D₄×C₁₂₇	—	2	2	no	—
113	C₂×D₄×C₁₂₇	—	2	2	no	—
114	C₂×D₄×C₁₂₇	—	2	1	no	—
115	C₂×D₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
116	C₂×D₄×C₁₂₇	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₁₂₇	—	2	1	no	—
118	C₂²×C₄×C₁₂₇	—	2	4	no	—
119	C₂²×C₄×C₁₂₇	—	2	2	no	—
120	C₂²×C₄×C₁₂₇	—	2	4	no	—
121	C₂²×C₄×C₁₂₇	—	2	2	no	—
122	C₂²×C₄×C₁₂₇	—	2	4	no	—
123	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
124	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
130	C₂²×C₄×C₁₂₇	#41	1	1	yes	characteristic
131	C₂²×C₄×C₁₂₇	#41	1	2	yes	—
132	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
133	M₁₆×C₁₂₇	—	2	4	no	—
134	M₁₆×C₁₂₇	—	2	2	no	—
135	M₁₆×C₁₂₇	—	2	8	no	—
136	M₁₆×C₁₂₇	—	2	8	no	—
137	M₁₆×C₁₂₇	—	2	4	no	—
138	M₁₆×C₁₂₇	—	2	4	no	—
139	M₁₆×C₁₂₇	—	4	4	no	—
140	M₁₆×C₁₂₇	C₂²×C₄	1	4	yes	—
141	C₂×C₈×C₁₂₇	—	2	4	no	—
142	C₂×C₈×C₁₂₇	—	2	2	no	—
143	C₂×C₈×C₁₂₇	—	2	8	no	—
144	C₂×C₈×C₁₂₇	—	2	8	no	—
145	C₂×C₈×C₁₂₇	—	2	4	no	—
146	C₂×C₈×C₁₂₇	—	4	4	no	—
147	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	4	yes	—
148	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	4	yes	—
149	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	4	yes	—
150	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	4	yes	—
151	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
152	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
153	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
154	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
155	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
156	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
157	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
158	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
159	C₄⋊C₄×C₁₂₇	—	2	4	no	—
160	C₄⋊C₄×C₁₂₇	—	2	4	no	—
161	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
162	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
163	C₄²×C₁₂₇	—	4	2	no	—
164	C₄²×C₁₂₇	—	4	2	no	—
165	C₂³×C₁₂₇	—	2	2	no	—
166	C₂³×C₁₂₇	—	2	4	no	—
167	C₂³×C₁₂₇	—	4	2	no	—
168	C₂³×C₁₂₇	—	4	4	no	—
169	C₂³×C₁₂₇	#117	1	2	yes	—
170	C₂³×C₁₂₇	#121	1	1	yes	characteristic
171	C₂³×C₁₂₇	#121	1	1	yes	characteristic
172	C₂³×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
173	C₂³×C₁₂₇	C₂×#41	1	1	yes	characteristic
174	Q₈×C₁₂₇	—	2	2	no	—
175	Q₈×C₁₂₇	—	2	4	no	—
176	Q₈×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
177	D₄×C₁₂₇	—	2	4	no	—
178	D₄×C₁₂₇	—	2	4	no	—
179	D₄×C₁₂₇	—	2	8	no	—
180	D₄×C₁₂₇	—	2	2	no	—
181	D₄×C₁₂₇	—	2	4	no	—
182	D₄×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
183	C₂×C₄×C₁₂₇	—	2	2	no	—
184	C₂×C₄×C₁₂₇	—	2	4	no	—
185	C₂×C₄×C₁₂₇	—	2	4	no	—
186	C₂×C₄×C₁₂₇	—	2	4	no	—
187	C₂×C₄×C₁₂₇	—	2	8	no	—
188	C₂×C₄×C₁₂₇	—	4	4	no	—
189	C₂×C₄×C₁₂₇	—	8	4	no	—
190	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
191	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
192	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
