Subgroups — C₂×#416×C₁₂₇ #30635844

Order 16256.

792 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#416×C₁₂₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×#103×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	C₂×#102×C₁₂₇	C₂	1	2	yes	maximal, complemented
7	C₂×#41×C₁₂₇	—	2	4	no	—
8	C₂×#41×C₁₂₇	—	2	2	no	—
9	C₂×#41×C₁₂₇	C₄	1	8	yes	complemented
10	C₂×#41×C₁₂₇	C₂²	1	4	yes	—
11	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₁₂₇	—	2	1	no	—
13	C₂²×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₂₇	—	2	1	no	—
15	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₁₂₇	—	2	4	no	—
17	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
18	C₂²×C₈×C₁₂₇	C₂²	1	4	yes	complemented
19	#117×C₁₂₇	—	2	2	no	—
20	#117×C₁₂₇	C₂²	1	4	yes	complemented
21	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₁₂₇	—	2	1	no	—
23	C₂×C₄²×C₁₂₇	—	2	1	no	—
24	#103×C₁₂₇	C₂²	1	8	yes	complemented
25	#102×C₁₂₇	C₂²	1	8	yes	complemented
26	C₂⁴×C₁₂₇	—	2	1	no	—
27	C₂⁴×C₁₂₇	D₄	1	1	yes	characteristic
28	#41×C₁₂₇	—	2	16	no	—
29	#41×C₁₂₇	—	2	8	no	—
30	#41×C₁₂₇	C₂×C₄	1	16	yes	complemented
31	C₂×Q₈×C₁₂₇	—	2	2	no	—
32	C₂×Q₈×C₁₂₇	—	2	1	no	—
33	C₂×Q₈×C₁₂₇	C₂×C₄	1	4	yes	—
34	C₂×Q₈×C₁₂₇	C₂³	1	2	yes	—
35	C₂×D₄×C₁₂₇	—	2	8	no	—
36	C₂×D₄×C₁₂₇	—	2	4	no	—
37	C₂×D₄×C₁₂₇	—	2	1	no	—
38	C₂×D₄×C₁₂₇	—	2	2	no	—
39	C₂×D₄×C₁₂₇	C₂×C₄	1	4	yes	—
40	C₂×D₄×C₁₂₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₂₇	—	2	1	no	—
42	C₂²×C₄×C₁₂₇	—	2	8	no	—
43	C₂²×C₄×C₁₂₇	—	2	4	no	—
44	C₂²×C₄×C₁₂₇	—	2	2	no	—
45	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
46	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
47	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
48	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
51	C₂×C₈×C₁₂₇	—	2	4	no	—
52	C₂×C₈×C₁₂₇	—	2	8	no	—
53	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
54	C₄⋊C₄×C₁₂₇	—	2	2	no	—
55	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
56	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
57	C₄²×C₁₂₇	—	2	4	no	—
58	C₂³×C₁₂₇	—	2	4	no	—
59	C₂³×C₁₂₇	—	2	4	no	—
60	C₂³×C₁₂₇	—	2	2	no	—
61	C₂³×C₁₂₇	—	4	4	no	—
62	C₂³×C₁₂₇	C₂²⋊C₄	1	4	yes	—
63	C₂³×C₁₂₇	C₂×D₄	1	2	yes	—
64	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
65	Q₈×C₁₂₇	—	2	4	no	—
66	Q₈×C₁₂₇	—	2	2	no	—
67	Q₈×C₁₂₇	C₂²×C₄	1	4	yes	—
68	D₄×C₁₂₇	—	2	16	no	—
69	D₄×C₁₂₇	—	2	4	no	—
70	D₄×C₁₂₇	—	2	2	no	—
71	D₄×C₁₂₇	C₂²×C₄	1	4	yes	—
72	C₂×C₄×C₁₂₇	—	2	4	no	—
73	C₂×C₄×C₁₂₇	—	2	2	no	—
74	C₂×C₄×C₁₂₇	—	2	16	no	—
75	C₂×C₄×C₁₂₇	—	2	4	no	—
76	C₂×C₄×C₁₂₇	—	4	4	no	—
77	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
78	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	8	yes	—
79	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
81	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
82	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
85	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
86	C₈×C₁₂₇	—	2	4	no	—
