Subgroups — C₁₂₇×#19129 #44582802

Order 32512.

620 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₂₇×#3107	C₂	1	2	yes	maximal
3	C₈×#41×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1680	C₂	1	2	yes	maximal, complemented
5	C₁₂₇×#1679	C₂	1	2	yes	maximal, complemented
6	C₁₂₇×#1674	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₂₇×#1673	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₂₇×#1670	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₂₇×#1669	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₂₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₂₇×#1544	C₂	1	2	yes	maximal, complemented
12	#533×C₁₂₇	C₂²	1	2	yes	—
13	C₄×#41×C₁₂₇	C₂²	1	1	yes	characteristic
14	#500×C₁₂₇	C₂²	1	1	yes	characteristic
15	#494×C₁₂₇	C₂²	1	1	yes	characteristic
16	#487×C₁₂₇	C₂²	1	2	yes	—
17	#485×C₁₂₇	—	2	2	no	—
18	#475×C₁₂₇	C₂²	1	2	yes	—
19	#474×C₁₂₇	C₂²	1	2	yes	—
20	#467×C₁₂₇	—	2	1	no	—
21	#467×C₁₂₇	—	2	1	no	—
22	#462×C₁₂₇	C₂²	1	2	yes	—
23	#461×C₁₂₇	C₂²	1	2	yes	—
24	#452×C₁₂₇	—	2	2	no	—
25	C₈×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
26	C₈×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
27	C₄×Q₁₆×C₁₂₇	—	2	2	no	—
28	C₄×Q₁₆×C₁₂₇	—	2	1	no	—
29	C₄×Q₁₆×C₁₂₇	—	2	1	no	—
30	C₄×SD₁₆×C₁₂₇	—	2	2	no	—
31	C₈×D₄×C₁₂₇	—	2	1	no	—
32	C₈×D₄×C₁₂₇	—	2	1	no	—
33	C₈×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
34	C₈×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
35	#431×C₁₂₇	—	2	1	no	—
36	#431×C₁₂₇	C₂²	1	1	yes	characteristic
37	#431×C₁₂₇	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₁₂₇	—	2	1	no	—
39	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
40	#335×C₁₂₇	C₂²	1	1	yes	characteristic
41	#335×C₁₂₇	C₂²	1	1	yes	characteristic
42	#334×C₁₂₇	C₂²	1	2	yes	—
43	#332×C₁₂₇	C₂²	1	2	yes	—
44	#332×C₁₂₇	C₂²	1	2	yes	—
45	#332×C₁₂₇	C₂²	1	2	yes	—
46	#332×C₁₂₇	C₂²	1	2	yes	—
47	C₂×#41×C₁₂₇	D₄	1	1	yes	characteristic
48	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
49	C₂×Q₁₆×C₁₂₇	—	2	1	no	—
50	C₂×Q₁₆×C₁₂₇	—	2	1	no	—
51	C₂×SD₁₆×C₁₂₇	—	2	2	no	—
52	C₂²×C₈×C₁₂₇	—	2	1	no	—
53	C₂²×C₈×C₁₂₇	—	2	1	no	—
54	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
56	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
57	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
58	#125×C₁₂₇	—	2	2	no	—
59	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
60	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
61	C₄×Q₈×C₁₂₇	—	2	2	no	—
62	C₄×Q₈×C₁₂₇	—	2	2	no	—
63	C₄×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
64	C₄×Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
65	C₄×D₄×C₁₂₇	—	2	1	no	—
66	C₄×D₄×C₁₂₇	—	2	1	no	—
67	C₄×D₄×C₁₂₇	D₄	1	1	yes	characteristic
68	C₄×D₄×C₁₂₇	C₂³	1	1	yes	characteristic
69	#117×C₁₂₇	—	2	1	no	—
70	#117×C₁₂₇	—	2	2	no	—
71	#117×C₁₂₇	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₁₂₇	—	2	1	no	—
73	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
74	#107×C₁₂₇	—	2	1	no	—
