Subgroups — Q₈×#95×C₁₂₇ #44577484

Order 32512.

2434 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₁₂₇	C₂	1	3	yes	maximal
2	C₁₂₇×C₄⋊#95	C₂	1	9	yes	maximal
3	C₄×#95×C₁₂₇	C₂	1	3	yes	maximal
4	C₂³×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
5	C₂×C₄×Q₈×C₁₂₇	—	2	24	no	—
6	C₂×C₄×Q₈×C₁₂₇	C₄	1	24	yes	complemented
7	C₂²×C₄⋊C₄×C₁₂₇	C₂²	1	9	yes	—
8	C₂²×C₄²×C₁₂₇	C₂²	1	9	yes	—
9	C₂×#95×C₁₂₇	C₂²	1	9	yes	—
10	C₂×#95×C₁₂₇	C₂²	1	6	yes	—
11	C₂×#95×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂²×Q₈×C₁₂₇	C₂×C₄	1	12	yes	—
13	C₂²×Q₈×C₁₂₇	D₄	1	12	yes	—
14	C₂²×Q₈×C₁₂₇	Q₈	1	4	yes	—
15	C₂³×C₄×C₁₂₇	C₂³	1	9	yes	—
16	C₂³×C₄×C₁₂₇	C₂³	1	3	yes	—
17	C₂³×C₄×C₁₂₇	C₂³	1	3	yes	—
18	C₄×Q₈×C₁₂₇	—	2	96	no	—
19	C₂×C₄⋊C₄×C₁₂₇	—	2	36	no	—
20	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	36	yes	—
21	C₂×C₄²×C₁₂₇	—	2	36	no	—
22	C₂×C₄²×C₁₂₇	C₂×C₄	1	36	yes	—
23	#95×C₁₂₇	—	2	18	no	—
24	#95×C₁₂₇	—	4	6	no	—
25	#95×C₁₂₇	Q₈	1	4	yes	direct product factor
26	C₂⁴×C₁₂₇	C₂⁴	1	1	yes	characteristic
27	C₂×Q₈×C₁₂₇	C₄²	1	16	yes	—
28	C₂×Q₈×C₁₂₇	C₂²⋊C₄	1	48	yes	—
29	C₂×Q₈×C₁₂₇	C₄⋊C₄	1	48	yes	—
30	C₂²×C₄×C₁₂₇	—	2	6	no	—
31	C₂²×C₄×C₁₂₇	—	2	6	no	—
32	C₂²×C₄×C₁₂₇	—	2	18	no	—
33	C₂²×C₄×C₁₂₇	—	2	18	no	—
34	C₂²×C₄×C₁₂₇	—	2	9	no	—
35	C₂²×C₄×C₁₂₇	—	2	9	no	—
36	C₂²×C₄×C₁₂₇	C₂²×C₄	1	18	yes	—
37	C₂²×C₄×C₁₂₇	C₂²×C₄	1	6	yes	—
38	C₂²×C₄×C₁₂₇	C₂²×C₄	1	18	yes	—
39	C₂²×C₄×C₁₂₇	C₂×D₄	1	18	yes	—
40	C₂²×C₄×C₁₂₇	C₂×Q₈	1	6	yes	—
41	C₂²×C₄×C₁₂₇	C₂×Q₈	1	6	yes	—
42	C₂²×C₄×C₁₂₇	#41	1	6	yes	—
43	C₄⋊C₄×C₁₂₇	—	2	72	no	—
44	C₄²×C₁₂₇	—	2	72	no	—
45	C₂³×C₁₂₇	C₂³×C₄	1	3	yes	—
46	C₂³×C₁₂₇	C₂²×D₄	1	3	yes	—
47	C₂³×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
48	C₂³×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
49	C₂³×C₁₂₇	C₂×#41	1	3	yes	—
50	C₂³×C₁₂₇	H(5,𝔽₂)	1	1	yes	characteristic
51	C₂³×C₁₂₇	2₋¹⁺⁴	1	3	yes	—
52	Q₈×C₁₂₇	#95	1	64	yes	direct product factor
53	C₂×C₄×C₁₂₇	—	2	6	no	—
54	C₂×C₄×C₁₂₇	—	2	6	no	—
55	C₂×C₄×C₁₂₇	—	2	12	no	—
56	C₂×C₄×C₁₂₇	—	2	12	no	—
57	C₂×C₄×C₁₂₇	—	2	18	no	—
58	C₂×C₄×C₁₂₇	—	2	18	no	—
59	C₂×C₄×C₁₂₇	—	2	18	no	—
60	C₂×C₄×C₁₂₇	—	4	18	no	—
61	C₂×C₄×C₁₂₇	—	4	18	no	—
62	C₂×C₄×C₁₂₇	C₂×C₄²	1	12	yes	—
63	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	36	yes	—