193	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
194	C₂×C₄×C₁₂₇	#117	1	2	yes	—
195	C₂×C₄×C₁₂₇	#121	1	2	yes	—
196	C₂×C₄×C₁₂₇	#121	1	2	yes	—
197	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
198	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
199	C₂×C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
200	C₂×C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
201	C₂×C₄×C₁₂₇	C₂²⋊Q₈	1	2	yes	—
202	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
203	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
204	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
205	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
206	C₂×C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
207	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
208	C₈×C₁₂₇	—	2	4	no	—
209	C₈×C₁₂₇	—	4	4	no	—
210	C₈×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
211	C₈×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
212	C₈×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
213	C₈×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
214	C₂²×C₁₂₇	—	2	2	no	—
215	C₂²×C₁₂₇	—	2	2	no	—
216	C₂²×C₁₂₇	—	2	2	no	—
217	C₂²×C₁₂₇	—	2	4	no	—
218	C₂²×C₁₂₇	—	4	2	no	—
219	C₂²×C₁₂₇	—	4	2	no	—
220	C₂²×C₁₂₇	—	4	8	no	—
221	C₂²×C₁₂₇	—	4	4	no	—
222	C₂²×C₁₂₇	#380	1	2	yes	—
223	C₂²×C₁₂₇	#380	1	2	yes	—
224	C₂²×C₁₂₇	C₂×#117	1	1	yes	characteristic
225	C₂²×C₁₂₇	C₂×#121	1	1	yes	characteristic
226	C₂²×C₁₂₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
227	C₄×C₁₂₇	—	2	4	no	—
228	C₄×C₁₂₇	—	8	4	no	—
229	C₄×C₁₂₇	#380	1	2	yes	—
230	C₄×C₁₂₇	#380	1	2	yes	—
231	C₄×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
232	C₄×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
233	C₄×C₁₂₇	C₂×#121	1	1	yes	characteristic
234	C₄×C₁₂₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
235	#20678	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
236	C₂×C₁₂₇	—	2	2	no	—
237	C₂×C₁₂₇	—	2	1	no	—
238	C₂×C₁₂₇	—	2	1	no	—
239	C₂×C₁₂₇	—	2	2	no	—
240	C₂×C₁₂₇	—	4	4	no	—
241	C₂×C₁₂₇	C₂×#380	1	1	yes	characteristic
242	C₂×C₁₂₇	#3067	1	2	yes	—
243	C₂²×#131	C₂×C₁₂₇	1	1	yes	characteristic
244	C₂²×#108	C₂×C₁₂₇	1	1	yes	characteristic, complemented
245	C₂×#427	C₂×C₁₂₇	1	1	yes	characteristic, complemented
246	#2896	C₂×C₁₂₇	1	2	yes	—
247	#1816	C₂×C₁₂₇	1	4	yes	complemented
248	#1815	C₂×C₁₂₇	1	4	yes	complemented
249	#1742	C₂×C₁₂₇	1	2	yes	complemented
250	C₁₂₇	#20678	1	1	yes	characteristic, direct product factor, Sylow
251	C₂²×#41	C₂²×C₁₂₇	1	1	yes	characteristic
252	C₂×#131	—	2	4	no	—
253	C₂×#131	—	2	1	no	—
254	C₂×#131	—	2	1	no	—
255	C₂×#131	C₄×C₁₂₇	1	8	yes	complemented
256	C₂×#131	C₂²×C₁₂₇	1	2	yes	—
257	C₂×#131	C₂²×C₁₂₇	1	2	yes	—
258	C₂²×M₁₆	—	2	1	no	—
259	C₂²×M₁₆	C₂²×C₁₂₇	1	2	yes	—
260	C₂²×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
261	C₂³×C₈	—	2	1	no	—
262	C₂³×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
263	C₂×#117	C₂²×C₁₂₇	1	2	yes	—
