Subgroups — C₁₂₇×#3400 #30636352

Order 16256.

362 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	#559×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#557×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#497×C₁₂₇	C₂	1	2	yes	maximal, complemented
4	#492×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	#464×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	#462×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#461×C₁₂₇	C₂	1	2	yes	maximal, complemented
8	#460×C₁₂₇	C₂	1	1	yes	maximal, characteristic
9	#446×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#440×C₁₂₇	C₂	1	2	yes	maximal, complemented
11	C₄×Q₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×Q₁₆×C₁₂₇	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₁₂₇	C₂²	1	2	yes	—
15	C₂×SD₁₆×C₁₂₇	C₂²	1	2	yes	—
16	C₂×SD₁₆×C₁₂₇	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
18	C₄⋊Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	#127×C₁₂₇	C₂²	1	1	yes	characteristic
20	#125×C₁₂₇	—	2	1	no	—
21	#125×C₁₂₇	—	2	2	no	—
22	#124×C₁₂₇	—	2	1	no	—
23	#124×C₁₂₇	C₂²	1	2	yes	—
24	#123×C₁₂₇	—	2	2	no	—
25	C₄×Q₈×C₁₂₇	—	2	1	no	—
26	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
27	C₄×Q₈×C₁₂₇	C₂²	1	2	yes	—
28	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
29	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
30	C₄×D₄×C₁₂₇	—	2	1	no	—
31	C₄×D₄×C₁₂₇	C₂²	1	2	yes	—
32	#107×C₁₂₇	C₂²	1	1	yes	characteristic
33	C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
34	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
35	#103×C₁₂₇	C₂²	1	1	yes	characteristic
36	#103×C₁₂₇	C₂²	1	1	yes	characteristic
37	#103×C₁₂₇	C₂²	1	2	yes	—
38	#102×C₁₂₇	C₂²	1	2	yes	—
39	#97×C₁₂₇	C₂²	1	2	yes	—
40	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₁₂₇	C₂³	1	2	yes	—
44	C₂×Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₁₂₇	—	2	1	no	—
46	C₂×D₄×C₁₂₇	—	2	1	no	—
47	C₂×D₄×C₁₂₇	C₂³	1	2	yes	—
48	C₂²×C₄×C₁₂₇	—	2	2	no	—
49	Q₁₆×C₁₂₇	—	2	2	no	—
50	Q₁₆×C₁₂₇	—	4	2	no	—
51	SD₁₆×C₁₂₇	—	2	4	no	—
52	SD₁₆×C₁₂₇	—	4	2	no	—
53	SD₁₆×C₁₂₇	—	4	2	no	—
54	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
56	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₁₂₇	—	2	2	no	—
58	C₄⋊C₄×C₁₂₇	—	2	1	no	—
59	C₄⋊C₄×C₁₂₇	—	2	1	no	—
60	C₄⋊C₄×C₁₂₇	—	2	2	no	—
61	C₄⋊C₄×C₁₂₇	—	2	1	no	—
62	C₄⋊C₄×C₁₂₇	—	2	2	no	—
63	C₄⋊C₄×C₁₂₇	—	2	1	no	—
64	C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₁₂₇	D₄	1	2	yes	—
66	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
68	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
69	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
70	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
71	C₄²×C₁₂₇	—	2	1	no	—
72	C₄²×C₁₂₇	—	2	2	no	—
73	C₄²×C₁₂₇	—	2	1	no	—
74	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
75	C₄²×C₁₂₇	C₂³	1	2	yes	—
76	C₂³×C₁₂₇	—	2	2	no	—
77	Q₈×C₁₂₇	—	2	2	no	—
78	Q₈×C₁₂₇	—	2	2	no	—
