Subgroups — C₁₂₇×#21736 #44585409

Order 32512.

670 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3162	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#2961	C₂	1	2	yes	maximal
3	C₁₂₇×#1994	C₂	1	2	yes	maximal
4	C₁₂₇×#1946	C₂	1	2	yes	maximal
5	C₁₂₇×#1945	C₂	1	2	yes	maximal
6	C₁₂₇×#1849	C₂	1	1	yes	maximal, characteristic
7	C₁₂₇×#1773	C₂	1	2	yes	maximal
8	C₁₂₇×#1772	C₂	1	2	yes	maximal
9	C₄×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic
10	#533×C₁₂₇	C₂²	1	2	yes	—
11	#501×C₁₂₇	C₄	1	8	yes	complemented
12	#446×C₁₂₇	—	2	8	no	—
13	C₂×#108×C₁₂₇	C₂²	1	2	yes	—
14	C₂×#108×C₁₂₇	C₂²	1	2	yes	—
15	#427×C₁₂₇	C₂²	1	2	yes	—
16	#427×C₁₂₇	C₂²	1	2	yes	—
17	#424×C₁₂₇	C₂²	1	4	yes	—
18	#423×C₁₂₇	C₂²	1	2	yes	—
19	#405×C₁₂₇	C₂²	1	4	yes	—
20	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
26	C₄³×C₁₂₇	C₂²	1	1	yes	characteristic
27	#339×C₁₂₇	C₂²	1	4	yes	—
28	#339×C₁₂₇	C₂²	1	4	yes	—
29	C₂×M₁₆×C₁₂₇	C₂³	1	4	yes	—
30	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈×C₁₂₇	C₂×C₄	1	4	yes	—
35	#125×C₁₂₇	C₂×C₄	1	4	yes	—
36	C₂²⋊Q₈×C₁₂₇	—	2	4	no	—
37	C₄×Q₈×C₁₂₇	—	2	4	no	—
38	#117×C₁₂₇	C₂³	1	4	yes	—
39	C₂×C₄²×C₁₂₇	—	2	2	no	—
40	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
43	#108×C₁₂₇	—	2	4	no	—
44	#108×C₁₂₇	—	2	4	no	—
45	#107×C₁₂₇	C₂×C₄	1	4	yes	—
46	#107×C₁₂₇	C₂×C₄	1	4	yes	—
47	#106×C₁₂₇	C₂×C₄	1	4	yes	—
48	#106×C₁₂₇	C₂×C₄	1	4	yes	—
49	C₄⋊C₈×C₁₂₇	—	2	8	no	—
50	C₄⋊C₈×C₁₂₇	—	2	4	no	—
51	#97×C₁₂₇	—	2	4	no	—
52	#97×C₁₂₇	—	2	2	no	—
53	#97×C₁₂₇	C₂×C₄	1	4	yes	—
54	#97×C₁₂₇	D₄	1	4	yes	—
55	#97×C₁₂₇	Q₈	1	4	yes	complemented
56	C₄×C₈×C₁₂₇	—	2	4	no	—
57	C₄×C₈×C₁₂₇	—	2	2	no	—
58	C₄×C₈×C₁₂₇	Q₈	1	4	yes	complemented
59	C₂×Q₈×C₁₂₇	—	2	4	no	—
60	C₂²×C₄×C₁₂₇	—	2	1	no	—
61	C₂²×C₄×C₁₂₇	—	2	1	no	—
62	C₂²×C₄×C₁₂₇	#41	1	2	yes	—
63	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
64	M₁₆×C₁₂₇	—	2	4	no	—
65	M₁₆×C₁₂₇	—	2	4	no	—
66	C₂×C₈×C₁₂₇	—	2	1	no	—
67	C₂×C₈×C₁₂₇	—	2	1	no	—
68	C₂×C₈×C₁₂₇	—	2	4	no	—
69	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
75	C₂×C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
76	C₂×C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
77	C₂×C₈×C₁₂₇	#41	1	2	yes	—
78	C₂×C₈×C₁₂₇	#41	1	2	yes	—
79	C₄⋊C₄×C₁₂₇	—	2	8	no	—
80	C₄⋊C₄×C₁₂₇	—	2	4	no	—
81	C₄⋊C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
82	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
83	C₄²×C₁₂₇	—	2	4	no	—
84	C₄²×C₁₂₇	—	2	1	no	—
85	C₄²×C₁₂₇	—	2	1	no	—
86	C₄²×C₁₂₇	—	4	4	no	—
87	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
88	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
89	C₄²×C₁₂₇	C₂×Q₈	1	2	yes	—
90	C₄²×C₁₂₇	C₂×Q₈	1	2	yes	—
91	C₂³×C₁₂₇	C₂×#41	1	1	yes	characteristic
92	Q₈×C₁₂₇	—	8	4	no	—
93	C₂×C₄×C₁₂₇	—	2	4	no	—
94	C₂×C₄×C₁₂₇	—	2	2	no	—
95	C₂×C₄×C₁₂₇	—	2	8	no	—
96	C₂×C₄×C₁₂₇	—	2	2	no	—
97	C₂×C₄×C₁₂₇	—	4	2	no	—
98	C₂×C₄×C₁₂₇	—	4	2	no	—
99	C₂×C₄×C₁₂₇	C₄×D₄	1	4	yes	—
