Subgroups — C₁₂₇×#11930 #44575603

Order 32512.

418 conjugacy classes of subgroups.

The subgroup lattice has 276 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	C₁₂₇×#2150	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#335×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₂₇×#1362	C₂	1	2	yes	maximal, complemented
5	C₁₂₇×#1361	C₂	1	2	yes	maximal, complemented
6	C₂×#135×C₁₂₇	—	2	1	no	—
7	C₂×#135×C₁₂₇	C₄	1	2	yes	—
8	#532×C₁₂₇	—	2	1	no	—
9	C₂×M₃₂×C₁₂₇	—	2	1	no	—
10	#495×C₁₂₇	—	2	1	no	—
11	#495×C₁₂₇	C₄	1	2	yes	—
12	#494×C₁₂₇	C₂²	1	1	yes	characteristic
13	#493×C₁₂₇	C₂²	1	1	yes	characteristic
14	#492×C₁₂₇	—	2	1	no	—
15	C₂×C₄⋊C₈×C₁₂₇	—	2	1	no	—
16	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
18	C₄⋊C₁₆×C₁₂₇	C₂²	1	2	yes	—
19	#335×C₁₂₇	—	2	1	no	—
20	#335×C₁₂₇	C₂²	1	2	yes	—
21	C₂×#41×C₁₂₇	—	2	2	no	—
22	#135×C₁₂₇	—	4	4	no	—
23	#135×C₁₂₇	—	4	2	no	—
24	#135×C₁₂₇	—	4	2	no	—
25	C₂×Q₁₆×C₁₂₇	—	2	1	no	—
26	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
27	C₂×SD₁₆×C₁₂₇	—	2	2	no	—
28	C₂×SD₁₆×C₁₂₇	—	2	2	no	—
29	C₂×D₈×C₁₂₇	—	2	1	no	—
30	C₂×D₈×C₁₂₇	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₂₇	—	4	1	no	—
32	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
34	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
35	#127×C₁₂₇	—	2	1	no	—
36	#124×C₁₂₇	—	2	2	no	—
37	#121×C₁₂₇	—	4	1	no	—
38	#121×C₁₂₇	—	4	1	no	—
39	C₄×D₄×C₁₂₇	—	4	1	no	—
40	C₄×D₄×C₁₂₇	—	4	1	no	—
41	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
42	M₃₂×C₁₂₇	—	4	2	no	—
43	C₂×C₁₆×C₁₂₇	—	2	2	no	—
44	C₄⋊C₈×C₁₂₇	—	2	2	no	—
45	C₄⋊C₈×C₁₂₇	—	4	1	no	—
46	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
49	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
50	#41×C₁₂₇	—	4	4	no	—
51	#41×C₁₂₇	—	4	4	no	—
52	C₂×Q₈×C₁₂₇	—	2	2	no	—
53	C₂×D₄×C₁₂₇	—	2	2	no	—
54	C₂×D₄×C₁₂₇	—	4	2	no	—
55	C₂×D₄×C₁₂₇	—	4	1	no	—
56	C₂×D₄×C₁₂₇	—	4	1	no	—
57	C₂²×C₄×C₁₂₇	—	4	2	no	—
58	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
61	Q₁₆×C₁₂₇	—	4	2	no	—
62	Q₁₆×C₁₂₇	—	4	1	no	—
63	Q₁₆×C₁₂₇	—	4	1	no	—
64	SD₁₆×C₁₂₇	—	4	4	no	—
65	SD₁₆×C₁₂₇	—	4	4	no	—
66	D₈×C₁₂₇	—	4	2	no	—
67	D₈×C₁₂₇	—	4	1	no	—
68	D₈×C₁₂₇	—	4	1	no	—
69	C₂×C₈×C₁₂₇	—	2	2	no	—
70	C₂×C₈×C₁₂₇	—	4	2	no	—
71	C₂×C₈×C₁₂₇	—	8	2	no	—
72	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
75	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
76	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
78	C₄⋊C₄×C₁₂₇	—	4	1	no	—
79	C₄⋊C₄×C₁₂₇	—	4	1	no	—
80	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
81	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
82	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
84	C₄²×C₁₂₇	SD₁₆	1	1	yes	characteristic
85	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
86	C₁₆×C₁₂₇	—	4	2	no	—
87	C₂³×C₁₂₇	—	4	2	no	—
88	C₂³×C₁₂₇	#102	1	1	yes	characteristic
89	Q₈×C₁₂₇	—	4	2	no	—
90	Q₈×C₁₂₇	—	4	2	no	—
91	D₄×C₁₂₇	—	4	2	no	—
92	D₄×C₁₂₇	—	4	2	no	—
93	D₄×C₁₂₇	—	8	2	no	—
94	D₄×C₁₂₇	—	8	4	no	—
95	D₄×C₁₂₇	—	8	1	no	—
96	D₄×C₁₂₇	—	8	1	no	—
97	C₂×C₄×C₁₂₇	—	2	1	no	—
98	C₂×C₄×C₁₂₇	—	2	1	no	—
99	C₂×C₄×C₁₂₇	—	4	1	no	—
100	C₂×C₄×C₁₂₇	—	4	2	no	—
