Subgroups — C₁₂₇×#13212 #44576885

Order 32512.

470 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#362×C₁₂₇	C₂	1	4	yes	maximal
3	C₄×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₁₂₇	—	2	1	no	—
6	C₂²×Q₁₆×C₁₂₇	C₂²	1	2	yes	—
7	C₂×C₄⋊Q₈×C₁₂₇	—	2	1	no	—
8	C₂×M₃₂×C₁₂₇	C₂²	1	2	yes	—
9	#494×C₁₂₇	—	2	1	no	—
10	#494×C₁₂₇	—	2	1	no	—
11	#494×C₁₂₇	C₄	1	2	yes	—
12	#494×C₁₂₇	C₄	1	2	yes	—
13	C₂×#108×C₁₂₇	—	2	1	no	—
14	C₂×#108×C₁₂₇	C₂²	1	2	yes	—
15	#423×C₁₂₇	—	2	1	no	—
16	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	#362×C₁₂₇	—	2	8	no	—
18	#346×C₁₂₇	—	2	1	no	—
19	C₄×C₁₆×C₁₂₇	—	2	1	no	—
20	C₂²×Q₈×C₁₂₇	—	2	2	no	—
21	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
22	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
23	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
24	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
25	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
26	C₂×Q₁₆×C₁₂₇	—	4	2	no	—
27	C₂×Q₁₆×C₁₂₇	—	4	2	no	—
28	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
29	C₂×M₁₆×C₁₂₇	—	2	2	no	—
30	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
32	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
33	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
34	C₄⋊Q₈×C₁₂₇	—	4	2	no	—
35	C₂×C₄⋊C₄×C₁₂₇	—	4	2	no	—
36	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
37	M₃₂×C₁₂₇	—	2	2	no	—
38	M₃₂×C₁₂₇	D₄	1	4	yes	—
39	C₂×C₁₆×C₁₂₇	—	2	2	no	—
40	#108×C₁₂₇	—	2	4	no	—
41	#108×C₁₂₇	—	4	2	no	—
42	C₄⋊C₈×C₁₂₇	—	4	2	no	—
43	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
47	C₂×Q₈×C₁₂₇	—	2	4	no	—
48	C₂×Q₈×C₁₂₇	—	4	2	no	—
49	C₂×Q₈×C₁₂₇	—	4	2	no	—
50	C₂×Q₈×C₁₂₇	—	4	2	no	—
51	C₂×Q₈×C₁₂₇	—	4	2	no	—
52	C₂×Q₈×C₁₂₇	—	4	2	no	—
53	C₂²×C₄×C₁₂₇	—	4	2	no	—
54	C₂²×C₄×C₁₂₇	SD₁₆	1	2	yes	—
55	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
56	Q₁₆×C₁₂₇	—	2	8	no	—
57	Q₁₆×C₁₂₇	—	4	4	no	—
58	Q₁₆×C₁₂₇	—	4	4	no	—
59	Q₁₆×C₁₂₇	—	4	4	no	—
60	M₁₆×C₁₂₇	—	4	2	no	—
61	M₁₆×C₁₂₇	—	4	2	no	—
62	C₂×C₈×C₁₂₇	—	4	2	no	—
63	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
66	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
67	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₂₇	#41	1	2	yes	—
71	C₄⋊C₄×C₁₂₇	—	4	4	no	—
72	C₄⋊C₄×C₁₂₇	—	4	4	no	—
73	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₁₂₇	D₈	1	2	yes	—
76	C₁₆×C₁₂₇	—	2	4	no	—
77	C₂³×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
78	Q₈×C₁₂₇	—	4	4	no	—
79	Q₈×C₁₂₇	—	4	4	no	—
80	Q₈×C₁₂₇	—	8	2	no	—
81	Q₈×C₁₂₇	—	8	2	no	—
82	C₂×C₄×C₁₂₇	—	2	4	no	—
83	C₂×C₄×C₁₂₇	—	4	4	no	—
84	C₂×C₄×C₁₂₇	—	8	2	no	—
85	C₂×C₄×C₁₂₇	—	8	2	no	—
86	C₂×C₄×C₁₂₇	#102	1	2	yes	—
87	C₂×C₄×C₁₂₇	#102	1	2	yes	—
88	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₂₇	#124	1	1	yes	characteristic
90	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
91	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
94	C₈×C₁₂₇	—	8	2	no	—
95	C₈×C₁₂₇	#102	1	2	yes	—
96	C₈×C₁₂₇	#102	1	2	yes	—
97	C₈×C₁₂₇	C₄×D₄	1	2	yes	—
98	C₈×C₁₂₇	#121	1	2	yes	—
99	C₂²×C₁₂₇	—	2	2	no	—
100	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
101	C₂²×C₁₂₇	#486	1	1	yes	characteristic
102	C₂²×C₁₂₇	#492	1	1	yes	characteristic
103	C₄×C₁₂₇	—	2	2	no	—
104	C₄×C₁₂₇	—	8	4	no	—
105	C₄×C₁₂₇	#362	1	4	yes	complemented
106	C₄×C₁₂₇	#390	1	1	yes	characteristic
107	C₄×C₁₂₇	C₂×#102	1	1	yes	characteristic
108	C₄×C₁₂₇	#486	1	1	yes	characteristic
109	C₄×C₁₂₇	#493	1	1	yes	characteristic
110	#13212	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂×C₁₂₇	—	2	1	no	—
112	C₂×C₁₂₇	—	2	1	no	—
113	C₂×C₁₂₇	#1957	1	1	yes	characteristic
114	C₂×C₁₂₇	C₂×#362	1	2	yes	—
115	C₂×#494	C₂×C₁₂₇	1	1	yes	characteristic
116	C₂×#362	C₂×C₁₂₇	1	4	yes	—
117	C₄×M₃₂	C₂×C₁₂₇	1	1	yes	characteristic
118	#1848	C₂×C₁₂₇	1	1	yes	characteristic
119	C₁₂₇	#13212	1	1	yes	characteristic, direct product factor, Sylow
120	C₂²×Q₁₆	—	2	1	no	—
121	C₂²×Q₁₆	C₂²×C₁₂₇	1	2	yes	—
122	C₂×C₄⋊Q₈	—	2	1	no	—
123	C₂×M₃₂	C₂²×C₁₂₇	1	2	yes	—