75	#107×C₁₂₇	—	2	1	no	—
76	#107×C₁₂₇	—	2	2	no	—
77	#106×C₁₂₇	—	2	2	no	—
78	C₄⋊C₈×C₁₂₇	—	2	1	no	—
79	C₄⋊C₈×C₁₂₇	—	2	1	no	—
80	C₄⋊C₈×C₁₂₇	C₂³	1	1	yes	characteristic
81	C₄⋊C₈×C₁₂₇	C₂³	1	1	yes	characteristic
82	C₄⋊C₈×C₁₂₇	C₂³	1	2	yes	—
83	C₄⋊C₈×C₁₂₇	C₂³	1	2	yes	—
84	#103×C₁₂₇	—	2	2	no	—
85	#103×C₁₂₇	—	2	2	no	—
86	#103×C₁₂₇	—	2	2	no	—
87	#103×C₁₂₇	—	2	2	no	—
88	#103×C₁₂₇	—	2	2	no	—
89	#102×C₁₂₇	—	2	2	no	—
90	C₂²⋊C₈×C₁₂₇	—	2	1	no	—
91	C₂²⋊C₈×C₁₂₇	—	2	2	no	—
92	C₂²⋊C₈×C₁₂₇	—	2	1	no	—
93	C₂²⋊C₈×C₁₂₇	—	2	1	no	—
94	C₄×C₈×C₁₂₇	—	2	1	no	—
95	C₄×C₈×C₁₂₇	—	2	1	no	—
96	C₄×C₈×C₁₂₇	—	2	1	no	—
97	C₄×C₈×C₁₂₇	—	2	2	no	—
98	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
99	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
100	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
102	#41×C₁₂₇	—	2	2	no	—
103	#41×C₁₂₇	—	2	2	no	—
104	C₂×Q₈×C₁₂₇	—	2	2	no	—
105	C₂×Q₈×C₁₂₇	—	2	2	no	—
106	C₂×Q₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
107	C₂×D₄×C₁₂₇	—	2	1	no	—
108	C₂×D₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₁₂₇	—	2	1	no	—
110	C₂²×C₄×C₁₂₇	—	2	1	no	—
111	C₂²×C₄×C₁₂₇	—	2	1	no	—
112	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₁₂₇	#41	1	1	yes	characteristic
115	Q₁₆×C₁₂₇	—	4	2	no	—
116	Q₁₆×C₁₂₇	—	4	4	no	—
117	Q₁₆×C₁₂₇	—	4	2	no	—
118	SD₁₆×C₁₂₇	—	4	4	no	—
119	C₂×C₈×C₁₂₇	—	2	2	no	—
120	C₂×C₈×C₁₂₇	—	2	1	no	—
121	C₂×C₈×C₁₂₇	—	2	1	no	—
122	C₂×C₈×C₁₂₇	—	2	2	no	—
123	C₂×C₈×C₁₂₇	—	2	2	no	—
124	C₂×C₈×C₁₂₇	—	4	1	no	—
125	C₂×C₈×C₁₂₇	—	4	2	no	—
126	C₂×C₈×C₁₂₇	—	4	2	no	—
127	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
131	C₄⋊C₄×C₁₂₇	—	2	1	no	—
132	C₄⋊C₄×C₁₂₇	—	2	1	no	—
133	C₄⋊C₄×C₁₂₇	—	2	2	no	—
134	C₄⋊C₄×C₁₂₇	—	2	2	no	—
135	C₄⋊C₄×C₁₂₇	—	2	2	no	—
136	C₄⋊C₄×C₁₂₇	—	2	2	no	—
137	C₄⋊C₄×C₁₂₇	—	4	2	no	—
138	C₄⋊C₄×C₁₂₇	—	4	2	no	—
139	C₄⋊C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
140	C₄⋊C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
141	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
142	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
143	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
144	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
145	C₄²×C₁₂₇	—	2	1	no	—
146	C₄²×C₁₂₇	—	2	1	no	—
147	C₄²×C₁₂₇	—	2	1	no	—
148	C₄²×C₁₂₇	—	4	2	no	—
149	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
150	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
151	C₄²×C₁₂₇	#41	1	1	yes	characteristic
152	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
153	C₂³×C₁₂₇	—	2	1	no	—
154	C₂³×C₁₂₇	#121	1	1	yes	characteristic
155	Q₈×C₁₂₇	—	2	1	no	—
156	Q₈×C₁₂₇	—	4	2	no	—
157	Q₈×C₁₂₇	—	4	2	no	—
158	Q₈×C₁₂₇	—	4	2	no	—
159	Q₈×C₁₂₇	—	4	2	no	—
160	Q₈×C₁₂₇	#135	1	2	yes	—
161	D₄×C₁₂₇	—	2	4	no	—
162	D₄×C₁₂₇	—	2	1	no	—
163	D₄×C₁₂₇	#135	1	2	yes	—
164	C₂×C₄×C₁₂₇	—	2	4	no	—
165	C₂×C₄×C₁₂₇	—	2	1	no	—
166	C₂×C₄×C₁₂₇	—	2	1	no	—
167	C₂×C₄×C₁₂₇	—	2	1	no	—
168	C₂×C₄×C₁₂₇	—	2	1	no	—
169	C₂×C₄×C₁₂₇	—	2	1	no	—
170	C₂×C₄×C₁₂₇	—	2	2	no	—
