Subgroups — C₁₂₇×#11932 #44575605

Order 32512.

398 conjugacy classes of subgroups.

The subgroup lattice has 228 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	C₁₂₇×#2150	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1564	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1362	C₂	1	4	yes	maximal, complemented
5	C₂²×Q₁₆×C₁₂₇	—	2	1	no	—
6	C₂²×Q₁₆×C₁₂₇	C₄	1	2	yes	—
7	C₂×C₄⋊Q₈×C₁₂₇	—	2	1	no	—
8	C₂×M₃₂×C₁₂₇	—	2	1	no	—
9	#494×C₁₂₇	—	2	1	no	—
10	#494×C₁₂₇	—	2	1	no	—
11	#494×C₁₂₇	C₄	1	2	yes	—
12	#494×C₁₂₇	C₂²	1	2	yes	—
13	#431×C₁₂₇	—	2	1	no	—
14	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
16	C₄⋊C₁₆×C₁₂₇	C₂²	1	2	yes	—
17	#335×C₁₂₇	C₂²	1	2	yes	—
18	#334×C₁₂₇	—	2	1	no	—
19	C₂²×Q₈×C₁₂₇	—	2	2	no	—
20	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
21	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
22	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
23	C₂×Q₁₆×C₁₂₇	—	4	4	no	—
24	C₂×Q₁₆×C₁₂₇	—	4	2	no	—
25	C₂×Q₁₆×C₁₂₇	—	4	2	no	—
26	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
29	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
30	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
31	C₄⋊Q₈×C₁₂₇	—	4	1	no	—
32	C₄⋊Q₈×C₁₂₇	—	4	1	no	—
33	C₂×C₄⋊C₄×C₁₂₇	—	4	1	no	—
34	C₂×C₄⋊C₄×C₁₂₇	—	4	1	no	—
35	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
36	M₃₂×C₁₂₇	—	4	2	no	—
37	C₂×C₁₆×C₁₂₇	—	2	2	no	—
38	C₄⋊C₈×C₁₂₇	—	2	2	no	—
39	C₂²⋊C₈×C₁₂₇	—	4	1	no	—
40	C₄×C₈×C₁₂₇	—	4	1	no	—
41	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₁₂₇	—	2	4	no	—
46	C₂×Q₈×C₁₂₇	—	4	2	no	—
47	C₂×Q₈×C₁₂₇	—	4	4	no	—
48	C₂×Q₈×C₁₂₇	—	4	4	no	—
49	C₂²×C₄×C₁₂₇	—	4	2	no	—
50	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
53	Q₁₆×C₁₂₇	—	4	4	no	—
54	Q₁₆×C₁₂₇	—	4	4	no	—
55	Q₁₆×C₁₂₇	—	4	4	no	—
56	Q₁₆×C₁₂₇	—	4	2	no	—
57	Q₁₆×C₁₂₇	—	4	2	no	—
58	C₂×C₈×C₁₂₇	—	2	2	no	—
59	C₂×C₈×C₁₂₇	—	4	2	no	—
60	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
63	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
64	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₁₂₇	—	4	2	no	—
67	C₄⋊C₄×C₁₂₇	—	4	2	no	—
68	C₄⋊C₄×C₁₂₇	—	4	2	no	—
69	C₄⋊C₄×C₁₂₇	—	4	2	no	—
70	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
72	C₄²×C₁₂₇	SD₁₆	1	1	yes	characteristic
73	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
74	C₁₆×C₁₂₇	—	4	2	no	—
75	C₂³×C₁₂₇	#102	1	1	yes	characteristic
76	Q₈×C₁₂₇	—	4	4	no	—
77	Q₈×C₁₂₇	—	4	4	no	—
78	Q₈×C₁₂₇	—	8	4	no	—
79	C₂×C₄×C₁₂₇	—	2	2	no	—
80	C₂×C₄×C₁₂₇	—	2	2	no	—
81	C₂×C₄×C₁₂₇	—	4	4	no	—
82	C₂×C₄×C₁₂₇	—	8	4	no	—
83	C₂×C₄×C₁₂₇	#100	1	2	yes	—
84	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
85	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
86	C₂×C₄×C₁₂₇	#102	1	2	yes	—
87	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
88	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
91	C₈×C₁₂₇	—	2	1	no	—
92	C₈×C₁₂₇	—	2	1	no	—
93	C₈×C₁₂₇	—	8	2	no	—
94	C₈×C₁₂₇	#102	1	2	yes	—
95	C₈×C₁₂₇	C₂×D₈	1	1	yes	characteristic
96	C₈×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
97	C₂²×C₁₂₇	—	2	1	no	—
98	C₂²×C₁₂₇	—	2	1	no	—
99	C₂²×C₁₂₇	#331	1	2	yes	—
100	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
101	C₄×C₁₂₇	—	2	1	no	—
102	C₄×C₁₂₇	—	2	1	no	—
103	C₄×C₁₂₇	—	8	4	no	—
104	C₄×C₁₂₇	#331	1	2	yes	—
105	C₄×C₁₂₇	#362	1	2	yes	—
106	C₄×C₁₂₇	C₂×#100	1	1	yes	characteristic
107	C₄×C₁₂₇	C₂×#102	1	1	yes	characteristic
108	C₄×C₁₂₇	#510	1	2	yes	—
109	#11932	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂×C₁₂₇	—	2	2	no	—
111	C₂×C₁₂₇	C₂×#331	1	1	yes	characteristic
112	C₂×C₁₂₇	#2141	1	1	yes	characteristic
113	C₂×C₁₂₇	C₂×#362	1	1	yes	characteristic
114	C₂×#494	C₂×C₁₂₇	1	1	yes	characteristic
115	#2150	C₂×C₁₂₇	1	1	yes	characteristic
116	#1564	C₂×C₁₂₇	1	1	yes	characteristic
117	#1362	C₂×C₁₂₇	1	4	yes	complemented
118	C₁₂₇	#11932	1	1	yes	characteristic, direct product factor, Sylow