79	Q₈×C₁₂₇	—	2	1	no	—
80	Q₈×C₁₂₇	—	2	1	no	—
81	Q₈×C₁₂₇	—	2	1	no	—
82	Q₈×C₁₂₇	—	2	2	no	—
83	Q₈×C₁₂₇	—	2	1	no	—
84	D₄×C₁₂₇	—	2	2	no	—
85	D₄×C₁₂₇	—	2	2	no	—
86	D₄×C₁₂₇	—	4	2	no	—
87	D₄×C₁₂₇	—	8	1	no	—
88	C₂×C₄×C₁₂₇	—	2	1	no	—
89	C₂×C₄×C₁₂₇	—	2	1	no	—
90	C₂×C₄×C₁₂₇	—	2	1	no	—
91	C₂×C₄×C₁₂₇	—	2	2	no	—
92	C₂×C₄×C₁₂₇	—	2	1	no	—
93	C₂×C₄×C₁₂₇	—	8	2	no	—
94	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
99	C₈×C₁₂₇	—	2	1	no	—
100	C₈×C₁₂₇	—	2	2	no	—
101	C₈×C₁₂₇	#41	1	2	yes	—
102	C₂²×C₁₂₇	—	4	2	no	—
103	C₂²×C₁₂₇	—	4	2	no	—
104	C₂²×C₁₂₇	—	4	2	no	—
105	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₁₂₇	—	2	2	no	—
107	C₄×C₁₂₇	—	2	2	no	—
108	C₄×C₁₂₇	—	2	1	no	—
109	C₄×C₁₂₇	—	4	1	no	—
110	C₄×C₁₂₇	—	4	2	no	—
111	C₄×C₁₂₇	—	4	1	no	—
112	C₄×C₁₂₇	—	4	1	no	—
113	C₄×C₁₂₇	—	4	1	no	—
114	C₄×C₁₂₇	#137	1	2	yes	—
115	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
116	C₄×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂×C₁₂₇	—	8	2	no	—
118	C₂×C₁₂₇	#550	1	1	yes	characteristic
119	C₂×C₁₂₇	C₂×#137	1	1	yes	characteristic
120	C₂×C₁₂₇	#577	1	1	yes	characteristic
121	#3400	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₁₂₇	#3400	1	1	yes	characteristic, direct product factor, Sylow
123	#559	C₂×C₁₂₇	1	1	yes	characteristic
124	#557	C₂×C₁₂₇	1	1	yes	characteristic, complemented
125	#497	C₂×C₁₂₇	1	2	yes	complemented
126	#492	C₂×C₁₂₇	1	1	yes	characteristic
127	#464	C₂×C₁₂₇	1	2	yes	complemented
128	#462	C₂×C₁₂₇	1	1	yes	characteristic, complemented
129	#461	C₂×C₁₂₇	1	2	yes	complemented
130	#460	C₂×C₁₂₇	1	1	yes	characteristic
131	#446	C₂×C₁₂₇	1	1	yes	characteristic, complemented
132	#440	C₂×C₁₂₇	1	2	yes	complemented
133	C₄×Q₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
134	C₂×Q₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
135	C₂×Q₁₆	C₂²×C₁₂₇	1	2	yes	—
136	C₂×SD₁₆	C₂²×C₁₂₇	1	2	yes	—
137	C₂×SD₁₆	C₂²×C₁₂₇	1	2	yes	—
138	C₂×SD₁₆	C₂²×C₁₂₇	1	2	yes	—
139	C₄⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
140	C₄⋊Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
141	#127	C₂²×C₁₂₇	1	1	yes	characteristic
142	#125	—	2	1	no	—
143	#125	—	2	2	no	—
144	#124	—	2	1	no	—
145	#124	C₂²×C₁₂₇	1	2	yes	—
146	#123	—	2	2	no	—
147	C₄×Q₈	—	2	1	no	—
148	C₄×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
149	C₄×Q₈	C₂²×C₁₂₇	1	2	yes	—
150	C₄×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
151	C₄×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
152	C₄×D₄	—	2	1	no	—
153	C₄×D₄	C₂²×C₁₂₇	1	2	yes	—
154	#107	C₂²×C₁₂₇	1	1	yes	characteristic
155	C₄⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
156	C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
157	#103	C₂²×C₁₂₇	1	1	yes	characteristic
158	#103	C₂²×C₁₂₇	1	1	yes	characteristic
159	#103	C₂²×C₁₂₇	1	2	yes	—
160	#102	C₂²×C₁₂₇	1	2	yes	—
161	#97	C₂²×C₁₂₇	1	2	yes	—
162	C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
163	C₂×Q₈	D₄×C₁₂₇	1	1	yes	characteristic