171	C₂×C₄×C₁₂₇	—	2	1	no	—
172	C₂×C₄×C₁₂₇	—	2	1	no	—
173	C₂×C₄×C₁₂₇	—	4	1	no	—
174	C₂×C₄×C₁₂₇	—	4	1	no	—
175	C₂×C₄×C₁₂₇	—	4	2	no	—
176	C₂×C₄×C₁₂₇	—	4	2	no	—
177	C₂×C₄×C₁₂₇	—	4	2	no	—
178	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
179	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
180	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
181	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
184	C₈×C₁₂₇	—	2	1	no	—
185	C₈×C₁₂₇	—	2	1	no	—
186	C₈×C₁₂₇	—	2	1	no	—
187	C₈×C₁₂₇	—	4	1	no	—
188	C₈×C₁₂₇	—	4	2	no	—
189	C₈×C₁₂₇	—	4	2	no	—
190	C₈×C₁₂₇	—	4	1	no	—
191	C₈×C₁₂₇	#135	1	2	yes	—
192	C₂²×C₁₂₇	—	2	1	no	—
193	C₂²×C₁₂₇	—	2	1	no	—
194	C₂²×C₁₂₇	—	2	2	no	—
195	C₂²×C₁₂₇	—	2	1	no	—
196	C₂²×C₁₂₇	—	4	1	no	—
197	C₂²×C₁₂₇	—	4	1	no	—
198	C₂²×C₁₂₇	C₂×#121	1	1	yes	characteristic
199	C₄×C₁₂₇	—	2	1	no	—
200	C₄×C₁₂₇	—	2	1	no	—
201	C₄×C₁₂₇	—	2	2	no	—
202	C₄×C₁₂₇	—	2	1	no	—
203	C₄×C₁₂₇	—	4	1	no	—
204	C₄×C₁₂₇	—	4	1	no	—
205	C₄×C₁₂₇	—	8	2	no	—
206	C₄×C₁₂₇	—	8	2	no	—
207	C₄×C₁₂₇	#473	1	2	yes	—
208	C₄×C₁₂₇	C₂×#135	1	1	yes	characteristic
209	C₄×C₁₂₇	C₂×#135	1	1	yes	characteristic
210	C₄×C₁₂₇	C₂×#135	1	1	yes	characteristic
211	C₄×C₁₂₇	#575	1	1	yes	characteristic
212	#19129	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂×C₁₂₇	—	4	1	no	—
214	C₂×C₁₂₇	—	4	2	no	—
215	C₂×C₁₂₇	#3035	1	1	yes	characteristic
216	C₂×C₁₂₇	#3050	1	1	yes	characteristic
217	C₂×C₁₂₇	C₂×#473	1	1	yes	characteristic
218	#3170	C₂×C₁₂₇	1	1	yes	characteristic, complemented
219	#3107	C₂×C₁₂₇	1	2	yes	—
220	C₈×#41	C₂×C₁₂₇	1	1	yes	characteristic
221	#1680	C₂×C₁₂₇	1	2	yes	complemented
222	#1679	C₂×C₁₂₇	1	2	yes	complemented
223	#1674	C₂×C₁₂₇	1	1	yes	characteristic, complemented
224	#1673	C₂×C₁₂₇	1	1	yes	characteristic, complemented
225	#1670	C₂×C₁₂₇	1	1	yes	characteristic, complemented
226	#1669	C₂×C₁₂₇	1	1	yes	characteristic, complemented
227	#1564	C₂×C₁₂₇	1	1	yes	characteristic, complemented
228	#1544	C₂×C₁₂₇	1	2	yes	complemented
229	C₁₂₇	#19129	1	1	yes	characteristic, direct product factor, Sylow
230	#533	C₂²×C₁₂₇	1	2	yes	—
231	C₄×#41	C₂²×C₁₂₇	1	1	yes	characteristic
232	#500	C₂²×C₁₂₇	1	1	yes	characteristic
233	#494	C₂²×C₁₂₇	1	1	yes	characteristic
234	#487	C₂²×C₁₂₇	1	2	yes	—
235	#485	—	2	2	no	—
236	#475	C₂²×C₁₂₇	1	2	yes	—
237	#474	C₂²×C₁₂₇	1	2	yes	—
238	#467	—	2	1	no	—
239	#467	—	2	1	no	—
240	#462	C₂²×C₁₂₇	1	2	yes	—
241	#461	C₂²×C₁₂₇	1	2	yes	—
242	#452	—	2	2	no	—
243	C₈×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
244	C₈×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
245	C₄×Q₁₆	—	2	2	no	—
246	C₄×Q₁₆	—	2	1	no	—
247	C₄×Q₁₆	—	2	1	no	—
248	C₄×SD₁₆	—	2	2	no	—
249	C₈×D₄	—	2	1	no	—
250	C₈×D₄	—	2	1	no	—
251	C₈×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
252	C₈×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
253	#431	—	2	1	no	—
254	#431	C₂²×C₁₂₇	1	1	yes	characteristic
255	#431	C₂²×C₁₂₇	1	2	yes	—
256	C₂×C₄×C₈	—	2	1	no	—
257	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