101	C₂×C₄×C₁₂₇	—	8	4	no	—
102	C₂×C₄×C₁₂₇	#100	1	2	yes	—
103	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
104	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
105	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
106	C₂×C₄×C₁₂₇	#102	1	2	yes	—
107	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
110	C₈×C₁₂₇	—	2	1	no	—
111	C₈×C₁₂₇	—	2	1	no	—
112	C₈×C₁₂₇	—	8	2	no	—
113	C₈×C₁₂₇	#102	1	2	yes	—
114	C₈×C₁₂₇	C₂×D₈	1	1	yes	characteristic
115	C₈×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
116	C₂²×C₁₂₇	—	2	1	no	—
117	C₂²×C₁₂₇	—	2	1	no	—
118	C₂²×C₁₂₇	—	8	2	no	—
119	C₂²×C₁₂₇	—	8	2	no	—
120	C₂²×C₁₂₇	—	8	2	no	—
121	C₂²×C₁₂₇	#331	1	2	yes	—
122	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
123	C₄×C₁₂₇	—	2	1	no	—
124	C₄×C₁₂₇	—	2	1	no	—
125	C₄×C₁₂₇	—	2	1	no	—
126	C₄×C₁₂₇	—	8	2	no	—
127	C₄×C₁₂₇	#331	1	2	yes	—
128	C₄×C₁₂₇	C₂×#100	1	1	yes	characteristic
129	C₄×C₁₂₇	C₂×#102	1	1	yes	characteristic
130	C₄×C₁₂₇	#509	1	1	yes	characteristic
131	C₄×C₁₂₇	#510	1	1	yes	characteristic
132	#11930	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂×C₁₂₇	—	2	2	no	—
134	C₂×C₁₂₇	—	16	2	no	—
135	C₂×C₁₂₇	C₂×#331	1	1	yes	characteristic
136	C₂×C₁₂₇	#2139	1	1	yes	characteristic
137	C₂×C₁₂₇	#2148	1	1	yes	characteristic
138	#3147	C₂×C₁₂₇	1	1	yes	characteristic
139	#2150	C₂×C₁₂₇	1	1	yes	characteristic, complemented
140	C₂×#335	C₂×C₁₂₇	1	1	yes	characteristic, complemented
141	#1362	C₂×C₁₂₇	1	2	yes	complemented
142	#1361	C₂×C₁₂₇	1	2	yes	complemented
143	C₁₂₇	#11930	1	1	yes	characteristic, direct product factor, Sylow
144	C₂×#135	—	2	1	no	—
145	C₂×#135	C₄×C₁₂₇	1	2	yes	—
146	#532	—	2	1	no	—
147	C₂×M₃₂	—	2	1	no	—
148	#495	—	2	1	no	—
149	#495	C₄×C₁₂₇	1	2	yes	—
150	#494	C₂²×C₁₂₇	1	1	yes	characteristic
151	#493	C₂²×C₁₂₇	1	1	yes	characteristic
152	#492	—	2	1	no	—
153	C₂×C₄⋊C₈	—	2	1	no	—
154	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
155	C₄⋊C₁₆	—	2	1	no	—
156	C₄⋊C₁₆	C₂²×C₁₂₇	1	2	yes	—
157	#335	—	2	1	no	—
158	#335	C₂²×C₁₂₇	1	2	yes	—
159	C₂×#41	—	2	2	no	—
160	#135	—	4	4	no	—
161	#135	—	4	2	no	—
162	#135	—	4	2	no	—
163	C₂×Q₁₆	—	2	1	no	—
164	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
165	C₂×SD₁₆	—	2	2	no	—
166	C₂×SD₁₆	—	2	2	no	—
167	C₂×D₈	—	2	1	no	—
168	C₂×D₈	C₂×C₄×C₁₂₇	1	2	yes	—
169	C₂²×C₈	—	4	1	no	—
170	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
171	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
172	C₄⋊Q₈	—	2	1	no	—
173	#127	—	2	1	no	—
174	#124	—	2	2	no	—
175	#121	—	4	1	no	—
176	#121	—	4	1	no	—
177	C₄×D₄	—	4	1	no	—
178	C₄×D₄	—	4	1	no	—
179	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
180	M₃₂	—	4	2	no	—
181	C₂×C₁₆	—	2	2	no	—
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	—	4	1	no	—
184	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
185	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
186	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
187	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
188	#41	—	4	4	no	—
189	#41	—	4	4	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	4	2	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂×D₄	—	4	1	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