64	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	36	yes	—
65	C₂×C₄×C₁₂₇	C₄×Q₈	1	12	yes	complemented
66	C₂²×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
67	C₂²×C₁₂₇	C₂²×C₂²⋊C₄	1	3	yes	—
68	C₂²×C₁₂₇	C₂²×C₄⋊C₄	1	3	yes	—
69	C₂²×C₁₂₇	C₂×C₄×Q₈	1	3	yes	—
70	C₂²×C₁₂₇	C₄×#41	1	3	yes	—
71	C₂²×C₁₂₇	#519	1	3	yes	—
72	C₂²×C₁₂₇	#520	1	3	yes	—
73	C₂²×C₁₂₇	#548	1	6	yes	—
74	C₂²×C₁₂₇	D₄×Q₈	1	3	yes	—
75	C₂²×C₁₂₇	#550	1	3	yes	—
76	C₂²×C₁₂₇	#557	1	3	yes	—
77	C₂²×C₁₂₇	Q₈²	1	1	yes	characteristic
78	C₄×C₁₂₇	—	2	12	no	—
79	C₄×C₁₂₇	—	4	18	no	—
80	C₄×C₁₂₇	C₂×#95	1	24	yes	—
81	Q₈×#95	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₁₂₇	C₂²×#95	1	1	yes	characteristic
83	C₂×C₁₂₇	C₄²×Q₈	1	1	yes	characteristic
84	C₂×C₁₂₇	#2276	1	1	yes	characteristic
85	C₂×C₁₂₇	Q₈×C₂²⋊C₄	1	3	yes	—
86	C₂×C₁₂₇	#2341	1	3	yes	—
87	C₂×C₁₂₇	Q₈×C₄⋊C₄	1	3	yes	—
88	C₂×C₁₂₇	#2351	1	3	yes	—
89	C₂²×C₄×Q₈	C₂×C₁₂₇	1	3	yes	—
90	C₄⋊#95	C₂×C₁₂₇	1	9	yes	—
91	C₄×#95	C₂×C₁₂₇	1	3	yes	—
92	C₁₂₇	Q₈×#95	1	1	yes	characteristic, direct product factor, Sylow
93	C₂³×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
94	C₂×C₄×Q₈	—	2	24	no	—
95	C₂×C₄×Q₈	C₄×C₁₂₇	1	24	yes	complemented
96	C₂²×C₄⋊C₄	C₂²×C₁₂₇	1	9	yes	—
97	C₂²×C₄²	C₂²×C₁₂₇	1	9	yes	—
98	C₂×#95	C₂²×C₁₂₇	1	9	yes	—
99	C₂×#95	C₂²×C₁₂₇	1	6	yes	—
100	C₂×#95	C₂²×C₁₂₇	1	1	yes	characteristic
101	C₂²×Q₈	C₂×C₄×C₁₂₇	1	12	yes	—
102	C₂²×Q₈	D₄×C₁₂₇	1	12	yes	—
103	C₂²×Q₈	Q₈×C₁₂₇	1	4	yes	—
104	C₂³×C₄	C₂³×C₁₂₇	1	9	yes	—
105	C₂³×C₄	C₂³×C₁₂₇	1	3	yes	—
106	C₂³×C₄	C₂³×C₁₂₇	1	3	yes	—
107	C₄×Q₈	—	2	96	no	—
108	C₂×C₄⋊C₄	—	2	36	no	—
109	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	36	yes	—
110	C₂×C₄²	—	2	36	no	—
111	C₂×C₄²	C₂×C₄×C₁₂₇	1	36	yes	—
112	#95	—	2	18	no	—
113	#95	—	4	6	no	—
114	#95	Q₈×C₁₂₇	1	4	yes	direct product factor
115	C₂⁴	C₂⁴×C₁₂₇	1	1	yes	characteristic
116	C₂×Q₈	C₄²×C₁₂₇	1	16	yes	—
117	C₂×Q₈	C₂²⋊C₄×C₁₂₇	1	48	yes	—
118	C₂×Q₈	C₄⋊C₄×C₁₂₇	1	48	yes	—
119	C₂²×C₄	—	2	6	no	—
120	C₂²×C₄	—	2	6	no	—
121	C₂²×C₄	—	2	18	no	—
122	C₂²×C₄	—	2	18	no	—
123	C₂²×C₄	—	2	9	no	—
124	C₂²×C₄	—	2	9	no	—
125	C₂²×C₄	C₂²×C₄×C₁₂₇	1	18	yes	—
126	C₂²×C₄	C₂²×C₄×C₁₂₇	1	6	yes	—
127	C₂²×C₄	C₂²×C₄×C₁₂₇	1	18	yes	—
128	C₂²×C₄	C₂×D₄×C₁₂₇	1	18	yes	—
129	C₂²×C₄	C₂×Q₈×C₁₂₇	1	6	yes	—
130	C₂²×C₄	C₂×Q₈×C₁₂₇	1	6	yes	—
131	C₂²×C₄	#41×C₁₂₇	1	6	yes	—
