Subgroups — C₁₂₇×#21709 #44585382

Order 32512.

1006 conjugacy classes of subgroups.

The subgroup lattice has 242 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×#444×C₁₂₇	C₂	1	4	yes	maximal, complemented
3	C₁₂₇×#2931	C₂	1	2	yes	maximal, complemented
4	C₁₂₇×#1974	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	#444×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	1	yes	characteristic
21	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
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	C₂×C₄	1	4	yes	—
50	#97×C₁₂₇	D₄	1	8	yes	complemented
51	C₄×C₈×C₁₂₇	—	2	4	no	—
52	C₄×C₈×C₁₂₇	D₄	1	4	yes	—
53	#95×C₁₂₇	—	2	2	no	—
54	#41×C₁₂₇	—	2	8	no	—
55	#41×C₁₂₇	—	2	8	no	—
56	C₂×Q₈×C₁₂₇	C₂²×C₄	1	4	yes	—
57	C₂×D₄×C₁₂₇	—	2	4	no	—
58	C₂×D₄×C₁₂₇	—	2	4	no	—
59	C₂×D₄×C₁₂₇	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₁₂₇	—	2	2	no	—
61	C₂²×C₄×C₁₂₇	—	2	4	no	—
62	C₂²×C₄×C₁₂₇	#41	1	2	yes	—
63	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
64	Q₁₆×C₁₂₇	—	2	8	no	—
65	SD₁₆×C₁₂₇	—	2	16	no	—
66	D₈×C₁₂₇	—	2	8	no	—
67	M₁₆×C₁₂₇	—	2	8	no	—
68	M₁₆×C₁₂₇	—	4	8	no	—
69	C₂×C₈×C₁₂₇	—	2	2	no	—
70	C₂×C₈×C₁₂₇	—	2	4	no	—
71	C₂×C₈×C₁₂₇	—	4	8	no	—
72	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
75	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
76	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₁₂₇	#41	1	4	yes	—
78	C₄⋊C₄×C₁₂₇	—	2	4	no	—
79	C₂²⋊C₄×C₁₂₇	—	2	8	no	—
80	C₄²×C₁₂₇	—	2	2	no	—
81	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
82	C₄²×C₁₂₇	C₂×D₄	1	4	yes	—
83	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
84	C₂³×C₁₂₇	—	2	4	no	—
85	C₂³×C₁₂₇	C₂×#41	1	1	yes	characteristic
86	Q₈×C₁₂₇	—	2	8	no	—
87	D₄×C₁₂₇	—	2	8	no	—
88	D₄×C₁₂₇	—	4	8	no	—
89	D₄×C₁₂₇	—	4	8	no	—
90	C₂×C₄×C₁₂₇	—	2	4	no	—
91	C₂×C₄×C₁₂₇	—	2	4	no	—
92	C₂×C₄×C₁₂₇	—	4	2	no	—
93	C₂×C₄×C₁₂₇	—	4	4	no	—
94	C₂×C₄×C₁₂₇	—	4	4	no	—
95	C₂×C₄×C₁₂₇	—	4	4	no	—
96	C₂×C₄×C₁₂₇	C₄×D₄	1	4	yes	—
97	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
99	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
101	C₈×C₁₂₇	—	2	2	no	—
102	C₈×C₁₂₇	—	4	4	no	—
103	C₈×C₁₂₇	C₄×D₄	1	8	yes	—
104	C₈×C₁₂₇	#127	1	4	yes	—
105	C₂²×C₁₂₇	—	2	2	no	—
106	C₂²×C₁₂₇	—	4	4	no	—
107	C₂²×C₁₂₇	—	4	8	no	—
108	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
109	C₂²×C₁₂₇	#532	1	2	yes	—
110	C₄×C₁₂₇	—	2	4	no	—
111	C₄×C₁₂₇	—	4	4	no	—
112	C₄×C₁₂₇	—	4	4	no	—
113	C₄×C₁₂₇	C₂×C₄×D₄	1	2	yes	—
114	C₄×C₁₂₇	C₂×#127	1	1	yes	characteristic
115	C₄×C₁₂₇	#532	1	1	yes	characteristic
116	#21709	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂×C₁₂₇	—	2	2	no	—
118	C₂×C₁₂₇	—	8	4	no	—
119	C₂×C₁₂₇	C₄×#127	1	1	yes	characteristic
120	C₂×C₁₂₇	#2955	1	2	yes	—
121	#3150	C₂×C₁₂₇	1	1	yes	characteristic
122	C₂×#444	C₂×C₁₂₇	1	4	yes	complemented
123	#2931	C₂×C₁₂₇	1	2	yes	complemented
124	#1974	C₂×C₁₂₇	1	4	yes	complemented
125	#1963	C₂×C₁₂₇	1	2	yes	complemented
126	#1849	C₂×C₁₂₇	1	1	yes	characteristic, complemented
127	#1726	C₂×C₁₂₇	1	1	yes	characteristic, complemented
128	C₁₂₇	#21709	1	1	yes	characteristic, direct product factor, Sylow
129	C₂×#135	C₂²×C₁₂₇	1	4	yes	—
130	C₂×#131	C₂²×C₁₂₇	1	4	yes	—
131	#532	C₂²×C₁₂₇	1	2	yes	—
132	#497	C₄×C₁₂₇	1	4	yes	complemented
133	#496	C₄×C₁₂₇	1	4	yes	complemented
134	#444	—	2	16	no	—
135	#436	—	2	4	no	—
136	C₈×D₄	—	2	4	no	—
137	C₂×#108	C₂²×C₁₂₇	1	4	yes	—
138	#423	C₂²×C₁₂₇	1	2	yes	—
139	C₂×#104	C₂²×C₁₂₇	1	8	yes	—
140	#408	C₂²×C₁₂₇	1	4	yes	—
141	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
142	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
143	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
144	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
145	#376	C₂²×C₁₂₇	1	1	yes	characteristic
