Subgroups — C₁₂₇×#21706 #44585379

Order 32512.

1022 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3147	C₂	1	1	yes	maximal, characteristic
2	C₂×#444×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×#1962	C₂	1	2	yes	maximal, complemented
6	C₁₂₇×#1848	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#97×C₁₂₇	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	#495×C₁₂₇	C₄	1	4	yes	complemented
12	#494×C₁₂₇	C₄	1	1	yes	characteristic, complemented
13	#493×C₁₂₇	C₄	1	1	yes	characteristic, complemented
14	#492×C₁₂₇	C₄	1	2	yes	complemented
15	#444×C₁₂₇	—	2	16	no	—
16	#436×C₁₂₇	—	2	4	no	—
17	C₈×D₄×C₁₂₇	—	2	4	no	—
18	C₂×#108×C₁₂₇	C₂²	1	4	yes	—
19	#423×C₁₂₇	C₂²	1	2	yes	—
20	C₂×#104×C₁₂₇	C₂²	1	8	yes	—
21	#408×C₁₂₇	C₂²	1	4	yes	—
22	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
23	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
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	C₂×#41×C₁₂₇	C₂³	1	4	yes	—
28	#135×C₁₂₇	—	2	8	no	—
29	#135×C₁₂₇	—	2	8	no	—
30	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
31	C₂×SD₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
32	C₂×D₈×C₁₂₇	C₂×C₄	1	4	yes	—
33	#131×C₁₂₇	—	2	16	no	—
34	C₂×M₁₆×C₁₂₇	—	2	4	no	—
35	C₂×M₁₆×C₁₂₇	C₂³	1	4	yes	—
36	C₂²×C₈×C₁₂₇	—	2	4	no	—
37	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
41	#127×C₁₂₇	C₂×C₄	1	2	yes	—
42	#124×C₁₂₇	C₂×C₄	1	4	yes	—
43	#121×C₁₂₇	—	2	4	no	—
44	C₄×D₄×C₁₂₇	—	2	4	no	—
45	C₂×C₄²×C₁₂₇	—	2	2	no	—
46	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
47	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
48	#108×C₁₂₇	—	2	8	no	—
49	C₄⋊C₈×C₁₂₇	—	2	4	no	—
50	#104×C₁₂₇	—	2	16	no	—
51	C₂²⋊C₈×C₁₂₇	—	2	8	no	—
52	#97×C₁₂₇	—	2	4	no	—
53	#97×C₁₂₇	D₄	1	8	yes	complemented
54	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₁₂₇	D₄	1	4	yes	—
58	#41×C₁₂₇	—	2	8	no	—
59	#41×C₁₂₇	—	2	8	no	—
60	C₂×Q₈×C₁₂₇	C₂²×C₄	1	4	yes	—
61	C₂×D₄×C₁₂₇	—	2	4	no	—
62	C₂×D₄×C₁₂₇	—	2	4	no	—
63	C₂×D₄×C₁₂₇	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₁₂₇	—	2	2	no	—
65	C₂²×C₄×C₁₂₇	—	2	4	no	—
66	C₂²×C₄×C₁₂₇	#41	1	2	yes	—
67	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
68	Q₁₆×C₁₂₇	—	2	8	no	—
69	SD₁₆×C₁₂₇	—	2	16	no	—
70	D₈×C₁₂₇	—	2	8	no	—
71	M₁₆×C₁₂₇	—	2	8	no	—
72	M₁₆×C₁₂₇	—	4	8	no	—
73	C₂×C₈×C₁₂₇	—	2	2	no	—
74	C₂×C₈×C₁₂₇	—	2	4	no	—
75	C₂×C₈×C₁₂₇	—	4	8	no	—
76	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
80	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
81	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₁₂₇	#41	1	4	yes	—
83	C₄⋊C₄×C₁₂₇	—	2	4	no	—
84	C₂²⋊C₄×C₁₂₇	—	2	8	no	—
85	C₄²×C₁₂₇	—	2	2	no	—
86	C₄²×C₁₂₇	—	4	4	no	—
87	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₂₇	C₂×D₄	1	4	yes	—
90	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
91	C₂³×C₁₂₇	—	2	4	no	—
92	C₂³×C₁₂₇	C₂×#41	1	1	yes	characteristic
93	Q₈×C₁₂₇	—	2	8	no	—
94	D₄×C₁₂₇	—	2	8	no	—
95	D₄×C₁₂₇	—	4	8	no	—
96	D₄×C₁₂₇	—	4	8	no	—
97	C₂×C₄×C₁₂₇	—	2	4	no	—
98	C₂×C₄×C₁₂₇	—	2	4	no	—
99	C₂×C₄×C₁₂₇	—	2	4	no	—
100	C₂×C₄×C₁₂₇	—	4	4	no	—
101	C₂×C₄×C₁₂₇	—	4	4	no	—
102	C₂×C₄×C₁₂₇	—	4	4	no	—
103	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
107	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