132	C₄⋊C₄	—	2	72	no	—
133	C₄²	—	2	72	no	—
134	C₂³	C₂³×C₄×C₁₂₇	1	3	yes	—
135	C₂³	C₂²×D₄×C₁₂₇	1	3	yes	—
136	C₂³	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
137	C₂³	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
138	C₂³	C₂×#41×C₁₂₇	1	3	yes	—
139	C₂³	H(5,𝔽₂)×C₁₂₇	1	1	yes	characteristic
140	C₂³	2₋¹⁺⁴×C₁₂₇	1	3	yes	—
141	Q₈	#95×C₁₂₇	1	64	yes	direct product factor
142	C₂×C₄	—	2	6	no	—
143	C₂×C₄	—	2	6	no	—
144	C₂×C₄	—	2	12	no	—
145	C₂×C₄	—	2	12	no	—
146	C₂×C₄	—	2	18	no	—
147	C₂×C₄	—	2	18	no	—
148	C₂×C₄	—	2	18	no	—
149	C₂×C₄	—	4	18	no	—
150	C₂×C₄	—	4	18	no	—
151	C₂×C₄	C₂×C₄²×C₁₂₇	1	12	yes	—
152	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	36	yes	—
153	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	36	yes	—
154	C₂×C₄	C₄×Q₈×C₁₂₇	1	12	yes	complemented
155	C₂²	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
156	C₂²	C₂²×C₂²⋊C₄×C₁₂₇	1	3	yes	—
157	C₂²	C₂²×C₄⋊C₄×C₁₂₇	1	3	yes	—
158	C₂²	C₂×C₄×Q₈×C₁₂₇	1	3	yes	—
159	C₂²	C₄×#41×C₁₂₇	1	3	yes	—
160	C₂²	#519×C₁₂₇	1	3	yes	—
161	C₂²	#520×C₁₂₇	1	3	yes	—
162	C₂²	#548×C₁₂₇	1	6	yes	—
163	C₂²	D₄×Q₈×C₁₂₇	1	3	yes	—
164	C₂²	#550×C₁₂₇	1	3	yes	—
165	C₂²	#557×C₁₂₇	1	3	yes	—
166	C₂²	Q₈²×C₁₂₇	1	1	yes	characteristic
167	C₄	—	2	12	no	—
168	C₄	—	4	18	no	—
169	C₄	C₂×#95×C₁₂₇	1	24	yes	—
170	C₂	C₂²×#95×C₁₂₇	1	1	yes	characteristic
171	C₂	C₄²×Q₈×C₁₂₇	1	1	yes	characteristic
172	C₂	C₁₂₇×#2276	1	1	yes	characteristic
173	C₂	Q₈×C₂²⋊C₄×C₁₂₇	1	3	yes	—
174	C₂	C₁₂₇×#2341	1	3	yes	—
175	C₂	Q₈×C₄⋊C₄×C₁₂₇	1	3	yes	—
176	C₂	C₁₂₇×#2351	1	3	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₂₇	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₂₇
Frattini, cosocle	C₂⁴	C₂⁴×C₁₂₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₂×C₄×Q₈×C₁₂₇	C₄	semidirect
25. #95×C₁₂₇	Q₈	direct (N × H)
52. Q₈×C₁₂₇	#95	direct (N × H)
65. C₂×C₄×C₁₂₇	C₄×Q₈	semidirect
81. Q₈×#95	C₁₂₇	direct (N × H)
92. C₁₂₇	Q₈×#95	direct (N × H)
95. C₂×C₄×Q₈	C₄×C₁₂₇	semidirect
114. #95	Q₈×C₁₂₇	direct (N × H)
141. Q₈	#95×C₁₂₇	direct (N × H)
154. C₂×C₄	C₄×Q₈×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₁₂₇	2
C₁₂₇×C₄⋊#95	2
C₄×#95×C₁₂₇	2
Q₈×#95	127

Subgroups — Q8×#95×C127 #44577484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#95×C₁₂₇ #44577484