146	C₂×#41	C₂³×C₁₂₇	1	4	yes	—
147	#135	—	2	8	no	—
148	#135	—	2	8	no	—
149	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
150	C₂×SD₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
151	C₂×D₈	C₂×C₄×C₁₂₇	1	4	yes	—
152	#131	—	2	16	no	—
153	C₂×M₁₆	—	2	4	no	—
154	C₂×M₁₆	C₂³×C₁₂₇	1	4	yes	—
155	C₂²×C₈	—	2	4	no	—
156	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
157	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
158	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
159	C₄⋊Q₈	C₂×C₄×C₁₂₇	1	2	yes	—
160	#127	C₂×C₄×C₁₂₇	1	2	yes	—
161	#124	C₂×C₄×C₁₂₇	1	4	yes	—
162	#121	—	2	4	no	—
163	C₄×D₄	—	2	4	no	—
164	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
165	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
166	#108	—	2	8	no	—
167	C₄⋊C₈	—	2	4	no	—
168	#104	—	2	16	no	—
169	C₂²⋊C₈	—	2	8	no	—
170	#97	C₂×C₄×C₁₂₇	1	4	yes	—
171	#97	D₄×C₁₂₇	1	8	yes	complemented
172	C₄×C₈	—	2	4	no	—
173	C₄×C₈	D₄×C₁₂₇	1	4	yes	—
174	#95	—	2	2	no	—
175	#41	—	2	8	no	—
176	#41	—	2	8	no	—
177	C₂×Q₈	C₂²×C₄×C₁₂₇	1	4	yes	—
178	C₂×D₄	—	2	4	no	—
179	C₂×D₄	—	2	4	no	—
180	C₂×D₄	C₂²×C₄×C₁₂₇	1	4	yes	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	#41×C₁₂₇	1	2	yes	—
184	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
185	Q₁₆	—	2	8	no	—
186	SD₁₆	—	2	16	no	—
187	D₈	—	2	8	no	—
188	M₁₆	—	2	8	no	—
189	M₁₆	—	4	8	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	4	8	no	—
193	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
194	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
195	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
196	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
197	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
198	C₂×C₈	#41×C₁₂₇	1	4	yes	—
199	C₄⋊C₄	—	2	4	no	—
200	C₂²⋊C₄	—	2	8	no	—
201	C₄²	—	2	2	no	—
202	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
203	C₄²	C₂×D₄×C₁₂₇	1	4	yes	—
204	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
205	C₂³	—	2	4	no	—
206	C₂³	C₂×#41×C₁₂₇	1	1	yes	characteristic
207	Q₈	—	2	8	no	—
208	D₄	—	2	8	no	—
209	D₄	—	4	8	no	—
210	D₄	—	4	8	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	C₄×D₄×C₁₂₇	1	4	yes	—
218	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
219	C₂×C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
220	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
221	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
222	C₈	—	2	2	no	—
223	C₈	—	4	4	no	—
224	C₈	C₄×D₄×C₁₂₇	1	8	yes	—
225	C₈	#127×C₁₂₇	1	4	yes	—
226	C₂²	—	2	2	no	—
227	C₂²	—	4	8	no	—
228	C₂²	—	4	4	no	—
229	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
230	C₂²	#532×C₁₂₇	1	2	yes	—
231	C₄	—	2	4	no	—
232	C₄	—	4	4	no	—
233	C₄	—	4	4	no	—
234	C₄	C₂×C₄×D₄×C₁₂₇	1	2	yes	—
235	C₄	C₂×#127×C₁₂₇	1	1	yes	characteristic
236	C₄	#532×C₁₂₇	1	1	yes	characteristic
237	C₂	—	2	2	no	—
238	C₂	—	8	4	no	—
239	C₂	C₄×#127×C₁₂₇	1	1	yes	characteristic
240	C₂	C₁₂₇×#2955	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×#3150	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×#1974	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
50. #97×C₁₂₇	D₄	semidirect
116. #21709	C₁₂₇	direct (N × H)
122. C₂×#444	C₂×C₁₂₇	semidirect
123. #2931	C₂×C₁₂₇	semidirect
124. #1974	C₂×C₁₂₇	semidirect
125. #1963	C₂×C₁₂₇	semidirect
126. #1849	C₂×C₁₂₇	semidirect
127. #1726	C₂×C₁₂₇	semidirect
128. C₁₂₇	#21709	direct (N × H)
132. #497	C₄×C₁₂₇	semidirect
133. #496	C₄×C₁₂₇	semidirect
171. #97	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×#444×C₁₂₇	2
C₁₂₇×#2931	2
C₁₂₇×#1974	2
C₁₂₇×#1963	2
C₁₂₇×#1849	2
C₁₂₇×#1726	2
#21709	127

Subgroups — C127×#21709 #44585382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21709 #44585382