Subgroups — C₂×C₁₂₇×#1830 #44584385

Order 32512.

1218 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#519×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#101×C₁₂₇	C₂	1	2	yes	maximal
3	C₁₂₇×#1830	C₂	1	8	yes	maximal, direct product factor
4	C₂×#344×C₁₂₇	C₂	1	4	yes	maximal
5	C₂³×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₂²×M₁₆×C₁₂₇	C₂²	1	4	yes	—
7	#519×C₁₂₇	—	2	4	no	—
8	#519×C₁₂₇	C₄	1	4	yes	—
9	#519×C₁₂₇	C₂²	1	4	yes	—
10	C₂×C₄×Q₈×C₁₂₇	—	2	4	no	—
11	C₂×#117×C₁₂₇	C₄	1	4	yes	—
12	C₂×#117×C₁₂₇	C₂²	1	2	yes	—
13	C₂×#101×C₁₂₇	—	2	4	no	—
14	C₂×#101×C₁₂₇	—	2	4	no	—
15	C₂×#101×C₁₂₇	C₂²	1	8	yes	—
16	#344×C₁₂₇	C₂²	1	16	yes	—
17	C₂²×Q₈×C₁₂₇	—	2	4	no	—
18	C₂²×Q₈×C₁₂₇	—	2	4	no	—
19	C₂²×Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
20	C₂²×Q₈×C₁₂₇	D₄	1	4	yes	—
21	C₂²×Q₈×C₁₂₇	D₄	1	4	yes	—
22	C₂²×Q₈×C₁₂₇	C₂³	1	2	yes	—
23	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
25	C₂×M₁₆×C₁₂₇	—	2	4	no	—
26	C₂×M₁₆×C₁₂₇	—	2	4	no	—
27	C₂×M₁₆×C₁₂₇	—	2	4	no	—
28	C₂×M₁₆×C₁₂₇	—	2	4	no	—
29	C₂×M₁₆×C₁₂₇	—	2	4	no	—
30	C₂×M₁₆×C₁₂₇	C₂³	1	8	yes	—
31	C₂²×C₈×C₁₂₇	—	2	4	no	—
32	C₄×Q₈×C₁₂₇	—	2	16	no	—
33	C₄×Q₈×C₁₂₇	—	2	16	no	—
34	#117×C₁₂₇	—	2	8	no	—
35	#117×C₁₂₇	—	2	2	no	—
36	#117×C₁₂₇	—	2	2	no	—
37	#117×C₁₂₇	C₂×C₄	1	16	yes	—
38	#117×C₁₂₇	C₂×C₄	1	4	yes	—
39	#117×C₁₂₇	C₂³	1	4	yes	—
40	C₂×C₄⋊C₄×C₁₂₇	—	2	4	no	—
41	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₁₂₇	—	2	4	no	—
44	C₂×C₄²×C₁₂₇	—	2	2	no	—
45	#101×C₁₂₇	—	2	16	no	—
46	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×Q₈×C₁₂₇	—	2	8	no	—
48	C₂×Q₈×C₁₂₇	—	2	8	no	—
49	C₂×Q₈×C₁₂₇	—	2	8	no	—
50	C₂×Q₈×C₁₂₇	—	2	8	no	—
51	C₂×Q₈×C₁₂₇	—	2	8	no	—
52	C₂×Q₈×C₁₂₇	—	2	8	no	—
53	C₂×Q₈×C₁₂₇	C₂²⋊C₄	1	8	yes	—
54	C₂×Q₈×C₁₂₇	C₂×D₄	1	8	yes	—
55	C₂²×C₄×C₁₂₇	—	2	4	no	—
56	C₂²×C₄×C₁₂₇	—	2	4	no	—
57	C₂²×C₄×C₁₂₇	—	2	4	no	—
58	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
60	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₂₇	#41	1	4	yes	—
65	C₂²×C₄×C₁₂₇	#41	1	4	yes	—
66	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
67	M₁₆×C₁₂₇	—	2	8	no	—
68	M₁₆×C₁₂₇	—	2	8	no	—
69	M₁₆×C₁₂₇	—	4	4	no	—
70	M₁₆×C₁₂₇	—	4	4	no	—
71	C₂×C₈×C₁₂₇	—	2	8	no	—
72	C₂×C₈×C₁₂₇	—	4	4	no	—
73	C₂×C₈×C₁₂₇	—	4	4	no	—
74	C₄⋊C₄×C₁₂₇	—	2	16	no	—
75	C₄⋊C₄×C₁₂₇	—	2	4	no	—
76	C₄⋊C₄×C₁₂₇	—	2	4	no	—
77	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
78	C₂²⋊C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
79	C₄²×C₁₂₇	—	2	16	no	—
80	C₄²×C₁₂₇	—	2	4	no	—
81	C₄²×C₁₂₇	—	2	4	no	—
82	C₂³×C₁₂₇	—	4	2	no	—
83	C₂³×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂³×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂³×C₁₂₇	#117	1	2	yes	—
86	C₂³×C₁₂₇	#117	1	2	yes	—
87	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
88	Q₈×C₁₂₇	—	2	16	no	—
89	Q₈×C₁₂₇	—	2	16	no	—
90	C₂×C₄×C₁₂₇	—	2	8	no	—
