Subgroups — C₁₂₇×#21673 #44585346

Order 32512.

694 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#125×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#1939	C₂	1	2	yes	maximal
3	C₁₂₇×#1916	C₂	1	4	yes	maximal
4	C₁₂₇×#1831	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1771	C₂	1	4	yes	maximal
6	C₁₂₇×#1770	C₂	1	2	yes	maximal
7	C₄²×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#125×C₁₂₇	C₄	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
10	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	4	yes	—
11	C₂×#97×C₁₂₇	C₂²	1	4	yes	—
12	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	#389×C₁₂₇	C₄	1	2	yes	—
17	#387×C₁₂₇	C₄	1	4	yes	—
18	#382×C₁₂₇	C₄	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	4	yes	—
20	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
21	C₄³×C₁₂₇	C₂²	1	1	yes	characteristic
22	#336×C₁₂₇	C₂²	1	4	yes	—
23	#336×C₁₂₇	C₂²	1	4	yes	—
24	#336×C₁₂₇	C₂²	1	4	yes	—
25	C₂²×C₈×C₁₂₇	C₂³	1	4	yes	—
26	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
28	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
29	#125×C₁₂₇	—	2	4	no	—
30	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
38	C₂×C₄²×C₁₂₇	C₂³	1	4	yes	—
39	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₈×C₁₂₇	—	2	8	no	—
41	C₄⋊C₈×C₁₂₇	—	2	4	no	—
42	#97×C₁₂₇	—	2	8	no	—
43	C₄×C₈×C₁₂₇	—	2	4	no	—
44	C₄×C₈×C₁₂₇	—	2	4	no	—
45	C₄×C₈×C₁₂₇	—	2	2	no	—
46	C₄×C₈×C₁₂₇	Q₈	1	4	yes	—
47	#95×C₁₂₇	C₂×C₄	1	4	yes	—
48	#95×C₁₂₇	C₂×C₄	1	4	yes	—
49	#95×C₁₂₇	C₂×C₄	1	4	yes	—
50	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
57	C₂×C₈×C₁₂₇	—	2	2	no	—
58	C₂×C₈×C₁₂₇	—	2	2	no	—
59	C₂×C₈×C₁₂₇	—	2	2	no	—
60	C₂×C₈×C₁₂₇	—	2	4	no	—
61	C₂×C₈×C₁₂₇	—	2	4	no	—
62	C₂×C₈×C₁₂₇	—	2	4	no	—
63	C₂×C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
64	C₂×C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
65	C₂×C₈×C₁₂₇	#41	1	4	yes	—
66	C₂×C₈×C₁₂₇	#41	1	4	yes	—
67	C₄⋊C₄×C₁₂₇	—	2	8	no	—
68	C₄⋊C₄×C₁₂₇	—	2	4	no	—
69	C₄⋊C₄×C₁₂₇	—	2	8	no	—
70	C₄⋊C₄×C₁₂₇	—	2	4	no	—
71	C₄²×C₁₂₇	—	2	4	no	—
72	C₄²×C₁₂₇	—	2	2	no	—
73	C₄²×C₁₂₇	—	2	2	no	—
74	C₄²×C₁₂₇	—	2	2	no	—
75	C₄²×C₁₂₇	—	2	4	no	—
76	C₄²×C₁₂₇	—	2	4	no	—
77	C₄²×C₁₂₇	M₁₆	1	4	yes	—
78	C₄²×C₁₂₇	C₂×Q₈	1	2	yes	—
79	C₄²×C₁₂₇	#41	1	2	yes	—
80	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₂₇	—	2	2	no	—
82	C₂×C₄×C₁₂₇	—	2	2	no	—
83	C₂×C₄×C₁₂₇	—	2	2	no	—
84	C₂×C₄×C₁₂₇	—	2	2	no	—
85	C₂×C₄×C₁₂₇	—	2	2	no	—
86	C₂×C₄×C₁₂₇	—	2	4	no	—
87	C₂×C₄×C₁₂₇	—	2	4	no	—
88	C₂×C₄×C₁₂₇	—	2	4	no	—
89	C₂×C₄×C₁₂₇	—	2	4	no	—
90	C₂×C₄×C₁₂₇	—	2	4	no	—
91	C₂×C₄×C₁₂₇	—	2	4	no	—
92	C₂×C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
93	C₂×C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
94	C₂×C₄×C₁₂₇	C₂×M₁₆	1	4	yes	—
95	C₂×C₄×C₁₂₇	C₂×M₁₆	1	2	yes	—
96	C₂×C₄×C₁₂₇	#131	1	4	yes	—