264	C₂×#108	—	2	4	no	—
265	C₂×#108	C₂²×C₁₂₇	1	2	yes	complemented
266	C₂×#108	C₂²×C₁₂₇	1	2	yes	complemented
267	#427	—	2	1	no	—
268	#427	—	2	1	no	—
269	#427	—	2	1	no	—
270	#427	—	2	1	no	—
271	C₂×#107	C₂²×C₁₂₇	1	2	yes	complemented
272	C₂×#106	C₂²×C₁₂₇	1	2	yes	complemented
273	#419	—	2	2	no	—
274	#418	—	2	2	no	—
275	#417	C₂²×C₁₂₇	1	8	yes	—
276	#416	—	2	4	no	—
277	#343	C₂²×C₁₂₇	1	8	yes	complemented
278	C₂×#41	—	2	2	no	—
279	C₂×#41	—	2	1	no	—
280	C₂×#41	—	2	1	no	—
281	C₂×#41	—	2	2	no	—
282	C₂×#41	C₂×C₄×C₁₂₇	1	4	yes	—
283	C₂×#41	D₄×C₁₂₇	1	2	yes	—
284	C₂×#41	Q₈×C₁₂₇	1	4	yes	—
285	C₂×#41	C₂³×C₁₂₇	1	2	yes	—
286	C₂²×Q₈	D₄×C₁₂₇	1	1	yes	characteristic
287	C₂²×D₄	—	2	1	no	—
288	C₂²×D₄	D₄×C₁₂₇	1	1	yes	characteristic
289	C₂³×C₄	—	2	1	no	—
290	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
291	#131	—	2	8	no	—
292	#131	—	2	8	no	—
293	#131	—	2	4	no	—
294	#131	—	2	4	no	—
295	#131	C₂×C₄×C₁₂₇	1	8	yes	—
296	#131	C₂×C₄×C₁₂₇	1	8	yes	—
297	C₂×M₁₆	—	2	2	no	—
298	C₂×M₁₆	—	2	1	no	—
299	C₂×M₁₆	—	2	2	no	—
300	C₂×M₁₆	—	2	8	no	—
301	C₂×M₁₆	—	2	2	no	—
302	C₂×M₁₆	—	2	2	no	—
303	C₂×M₁₆	—	2	4	no	—
304	C₂×M₁₆	—	2	4	no	—
305	C₂×M₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
306	C₂×M₁₆	C₂³×C₁₂₇	1	4	yes	—
307	C₂×M₁₆	C₂³×C₁₂₇	1	2	yes	—
308	C₂²×C₈	—	2	1	no	—
309	C₂²×C₈	—	2	8	no	—
310	C₂²×C₈	—	2	2	no	—
311	C₂²×C₈	—	2	2	no	—
312	C₂²×C₈	—	2	2	no	—
313	C₂²×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
314	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
315	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
316	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
317	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
318	C₂²×C₈	D₄×C₁₂₇	1	4	yes	—
319	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
320	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
321	#117	—	2	2	no	—
322	#117	—	2	2	no	—
323	#117	—	2	2	no	—
324	#117	—	2	2	no	—
325	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	4	yes	—
326	C₂×C₂²⋊C₄	—	2	2	no	—
327	C₂×C₄²	—	2	2	no	—
328	#108	—	2	4	no	—
329	#108	—	2	4	no	—
330	#107	—	2	2	no	—
331	#107	—	2	2	no	—
332	#106	—	2	2	no	—
333	#106	—	2	2	no	—
334	#103	—	2	8	no	—
335	#102	—	2	8	no	—
336	C₂⁴	—	2	1	no	—
337	C₂⁴	#41×C₁₂₇	1	1	yes	characteristic
338	#41	—	2	4	no	—
339	#41	—	2	4	no	—
340	#41	—	2	4	no	—
341	#41	—	2	4	no	—
342	#41	—	2	4	no	—
343	#41	—	2	2	no	—
344	#41	—	2	2	no	—
345	#41	C₄⋊C₄×C₁₂₇	1	8	yes	—
346	#41	C₂²×C₄×C₁₂₇	1	4	yes	—
347	#41	C₂×Q₈×C₁₂₇	1	4	yes	—
348	C₂×Q₈	—	2	1	no	—
349	C₂×Q₈	—	2	2	no	—
350	C₂×Q₈	C₄⋊C₄×C₁₂₇	1	4	yes	—
351	C₂×Q₈	C₂×D₄×C₁₂₇	1	2	yes	—
352	C₂×D₄	—	2	4	no	—
353	C₂×D₄	—	2	2	no	—
354	C₂×D₄	—	2	4	no	—
355	C₂×D₄	—	2	2	no	—
356	C₂×D₄	—	2	2	no	—
357	C₂×D₄	—	2	1	no	—
358	C₂×D₄	C₄⋊C₄×C₁₂₇	1	4	yes	—
359	C₂×D₄	C₂×D₄×C₁₂₇	1	2	yes	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	4	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	2	4	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	2	4	no	—