87	C₂²×C₁₂₇	—	2	4	no	—
88	C₂²×C₁₂₇	—	2	8	no	—
89	C₂²×C₁₂₇	—	2	4	no	—
90	C₂²×C₁₂₇	—	4	4	no	—
91	C₂²×C₁₂₇	—	4	8	no	—
92	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
93	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
94	C₂²×C₁₂₇	#135	1	4	yes	—
95	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
96	C₄×C₁₂₇	—	2	4	no	—
97	C₄×C₁₂₇	—	4	4	no	—
98	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
99	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
100	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
101	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₁₂₇	—	2	4	no	—
103	C₂×C₁₂₇	—	4	4	no	—
104	C₂×C₁₂₇	#416	1	4	yes	direct product factor
105	C₂×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₁₂₇	C₂×#135	1	2	yes	—
107	C₂×#416	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₁₂₇	C₂×#416	1	1	yes	characteristic, direct product factor, Sylow
109	C₂²×#41	C₂×C₁₂₇	1	1	yes	characteristic
110	C₂³×C₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
111	C₂×#117	C₂×C₁₂₇	1	1	yes	characteristic, complemented
112	#416	C₂×C₁₂₇	1	8	yes	direct product factor
113	C₂×#103	C₂×C₁₂₇	1	2	yes	complemented
114	C₂×#102	C₂×C₁₂₇	1	2	yes	complemented
115	C₂×#41	—	2	4	no	—
116	C₂×#41	—	2	2	no	—
117	C₂×#41	C₄×C₁₂₇	1	8	yes	complemented
118	C₂×#41	C₂²×C₁₂₇	1	4	yes	—
119	C₂²×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
120	C₂²×D₄	—	2	1	no	—
121	C₂²×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
122	C₂³×C₄	—	2	1	no	—
123	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
124	C₂²×C₈	—	2	4	no	—
125	C₂²×C₈	C₂²×C₁₂₇	1	2	yes	—
126	C₂²×C₈	C₂²×C₁₂₇	1	4	yes	complemented
127	#117	—	2	2	no	—
128	#117	C₂²×C₁₂₇	1	4	yes	complemented
129	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
130	C₂×C₂²⋊C₄	—	2	1	no	—
131	C₂×C₄²	—	2	1	no	—
132	#103	C₂²×C₁₂₇	1	8	yes	complemented
133	#102	C₂²×C₁₂₇	1	8	yes	complemented
134	C₂⁴	—	2	1	no	—
135	C₂⁴	D₄×C₁₂₇	1	1	yes	characteristic
136	#41	—	2	16	no	—
137	#41	—	2	8	no	—
138	#41	C₂×C₄×C₁₂₇	1	16	yes	complemented
139	C₂×Q₈	—	2	2	no	—
140	C₂×Q₈	—	2	1	no	—
141	C₂×Q₈	C₂×C₄×C₁₂₇	1	4	yes	—
142	C₂×Q₈	C₂³×C₁₂₇	1	2	yes	—
143	C₂×D₄	—	2	8	no	—
144	C₂×D₄	—	2	4	no	—
145	C₂×D₄	—	2	1	no	—
146	C₂×D₄	—	2	2	no	—
147	C₂×D₄	C₂×C₄×C₁₂₇	1	4	yes	—
148	C₂×D₄	C₂³×C₁₂₇	1	2	yes	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	8	no	—
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	—	2	2	no	—
153	C₂²×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
154	C₂²×C₄	D₄×C₁₂₇	1	2	yes	—
155	C₂²×C₄	D₄×C₁₂₇	1	1	yes	characteristic
156	C₂²×C₄	D₄×C₁₂₇	1	4	yes	—
157	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	8	no	—
161	C₂×C₈	C₂³×C₁₂₇	1	4	yes	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	C₂³×C₁₂₇	1	4	yes	—
164	C₂²⋊C₄	—	2	4	no	—
165	C₄²	—	2	4	no	—
166	C₂³	—	2	4	no	—
167	C₂³	—	2	4	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	4	4	no	—
170	C₂³	C₂²⋊C₄×C₁₂₇	1	4	yes	—
171	C₂³	C₂×D₄×C₁₂₇	1	2	yes	—
