Subgroups — C₁₂₇×#21804 #44585477

Order 32512.

1366 conjugacy classes of subgroups.

The subgroup lattice has 346 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×#1981	C₂	1	4	yes	maximal, complemented
5	C₁₂₇×#1956	C₂	1	4	yes	maximal, complemented
6	C₁₂₇×#1787	C₂	1	4	yes	maximal
7	C₂³×Q₈×C₁₂₇	C₂²	1	2	yes	—
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	—	2	1	no	—
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×#103×C₁₂₇	C₂²	1	16	yes	—
21	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
23	#384×C₁₂₇	C₂²	1	2	yes	—
24	#382×C₁₂₇	—	2	1	no	—
25	#380×C₁₂₇	—	2	1	no	—
26	#380×C₁₂₇	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₁₂₇	—	2	1	no	—
28	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₂₇	—	2	4	no	—
30	C₂²×Q₈×C₁₂₇	—	2	2	no	—
31	C₂²×Q₈×C₁₂₇	—	4	8	no	—
32	C₂²×Q₈×C₁₂₇	C₂×C₄	1	8	yes	—
33	C₂²×Q₈×C₁₂₇	C₂³	1	4	yes	—
34	C₂³×C₄×C₁₂₇	—	2	2	no	—
35	C₂³×C₄×C₁₂₇	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₁₂₇	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₂₇	C₂³	1	4	yes	—
39	C₄⋊Q₈×C₁₂₇	—	2	8	no	—
40	C₄⋊Q₈×C₁₂₇	—	2	8	no	—
41	C₄⋊Q₈×C₁₂₇	—	2	4	no	—
42	C₄⋊Q₈×C₁₂₇	—	2	4	no	—
43	C₄⋊Q₈×C₁₂₇	C₂×C₄	1	8	yes	—
44	C₄⋊Q₈×C₁₂₇	C₂×C₄	1	8	yes	—
45	#117×C₁₂₇	—	4	4	no	—
46	C₂×C₄⋊C₄×C₁₂₇	—	2	8	no	—
47	C₂×C₄⋊C₄×C₁₂₇	—	2	8	no	—
48	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
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	#103×C₁₂₇	—	4	16	no	—
65	C₂²⋊C₈×C₁₂₇	—	2	4	no	—
66	C₄×C₈×C₁₂₇	—	2	4	no	—
67	#95×C₁₂₇	—	2	2	no	—
68	#95×C₁₂₇	—	2	2	no	—
69	C₂⁴×C₁₂₇	C₂×D₄	1	1	yes	characteristic
70	C₂×Q₈×C₁₂₇	—	2	8	no	—
71	C₂×Q₈×C₁₂₇	—	2	4	no	—
72	C₂×Q₈×C₁₂₇	—	2	16	no	—
73	C₂×Q₈×C₁₂₇	—	4	16	no	—
74	C₂×Q₈×C₁₂₇	—	4	8	no	—
75	C₂×Q₈×C₁₂₇	—	4	16	no	—
76	C₂×Q₈×C₁₂₇	C₂²×C₄	1	8	yes	—
77	C₂²×C₄×C₁₂₇	—	2	4	no	—
78	C₂²×C₄×C₁₂₇	—	2	4	no	—
79	C₂²×C₄×C₁₂₇	—	2	8	no	—
80	C₂²×C₄×C₁₂₇	—	2	4	no	—
81	C₂²×C₄×C₁₂₇	—	2	4	no	—
82	C₂²×C₄×C₁₂₇	—	4	8	no	—
83	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
84	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
85	C₂²×C₄×C₁₂₇	SD₁₆	1	4	yes	—
86	C₂²×C₄×C₁₂₇	Q₁₆	1	4	yes	—
87	C₂²×C₄×C₁₂₇	C₂²×C₄	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	2	yes	—
91	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
95	C₂×C₈×C₁₂₇	—	2	4	no	—
96	C₂×C₈×C₁₂₇	—	2	8	no	—
97	C₄⋊C₄×C₁₂₇	—	2	8	no	—
98	C₄⋊C₄×C₁₂₇	—	2	8	no	—
99	C₄⋊C₄×C₁₂₇	—	2	8	no	—
100	C₄⋊C₄×C₁₂₇	—	2	8	no	—
101	C₄⋊C₄×C₁₂₇	—	2	8	no	—
102	C₄⋊C₄×C₁₂₇	—	4	4	no	—
103	C₄⋊C₄×C₁₂₇	—	4	4	no	—
104	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
105	C₄²×C₁₂₇	—	4	4	no	—
106	C₄²×C₁₂₇	C₂²⋊C₄	1	4	yes	—
107	C₄²×C₁₂₇	C₂²⋊C₄	1	4	yes	—
108	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
109	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
110	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