109	C₈×C₁₂₇	—	2	2	no	—
110	C₈×C₁₂₇	—	4	4	no	—
111	C₈×C₁₂₇	C₄×D₄	1	4	yes	—
112	C₈×C₁₂₇	C₄×D₄	1	4	yes	—
113	C₈×C₁₂₇	#127	1	4	yes	—
114	C₂²×C₁₂₇	—	2	2	no	—
115	C₂²×C₁₂₇	—	4	4	no	—
116	C₂²×C₁₂₇	—	4	8	no	—
117	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
118	C₂²×C₁₂₇	#532	1	2	yes	—
119	C₄×C₁₂₇	—	2	2	no	—
120	C₄×C₁₂₇	—	4	4	no	—
121	C₄×C₁₂₇	—	4	4	no	—
122	C₄×C₁₂₇	#444	1	4	yes	complemented
123	C₄×C₁₂₇	C₂×C₄×D₄	1	2	yes	—
124	C₄×C₁₂₇	C₂×#127	1	1	yes	characteristic
125	C₄×C₁₂₇	#532	1	1	yes	characteristic
126	#21706	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂×C₁₂₇	—	2	2	no	—
128	C₂×C₁₂₇	—	8	4	no	—
129	C₂×C₁₂₇	C₄×#127	1	1	yes	characteristic
130	C₂×C₁₂₇	C₂×#444	1	2	yes	—
131	#3147	C₂×C₁₂₇	1	1	yes	characteristic
132	C₂×#444	C₂×C₁₂₇	1	4	yes	complemented
133	#2931	C₂×C₁₂₇	1	2	yes	complemented
134	#1975	C₂×C₁₂₇	1	4	yes	complemented
135	#1962	C₂×C₁₂₇	1	2	yes	complemented
136	#1848	C₂×C₁₂₇	1	1	yes	characteristic, complemented
137	C₄×#97	C₂×C₁₂₇	1	1	yes	characteristic, complemented
138	C₁₂₇	#21706	1	1	yes	characteristic, direct product factor, Sylow
139	C₂×#135	C₂²×C₁₂₇	1	4	yes	—
140	C₂×#131	C₂²×C₁₂₇	1	4	yes	—
141	#532	C₂²×C₁₂₇	1	2	yes	—
142	#495	C₄×C₁₂₇	1	4	yes	complemented
143	#494	C₄×C₁₂₇	1	1	yes	characteristic, complemented
144	#493	C₄×C₁₂₇	1	1	yes	characteristic, complemented
145	#492	C₄×C₁₂₇	1	2	yes	complemented
146	#444	—	2	16	no	—
147	#436	—	2	4	no	—
148	C₈×D₄	—	2	4	no	—
149	C₂×#108	C₂²×C₁₂₇	1	4	yes	—
150	#423	C₂²×C₁₂₇	1	2	yes	—
151	C₂×#104	C₂²×C₁₂₇	1	8	yes	—
152	#408	C₂²×C₁₂₇	1	4	yes	—
153	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
154	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
155	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
156	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
157	C₄³	C₂²×C₁₂₇	1	1	yes	characteristic
158	C₂×#41	C₂³×C₁₂₇	1	4	yes	—
159	#135	—	2	8	no	—
160	#135	—	2	8	no	—
161	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
162	C₂×SD₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
163	C₂×D₈	C₂×C₄×C₁₂₇	1	4	yes	—
164	#131	—	2	16	no	—
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	C₂³×C₁₂₇	1	4	yes	—
167	C₂²×C₈	—	2	4	no	—
168	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
169	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
170	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
171	C₄⋊Q₈	C₂×C₄×C₁₂₇	1	2	yes	—
172	#127	C₂×C₄×C₁₂₇	1	2	yes	—
173	#124	C₂×C₄×C₁₂₇	1	4	yes	—
174	#121	—	2	4	no	—
175	C₄×D₄	—	2	4	no	—
176	C₂×C₄²	—	2	2	no	—
177	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
178	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
179	#108	—	2	8	no	—
180	C₄⋊C₈	—	2	4	no	—
181	#104	—	2	16	no	—
182	C₂²⋊C₈	—	2	8	no	—
183	#97	—	2	4	no	—
184	#97	D₄×C₁₂₇	1	8	yes	complemented
185	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
186	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
187	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
188	C₄×C₈	D₄×C₁₂₇	1	4	yes	—
189	#41	—	2	8	no	—
190	#41	—	2	8	no	—
191	C₂×Q₈	C₂²×C₄×C₁₂₇	1	4	yes	—
192	C₂×D₄	—	2	4	no	—
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	C₂²×C₄×C₁₂₇	1	4	yes	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	#41×C₁₂₇	1	2	yes	—