197	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
198	C₂²×C₄	SD₁₆×C₁₂₇	1	1	yes	characteristic
199	Q₁₆	—	4	2	no	—
200	Q₁₆	—	4	1	no	—
201	Q₁₆	—	4	1	no	—
202	SD₁₆	—	4	4	no	—
203	SD₁₆	—	4	4	no	—
204	D₈	—	4	2	no	—
205	D₈	—	4	1	no	—
206	D₈	—	4	1	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	8	2	no	—
210	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
211	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
212	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
213	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
214	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
215	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
216	C₄⋊C₄	—	4	1	no	—
217	C₄⋊C₄	—	4	1	no	—
218	C₂²⋊C₄	—	4	2	no	—
219	C₂²⋊C₄	—	4	2	no	—
220	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
221	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
222	C₄²	SD₁₆×C₁₂₇	1	1	yes	characteristic
223	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
224	C₁₆	—	4	2	no	—
225	C₂³	—	4	2	no	—
226	C₂³	#102×C₁₂₇	1	1	yes	characteristic
227	Q₈	—	4	2	no	—
228	Q₈	—	4	2	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	8	2	no	—
232	D₄	—	8	4	no	—
233	D₄	—	8	1	no	—
234	D₄	—	8	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	4	no	—
240	C₂×C₄	#100×C₁₂₇	1	2	yes	—
241	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
242	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
243	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
244	C₂×C₄	#102×C₁₂₇	1	2	yes	—
245	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
246	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
247	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
248	C₈	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	—	8	2	no	—
251	C₈	#102×C₁₂₇	1	2	yes	—
252	C₈	C₂×D₈×C₁₂₇	1	1	yes	characteristic
253	C₈	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	8	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	#331×C₁₂₇	1	2	yes	—
260	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	8	2	no	—
265	C₄	#331×C₁₂₇	1	2	yes	—
266	C₄	C₂×#100×C₁₂₇	1	1	yes	characteristic
267	C₄	C₂×#102×C₁₂₇	1	1	yes	characteristic
268	C₄	#509×C₁₂₇	1	1	yes	characteristic
269	C₄	#510×C₁₂₇	1	1	yes	characteristic
270	C₂	—	2	2	no	—
271	C₂	—	16	2	no	—
272	C₂	C₂×#331×C₁₂₇	1	1	yes	characteristic
273	C₂	C₁₂₇×#2139	1	1	yes	characteristic
274	C₂	C₁₂₇×#2148	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	C₁₂₇×#3147	C₂
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
2. C₁₂₇×#2150	C₂	semidirect
3. C₂×#335×C₁₂₇	C₂	semidirect
4. C₁₂₇×#1362	C₂	semidirect
5. C₁₂₇×#1361	C₂	semidirect
132. #11930	C₁₂₇	direct (N × H)
139. #2150	C₂×C₁₂₇	semidirect
140. C₂×#335	C₂×C₁₂₇	semidirect
141. #1362	C₂×C₁₂₇	semidirect
142. #1361	C₂×C₁₂₇	semidirect
143. C₁₂₇	#11930	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
C₁₂₇×#2150	2
C₂×#335×C₁₂₇	2
C₁₂₇×#1362	2
C₁₂₇×#1361	2
#11930	127

Subgroups — C127×#11930 #44575603

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#11930 #44575603