111	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
112	C₂³×C₁₂₇	—	2	2	no	—
113	C₂³×C₁₂₇	—	2	2	no	—
114	C₂³×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
115	C₂³×C₁₂₇	C₂×SD₁₆	1	2	yes	—
116	C₂³×C₁₂₇	C₂×Q₁₆	1	2	yes	—
117	C₂³×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
118	Q₈×C₁₂₇	—	2	16	no	—
119	Q₈×C₁₂₇	—	4	16	no	—
120	Q₈×C₁₂₇	—	4	8	no	—
121	C₂×C₄×C₁₂₇	—	2	2	no	—
122	C₂×C₄×C₁₂₇	—	2	2	no	—
123	C₂×C₄×C₁₂₇	—	2	4	no	—
124	C₂×C₄×C₁₂₇	—	2	4	no	—
125	C₂×C₄×C₁₂₇	—	2	8	no	—
126	C₂×C₄×C₁₂₇	—	2	8	no	—
127	C₂×C₄×C₁₂₇	—	4	16	no	—
128	C₂×C₄×C₁₂₇	—	4	8	no	—
129	C₂×C₄×C₁₂₇	—	4	4	no	—
130	C₂×C₄×C₁₂₇	—	4	4	no	—
131	C₂×C₄×C₁₂₇	—	4	4	no	—
132	C₂×C₄×C₁₂₇	#103	1	16	yes	complemented
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×C₂²⋊C₄	1	2	yes	—
136	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
137	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
138	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	2	yes	—
139	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	2	yes	—
140	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	2	yes	—
141	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	2	yes	—
142	C₂×C₄×C₁₂₇	#137	1	4	yes	—
143	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₂₇	H(5,𝔽₂)	1	4	yes	—
146	C₈×C₁₂₇	—	4	4	no	—
147	C₂²×C₁₂₇	—	2	2	no	—
148	C₂²×C₁₂₇	—	2	2	no	—
149	C₂²×C₁₂₇	—	2	4	no	—
150	C₂²×C₁₂₇	—	2	4	no	—
151	C₂²×C₁₂₇	C₂×#103	1	4	yes	—
152	C₂²×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₁₂₇	#535	1	2	yes	—
154	C₂²×C₁₂₇	C₂²×SD₁₆	1	1	yes	characteristic
155	C₂²×C₁₂₇	C₂²×Q₁₆	1	1	yes	characteristic
156	C₂²×C₁₂₇	C₂×#137	1	2	yes	—
157	C₄×C₁₂₇	—	2	2	no	—
158	C₄×C₁₂₇	—	2	2	no	—
159	C₄×C₁₂₇	—	4	8	no	—
160	C₄×C₁₂₇	—	8	4	no	—
161	C₄×C₁₂₇	C₂×#103	1	4	yes	—
162	C₄×C₁₂₇	C₂×#103	1	4	yes	—
163	C₄×C₁₂₇	#419	1	4	yes	—
164	C₄×C₁₂₇	#518	1	2	yes	—
165	C₄×C₁₂₇	#535	1	2	yes	—
166	#21804	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂×C₁₂₇	—	2	4	no	—
168	C₂×C₁₂₇	#2309	1	1	yes	characteristic
169	C₂×C₁₂₇	C₂²×#103	1	1	yes	characteristic
170	C₂×C₁₂₇	C₂×#419	1	1	yes	characteristic
171	C₂×C₁₂₇	#3206	1	2	yes	—
172	C₂×C₁₂₇	#3209	1	2	yes	—
173	C₂²×C₄⋊Q₈	C₂×C₁₂₇	1	1	yes	characteristic
174	C₂×#431	C₂×C₁₂₇	1	1	yes	characteristic
175	#2285	C₂×C₁₂₇	1	1	yes	characteristic
176	#1981	C₂×C₁₂₇	1	4	yes	complemented
177	#1956	C₂×C₁₂₇	1	4	yes	complemented
178	#1787	C₂×C₁₂₇	1	4	yes	—
179	C₁₂₇	#21804	1	1	yes	characteristic, direct product factor, Sylow
180	C₂³×Q₈	C₂²×C₁₂₇	1	2	yes	—
181	C₂×C₄⋊Q₈	—	2	1	no	—
182	C₂×C₄⋊Q₈	—	2	1	no	—
183	C₂×C₄⋊Q₈	—	2	4	no	—
184	C₂×C₄⋊Q₈	C₄×C₁₂₇	1	8	yes	complemented
185	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
186	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
187	C₂×#117	—	2	1	no	—