366	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
367	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
368	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
369	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
370	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
371	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
372	C₂²×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
373	C₂²×C₄	#41×C₁₂₇	1	1	yes	characteristic
374	C₂²×C₄	#41×C₁₂₇	1	2	yes	—
375	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
376	M₁₆	—	2	4	no	—
377	M₁₆	—	2	2	no	—
378	M₁₆	—	2	8	no	—
379	M₁₆	—	2	8	no	—
380	M₁₆	—	2	4	no	—
381	M₁₆	—	2	4	no	—
382	M₁₆	—	4	4	no	—
383	M₁₆	C₂²×C₄×C₁₂₇	1	4	yes	—
384	C₂×C₈	—	2	4	no	—
385	C₂×C₈	—	2	2	no	—
386	C₂×C₈	—	2	8	no	—
387	C₂×C₈	—	2	8	no	—
388	C₂×C₈	—	2	4	no	—
389	C₂×C₈	—	4	4	no	—
390	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	4	yes	—
391	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	4	yes	—
392	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	4	yes	—
393	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	4	yes	—
394	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
395	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
396	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
397	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
398	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
399	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
400	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
401	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
402	C₄⋊C₄	—	2	4	no	—
403	C₄⋊C₄	—	2	4	no	—
404	C₂²⋊C₄	—	4	2	no	—
405	C₂²⋊C₄	—	4	2	no	—
406	C₄²	—	4	2	no	—
407	C₄²	—	4	2	no	—
408	C₂³	—	2	2	no	—
409	C₂³	—	2	4	no	—
410	C₂³	—	4	2	no	—
411	C₂³	—	4	4	no	—
412	C₂³	#117×C₁₂₇	1	2	yes	—
413	C₂³	#121×C₁₂₇	1	1	yes	characteristic
414	C₂³	#121×C₁₂₇	1	1	yes	characteristic
415	C₂³	C₂²⋊Q₈×C₁₂₇	1	2	yes	—
416	C₂³	C₂×#41×C₁₂₇	1	1	yes	characteristic
417	Q₈	—	2	2	no	—
418	Q₈	—	2	4	no	—
419	Q₈	C₂×C₄⋊C₄×C₁₂₇	1	4	yes	—
420	D₄	—	2	4	no	—
421	D₄	—	2	4	no	—
422	D₄	—	2	8	no	—
423	D₄	—	2	2	no	—
424	D₄	—	2	4	no	—
425	D₄	C₂×C₄⋊C₄×C₁₂₇	1	4	yes	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	4	no	—
428	C₂×C₄	—	2	4	no	—
429	C₂×C₄	—	2	4	no	—
430	C₂×C₄	—	2	8	no	—
431	C₂×C₄	—	4	4	no	—
432	C₂×C₄	—	8	4	no	—
433	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
434	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
435	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
436	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
437	C₂×C₄	#117×C₁₂₇	1	2	yes	—
438	C₂×C₄	#121×C₁₂₇	1	2	yes	—
439	C₂×C₄	#121×C₁₂₇	1	2	yes	—
440	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
441	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
442	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	2	yes	—
