Subgroups — C₄⋊C₄×M₁₆×C₁₂₇ #44584113

Order 32512.

1242 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#423×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₁₂₇	C₂	1	2	yes	maximal
3	C₂×C₄×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1771	C₂	1	4	yes	maximal, complemented
6	C₁₂₇×#1770	C₂	1	2	yes	maximal, complemented
7	C₈×C₄⋊C₄×C₁₂₇	C₂	1	2	yes	maximal, complemented
8	C₁₂₇×#1728	C₂	1	2	yes	maximal
9	C₂²×M₁₆×C₁₂₇	C₂²	1	2	yes	—
10	C₂²×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₁₂₇	C₄	1	2	yes	—
13	C₂²×C₄²×C₁₂₇	C₂²	1	2	yes	—
14	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
15	#423×C₁₂₇	C₄	1	8	yes	complemented
16	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
18	C₄×M₁₆×C₁₂₇	—	2	8	no	—
19	C₄×M₁₆×C₁₂₇	C₄	1	8	yes	complemented
20	C₂×#97×C₁₂₇	C₂²	1	4	yes	—
21	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₁₂₇	C₂²	1	4	yes	—
23	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₁₂₇	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₁₂₇	C₄	1	2	yes	—
26	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
29	C₂×#95×C₁₂₇	C₄	1	2	yes	—
30	#336×C₁₂₇	C₂²	1	4	yes	—
31	#336×C₁₂₇	C₂²	1	4	yes	—
32	C₂³×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
33	C₂³×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₁₂₇	—	2	4	no	—
37	C₂×M₁₆×C₁₂₇	—	2	2	no	—
38	C₂×M₁₆×C₁₂₇	—	2	2	no	—
39	C₂×M₁₆×C₁₂₇	—	2	4	no	—
40	C₂×M₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
41	C₂×M₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
42	C₂×M₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
43	C₂×M₁₆×C₁₂₇	D₄	1	4	yes	—
44	C₂×M₁₆×C₁₂₇	Q₈	1	4	yes	—
45	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₁₂₇	C₂³	1	4	yes	—
47	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₁₂₇	—	2	8	no	—
49	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
51	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₁₂₇	—	2	4	no	—
53	C₂×C₄²×C₁₂₇	—	2	4	no	—
54	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
55	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
58	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
59	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
60	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
61	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	complemented
64	C₄⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	complemented
65	#97×C₁₂₇	—	2	8	no	—
66	#97×C₁₂₇	C₂×C₄	1	8	yes	complemented
67	C₄×C₈×C₁₂₇	—	2	8	no	—
68	C₄×C₈×C₁₂₇	C₂×C₄	1	8	yes	complemented
69	#95×C₁₂₇	C₂×C₄	1	2	yes	—
70	#95×C₁₂₇	C₂×C₄	1	4	yes	—
71	#95×C₁₂₇	C₂×C₄	1	2	yes	—
72	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₂₇	—	2	4	no	—
74	C₂²×C₄×C₁₂₇	—	2	8	no	—
75	C₂²×C₄×C₁₂₇	—	2	8	no	—
76	C₂²×C₄×C₁₂₇	—	4	2	no	—
77	C₂²×C₄×C₁₂₇	—	4	4	no	—
78	C₂²×C₄×C₁₂₇	C₄²	1	4	yes	—
79	C₂²×C₄×C₁₂₇	C₄²	1	4	yes	—
80	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
85	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
91	C₂²×C₄×C₁₂₇	#41	1	2	yes	—
92	C₂²×C₄×C₁₂₇	#41	1	2	yes	—
93	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
94	M₁₆×C₁₂₇	—	2	8	no	—
95	M₁₆×C₁₂₇	—	2	16	no	—
96	M₁₆×C₁₂₇	C₄⋊C₄	1	16	yes	direct product factor
97	C₂×C₈×C₁₂₇	—	2	4	no	—
98	C₂×C₈×C₁₂₇	—	2	2	no	—
99	C₂×C₈×C₁₂₇	—	2	2	no	—
100	C₂×C₈×C₁₂₇	—	2	4	no	—
101	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
102	C₂×C₈×C₁₂₇	C₂²×C₄	1	8	yes	—
103	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
104	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
105	C₂×C₈×C₁₂₇	C₂×Q₈	1	4	yes	—
106	C₄⋊C₄×C₁₂₇	—	2	8	no	—
107	C₄⋊C₄×C₁₂₇	—	2	4	no	—
108	C₄⋊C₄×C₁₂₇	M₁₆	1	8	yes	direct product factor
109	C₄²×C₁₂₇	—	2	8	no	—
110	C₄²×C₁₂₇	—	2	2	no	—
111	C₄²×C₁₂₇	—	2	2	no	—
112	C₄²×C₁₂₇	—	2	2	no	—
113	C₄²×C₁₂₇	—	2	2	no	—
114	C₄²×C₁₂₇	C₄²	1	4	yes	—
115	C₄²×C₁₂₇	C₄²	1	4	yes	—