188	C₂²×C₄⋊C₄	—	2	1	no	—
189	C₂²×C₄⋊C₄	—	2	1	no	—
190	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
191	#431	—	2	4	no	—
192	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
193	C₂×#103	C₂²×C₁₂₇	1	16	yes	—
194	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
195	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
196	#384	C₂²×C₁₂₇	1	2	yes	—
197	#382	—	2	1	no	—
198	#380	—	2	1	no	—
199	#380	C₂²×C₁₂₇	1	2	yes	—
200	C₄×C₄⋊C₄	—	2	1	no	—
201	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
202	C₂²×Q₈	—	2	4	no	—
203	C₂²×Q₈	—	2	2	no	—
204	C₂²×Q₈	—	4	8	no	—
205	C₂²×Q₈	C₂×C₄×C₁₂₇	1	8	yes	—
206	C₂²×Q₈	C₂³×C₁₂₇	1	4	yes	—
207	C₂³×C₄	—	2	2	no	—
208	C₂³×C₄	D₄×C₁₂₇	1	1	yes	characteristic
209	C₂³×C₄	D₄×C₁₂₇	1	1	yes	characteristic
210	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
211	C₂²×C₈	C₂³×C₁₂₇	1	4	yes	—
212	C₄⋊Q₈	—	2	8	no	—
213	C₄⋊Q₈	—	2	8	no	—
214	C₄⋊Q₈	—	2	4	no	—
215	C₄⋊Q₈	—	2	4	no	—
216	C₄⋊Q₈	C₂×C₄×C₁₂₇	1	8	yes	—
217	C₄⋊Q₈	C₂×C₄×C₁₂₇	1	8	yes	—
218	#117	—	4	4	no	—
219	C₂×C₄⋊C₄	—	2	8	no	—
220	C₂×C₄⋊C₄	—	2	8	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	4	yes	—
228	C₂×C₂²⋊C₄	—	2	2	no	—
229	C₂×C₄²	—	2	2	no	—
230	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
231	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
232	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
233	C₂×C₄²	D₄×C₁₂₇	1	4	yes	—
234	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
235	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
236	C₄⋊C₈	—	2	4	no	—
237	#103	—	4	16	no	—
238	C₂²⋊C₈	—	2	4	no	—
239	C₄×C₈	—	2	4	no	—
240	#95	—	2	2	no	—
241	#95	—	2	2	no	—
242	C₂⁴	C₂×D₄×C₁₂₇	1	1	yes	characteristic
243	C₂×Q₈	—	2	8	no	—
244	C₂×Q₈	—	2	4	no	—
245	C₂×Q₈	—	2	16	no	—
246	C₂×Q₈	—	4	16	no	—
247	C₂×Q₈	—	4	8	no	—
248	C₂×Q₈	—	4	16	no	—
249	C₂×Q₈	C₂²×C₄×C₁₂₇	1	8	yes	—
250	C₂²×C₄	—	2	4	no	—
251	C₂²×C₄	—	2	4	no	—
252	C₂²×C₄	—	2	8	no	—
253	C₂²×C₄	—	2	4	no	—
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	—	4	8	no	—
256	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
257	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
258	C₂²×C₄	SD₁₆×C₁₂₇	1	4	yes	—
259	C₂²×C₄	Q₁₆×C₁₂₇	1	4	yes	—
260	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
261	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
264	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
266	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
267	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	2	8	no	—
270	C₄⋊C₄	—	2	8	no	—
271	C₄⋊C₄	—	2	8	no	—
272	C₄⋊C₄	—	2	8	no	—
273	C₄⋊C₄	—	2	8	no	—
274	C₄⋊C₄	—	2	8	no	—
275	C₄⋊C₄	—	4	4	no	—
276	C₄⋊C₄	—	4	4	no	—
277	C₂²⋊C₄	—	4	4	no	—
278	C₄²	—	4	4	no	—
279	C₄²	C₂²⋊C₄×C₁₂₇	1	4	yes	—
280	C₄²	C₂²⋊C₄×C₁₂₇	1	4	yes	—
281	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
