Subgroups — C₁₂₇×#21708 #44585381

Order 32512.

1006 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3150	C₂	1	1	yes	maximal, characteristic
2	C₂×#443×C₁₂₇	C₂	1	4	yes	maximal, complemented
3	C₁₂₇×#2931	C₂	1	2	yes	maximal, complemented
4	C₁₂₇×#1975	C₂	1	4	yes	maximal, complemented
5	C₁₂₇×#1963	C₂	1	2	yes	maximal, complemented
6	C₁₂₇×#1849	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₂₇×#1726	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₁₂₇	C₂²	1	4	yes	—
9	C₂×#131×C₁₂₇	C₂²	1	4	yes	—
10	#532×C₁₂₇	C₂²	1	2	yes	—
11	#497×C₁₂₇	C₄	1	4	yes	complemented
12	#496×C₁₂₇	C₄	1	4	yes	complemented
13	#443×C₁₂₇	—	2	16	no	—
14	#436×C₁₂₇	—	2	4	no	—
15	C₈×D₄×C₁₂₇	—	2	4	no	—
16	C₂×#108×C₁₂₇	C₂²	1	4	yes	—
17	#423×C₁₂₇	C₂²	1	2	yes	—
18	C₂×#104×C₁₂₇	C₂²	1	8	yes	—
19	#408×C₁₂₇	C₂²	1	4	yes	—
20	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
21	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
24	#376×C₁₂₇	C₂²	1	1	yes	characteristic
25	C₂×#41×C₁₂₇	C₂³	1	4	yes	—
26	#135×C₁₂₇	—	2	8	no	—
27	#135×C₁₂₇	—	2	8	no	—
28	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
29	C₂×SD₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
30	C₂×D₈×C₁₂₇	C₂×C₄	1	4	yes	—
31	#131×C₁₂₇	—	2	16	no	—
32	C₂×M₁₆×C₁₂₇	—	2	4	no	—
33	C₂×M₁₆×C₁₂₇	C₂³	1	4	yes	—
34	C₂²×C₈×C₁₂₇	—	2	4	no	—
35	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
38	C₄⋊Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
39	#127×C₁₂₇	C₂×C₄	1	2	yes	—
40	#124×C₁₂₇	C₂×C₄	1	4	yes	—
41	#121×C₁₂₇	—	2	4	no	—
42	C₄×D₄×C₁₂₇	—	2	4	no	—
43	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
45	#108×C₁₂₇	—	2	8	no	—
46	C₄⋊C₈×C₁₂₇	—	2	4	no	—
47	#104×C₁₂₇	—	2	16	no	—
48	C₂²⋊C₈×C₁₂₇	—	2	8	no	—
49	#97×C₁₂₇	—	2	4	no	—
50	#97×C₁₂₇	C₂×C₄	1	4	yes	—
51	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
52	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
53	C₄×C₈×C₁₂₇	D₄	1	8	yes	complemented
54	#95×C₁₂₇	—	2	2	no	—
55	#41×C₁₂₇	—	2	8	no	—
56	#41×C₁₂₇	—	2	8	no	—
57	C₂×Q₈×C₁₂₇	C₂²×C₄	1	4	yes	—
58	C₂×D₄×C₁₂₇	—	2	4	no	—
59	C₂×D₄×C₁₂₇	—	2	4	no	—
60	C₂×D₄×C₁₂₇	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₁₂₇	—	2	2	no	—
62	C₂²×C₄×C₁₂₇	—	2	4	no	—
63	C₂²×C₄×C₁₂₇	#41	1	2	yes	—
64	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
65	Q₁₆×C₁₂₇	—	2	8	no	—
66	SD₁₆×C₁₂₇	—	2	16	no	—
67	D₈×C₁₂₇	—	2	8	no	—
68	M₁₆×C₁₂₇	—	2	8	no	—
69	M₁₆×C₁₂₇	—	4	8	no	—
70	C₂×C₈×C₁₂₇	—	2	2	no	—
71	C₂×C₈×C₁₂₇	—	2	4	no	—
72	C₂×C₈×C₁₂₇	—	4	8	no	—
73	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
76	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
78	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₂₇	#41	1	4	yes	—
80	C₄⋊C₄×C₁₂₇	—	2	4	no	—
81	C₂²⋊C₄×C₁₂₇	—	2	8	no	—
82	C₄²×C₁₂₇	—	2	2	no	—
83	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
84	C₄²×C₁₂₇	C₂×D₄	1	4	yes	—
85	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
86	C₂³×C₁₂₇	—	2	4	no	—
87	C₂³×C₁₂₇	C₂×#41	1	1	yes	characteristic
88	Q₈×C₁₂₇	—	2	8	no	—
89	D₄×C₁₂₇	—	2	8	no	—
90	D₄×C₁₂₇	—	4	8	no	—
91	D₄×C₁₂₇	—	4	8	no	—
