Subgroups — C₁₂₇×#11924 #44575597

Order 32512.

414 conjugacy classes of subgroups.

The subgroup lattice has 264 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×C₄⋊C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₂₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₂₇×#1360	C₂	1	4	yes	maximal, complemented
5	C₂×#135×C₁₂₇	—	2	1	no	—
6	C₂×#135×C₁₂₇	C₄	1	2	yes	—
7	#532×C₁₂₇	—	2	1	no	—
8	C₂²×C₁₆×C₁₂₇	—	2	1	no	—
9	#495×C₁₂₇	—	2	1	no	—
10	#495×C₁₂₇	C₂²	1	2	yes	—
11	#494×C₁₂₇	C₄	1	1	yes	characteristic
12	#493×C₁₂₇	C₄	1	1	yes	characteristic
13	#492×C₁₂₇	—	2	1	no	—
14	#431×C₁₂₇	—	2	1	no	—
15	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
17	C₄⋊C₁₆×C₁₂₇	C₂²	1	2	yes	—
18	#335×C₁₂₇	—	2	1	no	—
19	#334×C₁₂₇	C₂²	1	2	yes	—
20	C₂×#41×C₁₂₇	—	2	2	no	—
21	#135×C₁₂₇	—	4	4	no	—
22	#135×C₁₂₇	—	4	2	no	—
23	#135×C₁₂₇	—	4	2	no	—
24	C₂×Q₁₆×C₁₂₇	—	2	1	no	—
25	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
26	C₂×SD₁₆×C₁₂₇	—	2	2	no	—
27	C₂×SD₁₆×C₁₂₇	—	2	2	no	—
28	C₂×D₈×C₁₂₇	—	2	1	no	—
29	C₂×D₈×C₁₂₇	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
32	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
33	#127×C₁₂₇	—	2	1	no	—
34	#124×C₁₂₇	—	2	2	no	—
35	#121×C₁₂₇	—	4	1	no	—
36	#121×C₁₂₇	—	4	1	no	—
37	C₄×D₄×C₁₂₇	—	4	1	no	—
38	C₄×D₄×C₁₂₇	—	4	1	no	—
39	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
40	C₂×C₁₆×C₁₂₇	—	2	2	no	—
41	C₂×C₁₆×C₁₂₇	—	4	2	no	—
42	C₄⋊C₈×C₁₂₇	—	2	2	no	—
43	C₂²⋊C₈×C₁₂₇	—	4	1	no	—
44	C₄×C₈×C₁₂₇	—	4	1	no	—
45	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
49	#41×C₁₂₇	—	4	4	no	—
50	#41×C₁₂₇	—	4	4	no	—
51	C₂×Q₈×C₁₂₇	—	2	2	no	—
52	C₂×D₄×C₁₂₇	—	2	2	no	—
53	C₂×D₄×C₁₂₇	—	4	2	no	—
54	C₂×D₄×C₁₂₇	—	4	1	no	—
55	C₂×D₄×C₁₂₇	—	4	1	no	—
56	C₂²×C₄×C₁₂₇	—	4	2	no	—
57	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
60	Q₁₆×C₁₂₇	—	4	2	no	—
61	Q₁₆×C₁₂₇	—	4	2	no	—
62	SD₁₆×C₁₂₇	—	4	4	no	—
63	SD₁₆×C₁₂₇	—	4	2	no	—
64	SD₁₆×C₁₂₇	—	4	2	no	—
65	D₈×C₁₂₇	—	4	2	no	—
66	D₈×C₁₂₇	—	4	2	no	—
67	C₂×C₈×C₁₂₇	—	2	2	no	—
68	C₂×C₈×C₁₂₇	—	4	2	no	—
69	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
72	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
73	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₁₂₇	—	4	1	no	—
76	C₄⋊C₄×C₁₂₇	—	4	1	no	—
77	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
78	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
79	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
81	C₄²×C₁₂₇	SD₁₆	1	1	yes	characteristic
82	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
83	C₁₆×C₁₂₇	—	4	2	no	—
84	C₂³×C₁₂₇	—	4	2	no	—
85	C₂³×C₁₂₇	#102	1	1	yes	characteristic
86	Q₈×C₁₂₇	—	4	2	no	—
87	Q₈×C₁₂₇	—	4	2	no	—
88	D₄×C₁₂₇	—	4	2	no	—
89	D₄×C₁₂₇	—	4	2	no	—
90	D₄×C₁₂₇	—	8	2	no	—
91	D₄×C₁₂₇	—	8	4	no	—
92	D₄×C₁₂₇	—	8	2	no	—
93	C₂×C₄×C₁₂₇	—	2	2	no	—
94	C₂×C₄×C₁₂₇	—	4	1	no	—
95	C₂×C₄×C₁₂₇	—	4	2	no	—
96	C₂×C₄×C₁₂₇	—	8	4	no	—