100	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
106	C₈×C₁₂₇	—	2	2	no	—
107	C₈×C₁₂₇	—	4	4	no	—
108	C₈×C₁₂₇	C₄×Q₈	1	4	yes	—
109	C₈×C₁₂₇	C₄×Q₈	1	4	yes	—
110	C₈×C₁₂₇	C₄⋊Q₈	1	4	yes	—
111	C₂²×C₁₂₇	—	2	2	no	—
112	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
113	C₂²×C₁₂₇	#533	1	1	yes	characteristic
114	C₂²×C₁₂₇	#533	1	1	yes	characteristic
115	C₄×C₁₂₇	—	2	4	no	—
116	C₄×C₁₂₇	—	2	4	no	—
117	C₄×C₁₂₇	—	4	2	no	—
118	C₄×C₁₂₇	—	4	8	no	—
119	C₄×C₁₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
120	C₄×C₁₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
121	C₄×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₄×C₁₂₇	#532	1	1	yes	characteristic
123	#21736	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂×C₁₂₇	—	2	1	no	—
125	C₂×C₁₂₇	—	2	1	no	—
126	C₂×C₁₂₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
127	C₂×C₁₂₇	#2956	1	2	yes	—
128	#3162	C₂×C₁₂₇	1	1	yes	characteristic
129	#2961	C₂×C₁₂₇	1	2	yes	—
130	#1994	C₂×C₁₂₇	1	2	yes	—
131	#1946	C₂×C₁₂₇	1	2	yes	—
132	#1945	C₂×C₁₂₇	1	2	yes	—
133	#1849	C₂×C₁₂₇	1	1	yes	characteristic
134	#1773	C₂×C₁₂₇	1	2	yes	—
135	#1772	C₂×C₁₂₇	1	2	yes	—
136	C₄×#97	C₂×C₁₂₇	1	1	yes	characteristic
137	C₁₂₇	#21736	1	1	yes	characteristic, direct product factor, Sylow
138	#533	C₂²×C₁₂₇	1	2	yes	—
139	#501	C₄×C₁₂₇	1	8	yes	complemented
140	#446	—	2	8	no	—
141	C₂×#108	C₂²×C₁₂₇	1	2	yes	—
142	C₂×#108	C₂²×C₁₂₇	1	2	yes	—
143	#427	C₂²×C₁₂₇	1	2	yes	—
144	#427	C₂²×C₁₂₇	1	2	yes	—
145	#424	C₂²×C₁₂₇	1	4	yes	—
146	#423	C₂²×C₁₂₇	1	2	yes	—
147	#405	C₂²×C₁₂₇	1	4	yes	—
148	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
149	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
150	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
151	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
154	C₄³	C₂²×C₁₂₇	1	1	yes	characteristic
155	#339	C₂²×C₁₂₇	1	4	yes	—
156	#339	C₂²×C₁₂₇	1	4	yes	—
157	C₂×M₁₆	C₂³×C₁₂₇	1	4	yes	—
158	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
159	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
160	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
161	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
162	C₄⋊Q₈	C₂×C₄×C₁₂₇	1	4	yes	—
163	#125	C₂×C₄×C₁₂₇	1	4	yes	—
164	C₂²⋊Q₈	—	2	4	no	—
165	C₄×Q₈	—	2	4	no	—
166	#117	C₂³×C₁₂₇	1	4	yes	—
167	C₂×C₄²	—	2	2	no	—
168	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
169	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
170	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
171	#108	—	2	4	no	—
172	#108	—	2	4	no	—
173	#107	C₂×C₄×C₁₂₇	1	4	yes	—
174	#107	C₂×C₄×C₁₂₇	1	4	yes	—
175	#106	C₂×C₄×C₁₂₇	1	4	yes	—
176	#106	C₂×C₄×C₁₂₇	1	4	yes	—
177	C₄⋊C₈	—	2	8	no	—
178	C₄⋊C₈	—	2	4	no	—
179	#97	—	2	4	no	—
180	#97	—	2	2	no	—
181	#97	C₂×C₄×C₁₂₇	1	4	yes	—
182	#97	D₄×C₁₂₇	1	4	yes	—
183	#97	Q₈×C₁₂₇	1	4	yes	complemented
184	C₄×C₈	—	2	4	no	—
185	C₄×C₈	—	2	2	no	—