443	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	2	yes	—
444	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	2	yes	—
445	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
446	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
447	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
448	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
449	C₂×C₄	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
450	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
451	C₈	—	2	4	no	—
452	C₈	—	4	4	no	—
453	C₈	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
454	C₈	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
455	C₈	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
456	C₈	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
457	C₂²	—	2	2	no	—
458	C₂²	—	2	2	no	—
459	C₂²	—	2	2	no	—
460	C₂²	—	2	4	no	—
461	C₂²	—	4	2	no	—
462	C₂²	—	4	2	no	—
463	C₂²	—	4	8	no	—
464	C₂²	—	4	4	no	—
465	C₂²	#380×C₁₂₇	1	2	yes	—
466	C₂²	#380×C₁₂₇	1	2	yes	—
467	C₂²	C₂×#117×C₁₂₇	1	1	yes	characteristic
468	C₂²	C₂×#121×C₁₂₇	1	1	yes	characteristic
469	C₂²	C₂×C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
470	C₄	—	2	4	no	—
471	C₄	—	8	4	no	—
472	C₄	#380×C₁₂₇	1	2	yes	—
473	C₄	#380×C₁₂₇	1	2	yes	—
474	C₄	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
475	C₄	C₂²×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
476	C₄	C₂×#121×C₁₂₇	1	1	yes	characteristic
477	C₄	C₂×C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
478	C₂	—	2	2	no	—
479	C₂	—	2	1	no	—
480	C₂	—	2	1	no	—
481	C₂	—	2	2	no	—
482	C₂	—	4	4	no	—
483	C₂	C₂×#380×C₁₂₇	1	1	yes	characteristic
484	C₂	C₁₂₇×#3067	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#121
commutator	C₂×C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂×#121
prime radical	C₂²×#41×C₁₂₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂³×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#108×C₁₂₇	C₂	semidirect
3. C₂×#427×C₁₂₇	C₂	semidirect
5. C₁₂₇×#1816	C₂	semidirect
6. C₁₂₇×#1815	C₂	semidirect
7. C₁₂₇×#1742	C₂	semidirect
12. C₂×#131×C₁₂₇	C₄	semidirect
22. C₂×#108×C₁₂₇	C₂²	semidirect
23. C₂×#108×C₁₂₇	C₂²	semidirect
28. C₂×#107×C₁₂₇	C₂²	semidirect
29. C₂×#106×C₁₂₇	C₂²	semidirect
34. #343×C₁₂₇	C₂²	semidirect
235. #20678	C₁₂₇	direct (N × H)
244. C₂²×#108	C₂×C₁₂₇	semidirect
245. C₂×#427	C₂×C₁₂₇	semidirect
247. #1816	C₂×C₁₂₇	semidirect
248. #1815	C₂×C₁₂₇	semidirect
249. #1742	C₂×C₁₂₇	semidirect
250. C₁₂₇	#20678	direct (N × H)
255. C₂×#131	C₄×C₁₂₇	semidirect
265. C₂×#108	C₂²×C₁₂₇	semidirect
266. C₂×#108	C₂²×C₁₂₇	semidirect
271. C₂×#107	C₂²×C₁₂₇	semidirect
272. C₂×#106	C₂²×C₁₂₇	semidirect
277. #343	C₂²×C₁₂₇	semidirect

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×#108×C₁₂₇	2
C₂×#427×C₁₂₇	2
C₁₂₇×#2896	2
C₁₂₇×#1816	2
C₁₂₇×#1815	2
C₁₂₇×#1742	2
#20678	127

Subgroups — C127×#20678 #44584351

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#20678 #44584351