172	C₂³	C₂×D₄×C₁₂₇	1	1	yes	characteristic
173	Q₈	—	2	4	no	—
174	Q₈	—	2	2	no	—
175	Q₈	C₂²×C₄×C₁₂₇	1	4	yes	—
176	D₄	—	2	16	no	—
177	D₄	—	2	4	no	—
178	D₄	—	2	2	no	—
179	D₄	C₂²×C₄×C₁₂₇	1	4	yes	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	16	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
186	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	8	yes	—
187	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
188	C₂×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
189	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
190	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
191	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
192	C₂×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
193	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
194	C₈	—	2	4	no	—
195	C₂²	—	2	4	no	—
196	C₂²	—	2	8	no	—
197	C₂²	—	2	4	no	—
198	C₂²	—	4	4	no	—
199	C₂²	—	4	8	no	—
200	C₂²	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
201	C₂²	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
202	C₂²	#135×C₁₂₇	1	4	yes	—
203	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
204	C₄	—	2	4	no	—
205	C₄	—	4	4	no	—
206	C₄	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
207	C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
208	C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
209	C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
210	C₂	—	2	4	no	—
211	C₂	—	4	4	no	—
212	C₂	#416×C₁₂₇	1	4	yes	direct product factor
213	C₂	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
214	C₂	C₂×#135×C₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	D₄
commutator	C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
socle	C₂³×C₁₂₇	C₂×D₄
prime radical	C₂²×#41×C₁₂₇	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₂₇ ◁ C₂³×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₈×C₁₂₇	C₂	semidirect
3. C₂×#117×C₁₂₇	C₂	semidirect
4. #416×C₁₂₇	C₂	direct (N × H)
5. C₂×#103×C₁₂₇	C₂	semidirect
6. C₂×#102×C₁₂₇	C₂	semidirect
9. C₂×#41×C₁₂₇	C₄	semidirect
18. C₂²×C₈×C₁₂₇	C₂²	semidirect
20. #117×C₁₂₇	C₂²	semidirect
24. #103×C₁₂₇	C₂²	semidirect
25. #102×C₁₂₇	C₂²	semidirect
30. #41×C₁₂₇	C₂×C₄	semidirect
104. C₂×C₁₂₇	#416	direct (N × H)
107. C₂×#416	C₁₂₇	direct (N × H)
108. C₁₂₇	C₂×#416	direct (N × H)
110. C₂³×C₈	C₂×C₁₂₇	semidirect
111. C₂×#117	C₂×C₁₂₇	semidirect
112. #416	C₂×C₁₂₇	direct (N × H)
113. C₂×#103	C₂×C₁₂₇	semidirect
114. C₂×#102	C₂×C₁₂₇	semidirect
117. C₂×#41	C₄×C₁₂₇	semidirect
126. C₂²×C₈	C₂²×C₁₂₇	semidirect
128. #117	C₂²×C₁₂₇	semidirect
132. #103	C₂²×C₁₂₇	semidirect
133. #102	C₂²×C₁₂₇	semidirect
138. #41	C₂×C₄×C₁₂₇	semidirect
212. C₂	#416×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
C₂²×#41×C₁₂₇	2
C₂³×C₈×C₁₂₇	2
C₂×#117×C₁₂₇	2
#416×C₁₂₇	2
C₂×#103×C₁₂₇	2
C₂×#102×C₁₂₇	2
C₂×#416	127

Subgroups — C2×#416×C127 #30635844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#416×C₁₂₇ #30635844