92	C₂×C₄×C₁₂₇	—	2	4	no	—
93	C₂×C₄×C₁₂₇	—	2	4	no	—
94	C₂×C₄×C₁₂₇	—	4	2	no	—
95	C₂×C₄×C₁₂₇	—	4	4	no	—
96	C₂×C₄×C₁₂₇	—	4	4	no	—
97	C₂×C₄×C₁₂₇	—	4	4	no	—
98	C₂×C₄×C₁₂₇	C₄×D₄	1	4	yes	—
99	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
101	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
103	C₈×C₁₂₇	—	2	2	no	—
104	C₈×C₁₂₇	—	4	4	no	—
105	C₈×C₁₂₇	C₄×D₄	1	8	yes	—
106	C₈×C₁₂₇	#127	1	2	yes	—
107	C₈×C₁₂₇	#127	1	2	yes	—
108	C₂²×C₁₂₇	—	2	2	no	—
109	C₂²×C₁₂₇	—	4	4	no	—
110	C₂²×C₁₂₇	—	4	8	no	—
111	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
112	C₂²×C₁₂₇	#532	1	2	yes	—
113	C₄×C₁₂₇	—	2	4	no	—
114	C₄×C₁₂₇	—	4	4	no	—
115	C₄×C₁₂₇	—	4	4	no	—
116	C₄×C₁₂₇	C₂×C₄×D₄	1	2	yes	—
117	C₄×C₁₂₇	C₂×#127	1	1	yes	characteristic
118	C₄×C₁₂₇	#532	1	1	yes	characteristic
119	#21708	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂×C₁₂₇	—	2	2	no	—
121	C₂×C₁₂₇	—	8	4	no	—
122	C₂×C₁₂₇	C₄×#127	1	1	yes	characteristic
123	C₂×C₁₂₇	#2955	1	2	yes	—
124	#3150	C₂×C₁₂₇	1	1	yes	characteristic
125	C₂×#443	C₂×C₁₂₇	1	4	yes	complemented
126	#2931	C₂×C₁₂₇	1	2	yes	complemented
127	#1975	C₂×C₁₂₇	1	4	yes	complemented
128	#1963	C₂×C₁₂₇	1	2	yes	complemented
129	#1849	C₂×C₁₂₇	1	1	yes	characteristic, complemented
130	#1726	C₂×C₁₂₇	1	1	yes	characteristic, complemented
131	C₁₂₇	#21708	1	1	yes	characteristic, direct product factor, Sylow
132	C₂×#135	C₂²×C₁₂₇	1	4	yes	—
133	C₂×#131	C₂²×C₁₂₇	1	4	yes	—
134	#532	C₂²×C₁₂₇	1	2	yes	—
135	#497	C₄×C₁₂₇	1	4	yes	complemented
136	#496	C₄×C₁₂₇	1	4	yes	complemented
137	#443	—	2	16	no	—
138	#436	—	2	4	no	—
139	C₈×D₄	—	2	4	no	—
140	C₂×#108	C₂²×C₁₂₇	1	4	yes	—
141	#423	C₂²×C₁₂₇	1	2	yes	—
142	C₂×#104	C₂²×C₁₂₇	1	8	yes	—
143	#408	C₂²×C₁₂₇	1	4	yes	—
144	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
145	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
148	#376	C₂²×C₁₂₇	1	1	yes	characteristic
149	C₂×#41	C₂³×C₁₂₇	1	4	yes	—
150	#135	—	2	8	no	—
151	#135	—	2	8	no	—
152	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
153	C₂×SD₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
154	C₂×D₈	C₂×C₄×C₁₂₇	1	4	yes	—
155	#131	—	2	16	no	—
156	C₂×M₁₆	—	2	4	no	—
157	C₂×M₁₆	C₂³×C₁₂₇	1	4	yes	—
158	C₂²×C₈	—	2	4	no	—
159	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
160	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
161	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
162	C₄⋊Q₈	C₂×C₄×C₁₂₇	1	2	yes	—
163	#127	C₂×C₄×C₁₂₇	1	2	yes	—
164	#124	C₂×C₄×C₁₂₇	1	4	yes	—
165	#121	—	2	4	no	—
166	C₄×D₄	—	2	4	no	—
167	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
168	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
169	#108	—	2	8	no	—
170	C₄⋊C₈	—	2	4	no	—
171	#104	—	2	16	no	—
172	C₂²⋊C₈	—	2	8	no	—
173	#97	—	2	4	no	—
174	#97	C₂×C₄×C₁₂₇	1	4	yes	—
175	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
176	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
177	C₄×C₈	D₄×C₁₂₇	1	8	yes	complemented
178	#95	—	2	2	no	—
179	#41	—	2	8	no	—
180	#41	—	2	8	no	—
181	C₂×Q₈	C₂²×C₄×C₁₂₇	1	4	yes	—
182	C₂×D₄	—	2	4	no	—