97	C₂×C₄×C₁₂₇	#131	1	2	yes	—
98	C₂×C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
100	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
101	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
102	C₈×C₁₂₇	—	2	4	no	—
103	C₈×C₁₂₇	—	2	2	no	—
104	C₈×C₁₂₇	—	4	4	no	—
105	C₈×C₁₂₇	#125	1	4	yes	complemented
106	C₂²×C₁₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
107	C₂²×C₁₂₇	C₄×#41	1	2	yes	—
108	C₂²×C₁₂₇	C₂²×M₁₆	1	1	yes	characteristic
109	C₂²×C₁₂₇	C₂×#131	1	1	yes	characteristic
110	C₂²×C₁₂₇	C₂×#131	1	2	yes	—
111	C₄×C₁₂₇	—	2	2	no	—
112	C₄×C₁₂₇	—	2	4	no	—
113	C₄×C₁₂₇	—	2	4	no	—
114	C₄×C₁₂₇	—	4	4	no	—
115	C₄×C₁₂₇	—	4	4	no	—
116	C₄×C₁₂₇	#446	1	4	yes	—
117	C₄×C₁₂₇	C₂×#125	1	1	yes	characteristic
118	C₄×C₁₂₇	#529	1	2	yes	—
119	C₄×C₁₂₇	#533	1	1	yes	characteristic
120	#21673	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂×C₁₂₇	C₄×#125	1	1	yes	characteristic
122	C₂×C₁₂₇	C₂×#446	1	1	yes	characteristic
123	C₂×C₁₂₇	#2960	1	1	yes	characteristic
124	C₂×C₁₂₇	#2965	1	2	yes	—
125	C₂×C₁₂₇	#2966	1	2	yes	—
126	C₄×#125	C₂×C₁₂₇	1	1	yes	characteristic
127	#1939	C₂×C₁₂₇	1	2	yes	—
128	#1916	C₂×C₁₂₇	1	4	yes	—
129	#1831	C₂×C₁₂₇	1	1	yes	characteristic
130	#1771	C₂×C₁₂₇	1	4	yes	—
131	#1770	C₂×C₁₂₇	1	2	yes	—
132	C₄²×C₈	C₂×C₁₂₇	1	1	yes	characteristic
133	C₁₂₇	#21673	1	1	yes	characteristic, direct product factor, Sylow
134	C₂×#125	C₄×C₁₂₇	1	1	yes	characteristic
135	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
136	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	4	yes	—
137	C₂×#97	C₂²×C₁₂₇	1	4	yes	—
138	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
139	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
141	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
142	#389	C₄×C₁₂₇	1	2	yes	—
143	#387	C₄×C₁₂₇	1	4	yes	—
144	#382	C₄×C₁₂₇	1	1	yes	characteristic
145	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	4	yes	—
146	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
147	C₄³	C₂²×C₁₂₇	1	1	yes	characteristic
148	#336	C₂²×C₁₂₇	1	4	yes	—
149	#336	C₂²×C₁₂₇	1	4	yes	—
150	#336	C₂²×C₁₂₇	1	4	yes	—
151	C₂²×C₈	C₂³×C₁₂₇	1	4	yes	—
152	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
153	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
154	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
155	#125	—	2	4	no	—
156	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
157	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
158	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
159	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
160	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
161	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
162	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
163	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