119	C₂²×Q₁₆	—	2	1	no	—
120	C₂²×Q₁₆	C₄×C₁₂₇	1	2	yes	—
121	C₂×C₄⋊Q₈	—	2	1	no	—
122	C₂×M₃₂	—	2	1	no	—
123	#494	—	2	1	no	—
124	#494	—	2	1	no	—
125	#494	C₄×C₁₂₇	1	2	yes	—
126	#494	C₂²×C₁₂₇	1	2	yes	—
127	#431	—	2	1	no	—
128	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
129	C₄⋊C₁₆	—	2	1	no	—
130	C₄⋊C₁₆	C₂²×C₁₂₇	1	2	yes	—
131	#335	C₂²×C₁₂₇	1	2	yes	—
132	#334	—	2	1	no	—
133	C₂²×Q₈	—	2	2	no	—
134	C₂×Q₁₆	—	2	2	no	—
135	C₂×Q₁₆	—	2	2	no	—
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×Q₁₆	—	4	4	no	—
138	C₂×Q₁₆	—	4	2	no	—
139	C₂×Q₁₆	—	4	2	no	—
140	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
141	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
142	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
143	C₄⋊Q₈	—	2	2	no	—
144	C₄⋊Q₈	—	2	2	no	—
145	C₄⋊Q₈	—	4	1	no	—
146	C₄⋊Q₈	—	4	1	no	—
147	C₂×C₄⋊C₄	—	4	1	no	—
148	C₂×C₄⋊C₄	—	4	1	no	—
149	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
150	M₃₂	—	4	2	no	—
151	C₂×C₁₆	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₂²⋊C₈	—	4	1	no	—
154	C₄×C₈	—	4	1	no	—
155	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
156	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
157	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
158	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
159	C₂×Q₈	—	2	4	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	—	4	4	no	—
162	C₂×Q₈	—	4	4	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
165	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
166	C₂²×C₄	SD₁₆×C₁₂₇	1	1	yes	characteristic
167	Q₁₆	—	4	4	no	—
168	Q₁₆	—	4	4	no	—
169	Q₁₆	—	4	4	no	—
170	Q₁₆	—	4	2	no	—
171	Q₁₆	—	4	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
175	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
176	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
177	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
178	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
179	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
185	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
186	C₄²	SD₁₆×C₁₂₇	1	1	yes	characteristic
187	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
188	C₁₆	—	4	2	no	—
189	C₂³	#102×C₁₂₇	1	1	yes	characteristic
190	Q₈	—	4	4	no	—
191	Q₈	—	4	4	no	—
192	Q₈	—	8	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	8	4	no	—
197	C₂×C₄	#100×C₁₂₇	1	2	yes	—
198	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
199	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
200	C₂×C₄	#102×C₁₂₇	1	2	yes	—
201	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
202	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
203	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
204	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	8	2	no	—
208	C₈	#102×C₁₂₇	1	2	yes	—
209	C₈	C₂×D₈×C₁₂₇	1	1	yes	characteristic
210	C₈	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	#331×C₁₂₇	1	2	yes	—
214	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	8	4	no	—
218	C₄	#331×C₁₂₇	1	2	yes	—
219	C₄	#362×C₁₂₇	1	2	yes	—
220	C₄	C₂×#100×C₁₂₇	1	1	yes	characteristic
221	C₄	C₂×#102×C₁₂₇	1	1	yes	characteristic
222	C₄	#510×C₁₂₇	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	C₂×#331×C₁₂₇	1	1	yes	characteristic
225	C₂	C₁₂₇×#2141	1	1	yes	characteristic
226	C₂	C₂×#362×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
prime radical	C₂³×C₁₂₇	#102
LCS	G ▷ C₂×C₈ ▷ C₂×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
4. C₁₂₇×#1362	C₂	semidirect
109. #11932	C₁₂₇	direct (N × H)
117. #1362	C₂×C₁₂₇	semidirect
118. C₁₂₇	#11932	direct (N × H)

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
C₁₂₇×#2150	2
C₁₂₇×#1564	2
C₁₂₇×#1362	2
#11932	127

Subgroups — C127×#11932 #44575605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#11932 #44575605