164	C₂×Q₈	C₂³×C₁₂₇	1	1	yes	characteristic
165	C₂×Q₈	C₂³×C₁₂₇	1	2	yes	—
166	C₂×Q₈	C₂³×C₁₂₇	1	1	yes	characteristic
167	C₂×D₄	—	2	1	no	—
168	C₂×D₄	—	2	1	no	—
169	C₂×D₄	C₂³×C₁₂₇	1	2	yes	—
170	C₂²×C₄	—	2	2	no	—
171	Q₁₆	—	2	2	no	—
172	Q₁₆	—	4	2	no	—
173	SD₁₆	—	2	4	no	—
174	SD₁₆	—	4	2	no	—
175	SD₁₆	—	4	2	no	—
176	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
177	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
178	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	D₄×C₁₂₇	1	1	yes	characteristic
187	C₄⋊C₄	D₄×C₁₂₇	1	2	yes	—
188	C₄⋊C₄	C₂³×C₁₂₇	1	1	yes	characteristic
189	C₄⋊C₄	C₂³×C₁₂₇	1	1	yes	characteristic
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₄²	—	2	1	no	—
194	C₄²	—	2	2	no	—
195	C₄²	—	2	1	no	—
196	C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
197	C₄²	C₂³×C₁₂₇	1	2	yes	—
198	C₂³	—	2	2	no	—
199	Q₈	—	2	2	no	—
200	Q₈	—	2	2	no	—
201	Q₈	—	2	1	no	—
202	Q₈	—	2	1	no	—
203	Q₈	—	2	1	no	—
204	Q₈	—	2	2	no	—
205	Q₈	—	2	1	no	—
206	D₄	—	2	2	no	—
207	D₄	—	2	2	no	—
208	D₄	—	4	2	no	—
209	D₄	—	8	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	8	2	no	—
216	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
217	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
218	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
219	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
220	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	2	2	no	—
223	C₈	#41×C₁₂₇	1	2	yes	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	4	2	no	—
227	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	2	no	—
233	C₄	—	4	1	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	1	no	—
236	C₄	#137×C₁₂₇	1	2	yes	—
237	C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
238	C₄	H(5,𝔽₂)×C₁₂₇	1	1	yes	characteristic
239	C₂	—	8	2	no	—
240	C₂	#550×C₁₂₇	1	1	yes	characteristic
241	C₂	C₂×#137×C₁₂₇	1	1	yes	characteristic
242	C₂	#577×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
prime radical	#127×C₁₂₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₄×Q₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #557×C₁₂₇	C₂	semidirect
3. #497×C₁₂₇	C₂	semidirect
5. #464×C₁₂₇	C₂	semidirect
6. #462×C₁₂₇	C₂	semidirect
7. #461×C₁₂₇	C₂	semidirect
9. #446×C₁₂₇	C₂	semidirect
10. #440×C₁₂₇	C₂	semidirect
11. C₄×Q₁₆×C₁₂₇	C₂	semidirect
121. #3400	C₁₂₇	direct (N × H)
122. C₁₂₇	#3400	direct (N × H)
124. #557	C₂×C₁₂₇	semidirect
125. #497	C₂×C₁₂₇	semidirect
127. #464	C₂×C₁₂₇	semidirect
128. #462	C₂×C₁₂₇	semidirect
129. #461	C₂×C₁₂₇	semidirect
131. #446	C₂×C₁₂₇	semidirect
132. #440	C₂×C₁₂₇	semidirect
133. C₄×Q₁₆	C₂×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#559×C₁₂₇	2
#557×C₁₂₇	2
#497×C₁₂₇	2
#492×C₁₂₇	2
#464×C₁₂₇	2
#462×C₁₂₇	2
#461×C₁₂₇	2
#460×C₁₂₇	2
#446×C₁₂₇	2
#440×C₁₂₇	2
C₄×Q₁₆×C₁₂₇	2
#3400	127

Subgroups — C127×#3400 #30636352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#3400 #30636352