Subgroups — C₄×#359×C₁₂₇ #44576312

Order 32512.

450 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#135×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#359×C₁₂₇	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#108×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₂₇×#1382	C₂	1	2	yes	maximal, complemented
6	C₂×#135×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₄×#41×C₁₂₇	—	2	1	no	—
8	C₂²×C₁₆×C₁₂₇	C₂²	1	2	yes	—
9	C₄×Q₁₆×C₁₂₇	C₄	1	2	yes	—
10	C₄×SD₁₆×C₁₂₇	—	2	2	no	—
11	C₄×D₈×C₁₂₇	C₄	1	2	yes	—
12	C₂×#108×C₁₂₇	C₂²	1	2	yes	—
13	#427×C₁₂₇	C₂²	1	1	yes	characteristic
14	#416×C₁₂₇	—	2	1	no	—
15	C₄×M₁₆×C₁₂₇	—	2	1	no	—
16	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	#359×C₁₂₇	C₄	1	8	yes	direct product factor
18	C₄×C₁₆×C₁₂₇	—	2	2	no	—
19	#341×C₁₂₇	—	2	1	no	—
20	C₂×#41×C₁₂₇	—	2	1	no	—
21	#135×C₁₂₇	—	2	2	no	—
22	#135×C₁₂₇	C₂×C₄	1	4	yes	—
23	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₂×SD₁₆×C₁₂₇	—	2	1	no	—
25	C₂×D₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₂×M₁₆×C₁₂₇	—	2	2	no	—
27	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
29	C₄×Q₈×C₁₂₇	—	2	2	no	—
30	C₄×D₄×C₁₂₇	—	2	2	no	—
31	C₄×D₄×C₁₂₇	—	4	2	no	—
32	#117×C₁₂₇	—	2	1	no	—
33	#117×C₁₂₇	—	4	1	no	—
34	C₂×C₄²×C₁₂₇	—	4	1	no	—
35	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
36	C₂×C₁₆×C₁₂₇	—	2	2	no	—
37	C₂×C₁₆×C₁₂₇	—	2	2	no	—
38	C₂×C₁₆×C₁₂₇	C₂×C₄	1	4	yes	complemented
39	#108×C₁₂₇	—	2	2	no	—
40	#108×C₁₂₇	C₂×C₄	1	4	yes	complemented
41	#107×C₁₂₇	C₂×C₄	1	2	yes	—
42	#106×C₁₂₇	—	2	1	no	—
43	#103×C₁₂₇	—	2	2	no	—
44	#102×C₁₂₇	—	2	2	no	—
45	#97×C₁₂₇	—	4	1	no	—
46	C₄×C₈×C₁₂₇	—	4	1	no	—
47	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
49	#41×C₁₂₇	—	2	4	no	—
50	#41×C₁₂₇	—	4	2	no	—
51	C₂×Q₈×C₁₂₇	—	2	1	no	—
52	C₂×D₄×C₁₂₇	—	2	1	no	—
53	C₂×D₄×C₁₂₇	—	4	1	no	—
54	C₂²×C₄×C₁₂₇	—	4	1	no	—
55	C₂²×C₄×C₁₂₇	—	4	2	no	—
56	C₂²×C₄×C₁₂₇	SD₁₆	1	2	yes	—
57	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
58	Q₁₆×C₁₂₇	—	2	1	no	—
59	Q₁₆×C₁₂₇	C₄²	1	2	yes	—
60	SD₁₆×C₁₂₇	—	2	4	no	—
61	D₈×C₁₂₇	—	2	1	no	—
62	D₈×C₁₂₇	C₄²	1	2	yes	—
63	M₁₆×C₁₂₇	—	2	4	no	—
64	M₁₆×C₁₂₇	—	4	2	no	—
65	C₂×C₈×C₁₂₇	—	2	2	no	—
66	C₂×C₈×C₁₂₇	—	4	2	no	—
67	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₂₇	#41	1	2	yes	—
72	C₄⋊C₄×C₁₂₇	—	2	2	no	—
73	C₄⋊C₄×C₁₂₇	—	4	2	no	—
74	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
75	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
76	C₄²×C₁₂₇	—	4	2	no	—
77	C₄²×C₁₂₇	—	4	1	no	—
78	C₄²×C₁₂₇	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
80	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
81	C₁₆×C₁₂₇	—	2	4	no	—
82	C₂³×C₁₂₇	—	4	1	no	—
83	C₂³×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
84	Q₈×C₁₂₇	—	2	2	no	—
85	Q₈×C₁₂₇	—	4	1	no	—
86	D₄×C₁₂₇	—	2	2	no	—
87	D₄×C₁₂₇	—	4	4	no	—
88	D₄×C₁₂₇	—	4	1	no	—
89	C₂×C₄×C₁₂₇	—	2	4	no	—
90	C₂×C₄×C₁₂₇	—	4	1	no	—
91	C₂×C₄×C₁₂₇	—	4	4	no	—
92	C₂×C₄×C₁₂₇	—	4	2	no	—