282	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
283	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
284	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
285	C₂³	—	2	2	no	—
286	C₂³	—	2	2	no	—
287	C₂³	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
288	C₂³	C₂×SD₁₆×C₁₂₇	1	2	yes	—
289	C₂³	C₂×Q₁₆×C₁₂₇	1	2	yes	—
290	C₂³	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
291	Q₈	—	2	16	no	—
292	Q₈	—	4	16	no	—
293	Q₈	—	4	8	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	8	no	—
299	C₂×C₄	—	2	8	no	—
300	C₂×C₄	—	4	16	no	—
301	C₂×C₄	—	4	8	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	#103×C₁₂₇	1	16	yes	complemented
306	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
307	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
308	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
309	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
310	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
311	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	2	yes	—
312	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	2	yes	—
313	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	2	yes	—
314	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	2	yes	—
315	C₂×C₄	#137×C₁₂₇	1	4	yes	—
316	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
317	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
318	C₂×C₄	H(5,𝔽₂)×C₁₂₇	1	4	yes	—
319	C₈	—	4	4	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	4	no	—
322	C₂²	—	2	4	no	—
323	C₂²	—	2	2	no	—
324	C₂²	C₂×#103×C₁₂₇	1	4	yes	—
325	C₂²	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
326	C₂²	#535×C₁₂₇	1	2	yes	—
327	C₂²	C₂²×SD₁₆×C₁₂₇	1	1	yes	characteristic
328	C₂²	C₂²×Q₁₆×C₁₂₇	1	1	yes	characteristic
329	C₂²	C₂×#137×C₁₂₇	1	2	yes	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	4	8	no	—
333	C₄	—	8	4	no	—
334	C₄	C₂×#103×C₁₂₇	1	4	yes	—
335	C₄	C₂×#103×C₁₂₇	1	4	yes	—
336	C₄	#419×C₁₂₇	1	4	yes	—
337	C₄	#518×C₁₂₇	1	2	yes	—
338	C₄	#535×C₁₂₇	1	2	yes	—
339	C₂	—	2	4	no	—
340	C₂	C₁₂₇×#2309	1	1	yes	characteristic
341	C₂	C₂²×#103×C₁₂₇	1	1	yes	characteristic
342	C₂	C₂×#419×C₁₂₇	1	1	yes	characteristic
343	C₂	C₁₂₇×#3206	1	2	yes	—
344	C₂	C₁₂₇×#3209	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×#1981	C₂	semidirect
5. C₁₂₇×#1956	C₂	semidirect
11. C₂×C₄⋊Q₈×C₁₂₇	C₄	semidirect
132. C₂×C₄×C₁₂₇	#103	semidirect
166. #21804	C₁₂₇	direct (N × H)
176. #1981	C₂×C₁₂₇	semidirect
177. #1956	C₂×C₁₂₇	semidirect
179. C₁₂₇	#21804	direct (N × H)
184. C₂×C₄⋊Q₈	C₄×C₁₂₇	semidirect
305. C₂×C₄	#103×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×#1981	2
C₁₂₇×#1956	2
C₁₂₇×#1787	2
#21804	127

Subgroups — C127×#21804 #44585477

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21804 #44585477