91	C₂×C₄×C₁₂₇	—	2	4	no	—
92	C₂×C₄×C₁₂₇	—	2	8	no	—
93	C₂×C₄×C₁₂₇	—	2	4	no	—
94	C₂×C₄×C₁₂₇	—	2	8	no	—
95	C₂×C₄×C₁₂₇	—	4	8	no	—
96	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	8	yes	—
97	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₁₂₇	#117	1	4	yes	—
99	C₂×C₄×C₁₂₇	#123	1	4	yes	—
100	C₂×C₄×C₁₂₇	#123	1	4	yes	—
101	C₂×C₄×C₁₂₇	#123	1	4	yes	—
102	C₂×C₄×C₁₂₇	#123	1	4	yes	—
103	C₂×C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
104	C₂×C₄×C₁₂₇	C₂×#41	1	4	yes	—
105	C₈×C₁₂₇	—	4	8	no	—
106	C₂²×C₁₂₇	—	2	2	no	—
107	C₂²×C₁₂₇	—	2	4	no	—
108	C₂²×C₁₂₇	—	4	1	no	—
109	C₂²×C₁₂₇	—	4	2	no	—
110	C₂²×C₁₂₇	—	4	1	no	—
111	C₂²×C₁₂₇	#381	1	2	yes	—
112	C₂²×C₁₂₇	#381	1	1	yes	characteristic
113	C₂²×C₁₂₇	#381	1	1	yes	characteristic
114	C₂²×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×C₁₂₇	C₂×#117	1	2	yes	—
116	C₄×C₁₂₇	—	2	8	no	—
117	C₄×C₁₂₇	—	4	8	no	—
118	C₄×C₁₂₇	#381	1	4	yes	—
119	C₄×C₁₂₇	C₂×#123	1	4	yes	—
120	C₂×#1830	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂×C₁₂₇	—	2	1	no	—
122	C₂×C₁₂₇	—	2	2	no	—
123	C₂×C₁₂₇	—	2	1	no	—
124	C₂×C₁₂₇	—	2	2	no	—
125	C₂×C₁₂₇	#1830	1	2	yes	direct product factor
126	C₂×C₁₂₇	C₂×#381	1	1	yes	characteristic
127	C₂×#519	C₂×C₁₂₇	1	1	yes	characteristic
128	C₂²×#101	C₂×C₁₂₇	1	2	yes	—
129	#1830	C₂×C₁₂₇	1	8	yes	direct product factor
130	C₂×#344	C₂×C₁₂₇	1	4	yes	—
131	C₁₂₇	C₂×#1830	1	1	yes	characteristic, direct product factor, Sylow
132	C₂³×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
133	C₂²×M₁₆	C₂²×C₁₂₇	1	4	yes	—
134	#519	—	2	4	no	—
135	#519	C₄×C₁₂₇	1	4	yes	—
136	#519	C₂²×C₁₂₇	1	4	yes	—
137	C₂×C₄×Q₈	—	2	4	no	—
138	C₂×#117	C₄×C₁₂₇	1	4	yes	—
139	C₂×#117	C₂²×C₁₂₇	1	2	yes	—
140	C₂×#101	—	2	4	no	—
141	C₂×#101	—	2	4	no	—
142	C₂×#101	C₂²×C₁₂₇	1	8	yes	—
143	#344	C₂²×C₁₂₇	1	16	yes	—
144	C₂²×Q₈	—	2	4	no	—
145	C₂²×Q₈	—	2	4	no	—
146	C₂²×Q₈	C₂×C₄×C₁₂₇	1	2	yes	—
147	C₂²×Q₈	D₄×C₁₂₇	1	4	yes	—
148	C₂²×Q₈	D₄×C₁₂₇	1	4	yes	—
149	C₂²×Q₈	C₂³×C₁₂₇	1	2	yes	—
150	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
151	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
152	C₂×M₁₆	—	2	4	no	—
153	C₂×M₁₆	—	2	4	no	—
154	C₂×M₁₆	—	2	4	no	—
155	C₂×M₁₆	—	2	4	no	—
156	C₂×M₁₆	—	2	4	no	—
157	C₂×M₁₆	C₂³×C₁₂₇	1	8	yes	—
158	C₂²×C₈	—	2	4	no	—
159	C₄×Q₈	—	2	16	no	—
160	C₄×Q₈	—	2	16	no	—
161	#117	—	2	8	no	—
162	#117	—	2	2	no	—
163	#117	—	2	2	no	—
164	#117	C₂×C₄×C₁₂₇	1	16	yes	—
165	#117	C₂×C₄×C₁₂₇	1	4	yes	—
166	#117	C₂³×C₁₂₇	1	4	yes	—
167	C₂×C₄⋊C₄	—	2	4	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
170	C₂×C₄²	—	2	4	no	—
171	C₂×C₄²	—	2	2	no	—
172	#101	—	2	16	no	—
173	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
174	C₂×Q₈	—	2	8	no	—
175	C₂×Q₈	—	2	8	no	—
176	C₂×Q₈	—	2	8	no	—
177	C₂×Q₈	—	2	8	no	—
178	C₂×Q₈	—	2	8	no	—
179	C₂×Q₈	—	2	8	no	—
180	C₂×Q₈	C₂²⋊C₄×C₁₂₇	1	8	yes	—