198	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
199	Q₁₆	—	2	8	no	—
200	SD₁₆	—	2	16	no	—
201	D₈	—	2	8	no	—
202	M₁₆	—	2	8	no	—
203	M₁₆	—	4	8	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	4	8	no	—
207	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
208	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
209	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
210	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
211	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
212	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
213	C₂×C₈	#41×C₁₂₇	1	4	yes	—
214	C₄⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	—	2	8	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	4	4	no	—
218	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
219	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
220	C₄²	C₂×D₄×C₁₂₇	1	4	yes	—
221	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
222	C₂³	—	2	4	no	—
223	C₂³	C₂×#41×C₁₂₇	1	1	yes	characteristic
224	Q₈	—	2	8	no	—
225	D₄	—	2	8	no	—
226	D₄	—	4	8	no	—
227	D₄	—	4	8	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
235	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
236	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
237	C₂×C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
238	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
239	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
240	C₈	—	2	2	no	—
241	C₈	—	4	4	no	—
242	C₈	C₄×D₄×C₁₂₇	1	4	yes	—
243	C₈	C₄×D₄×C₁₂₇	1	4	yes	—
244	C₈	#127×C₁₂₇	1	4	yes	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	8	no	—
247	C₂²	—	4	4	no	—
248	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
249	C₂²	#532×C₁₂₇	1	2	yes	—
250	C₄	—	2	2	no	—
251	C₄	—	4	4	no	—
252	C₄	—	4	4	no	—
253	C₄	#444×C₁₂₇	1	4	yes	complemented
254	C₄	C₂×C₄×D₄×C₁₂₇	1	2	yes	—
255	C₄	C₂×#127×C₁₂₇	1	1	yes	characteristic
256	C₄	#532×C₁₂₇	1	1	yes	characteristic
257	C₂	—	2	2	no	—
258	C₂	—	8	4	no	—
259	C₂	C₄×#127×C₁₂₇	1	1	yes	characteristic
260	C₂	C₂×#444×C₁₂₇	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	C₁₂₇×#3147	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#444×C₁₂₇	C₂	semidirect
3. C₁₂₇×#2931	C₂	semidirect
4. C₁₂₇×#1975	C₂	semidirect
5. C₁₂₇×#1962	C₂	semidirect
6. C₁₂₇×#1848	C₂	semidirect
7. C₄×#97×C₁₂₇	C₂	semidirect
11. #495×C₁₂₇	C₄	semidirect
12. #494×C₁₂₇	C₄	semidirect
13. #493×C₁₂₇	C₄	semidirect
14. #492×C₁₂₇	C₄	semidirect
53. #97×C₁₂₇	D₄	semidirect
122. C₄×C₁₂₇	#444	semidirect
126. #21706	C₁₂₇	direct (N × H)
132. C₂×#444	C₂×C₁₂₇	semidirect
133. #2931	C₂×C₁₂₇	semidirect
134. #1975	C₂×C₁₂₇	semidirect
135. #1962	C₂×C₁₂₇	semidirect
136. #1848	C₂×C₁₂₇	semidirect
137. C₄×#97	C₂×C₁₂₇	semidirect
138. C₁₂₇	#21706	direct (N × H)
142. #495	C₄×C₁₂₇	semidirect
143. #494	C₄×C₁₂₇	semidirect
144. #493	C₄×C₁₂₇	semidirect
145. #492	C₄×C₁₂₇	semidirect
184. #97	D₄×C₁₂₇	semidirect
253. C₄	#444×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#3147	2
C₂×#444×C₁₂₇	2
C₁₂₇×#2931	2
C₁₂₇×#1975	2
C₁₂₇×#1962	2
C₁₂₇×#1848	2
C₄×#97×C₁₂₇	2
#21706	127

Subgroups — C127×#21706 #44585379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21706 #44585379