93	C₂×C₄×C₁₂₇	—	8	4	no	—
94	C₂×C₄×C₁₂₇	#102	1	2	yes	—
95	C₂×C₄×C₁₂₇	#102	1	2	yes	—
96	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₁₂₇	#135	1	2	yes	—
100	C₈×C₁₂₇	—	2	2	no	—
101	C₈×C₁₂₇	—	4	4	no	—
102	C₈×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
103	C₈×C₁₂₇	#117	1	1	yes	characteristic
104	C₈×C₁₂₇	C₄×D₄	1	2	yes	—
105	C₂²×C₁₂₇	—	2	2	no	—
106	C₂²×C₁₂₇	—	4	2	no	—
107	C₂²×C₁₂₇	—	8	2	no	—
108	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
109	C₂²×C₁₂₇	C₄×SD₁₆	1	2	yes	—
110	C₄×C₁₂₇	—	2	2	no	—
111	C₄×C₁₂₇	—	4	2	no	—
112	C₄×C₁₂₇	—	8	2	no	—
113	C₄×C₁₂₇	#359	1	4	yes	direct product factor
114	C₄×C₁₂₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
115	C₄×C₁₂₇	#416	1	1	yes	characteristic
116	C₄×C₁₂₇	C₄×D₈	1	2	yes	—
117	C₄×#359	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂×C₁₂₇	—	2	2	no	—
119	C₂×C₁₂₇	—	8	2	no	—
120	C₂×C₁₂₇	C₄×#102	1	1	yes	characteristic
121	C₂×C₁₂₇	C₂×#359	1	2	yes	—
122	C₄×#135	C₂×C₁₂₇	1	1	yes	characteristic
123	C₂×#359	C₂×C₁₂₇	1	2	yes	—
124	C₂×C₄×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
125	C₄×#108	C₂×C₁₂₇	1	1	yes	characteristic, complemented
126	#1382	C₂×C₁₂₇	1	2	yes	complemented
127	C₁₂₇	C₄×#359	1	1	yes	characteristic, direct product factor, Sylow
128	C₂×#135	C₂²×C₁₂₇	1	1	yes	characteristic
129	C₄×#41	—	2	1	no	—
130	C₂²×C₁₆	C₂²×C₁₂₇	1	2	yes	—
131	C₄×Q₁₆	C₄×C₁₂₇	1	2	yes	—
132	C₄×SD₁₆	—	2	2	no	—
133	C₄×D₈	C₄×C₁₂₇	1	2	yes	—
134	C₂×#108	C₂²×C₁₂₇	1	2	yes	—
135	#427	C₂²×C₁₂₇	1	1	yes	characteristic
136	#416	—	2	1	no	—
137	C₄×M₁₆	—	2	1	no	—
138	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
139	#359	C₄×C₁₂₇	1	8	yes	direct product factor
140	C₄×C₁₆	—	2	2	no	—
141	#341	—	2	1	no	—
142	C₂×#41	—	2	1	no	—
143	#135	—	2	2	no	—
144	#135	C₂×C₄×C₁₂₇	1	4	yes	—
145	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
146	C₂×SD₁₆	—	2	1	no	—
147	C₂×D₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
148	C₂×M₁₆	—	2	2	no	—
149	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
150	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
151	C₄×Q₈	—	2	2	no	—
152	C₄×D₄	—	2	2	no	—
153	C₄×D₄	—	4	2	no	—
154	#117	—	2	1	no	—
155	#117	—	4	1	no	—
156	C₂×C₄²	—	4	1	no	—
157	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
158	C₂×C₁₆	—	2	2	no	—
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	C₂×C₄×C₁₂₇	1	4	yes	complemented
161	#108	—	2	2	no	—
162	#108	C₂×C₄×C₁₂₇	1	4	yes	complemented
163	#107	C₂×C₄×C₁₂₇	1	2	yes	—
164	#106	—	2	1	no	—
165	#103	—	2	2	no	—
166	#102	—	2	2	no	—
167	#97	—	4	1	no	—
168	C₄×C₈	—	4	1	no	—
169	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
170	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
171	#41	—	2	4	no	—
172	#41	—	4	2	no	—
173	C₂×Q₈	—	2	1	no	—
174	C₂×D₄	—	2	1	no	—
175	C₂×D₄	—	4	1	no	—
176	C₂²×C₄	—	4	1	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	SD₁₆×C₁₂₇	1	2	yes	—
179	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
