Subgroups — C₁₂₇×#21372 #44585045

Order 32512.

714 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2921	C₂	1	1	yes	maximal, characteristic
2	C₂×#433×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#2873	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1950	C₂	1	4	yes	maximal, complemented
5	C₁₂₇×#1855	C₂	1	4	yes	maximal
6	C₁₂₇×#1778	C₂	1	2	yes	maximal, complemented
7	C₄×#108×C₁₂₇	C₂	1	2	yes	maximal, complemented
8	C₂²×M₁₆×C₁₂₇	C₂²	1	2	yes	—
9	C₂×#117×C₁₂₇	C₂²	1	1	yes	characteristic
10	#433×C₁₂₇	—	2	4	no	—
11	#433×C₁₂₇	C₄	1	8	yes	—
12	C₂×#108×C₁₂₇	C₂²	1	4	yes	—
13	C₂×#108×C₁₂₇	C₂²	1	4	yes	—
14	#423×C₁₂₇	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
16	#407×C₁₂₇	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
18	#405×C₁₂₇	—	2	4	no	—
19	C₄×M₁₆×C₁₂₇	C₂²	1	2	yes	—
20	C₄×M₁₆×C₁₂₇	C₂²	1	2	yes	—
21	C₄×M₁₆×C₁₂₇	C₂²	1	2	yes	—
22	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
24	#341×C₁₂₇	C₂²	1	4	yes	—
25	#341×C₁₂₇	C₂²	1	4	yes	—
26	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₂×M₁₆×C₁₂₇	—	2	2	no	—
28	C₂×M₁₆×C₁₂₇	—	2	8	no	—
29	C₂×M₁₆×C₁₂₇	C₂³	1	4	yes	—
30	C₂×M₁₆×C₁₂₇	C₂³	1	4	yes	—
31	C₂²×C₈×C₁₂₇	—	2	2	no	—
32	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
34	#117×C₁₂₇	C₂×C₄	1	4	yes	—
35	#117×C₁₂₇	D₄	1	4	yes	—
36	C₂×C₄⋊C₄×C₁₂₇	Q₈	1	2	yes	—
37	C₂×C₂²⋊C₄×C₁₂₇	Q₈	1	2	yes	—
38	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
40	#108×C₁₂₇	—	2	8	no	—
41	#108×C₁₂₇	—	2	8	no	—
42	C₄⋊C₈×C₁₂₇	—	2	4	no	—
43	C₄⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
44	C₂²⋊C₈×C₁₂₇	—	2	4	no	—
45	C₂²⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
46	#97×C₁₂₇	—	2	2	no	—
47	#97×C₁₂₇	—	2	2	no	—
48	#97×C₁₂₇	—	2	2	no	—
49	#97×C₁₂₇	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₁₂₇	—	2	2	no	—
51	C₄×C₈×C₁₂₇	—	2	2	no	—
52	C₄×C₈×C₁₂₇	—	2	2	no	—
53	C₄×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
54	C₂⁴×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₂₇	—	2	4	no	—
56	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
62	M₁₆×C₁₂₇	—	2	8	no	—
63	M₁₆×C₁₂₇	—	2	4	no	—
64	M₁₆×C₁₂₇	—	2	4	no	—
65	M₁₆×C₁₂₇	—	2	4	no	—
66	M₁₆×C₁₂₇	#41	1	8	yes	—
67	C₂×C₈×C₁₂₇	—	2	8	no	—
68	C₂×C₈×C₁₂₇	—	2	4	no	—
69	C₂×C₈×C₁₂₇	—	2	4	no	—
70	C₂×C₈×C₁₂₇	—	2	4	no	—
71	C₂×C₈×C₁₂₇	—	2	4	no	—
72	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
74	C₄⋊C₄×C₁₂₇	C₄⋊C₄	1	8	yes	—
75	C₂²⋊C₄×C₁₂₇	C₄⋊C₄	1	8	yes	—
76	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
77	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
78	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
79	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
80	C₂³×C₁₂₇	—	2	2	no	—
81	C₂³×C₁₂₇	—	4	2	no	—
82	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
83	C₂³×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₁₂₇	—	2	2	no	—
85	C₂×C₄×C₁₂₇	—	2	4	no	—
86	C₂×C₄×C₁₂₇	—	2	2	no	—
87	C₂×C₄×C₁₂₇	—	2	2	no	—
88	C₂×C₄×C₁₂₇	—	4	2	no	—
89	C₂×C₄×C₁₂₇	—	4	2	no	—
90	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
91	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₂₇	C₂×#41	1	4	yes	—
96	C₈×C₁₂₇	—	2	4	no	—
97	C₈×C₁₂₇	—	2	4	no	—
98	C₈×C₁₂₇	—	2	8	no	—
99	C₈×C₁₂₇	—	4	4	no	—
100	C₂²×C₁₂₇	—	2	2	no	—
101	C₂²×C₁₂₇	—	2	4	no	—
102	C₂²×C₁₂₇	—	2	2	no	—
103	C₂²×C₁₂₇	—	8	2	no	—
104	C₂²×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₁₂₇	#533	1	2	yes	—
106	C₄×C₁₂₇	—	2	4	no	—
107	C₄×C₁₂₇	—	2	4	no	—
108	C₄×C₁₂₇	—	2	2	no	—
109	C₄×C₁₂₇	C₄×#41	1	2	yes	—
110	C₄×C₁₂₇	#525	1	2	yes	—
111	#21372	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂×C₁₂₇	—	2	2	no	—
113	C₂×C₁₂₇	—	4	2	no	—
114	C₂×C₁₂₇	#2296	1	1	yes	characteristic
115	C₂×C₁₂₇	#2916	1	2	yes	—
116	#2921	C₂×C₁₂₇	1	1	yes	characteristic
117	C₂×#433	C₂×C₁₂₇	1	1	yes	characteristic
118	#2873	C₂×C₁₂₇	1	1	yes	characteristic
119	#1950	C₂×C₁₂₇	1	4	yes	complemented
120	#1855	C₂×C₁₂₇	1	4	yes	—
121	#1778	C₂×C₁₂₇	1	2	yes	complemented
122	C₄×#108	C₂×C₁₂₇	1	2	yes	complemented
123	C₁₂₇	#21372	1	1	yes	characteristic, direct product factor, Sylow
124	C₂²×M₁₆	C₂²×C₁₂₇	1	2	yes	—
125	C₂×#117	C₂²×C₁₂₇	1	1	yes	characteristic