183	C₂×D₄	—	2	4	no	—
184	C₂×D₄	C₂²×C₄×C₁₂₇	1	4	yes	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	4	no	—
187	C₂²×C₄	#41×C₁₂₇	1	2	yes	—
188	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
189	Q₁₆	—	2	8	no	—
190	SD₁₆	—	2	16	no	—
191	D₈	—	2	8	no	—
192	M₁₆	—	2	8	no	—
193	M₁₆	—	4	8	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	4	8	no	—
197	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
198	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
199	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
200	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
201	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
202	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
203	C₂×C₈	#41×C₁₂₇	1	4	yes	—
204	C₄⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	2	8	no	—
206	C₄²	—	2	2	no	—
207	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
208	C₄²	C₂×D₄×C₁₂₇	1	4	yes	—
209	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
210	C₂³	—	2	4	no	—
211	C₂³	C₂×#41×C₁₂₇	1	1	yes	characteristic
212	Q₈	—	2	8	no	—
213	D₄	—	2	8	no	—
214	D₄	—	4	8	no	—
215	D₄	—	4	8	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	C₄×D₄×C₁₂₇	1	4	yes	—
223	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
224	C₂×C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
225	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
226	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
227	C₈	—	2	2	no	—
228	C₈	—	4	4	no	—
229	C₈	C₄×D₄×C₁₂₇	1	8	yes	—
230	C₈	#127×C₁₂₇	1	2	yes	—
231	C₈	#127×C₁₂₇	1	2	yes	—
232	C₂²	—	2	2	no	—
233	C₂²	—	4	8	no	—
234	C₂²	—	4	4	no	—
235	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
236	C₂²	#532×C₁₂₇	1	2	yes	—
237	C₄	—	2	4	no	—
238	C₄	—	4	4	no	—
239	C₄	—	4	4	no	—
240	C₄	C₂×C₄×D₄×C₁₂₇	1	2	yes	—
241	C₄	C₂×#127×C₁₂₇	1	1	yes	characteristic
242	C₄	#532×C₁₂₇	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	—	8	4	no	—
245	C₂	C₄×#127×C₁₂₇	1	1	yes	characteristic
246	C₂	C₁₂₇×#2955	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₂₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂×C₄×D₄
prime radical	C₁₂₇×#3150	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#443×C₁₂₇	C₂	semidirect
3. C₁₂₇×#2931	C₂	semidirect
4. C₁₂₇×#1975	C₂	semidirect
5. C₁₂₇×#1963	C₂	semidirect
6. C₁₂₇×#1849	C₂	semidirect
7. C₁₂₇×#1726	C₂	semidirect
11. #497×C₁₂₇	C₄	semidirect
12. #496×C₁₂₇	C₄	semidirect
53. C₄×C₈×C₁₂₇	D₄	semidirect
119. #21708	C₁₂₇	direct (N × H)
125. C₂×#443	C₂×C₁₂₇	semidirect
126. #2931	C₂×C₁₂₇	semidirect
127. #1975	C₂×C₁₂₇	semidirect
128. #1963	C₂×C₁₂₇	semidirect
129. #1849	C₂×C₁₂₇	semidirect
130. #1726	C₂×C₁₂₇	semidirect
131. C₁₂₇	#21708	direct (N × H)
135. #497	C₄×C₁₂₇	semidirect
136. #496	C₄×C₁₂₇	semidirect
177. C₄×C₈	D₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#3150	2
C₂×#443×C₁₂₇	2
C₁₂₇×#2931	2
C₁₂₇×#1975	2
C₁₂₇×#1963	2
C₁₂₇×#1849	2
C₁₂₇×#1726	2
#21708	127

Subgroups — C127×#21708 #44585381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21708 #44585381