116	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
117	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
118	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
119	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
120	C₂³×C₁₂₇	—	2	2	no	—
121	C₂³×C₁₂₇	—	2	2	no	—
122	C₂³×C₁₂₇	C₂×C₄²	1	2	yes	—
123	C₂³×C₁₂₇	C₄×D₄	1	2	yes	—
124	C₂³×C₁₂₇	C₄×Q₈	1	2	yes	—
125	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₂₇	—	2	4	no	—
127	C₂×C₄×C₁₂₇	—	2	2	no	—
128	C₂×C₄×C₁₂₇	—	2	2	no	—
129	C₂×C₄×C₁₂₇	—	2	2	no	—
130	C₂×C₄×C₁₂₇	—	2	4	no	—
131	C₂×C₄×C₁₂₇	—	2	8	no	—
132	C₂×C₄×C₁₂₇	—	2	4	no	—
133	C₂×C₄×C₁₂₇	—	2	4	no	—
134	C₂×C₄×C₁₂₇	—	2	4	no	—
135	C₂×C₄×C₁₂₇	—	4	4	no	—
136	C₂×C₄×C₁₂₇	—	4	16	no	—
137	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
138	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
139	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
140	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
141	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
142	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
143	C₂×C₄×C₁₂₇	#117	1	4	yes	—
144	C₂×C₄×C₁₂₇	C₄×D₄	1	4	yes	—
145	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
146	C₂×C₄×C₁₂₇	C₄×Q₈	1	4	yes	—
147	C₂×C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
148	C₂×C₄×C₁₂₇	C₂×M₁₆	1	8	yes	—
149	C₂×C₄×C₁₂₇	C₂×M₁₆	1	4	yes	—
150	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
154	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
156	C₈×C₁₂₇	—	2	4	no	—
157	C₈×C₁₂₇	—	2	8	no	—
158	C₈×C₁₂₇	C₂×C₄⋊C₄	1	8	yes	complemented
159	C₂²×C₁₂₇	—	2	2	no	—
160	C₂²×C₁₂₇	—	2	2	no	—
161	C₂²×C₁₂₇	—	2	4	no	—
162	C₂²×C₁₂₇	—	2	4	no	—
163	C₂²×C₁₂₇	C₄×C₄⋊C₄	1	4	yes	—
164	C₂²×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
165	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
166	C₂²×C₁₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
167	C₂²×C₁₂₇	C₂²×M₁₆	1	2	yes	—
168	C₂²×C₁₂₇	#568	1	2	yes	—
169	C₄×C₁₂₇	—	2	2	no	—
170	C₄×C₁₂₇	—	2	2	no	—
171	C₄×C₁₂₇	—	2	2	no	—
172	C₄×C₁₂₇	—	2	8	no	—
173	C₄×C₁₂₇	—	4	4	no	—
174	C₄×C₁₂₇	C₄×C₄⋊C₄	1	4	yes	—
175	C₄×C₁₂₇	C₄×M₁₆	1	4	yes	complemented
176	C₄×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
177	C₄×C₁₂₇	C₂×#117	1	1	yes	characteristic
178	C₄×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
179	C₄×C₁₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
180	C₄⋊C₄×M₁₆	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
181	C₂×C₁₂₇	—	2	4	no	—
182	C₂×C₁₂₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
183	C₂×C₁₂₇	C₂×C₄×M₁₆	1	1	yes	characteristic
184	C₂×C₁₂₇	#2873	1	1	yes	characteristic
185	C₂×C₁₂₇	D₄×M₁₆	1	2	yes	—
186	C₂×C₁₂₇	Q₈×M₁₆	1	2	yes	—
187	C₂×#423	C₂×C₁₂₇	1	1	yes	characteristic
188	C₂×C₄×M₁₆	C₂×C₁₂₇	1	2	yes	—
189	C₂×C₄×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic
190	C₂×C₄×C₄⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic
191	#1771	C₂×C₁₂₇	1	4	yes	complemented
192	#1770	C₂×C₁₂₇	1	2	yes	complemented
193	C₈×C₄⋊C₄	C₂×C₁₂₇	1	2	yes	complemented
194	#1728	C₂×C₁₂₇	1	2	yes	—
195	C₁₂₇	C₄⋊C₄×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
196	C₂²×M₁₆	C₂²×C₁₂₇	1	2	yes	—
197	C₂²×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
198	C₂²×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
199	C₂²×C₄⋊C₄	C₄×C₁₂₇	1	2	yes	—
200	C₂²×C₄²	C₂²×C₁₂₇	1	2	yes	—
201	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
202	#423	C₄×C₁₂₇	1	8	yes	complemented
203	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
204	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
205	C₄×M₁₆	—	2	8	no	—
206	C₄×M₁₆	C₄×C₁₂₇	1	8	yes	complemented
207	C₂×#97	C₂²×C₁₂₇	1	4	yes	—
208	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
209	C₂×C₄×C₈	C₂²×C₁₂₇	1	4	yes	—
210	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