180	Q₁₆	—	2	1	no	—
181	Q₁₆	C₄²×C₁₂₇	1	2	yes	—
182	SD₁₆	—	2	4	no	—
183	D₈	—	2	1	no	—
184	D₈	C₄²×C₁₂₇	1	2	yes	—
185	M₁₆	—	2	4	no	—
186	M₁₆	—	4	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	2	yes	—
190	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
191	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
192	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
193	C₂×C₈	#41×C₁₂₇	1	2	yes	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₂²⋊C₄	—	4	1	no	—
197	C₂²⋊C₄	—	4	2	no	—
198	C₄²	—	4	2	no	—
199	C₄²	—	4	1	no	—
200	C₄²	C₂²⋊C₄×C₁₂₇	1	2	yes	—
201	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
202	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
203	C₁₆	—	2	4	no	—
204	C₂³	—	4	1	no	—
205	C₂³	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
206	Q₈	—	2	2	no	—
207	Q₈	—	4	1	no	—
208	D₄	—	2	2	no	—
209	D₄	—	4	4	no	—
210	D₄	—	4	1	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	8	4	no	—
216	C₂×C₄	#102×C₁₂₇	1	2	yes	—
217	C₂×C₄	#102×C₁₂₇	1	2	yes	—
218	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
219	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
220	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
221	C₂×C₄	#135×C₁₂₇	1	2	yes	—
222	C₈	—	2	2	no	—
223	C₈	—	4	4	no	—
224	C₈	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
225	C₈	#117×C₁₂₇	1	1	yes	characteristic
226	C₈	C₄×D₄×C₁₂₇	1	2	yes	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	—	8	2	no	—
230	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
231	C₂²	C₄×SD₁₆×C₁₂₇	1	2	yes	—
232	C₄	—	2	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	8	2	no	—
235	C₄	#359×C₁₂₇	1	4	yes	direct product factor
236	C₄	C₄×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
237	C₄	#416×C₁₂₇	1	1	yes	characteristic
238	C₄	C₄×D₈×C₁₂₇	1	2	yes	—
239	C₂	—	2	2	no	—
240	C₂	—	8	2	no	—
241	C₂	C₄×#102×C₁₂₇	1	1	yes	characteristic
242	C₂	C₂×#359×C₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₄²×C₁₂₇	D₈
commutator	C₈	C₂×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×#102
prime radical	C₂×#135×C₁₂₇	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×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
3. C₂×C₄×C₁₆×C₁₂₇	C₂	semidirect
4. C₄×#108×C₁₂₇	C₂	semidirect
5. C₁₂₇×#1382	C₂	semidirect
17. #359×C₁₂₇	C₄	direct (N × H)
38. C₂×C₁₆×C₁₂₇	C₂×C₄	semidirect
40. #108×C₁₂₇	C₂×C₄	semidirect
113. C₄×C₁₂₇	#359	direct (N × H)
117. C₄×#359	C₁₂₇	direct (N × H)
124. C₂×C₄×C₁₆	C₂×C₁₂₇	semidirect
125. C₄×#108	C₂×C₁₂₇	semidirect
126. #1382	C₂×C₁₂₇	semidirect
127. C₁₂₇	C₄×#359	direct (N × H)
139. #359	C₄×C₁₂₇	direct (N × H)
160. C₂×C₁₆	C₂×C₄×C₁₂₇	semidirect
162. #108	C₂×C₄×C₁₂₇	semidirect
235. C₄	#359×C₁₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#135×C₁₂₇	2
C₂×#359×C₁₂₇	2
C₂×C₄×C₁₆×C₁₂₇	2
C₄×#108×C₁₂₇	2
C₁₂₇×#1382	2
C₄×#359	127

Subgroups — C4×#359×C127 #44576312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#359×C₁₂₇ #44576312