124	#494	—	2	1	no	—
125	#494	—	2	1	no	—
126	#494	C₄×C₁₂₇	1	2	yes	—
127	#494	C₄×C₁₂₇	1	2	yes	—
128	C₂×#108	—	2	1	no	—
129	C₂×#108	C₂²×C₁₂₇	1	2	yes	—
130	#423	—	2	1	no	—
131	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
132	#362	—	2	8	no	—
133	#346	—	2	1	no	—
134	C₄×C₁₆	—	2	1	no	—
135	C₂²×Q₈	—	2	2	no	—
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×Q₁₆	—	2	2	no	—
138	C₂×Q₁₆	—	2	4	no	—
139	C₂×Q₁₆	—	2	2	no	—
140	C₂×Q₁₆	—	2	4	no	—
141	C₂×Q₁₆	—	4	2	no	—
142	C₂×Q₁₆	—	4	2	no	—
143	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
144	C₂×M₁₆	—	2	2	no	—
145	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
146	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	4	2	no	—
150	C₂×C₄⋊C₄	—	4	2	no	—
151	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
152	M₃₂	—	2	2	no	—
153	M₃₂	D₄×C₁₂₇	1	4	yes	—
154	C₂×C₁₆	—	2	2	no	—
155	#108	—	2	4	no	—
156	#108	—	4	2	no	—
157	C₄⋊C₈	—	4	2	no	—
158	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
159	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
160	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
161	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
162	C₂×Q₈	—	2	4	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	4	2	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂×Q₈	—	4	2	no	—
167	C₂×Q₈	—	4	2	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	SD₁₆×C₁₂₇	1	2	yes	—
170	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
171	Q₁₆	—	2	8	no	—
172	Q₁₆	—	4	4	no	—
173	Q₁₆	—	4	4	no	—
174	Q₁₆	—	4	4	no	—
175	M₁₆	—	4	2	no	—
176	M₁₆	—	4	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
179	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
180	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
181	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
182	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
183	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
184	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
185	C₂×C₈	#41×C₁₂₇	1	2	yes	—
186	C₄⋊C₄	—	4	4	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
189	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
190	C₄²	D₈×C₁₂₇	1	2	yes	—
191	C₁₆	—	2	4	no	—
192	C₂³	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
193	Q₈	—	4	4	no	—
194	Q₈	—	4	4	no	—
195	Q₈	—	8	2	no	—
196	Q₈	—	8	2	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	8	2	no	—
200	C₂×C₄	—	8	2	no	—
201	C₂×C₄	#102×C₁₂₇	1	2	yes	—
202	C₂×C₄	#102×C₁₂₇	1	2	yes	—
203	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
204	C₂×C₄	#124×C₁₂₇	1	1	yes	characteristic
205	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
206	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
207	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
208	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
209	C₈	—	8	2	no	—
210	C₈	#102×C₁₂₇	1	2	yes	—
211	C₈	#102×C₁₂₇	1	2	yes	—
212	C₈	C₄×D₄×C₁₂₇	1	2	yes	—
213	C₈	#121×C₁₂₇	1	2	yes	—
214	C₂²	—	2	2	no	—
215	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
216	C₂²	#486×C₁₂₇	1	1	yes	characteristic
217	C₂²	#492×C₁₂₇	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	8	4	no	—
220	C₄	#362×C₁₂₇	1	4	yes	complemented
221	C₄	#390×C₁₂₇	1	1	yes	characteristic
222	C₄	C₂×#102×C₁₂₇	1	1	yes	characteristic
223	C₄	#486×C₁₂₇	1	1	yes	characteristic
224	C₄	#493×C₁₂₇	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	2	1	no	—
227	C₂	C₁₂₇×#1957	1	1	yes	characteristic
228	C₂	C₂×#362×C₁₂₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
prime radical	C₂³×C₁₂₇	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
105. C₄×C₁₂₇	#362	semidirect
110. #13212	C₁₂₇	direct (N × H)
119. C₁₂₇	#13212	direct (N × H)
220. C₄	#362×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#494×C₁₂₇	2
C₂×#362×C₁₂₇	2
C₄×M₃₂×C₁₂₇	2
C₁₂₇×#1848	2
#13212	127

Subgroups — C127×#13212 #44576885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13212 #44576885