181	C₂×Q₈	C₂×D₄×C₁₂₇	1	8	yes	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
186	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
187	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
188	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
190	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
191	C₂²×C₄	#41×C₁₂₇	1	4	yes	—
192	C₂²×C₄	#41×C₁₂₇	1	4	yes	—
193	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
194	M₁₆	—	2	8	no	—
195	M₁₆	—	2	8	no	—
196	M₁₆	—	4	4	no	—
197	M₁₆	—	4	4	no	—
198	C₂×C₈	—	2	8	no	—
199	C₂×C₈	—	4	4	no	—
200	C₂×C₈	—	4	4	no	—
201	C₄⋊C₄	—	2	16	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	4	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	C₂²×C₄×C₁₂₇	1	8	yes	—
206	C₄²	—	2	16	no	—
207	C₄²	—	2	4	no	—
208	C₄²	—	2	4	no	—
209	C₂³	—	4	2	no	—
210	C₂³	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
211	C₂³	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
212	C₂³	#117×C₁₂₇	1	2	yes	—
213	C₂³	#117×C₁₂₇	1	2	yes	—
214	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
215	Q₈	—	2	16	no	—
216	Q₈	—	2	16	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	8	no	—
222	C₂×C₄	—	4	8	no	—
223	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	8	yes	—
224	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
225	C₂×C₄	#117×C₁₂₇	1	4	yes	—
226	C₂×C₄	#123×C₁₂₇	1	4	yes	—
227	C₂×C₄	#123×C₁₂₇	1	4	yes	—
228	C₂×C₄	#123×C₁₂₇	1	4	yes	—
229	C₂×C₄	#123×C₁₂₇	1	4	yes	—
230	C₂×C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
231	C₂×C₄	C₂×#41×C₁₂₇	1	4	yes	—
232	C₈	—	4	8	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	4	no	—
235	C₂²	—	4	1	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	1	no	—
238	C₂²	#381×C₁₂₇	1	2	yes	—
239	C₂²	#381×C₁₂₇	1	1	yes	characteristic
240	C₂²	#381×C₁₂₇	1	1	yes	characteristic
241	C₂²	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
242	C₂²	C₂×#117×C₁₂₇	1	2	yes	—
243	C₄	—	2	8	no	—
244	C₄	—	4	8	no	—
245	C₄	#381×C₁₂₇	1	4	yes	—
246	C₄	C₂×#123×C₁₂₇	1	4	yes	—
247	C₂	—	2	1	no	—
248	C₂	—	2	2	no	—
249	C₂	—	2	1	no	—
250	C₂	—	2	2	no	—
251	C₂	C₁₂₇×#1830	1	2	yes	direct product factor
252	C₂	C₂×#381×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	#381
commutator	C₂³	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
prime radical	C₂⁴×C₁₂₇	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂³×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₂₇×#1830	C₂	direct (N × H)
120. C₂×#1830	C₁₂₇	direct (N × H)
125. C₂×C₁₂₇	#1830	direct (N × H)
129. #1830	C₂×C₁₂₇	direct (N × H)
131. C₁₂₇	C₂×#1830	direct (N × H)
251. C₂	C₁₂₇×#1830	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#519×C₁₂₇	2
C₂²×#101×C₁₂₇	2
C₁₂₇×#1830	2
C₂×#344×C₁₂₇	2
C₂×#1830	127

Subgroups — C2×C127×#1830 #44584385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×#1830 #44584385