Subgroups — C₂×C₁₂₇×#1362 #44575601

Order 32512.

414 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#335×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1362	C₂	1	4	yes	maximal, direct product factor
5	C₂²×Q₁₆×C₁₂₇	—	2	1	no	—
6	C₂²×Q₁₆×C₁₂₇	C₄	1	2	yes	—
7	C₂×C₄⋊Q₈×C₁₂₇	—	2	1	no	—
8	C₂²×C₁₆×C₁₂₇	—	2	1	no	—
9	#494×C₁₂₇	—	2	1	no	—
10	#494×C₁₂₇	—	2	1	no	—
11	#494×C₁₂₇	C₄	1	2	yes	—
12	#494×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈×C₁₂₇	—	2	1	no	—
14	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₁₂₇	—	2	1	no	—
16	C₄⋊C₁₆×C₁₂₇	C₂²	1	2	yes	—
17	#335×C₁₂₇	—	2	1	no	—
18	#335×C₁₂₇	C₂²	1	2	yes	—
19	C₂²×Q₈×C₁₂₇	—	2	2	no	—
20	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
21	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
22	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
23	C₂×Q₁₆×C₁₂₇	—	4	4	no	—
24	C₂×Q₁₆×C₁₂₇	—	4	2	no	—
25	C₂×Q₁₆×C₁₂₇	—	4	2	no	—
26	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₂₇	—	4	1	no	—
28	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₂₇	D₄	1	1	yes	characteristic
30	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
31	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
32	C₄⋊Q₈×C₁₂₇	—	4	1	no	—
33	C₄⋊Q₈×C₁₂₇	—	4	1	no	—
34	C₂×C₄⋊C₄×C₁₂₇	—	4	1	no	—
35	C₂×C₄⋊C₄×C₁₂₇	—	4	1	no	—
36	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
37	C₂×C₁₆×C₁₂₇	—	2	2	no	—
38	C₂×C₁₆×C₁₂₇	—	4	2	no	—
39	C₄⋊C₈×C₁₂₇	—	2	2	no	—
40	C₄⋊C₈×C₁₂₇	—	4	1	no	—
41	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₁₂₇	—	2	4	no	—
46	C₂×Q₈×C₁₂₇	—	4	2	no	—
47	C₂×Q₈×C₁₂₇	—	4	4	no	—
48	C₂×Q₈×C₁₂₇	—	4	4	no	—
49	C₂²×C₄×C₁₂₇	—	4	2	no	—
50	C₂²×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₂₇	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₂₇	SD₁₆	1	1	yes	characteristic
53	Q₁₆×C₁₂₇	—	4	4	no	—
54	Q₁₆×C₁₂₇	—	4	4	no	—
55	Q₁₆×C₁₂₇	—	4	4	no	—
56	Q₁₆×C₁₂₇	—	4	2	no	—
57	Q₁₆×C₁₂₇	—	4	2	no	—
58	C₂×C₈×C₁₂₇	—	2	2	no	—
59	C₂×C₈×C₁₂₇	—	4	2	no	—
60	C₂×C₈×C₁₂₇	—	8	2	no	—
61	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₂₇	D₈	1	2	yes	—
64	C₂×C₈×C₁₂₇	SD₁₆	1	2	yes	—
65	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
67	C₄⋊C₄×C₁₂₇	—	4	2	no	—
68	C₄⋊C₄×C₁₂₇	—	4	2	no	—
69	C₄⋊C₄×C₁₂₇	—	4	2	no	—
70	C₄⋊C₄×C₁₂₇	—	4	2	no	—
71	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₁₂₇	D₈	1	1	yes	characteristic
73	C₄²×C₁₂₇	SD₁₆	1	1	yes	characteristic
74	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
75	C₁₆×C₁₂₇	—	4	2	no	—
76	C₂³×C₁₂₇	#102	1	1	yes	characteristic
77	Q₈×C₁₂₇	—	4	4	no	—
78	Q₈×C₁₂₇	—	4	4	no	—
79	Q₈×C₁₂₇	—	8	4	no	—
80	C₂×C₄×C₁₂₇	—	2	2	no	—
81	C₂×C₄×C₁₂₇	—	4	4	no	—
82	C₂×C₄×C₁₂₇	—	8	4	no	—
83	C₂×C₄×C₁₂₇	#100	1	2	yes	—
84	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