164	C₂×C₄²	C₂³×C₁₂₇	1	4	yes	—
165	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
166	C₄⋊C₈	—	2	8	no	—
167	C₄⋊C₈	—	2	4	no	—
168	#97	—	2	8	no	—
169	C₄×C₈	—	2	4	no	—
170	C₄×C₈	—	2	4	no	—
171	C₄×C₈	—	2	2	no	—
172	C₄×C₈	Q₈×C₁₂₇	1	4	yes	—
173	#95	C₂×C₄×C₁₂₇	1	4	yes	—
174	#95	C₂×C₄×C₁₂₇	1	4	yes	—
175	#95	C₂×C₄×C₁₂₇	1	4	yes	—
176	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
177	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
178	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
179	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
180	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
181	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
182	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	C₂×Q₈×C₁₂₇	1	2	yes	—
190	C₂×C₈	C₂×Q₈×C₁₂₇	1	2	yes	—
191	C₂×C₈	#41×C₁₂₇	1	4	yes	—
192	C₂×C₈	#41×C₁₂₇	1	4	yes	—
193	C₄⋊C₄	—	2	8	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	2	8	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄²	—	2	4	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	4	no	—
202	C₄²	—	2	4	no	—
203	C₄²	M₁₆×C₁₂₇	1	4	yes	—
204	C₄²	C₂×Q₈×C₁₂₇	1	2	yes	—
205	C₄²	#41×C₁₂₇	1	2	yes	—
206	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	C₄×Q₈×C₁₂₇	1	2	yes	—
219	C₂×C₄	C₄×Q₈×C₁₂₇	1	2	yes	—
220	C₂×C₄	C₂×M₁₆×C₁₂₇	1	4	yes	—
221	C₂×C₄	C₂×M₁₆×C₁₂₇	1	2	yes	—
222	C₂×C₄	#131×C₁₂₇	1	4	yes	—
223	C₂×C₄	#131×C₁₂₇	1	2	yes	—
224	C₂×C₄	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
225	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
226	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
227	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
228	C₈	—	2	4	no	—
229	C₈	—	2	2	no	—
230	C₈	—	4	4	no	—
231	C₈	#125×C₁₂₇	1	4	yes	complemented
232	C₂²	C₂×C₄×Q₈×C₁₂₇	1	1	yes	characteristic
233	C₂²	C₄×#41×C₁₂₇	1	2	yes	—
234	C₂²	C₂²×M₁₆×C₁₂₇	1	1	yes	characteristic
235	C₂²	C₂×#131×C₁₂₇	1	1	yes	characteristic
236	C₂²	C₂×#131×C₁₂₇	1	2	yes	—
237	C₄	—	2	2	no	—
238	C₄	—	2	4	no	—
239	C₄	—	2	4	no	—
240	C₄	—	4	4	no	—
241	C₄	—	4	4	no	—
242	C₄	#446×C₁₂₇	1	4	yes	—
243	C₄	C₂×#125×C₁₂₇	1	1	yes	characteristic
244	C₄	#529×C₁₂₇	1	2	yes	—
245	C₄	#533×C₁₂₇	1	1	yes	characteristic
246	C₂	C₄×#125×C₁₂₇	1	1	yes	characteristic
247	C₂	C₂×#446×C₁₂₇	1	1	yes	characteristic
248	C₂	C₁₂₇×#2960	1	1	yes	characteristic
249	C₂	C₁₂₇×#2965	1	2	yes	—
250	C₂	C₁₂₇×#2966	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
105. C₈×C₁₂₇	#125	semidirect
120. #21673	C₁₂₇	direct (N × H)
133. C₁₂₇	#21673	direct (N × H)
231. C₈	#125×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×#125×C₁₂₇	2
C₁₂₇×#1939	2
C₁₂₇×#1916	2
C₁₂₇×#1831	2
C₁₂₇×#1771	2
C₁₂₇×#1770	2
C₄²×C₈×C₁₂₇	2
#21673	127

Subgroups — C127×#21673 #44585346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21673 #44585346