186	C₄×C₈	Q₈×C₁₂₇	1	4	yes	complemented
187	C₂×Q₈	—	2	4	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	#41×C₁₂₇	1	2	yes	—
191	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
192	M₁₆	—	2	4	no	—
193	M₁₆	—	2	4	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
198	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
199	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
200	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
201	C₂×C₈	C₂×Q₈×C₁₂₇	1	2	yes	—
202	C₂×C₈	C₂×Q₈×C₁₂₇	1	2	yes	—
203	C₂×C₈	C₂×Q₈×C₁₂₇	1	2	yes	—
204	C₂×C₈	C₂×Q₈×C₁₂₇	1	2	yes	—
205	C₂×C₈	#41×C₁₂₇	1	2	yes	—
206	C₂×C₈	#41×C₁₂₇	1	2	yes	—
207	C₄⋊C₄	—	2	8	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	C₂²×C₄×C₁₂₇	1	8	yes	—
210	C₂²⋊C₄	—	2	4	no	—
211	C₄²	—	2	4	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	4	4	no	—
215	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
216	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
217	C₄²	C₂×Q₈×C₁₂₇	1	2	yes	—
218	C₄²	C₂×Q₈×C₁₂₇	1	2	yes	—
219	C₂³	C₂×#41×C₁₂₇	1	1	yes	characteristic
220	Q₈	—	8	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	C₄×D₄×C₁₂₇	1	4	yes	—
228	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
229	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
230	C₂×C₄	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
231	C₂×C₄	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
232	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
233	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
234	C₈	—	2	2	no	—
235	C₈	—	4	4	no	—
236	C₈	C₄×Q₈×C₁₂₇	1	4	yes	—
237	C₈	C₄×Q₈×C₁₂₇	1	4	yes	—
238	C₈	C₄⋊Q₈×C₁₂₇	1	4	yes	—
239	C₂²	—	2	2	no	—
240	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
241	C₂²	#533×C₁₂₇	1	1	yes	characteristic
242	C₂²	#533×C₁₂₇	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	—	2	4	no	—
245	C₄	—	4	2	no	—
246	C₄	—	4	8	no	—
247	C₄	C₂×C₄×Q₈×C₁₂₇	1	1	yes	characteristic
248	C₄	C₂×C₄×Q₈×C₁₂₇	1	1	yes	characteristic
249	C₄	C₂×C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
250	C₄	#532×C₁₂₇	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	C₄×C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
254	C₂	C₁₂₇×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂×C₄×D₄
prime radical	C₂³×C₁₂₇	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×#97×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
11. #501×C₁₂₇	C₄	semidirect
55. #97×C₁₂₇	Q₈	semidirect
58. C₄×C₈×C₁₂₇	Q₈	semidirect
123. #21736	C₁₂₇	direct (N × H)
137. C₁₂₇	#21736	direct (N × H)
139. #501	C₄×C₁₂₇	semidirect
183. #97	Q₈×C₁₂₇	semidirect
186. C₄×C₈	Q₈×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#3162	2
C₁₂₇×#2961	2
C₁₂₇×#1994	2
C₁₂₇×#1946	2
C₁₂₇×#1945	2
C₁₂₇×#1849	2
C₁₂₇×#1773	2
C₁₂₇×#1772	2
C₄×#97×C₁₂₇	2
#21736	127

Subgroups — C127×#21736 #44585409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21736 #44585409