258	#335	C₂²×C₁₂₇	1	1	yes	characteristic
259	#335	C₂²×C₁₂₇	1	1	yes	characteristic
260	#334	C₂²×C₁₂₇	1	2	yes	—
261	#332	C₂²×C₁₂₇	1	2	yes	—
262	#332	C₂²×C₁₂₇	1	2	yes	—
263	#332	C₂²×C₁₂₇	1	2	yes	—
264	#332	C₂²×C₁₂₇	1	2	yes	—
265	C₂×#41	D₄×C₁₂₇	1	1	yes	characteristic
266	C₂×Q₁₆	—	2	2	no	—
267	C₂×Q₁₆	—	2	1	no	—
268	C₂×Q₁₆	—	2	1	no	—
269	C₂×SD₁₆	—	2	2	no	—
270	C₂²×C₈	—	2	1	no	—
271	C₂²×C₈	—	2	1	no	—
272	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
273	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
274	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
275	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
276	#125	—	2	2	no	—
277	C₂²⋊Q₈	—	2	2	no	—
278	C₂²⋊Q₈	—	2	2	no	—
279	C₄×Q₈	—	2	2	no	—
280	C₄×Q₈	—	2	2	no	—
281	C₄×Q₈	D₄×C₁₂₇	1	1	yes	characteristic
282	C₄×Q₈	C₂³×C₁₂₇	1	1	yes	characteristic
283	C₄×D₄	—	2	1	no	—
284	C₄×D₄	—	2	1	no	—
285	C₄×D₄	D₄×C₁₂₇	1	1	yes	characteristic
286	C₄×D₄	C₂³×C₁₂₇	1	1	yes	characteristic
287	#117	—	2	1	no	—
288	#117	—	2	2	no	—
289	#117	D₄×C₁₂₇	1	1	yes	characteristic
290	C₂×C₄²	—	2	1	no	—
291	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
292	#107	—	2	1	no	—
293	#107	—	2	1	no	—
294	#107	—	2	2	no	—
295	#106	—	2	2	no	—
296	C₄⋊C₈	—	2	1	no	—
297	C₄⋊C₈	—	2	1	no	—
298	C₄⋊C₈	C₂³×C₁₂₇	1	1	yes	characteristic
299	C₄⋊C₈	C₂³×C₁₂₇	1	1	yes	characteristic
300	C₄⋊C₈	C₂³×C₁₂₇	1	2	yes	—
301	C₄⋊C₈	C₂³×C₁₂₇	1	2	yes	—
302	#103	—	2	2	no	—
303	#103	—	2	2	no	—
304	#103	—	2	2	no	—
305	#103	—	2	2	no	—
306	#103	—	2	2	no	—
307	#102	—	2	2	no	—
308	C₂²⋊C₈	—	2	1	no	—
309	C₂²⋊C₈	—	2	2	no	—
310	C₂²⋊C₈	—	2	1	no	—
311	C₂²⋊C₈	—	2	1	no	—
312	C₄×C₈	—	2	1	no	—
313	C₄×C₈	—	2	1	no	—
314	C₄×C₈	—	2	1	no	—
315	C₄×C₈	—	2	2	no	—
316	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
317	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
318	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
319	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
320	#41	—	2	2	no	—
321	#41	—	2	2	no	—
322	C₂×Q₈	—	2	2	no	—
323	C₂×Q₈	—	2	2	no	—
324	C₂×Q₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
325	C₂×D₄	—	2	1	no	—
326	C₂×D₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	1	no	—
330	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
331	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
332	C₂²×C₄	#41×C₁₂₇	1	1	yes	characteristic
333	Q₁₆	—	4	2	no	—
334	Q₁₆	—	4	4	no	—
335	Q₁₆	—	4	2	no	—
336	SD₁₆	—	4	4	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	2	1	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	4	1	no	—
343	C₂×C₈	—	4	2	no	—
344	C₂×C₈	—	4	2	no	—
345	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
346	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
347	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
348	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	4	2	no	—
356	C₄⋊C₄	—	4	2	no	—
357	C₄⋊C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