211	C₄×C₄⋊C₄	C₄×C₁₂₇	1	2	yes	—
212	C₄×C₄⋊C₄	C₄×C₁₂₇	1	2	yes	—
213	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
214	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
215	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
216	C₂×#95	C₄×C₁₂₇	1	2	yes	—
217	#336	C₂²×C₁₂₇	1	4	yes	—
218	#336	C₂²×C₁₂₇	1	4	yes	—
219	C₂³×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
220	C₂³×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
221	C₂³×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
222	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
223	C₂×M₁₆	—	2	4	no	—
224	C₂×M₁₆	—	2	2	no	—
225	C₂×M₁₆	—	2	2	no	—
226	C₂×M₁₆	—	2	4	no	—
227	C₂×M₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
228	C₂×M₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
229	C₂×M₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
230	C₂×M₁₆	D₄×C₁₂₇	1	4	yes	—
231	C₂×M₁₆	Q₈×C₁₂₇	1	4	yes	—
232	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
233	C₂²×C₈	C₂³×C₁₂₇	1	4	yes	—
234	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
235	C₂×C₄⋊C₄	—	2	8	no	—
236	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
237	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
238	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
239	C₂×C₄²	—	2	4	no	—
240	C₂×C₄²	—	2	4	no	—
241	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
242	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
243	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
244	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
245	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
246	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
247	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
248	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
249	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
250	C₄⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	complemented
251	C₄⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	complemented
252	#97	—	2	8	no	—
253	#97	C₂×C₄×C₁₂₇	1	8	yes	complemented
254	C₄×C₈	—	2	8	no	—
255	C₄×C₈	C₂×C₄×C₁₂₇	1	8	yes	complemented
256	#95	C₂×C₄×C₁₂₇	1	2	yes	—
257	#95	C₂×C₄×C₁₂₇	1	4	yes	—
258	#95	C₂×C₄×C₁₂₇	1	2	yes	—
259	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
260	C₂²×C₄	—	2	4	no	—
261	C₂²×C₄	—	2	8	no	—
262	C₂²×C₄	—	2	8	no	—
263	C₂²×C₄	—	4	2	no	—
264	C₂²×C₄	—	4	4	no	—
265	C₂²×C₄	C₄²×C₁₂₇	1	4	yes	—
266	C₂²×C₄	C₄²×C₁₂₇	1	4	yes	—
267	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
268	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
269	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
270	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
271	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
272	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
273	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
274	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
275	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
276	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
277	C₂²×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
278	C₂²×C₄	#41×C₁₂₇	1	2	yes	—
279	C₂²×C₄	#41×C₁₂₇	1	2	yes	—
280	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
281	M₁₆	—	2	8	no	—
282	M₁₆	—	2	16	no	—
283	M₁₆	C₄⋊C₄×C₁₂₇	1	16	yes	direct product factor
284	C₂×C₈	—	2	4	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	4	no	—
288	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
289	C₂×C₈	C₂²×C₄×C₁₂₇	1	8	yes	—
290	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
291	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
292	C₂×C₈	C₂×Q₈×C₁₂₇	1	4	yes	—
293	C₄⋊C₄	—	2	8	no	—
294	C₄⋊C₄	—	2	4	no	—
295	C₄⋊C₄	M₁₆×C₁₂₇	1	8	yes	direct product factor
296	C₄²	—	2	8	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	C₄²×C₁₂₇	1	4	yes	—