97	C₂×C₄×C₁₂₇	#100	1	2	yes	—
98	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
99	C₂×C₄×C₁₂₇	#102	1	2	yes	—
100	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
101	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
102	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
105	C₈×C₁₂₇	—	2	1	no	—
106	C₈×C₁₂₇	—	2	1	no	—
107	C₈×C₁₂₇	—	8	2	no	—
108	C₈×C₁₂₇	#102	1	2	yes	—
109	C₈×C₁₂₇	C₂×D₈	1	1	yes	characteristic
110	C₈×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
111	C₂²×C₁₂₇	—	2	1	no	—
112	C₂²×C₁₂₇	—	2	1	no	—
113	C₂²×C₁₂₇	—	8	2	no	—
114	C₂²×C₁₂₇	—	8	2	no	—
115	C₂²×C₁₂₇	—	8	2	no	—
116	C₂²×C₁₂₇	#331	1	2	yes	—
117	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
118	C₄×C₁₂₇	—	2	1	no	—
119	C₄×C₁₂₇	—	2	1	no	—
120	C₄×C₁₂₇	—	2	1	no	—
121	C₄×C₁₂₇	—	8	2	no	—
122	C₄×C₁₂₇	#331	1	2	yes	—
123	C₄×C₁₂₇	#359	1	2	yes	—
124	C₄×C₁₂₇	C₂×#100	1	1	yes	characteristic
125	C₄×C₁₂₇	C₂×#102	1	1	yes	characteristic
126	#11924	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂×C₁₂₇	—	2	2	no	—
128	C₂×C₁₂₇	—	16	2	no	—
129	C₂×C₁₂₇	C₂×#331	1	1	yes	characteristic
130	C₂×C₁₂₇	C₂×#359	1	1	yes	characteristic
131	C₂×C₁₂₇	#2145	1	1	yes	characteristic
132	#3147	C₂×C₁₂₇	1	1	yes	characteristic
133	C₂×C₄⋊C₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
134	#1564	C₂×C₁₂₇	1	1	yes	characteristic, complemented
135	#1360	C₂×C₁₂₇	1	4	yes	complemented
136	C₁₂₇	#11924	1	1	yes	characteristic, direct product factor, Sylow
137	C₂×#135	—	2	1	no	—
138	C₂×#135	C₄×C₁₂₇	1	2	yes	—
139	#532	—	2	1	no	—
140	C₂²×C₁₆	—	2	1	no	—
141	#495	—	2	1	no	—
142	#495	C₂²×C₁₂₇	1	2	yes	—
143	#494	C₄×C₁₂₇	1	1	yes	characteristic
144	#493	C₄×C₁₂₇	1	1	yes	characteristic
145	#492	—	2	1	no	—
146	#431	—	2	1	no	—
147	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
148	C₄⋊C₁₆	—	2	1	no	—
149	C₄⋊C₁₆	C₂²×C₁₂₇	1	2	yes	—
150	#335	—	2	1	no	—
151	#334	C₂²×C₁₂₇	1	2	yes	—
152	C₂×#41	—	2	2	no	—
153	#135	—	4	4	no	—
154	#135	—	4	2	no	—
155	#135	—	4	2	no	—
156	C₂×Q₁₆	—	2	1	no	—
157	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
158	C₂×SD₁₆	—	2	2	no	—
159	C₂×SD₁₆	—	2	2	no	—
160	C₂×D₈	—	2	1	no	—
161	C₂×D₈	C₂×C₄×C₁₂₇	1	2	yes	—
162	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
163	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
164	C₄⋊Q₈	—	2	1	no	—
165	#127	—	2	1	no	—
166	#124	—	2	2	no	—
167	#121	—	4	1	no	—
168	#121	—	4	1	no	—
169	C₄×D₄	—	4	1	no	—
170	C₄×D₄	—	4	1	no	—
171	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	4	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₂²⋊C₈	—	4	1	no	—
176	C₄×C₈	—	4	1	no	—
177	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
178	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
179	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
180	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
181	#41	—	4	4	no	—