85	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
86	C₂×C₄×C₁₂₇	#102	1	2	yes	—
87	C₂×C₄×C₁₂₇	#102	1	1	yes	characteristic
88	C₂×C₄×C₁₂₇	SD₃₂	1	2	yes	—
89	C₂×C₄×C₁₂₇	Q₃₂	1	2	yes	—
90	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
93	C₈×C₁₂₇	—	2	1	no	—
94	C₈×C₁₂₇	—	2	1	no	—
95	C₈×C₁₂₇	—	8	2	no	—
96	C₈×C₁₂₇	#102	1	2	yes	—
97	C₈×C₁₂₇	C₂×D₈	1	1	yes	characteristic
98	C₈×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
99	C₂²×C₁₂₇	#331	1	2	yes	—
100	C₂²×C₁₂₇	#358	1	2	yes	—
101	C₂²×C₁₂₇	#362	1	2	yes	—
102	C₂²×C₁₂₇	C₂×#102	1	1	yes	characteristic
103	C₄×C₁₂₇	—	2	1	no	—
104	C₄×C₁₂₇	—	2	1	no	—
105	C₄×C₁₂₇	—	8	4	no	—
106	C₄×C₁₂₇	#331	1	2	yes	—
107	C₄×C₁₂₇	#358	1	2	yes	—
108	C₄×C₁₂₇	C₂×#100	1	1	yes	characteristic
109	C₄×C₁₂₇	C₂×#102	1	1	yes	characteristic
110	C₄×C₁₂₇	C₂×SD₃₂	1	1	yes	characteristic
111	C₄×C₁₂₇	C₂×Q₃₂	1	1	yes	characteristic
112	C₂×#1362	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂×C₁₂₇	#1362	1	4	yes	direct product factor
114	C₂×C₁₂₇	C₂×#331	1	1	yes	characteristic
115	C₂×C₁₂₇	C₂×#358	1	1	yes	characteristic
116	C₂×C₁₂₇	C₂×#362	1	1	yes	characteristic
117	C₂×#494	C₂×C₁₂₇	1	1	yes	characteristic
118	C₂×C₄⋊C₁₆	C₂×C₁₂₇	1	1	yes	characteristic
119	C₂×#335	C₂×C₁₂₇	1	1	yes	characteristic
120	#1362	C₂×C₁₂₇	1	4	yes	direct product factor
121	C₁₂₇	C₂×#1362	1	1	yes	characteristic, direct product factor, Sylow
122	C₂²×Q₁₆	—	2	1	no	—
123	C₂²×Q₁₆	C₄×C₁₂₇	1	2	yes	—
124	C₂×C₄⋊Q₈	—	2	1	no	—
125	C₂²×C₁₆	—	2	1	no	—
126	#494	—	2	1	no	—
127	#494	—	2	1	no	—
128	#494	C₄×C₁₂₇	1	2	yes	—
129	#494	C₂²×C₁₂₇	1	2	yes	—
130	C₂×C₄⋊C₈	—	2	1	no	—
131	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
132	C₄⋊C₁₆	—	2	1	no	—
133	C₄⋊C₁₆	C₂²×C₁₂₇	1	2	yes	—
134	#335	—	2	1	no	—
135	#335	C₂²×C₁₂₇	1	2	yes	—
136	C₂²×Q₈	—	2	2	no	—
137	C₂×Q₁₆	—	2	2	no	—
138	C₂×Q₁₆	—	2	2	no	—
139	C₂×Q₁₆	—	2	2	no	—
140	C₂×Q₁₆	—	4	4	no	—
141	C₂×Q₁₆	—	4	2	no	—
142	C₂×Q₁₆	—	4	2	no	—
143	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	4	yes	—
144	C₂²×C₈	—	4	1	no	—
145	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
146	C₂²×C₈	D₄×C₁₂₇	1	1	yes	characteristic
147	C₄⋊Q₈	—	2	2	no	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	4	1	no	—
150	C₄⋊Q₈	—	4	1	no	—
151	C₂×C₄⋊C₄	—	4	1	no	—
152	C₂×C₄⋊C₄	—	4	1	no	—
153	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
154	C₂×C₁₆	—	2	2	no	—
155	C₂×C₁₆	—	4	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	4	1	no	—
158	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
159	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
160	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
161	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
162	C₂×Q₈	—	2	4	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	4	4	no	—