358	C₄⋊C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
359	C₂²⋊C₄	—	2	1	no	—
360	C₂²⋊C₄	—	2	1	no	—
361	C₂²⋊C₄	—	2	1	no	—
362	C₂²⋊C₄	—	4	2	no	—
363	C₄²	—	2	1	no	—
364	C₄²	—	2	1	no	—
365	C₄²	—	2	1	no	—
366	C₄²	—	4	2	no	—
367	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
368	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
369	C₄²	#41×C₁₂₇	1	1	yes	characteristic
370	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
371	C₂³	—	2	1	no	—
372	C₂³	#121×C₁₂₇	1	1	yes	characteristic
373	Q₈	—	2	1	no	—
374	Q₈	—	4	2	no	—
375	Q₈	—	4	2	no	—
376	Q₈	—	4	2	no	—
377	Q₈	—	4	2	no	—
378	Q₈	#135×C₁₂₇	1	2	yes	—
379	D₄	—	2	4	no	—
380	D₄	—	2	1	no	—
381	D₄	#135×C₁₂₇	1	2	yes	—
382	C₂×C₄	—	2	4	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	1	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
397	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
398	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
399	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
400	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
401	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
402	C₈	—	2	1	no	—
403	C₈	—	2	1	no	—
404	C₈	—	2	1	no	—
405	C₈	—	4	1	no	—
406	C₈	—	4	2	no	—
407	C₈	—	4	2	no	—
408	C₈	—	4	1	no	—
409	C₈	#135×C₁₂₇	1	2	yes	—
410	C₂²	—	2	1	no	—
411	C₂²	—	2	1	no	—
412	C₂²	—	2	2	no	—
413	C₂²	—	2	1	no	—
414	C₂²	—	4	1	no	—
415	C₂²	—	4	1	no	—
416	C₂²	C₂×#121×C₁₂₇	1	1	yes	characteristic
417	C₄	—	2	1	no	—
418	C₄	—	2	1	no	—
419	C₄	—	2	2	no	—
420	C₄	—	2	1	no	—
421	C₄	—	4	1	no	—
422	C₄	—	4	1	no	—
423	C₄	—	8	2	no	—
424	C₄	—	8	2	no	—
425	C₄	#473×C₁₂₇	1	2	yes	—
426	C₄	C₂×#135×C₁₂₇	1	1	yes	characteristic
427	C₄	C₂×#135×C₁₂₇	1	1	yes	characteristic
428	C₄	C₂×#135×C₁₂₇	1	1	yes	characteristic
429	C₄	#575×C₁₂₇	1	1	yes	characteristic
430	C₂	—	4	1	no	—
431	C₂	—	4	2	no	—
432	C₂	C₁₂₇×#3035	1	1	yes	characteristic
433	C₂	C₁₂₇×#3050	1	1	yes	characteristic
434	C₂	C₂×#473×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂×#121
prime radical	C₂×#41×C₁₂₇	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₂₇×#3170	C₂	semidirect
4. C₁₂₇×#1680	C₂	semidirect
5. C₁₂₇×#1679	C₂	semidirect
6. C₁₂₇×#1674	C₂	semidirect
7. C₁₂₇×#1673	C₂	semidirect
8. C₁₂₇×#1670	C₂	semidirect
9. C₁₂₇×#1669	C₂	semidirect
10. C₁₂₇×#1564	C₂	semidirect
11. C₁₂₇×#1544	C₂	semidirect
212. #19129	C₁₂₇	direct (N × H)
218. #3170	C₂×C₁₂₇	semidirect
221. #1680	C₂×C₁₂₇	semidirect
222. #1679	C₂×C₁₂₇	semidirect
223. #1674	C₂×C₁₂₇	semidirect
224. #1673	C₂×C₁₂₇	semidirect
225. #1670	C₂×C₁₂₇	semidirect
226. #1669	C₂×C₁₂₇	semidirect
227. #1564	C₂×C₁₂₇	semidirect
228. #1544	C₂×C₁₂₇	semidirect
229. C₁₂₇	#19129	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₂₇×#3170	2
C₁₂₇×#3107	2
C₈×#41×C₁₂₇	2
C₁₂₇×#1680	2
C₁₂₇×#1679	2
C₁₂₇×#1674	2
C₁₂₇×#1673	2
C₁₂₇×#1670	2
C₁₂₇×#1669	2
C₁₂₇×#1564	2
C₁₂₇×#1544	2
#19129	127

Subgroups — C127×#19129 #44582802

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#19129 #44582802