126	#433	—	2	4	no	—
127	#433	C₄×C₁₂₇	1	8	yes	—
128	C₂×#108	C₂²×C₁₂₇	1	4	yes	—
129	C₂×#108	C₂²×C₁₂₇	1	4	yes	—
130	#423	C₂²×C₁₂₇	1	2	yes	—
131	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
132	#407	C₂²×C₁₂₇	1	2	yes	—
133	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
134	#405	—	2	4	no	—
135	C₄×M₁₆	C₂²×C₁₂₇	1	2	yes	—
136	C₄×M₁₆	C₂²×C₁₂₇	1	2	yes	—
137	C₄×M₁₆	C₂²×C₁₂₇	1	2	yes	—
138	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
140	#341	C₂²×C₁₂₇	1	4	yes	—
141	#341	C₂²×C₁₂₇	1	4	yes	—
142	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
143	C₂×M₁₆	—	2	2	no	—
144	C₂×M₁₆	—	2	8	no	—
145	C₂×M₁₆	C₂³×C₁₂₇	1	4	yes	—
146	C₂×M₁₆	C₂³×C₁₂₇	1	4	yes	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
149	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
150	#117	C₂×C₄×C₁₂₇	1	4	yes	—
151	#117	D₄×C₁₂₇	1	4	yes	—
152	C₂×C₄⋊C₄	Q₈×C₁₂₇	1	2	yes	—
153	C₂×C₂²⋊C₄	Q₈×C₁₂₇	1	2	yes	—
154	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
155	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
156	#108	—	2	8	no	—
157	#108	—	2	8	no	—
158	C₄⋊C₈	—	2	4	no	—
159	C₄⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
160	C₂²⋊C₈	—	2	4	no	—
161	C₂²⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
162	#97	—	2	2	no	—
163	#97	—	2	2	no	—
164	#97	—	2	2	no	—
165	#97	C₂×C₄×C₁₂₇	1	4	yes	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	—	2	2	no	—
168	C₄×C₈	—	2	2	no	—
169	C₄×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
170	C₂⁴	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
173	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
174	C₂²×C₄	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
175	C₂²×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
176	C₂²×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
177	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
178	M₁₆	—	2	8	no	—
179	M₁₆	—	2	4	no	—
180	M₁₆	—	2	4	no	—
181	M₁₆	—	2	4	no	—
182	M₁₆	#41×C₁₂₇	1	8	yes	—
183	C₂×C₈	—	2	8	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
189	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
190	C₄⋊C₄	C₄⋊C₄×C₁₂₇	1	8	yes	—
191	C₂²⋊C₄	C₄⋊C₄×C₁₂₇	1	8	yes	—
192	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
193	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
194	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
195	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
196	C₂³	—	2	2	no	—
197	C₂³	—	4	2	no	—
198	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
199	C₂³	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	4	yes	—
207	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	4	yes	—
208	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
209	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
210	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
211	C₂×C₄	C₂×#41×C₁₂₇	1	4	yes	—
212	C₈	—	2	4	no	—
213	C₈	—	2	4	no	—
214	C₈	—	2	8	no	—
215	C₈	—	4	4	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	4	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	8	2	no	—
220	C₂²	C₂²×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
221	C₂²	#533×C₁₂₇	1	2	yes	—
222	C₄	—	2	4	no	—
223	C₄	—	2	4	no	—
224	C₄	—	2	2	no	—
225	C₄	C₄×#41×C₁₂₇	1	2	yes	—
226	C₄	#525×C₁₂₇	1	2	yes	—
227	C₂	—	2	2	no	—
228	C₂	—	4	2	no	—
229	C₂	C₁₂₇×#2296	1	1	yes	characteristic
230	C₂	C₁₂₇×#2916	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	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₂₇	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ C₂×#117×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₂₇×#1950	C₂	semidirect
6. C₁₂₇×#1778	C₂	semidirect
7. C₄×#108×C₁₂₇	C₂	semidirect
111. #21372	C₁₂₇	direct (N × H)
119. #1950	C₂×C₁₂₇	semidirect
121. #1778	C₂×C₁₂₇	semidirect
122. C₄×#108	C₂×C₁₂₇	semidirect
123. C₁₂₇	#21372	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₂₇×#2921	2
C₂×#433×C₁₂₇	2
C₁₂₇×#2873	2
C₁₂₇×#1950	2
C₁₂₇×#1855	2
C₁₂₇×#1778	2
C₄×#108×C₁₂₇	2
#21372	127

Subgroups — C127×#21372 #44585045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21372 #44585045