302	C₄²	C₄²×C₁₂₇	1	4	yes	—
303	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
304	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
305	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
306	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
307	C₂³	—	2	2	no	—
308	C₂³	—	2	2	no	—
309	C₂³	C₂×C₄²×C₁₂₇	1	2	yes	—
310	C₂³	C₄×D₄×C₁₂₇	1	2	yes	—
311	C₂³	C₄×Q₈×C₁₂₇	1	2	yes	—
312	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	8	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	4	4	no	—
323	C₂×C₄	—	4	16	no	—
324	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
325	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
326	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
327	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
328	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
329	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	4	yes	—
330	C₂×C₄	#117×C₁₂₇	1	4	yes	—
331	C₂×C₄	C₄×D₄×C₁₂₇	1	4	yes	—
332	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
333	C₂×C₄	C₄×Q₈×C₁₂₇	1	4	yes	—
334	C₂×C₄	C₄×Q₈×C₁₂₇	1	2	yes	—
335	C₂×C₄	C₂×M₁₆×C₁₂₇	1	8	yes	—
336	C₂×C₄	C₂×M₁₆×C₁₂₇	1	4	yes	—
337	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
338	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
339	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
340	C₂×C₄	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
341	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
342	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
343	C₈	—	2	4	no	—
344	C₈	—	2	8	no	—
345	C₈	C₂×C₄⋊C₄×C₁₂₇	1	8	yes	complemented
346	C₂²	—	2	2	no	—
347	C₂²	—	2	2	no	—
348	C₂²	—	2	4	no	—
349	C₂²	—	2	4	no	—
350	C₂²	C₄×C₄⋊C₄×C₁₂₇	1	4	yes	—
351	C₂²	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
352	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
353	C₂²	C₂×C₄×Q₈×C₁₂₇	1	1	yes	characteristic
354	C₂²	C₂²×M₁₆×C₁₂₇	1	2	yes	—
355	C₂²	#568×C₁₂₇	1	2	yes	—
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	2	2	no	—
359	C₄	—	2	8	no	—
360	C₄	—	4	4	no	—
361	C₄	C₄×C₄⋊C₄×C₁₂₇	1	4	yes	—
362	C₄	C₄×M₁₆×C₁₂₇	1	4	yes	complemented
363	C₄	C₂²×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
364	C₄	C₂×#117×C₁₂₇	1	1	yes	characteristic
365	C₄	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
366	C₄	C₂×C₄×Q₈×C₁₂₇	1	1	yes	characteristic
367	C₂	—	2	4	no	—
368	C₂	C₂×C₄×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
369	C₂	C₂×C₄×M₁₆×C₁₂₇	1	1	yes	characteristic
370	C₂	C₁₂₇×#2873	1	1	yes	characteristic
371	C₂	D₄×M₁₆×C₁₂₇	1	2	yes	—
372	C₂	Q₈×M₁₆×C₁₂₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
5. C₁₂₇×#1771	C₂	semidirect
6. C₁₂₇×#1770	C₂	semidirect
7. C₈×C₄⋊C₄×C₁₂₇	C₂	semidirect
15. #423×C₁₂₇	C₄	semidirect
19. C₄×M₁₆×C₁₂₇	C₄	semidirect
63. C₄⋊C₈×C₁₂₇	C₂×C₄	semidirect
64. C₄⋊C₈×C₁₂₇	C₂×C₄	semidirect
66. #97×C₁₂₇	C₂×C₄	semidirect
68. C₄×C₈×C₁₂₇	C₂×C₄	semidirect
96. M₁₆×C₁₂₇	C₄⋊C₄	direct (N × H)
108. C₄⋊C₄×C₁₂₇	M₁₆	direct (N × H)
158. C₈×C₁₂₇	C₂×C₄⋊C₄	semidirect
175. C₄×C₁₂₇	C₄×M₁₆	semidirect
180. C₄⋊C₄×M₁₆	C₁₂₇	direct (N × H)
191. #1771	C₂×C₁₂₇	semidirect
192. #1770	C₂×C₁₂₇	semidirect
193. C₈×C₄⋊C₄	C₂×C₁₂₇	semidirect
195. C₁₂₇	C₄⋊C₄×M₁₆	direct (N × H)
202. #423	C₄×C₁₂₇	semidirect
206. C₄×M₁₆	C₄×C₁₂₇	semidirect
250. C₄⋊C₈	C₂×C₄×C₁₂₇	semidirect
251. C₄⋊C₈	C₂×C₄×C₁₂₇	semidirect
253. #97	C₂×C₄×C₁₂₇	semidirect
255. C₄×C₈	C₂×C₄×C₁₂₇	semidirect
283. M₁₆	C₄⋊C₄×C₁₂₇	direct (N × H)
295. C₄⋊C₄	M₁₆×C₁₂₇	direct (N × H)
345. C₈	C₂×C₄⋊C₄×C₁₂₇	semidirect
362. C₄	C₄×M₁₆×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#423×C₁₂₇	2
C₂×C₄×M₁₆×C₁₂₇	2
C₂×C₄×C₄⋊C₄×C₁₂₇	2
C₁₂₇×#1771	2
C₁₂₇×#1770	2
C₈×C₄⋊C₄×C₁₂₇	2
C₁₂₇×#1728	2
C₄⋊C₄×M₁₆	127

Subgroups — C4⋊C4×M16×C127 #44584113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊C₄×M₁₆×C₁₂₇ #44584113