165	C₂×Q₈	—	4	4	no	—
166	C₂²×C₄	—	4	2	no	—
167	C₂²×C₄	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
168	C₂²×C₄	D₈×C₁₂₇	1	1	yes	characteristic
169	C₂²×C₄	SD₁₆×C₁₂₇	1	1	yes	characteristic
170	Q₁₆	—	4	4	no	—
171	Q₁₆	—	4	4	no	—
172	Q₁₆	—	4	4	no	—
173	Q₁₆	—	4	2	no	—
174	Q₁₆	—	4	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	8	2	no	—
178	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
179	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
180	C₂×C₈	D₈×C₁₂₇	1	2	yes	—
181	C₂×C₈	SD₁₆×C₁₂₇	1	2	yes	—
182	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
183	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
189	C₄²	D₈×C₁₂₇	1	1	yes	characteristic
190	C₄²	SD₁₆×C₁₂₇	1	1	yes	characteristic
191	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
192	C₁₆	—	4	2	no	—
193	C₂³	#102×C₁₂₇	1	1	yes	characteristic
194	Q₈	—	4	4	no	—
195	Q₈	—	4	4	no	—
196	Q₈	—	8	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	8	4	no	—
200	C₂×C₄	#100×C₁₂₇	1	2	yes	—
201	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
202	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
203	C₂×C₄	#102×C₁₂₇	1	2	yes	—
204	C₂×C₄	#102×C₁₂₇	1	1	yes	characteristic
205	C₂×C₄	SD₃₂×C₁₂₇	1	2	yes	—
206	C₂×C₄	Q₃₂×C₁₂₇	1	2	yes	—
207	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
208	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
209	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	8	2	no	—
213	C₈	#102×C₁₂₇	1	2	yes	—
214	C₈	C₂×D₈×C₁₂₇	1	1	yes	characteristic
215	C₈	C₂×SD₁₆×C₁₂₇	1	1	yes	characteristic
216	C₂²	#331×C₁₂₇	1	2	yes	—
217	C₂²	#358×C₁₂₇	1	2	yes	—
218	C₂²	#362×C₁₂₇	1	2	yes	—
219	C₂²	C₂×#102×C₁₂₇	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	8	4	no	—
223	C₄	#331×C₁₂₇	1	2	yes	—
224	C₄	#358×C₁₂₇	1	2	yes	—
225	C₄	C₂×#100×C₁₂₇	1	1	yes	characteristic
226	C₄	C₂×#102×C₁₂₇	1	1	yes	characteristic
227	C₄	C₂×SD₃₂×C₁₂₇	1	1	yes	characteristic
228	C₄	C₂×Q₃₂×C₁₂₇	1	1	yes	characteristic
229	C₂	C₁₂₇×#1362	1	4	yes	direct product factor
230	C₂	C₂×#331×C₁₂₇	1	1	yes	characteristic
231	C₂	C₂×#358×C₁₂₇	1	1	yes	characteristic
232	C₂	C₂×#362×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₂₇	#102
commutator	C₂×C₈	C₂²×C₄×C₁₂₇
Frattini, cosocle	C₄×C₈	C₂³×C₁₂₇
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
4. C₁₂₇×#1362	C₂	direct (N × H)
112. C₂×#1362	C₁₂₇	direct (N × H)
113. C₂×C₁₂₇	#1362	direct (N × H)
120. #1362	C₂×C₁₂₇	direct (N × H)
121. C₁₂₇	C₂×#1362	direct (N × H)
229. C₂	C₁₂₇×#1362	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×#494×C₁₂₇	2
C₂×C₄⋊C₁₆×C₁₂₇	2
C₂×#335×C₁₂₇	2
C₁₂₇×#1362	2
C₂×#1362	127

Subgroups — C2×C127×#1362 #44575601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×#1362 #44575601