182	#41	—	4	4	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	4	2	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂×D₄	—	4	1	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
190	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
191	C₂²×C₄	SD₁₆×C₁₂₇	1	1	yes	characteristic
192	Q₁₆	—	4	2	no	—
193	Q₁₆	—	4	2	no	—
194	SD₁₆	—	4	4	no	—
195	SD₁₆	—	4	2	no	—
196	SD₁₆	—	4	2	no	—
197	D₈	—	4	2	no	—
198	D₈	—	4	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
203	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
204	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
205	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
206	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
207	C₄⋊C₄	—	4	1	no	—
208	C₄⋊C₄	—	4	1	no	—
209	C₂²⋊C₄	—	4	2	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
212	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
213	C₄²	SD₁₆×C₁₂₇	1	1	yes	characteristic
214	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
215	C₁₆	—	4	2	no	—
216	C₂³	—	4	2	no	—
217	C₂³	#102×C₁₂₇	1	1	yes	characteristic
218	Q₈	—	4	2	no	—
219	Q₈	—	4	2	no	—
220	D₄	—	4	2	no	—
221	D₄	—	4	2	no	—
222	D₄	—	8	2	no	—
223	D₄	—	8	4	no	—
224	D₄	—	8	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	8	4	no	—
229	C₂×C₄	#100×C₁₂₇	1	2	yes	—
230	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
231	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
232	C₂×C₄	#102×C₁₂₇	1	2	yes	—
233	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
235	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
236	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	8	2	no	—
240	C₈	#102×C₁₂₇	1	2	yes	—
241	C₈	C₂×D₈×C₁₂₇	1	1	yes	characteristic
242	C₈	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	8	2	no	—
246	C₂²	—	8	2	no	—
247	C₂²	—	8	2	no	—
248	C₂²	#331×C₁₂₇	1	2	yes	—
249	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	8	2	no	—
254	C₄	#331×C₁₂₇	1	2	yes	—
255	C₄	#359×C₁₂₇	1	2	yes	—
256	C₄	C₂×#100×C₁₂₇	1	1	yes	characteristic
257	C₄	C₂×#102×C₁₂₇	1	1	yes	characteristic
258	C₂	—	2	2	no	—
259	C₂	—	16	2	no	—
260	C₂	C₂×#331×C₁₂₇	1	1	yes	characteristic
261	C₂	C₂×#359×C₁₂₇	1	1	yes	characteristic
262	C₂	C₁₂₇×#2145	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	#102
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×#102
prime radical	C₁₂₇×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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
2. C₂×C₄⋊C₁₆×C₁₂₇	C₂	semidirect
3. C₁₂₇×#1564	C₂	semidirect
4. C₁₂₇×#1360	C₂	semidirect
126. #11924	C₁₂₇	direct (N × H)
133. C₂×C₄⋊C₁₆	C₂×C₁₂₇	semidirect
134. #1564	C₂×C₁₂₇	semidirect
135. #1360	C₂×C₁₂₇	semidirect
136. C₁₂₇	#11924	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×#3147	2
C₂×C₄⋊C₁₆×C₁₂₇	2
C₁₂₇×#1564	2
C₁₂₇×#1360	2
#11924	127

Subgroups — C127×#11924 #44575597

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#11924 #44575597