Subgroups — #30647374

Order 16256.

400 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549937	C₂	1	4	yes	maximal, complemented
2	#22549929	C₂	1	4	yes	maximal, complemented
3	#22549903	C₂	1	2	yes	maximal, complemented
4	#22549843	C₂	1	2	yes	maximal
5	C₄×#14523232	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#41×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14523232	C₂²	1	2	yes	—
9	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
10	C₄⋊(C₁₂₇⋊C₈)	C₂²	1	8	yes	—
11	C₂²⋊(C₁₂₇⋊C₈)	C₂²	1	8	yes	—
12	#16861120	C₂²	1	2	yes	—
13	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
14	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
15	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
16	C₂×#41×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₁₂₇	C₄	1	2	yes	—
18	C₄×D₄×C₁₂₇	C₄	1	4	yes	—
19	C₄×D₄×C₁₂₇	C₄	1	2	yes	—
20	#117×C₁₂₇	C₂²	1	1	yes	characteristic
21	#117×C₁₂₇	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
24	#14523232	—	2	4	no	—
25	C₂×C₁₂₇⋊C₈	—	2	4	no	—
26	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
27	C₂×C₁₂₇⋊C₈	C₂³	1	4	yes	—
28	#41×C₁₂₇	—	2	4	no	—
29	C₂×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
30	C₂×D₄×C₁₂₇	C₂×C₄	1	2	yes	—
31	C₂×D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
35	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
37	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
38	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
39	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
40	C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
41	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
42	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
43	C₄²×C₁₂₇	C₂³	1	2	yes	—
44	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
45	C₁₂₇⋊C₈	—	2	4	no	—
46	C₁₂₇⋊C₈	—	2	4	no	—
47	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
48	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
49	Q₈×C₁₂₇	—	2	2	no	—
50	D₄×C₁₂₇	—	2	4	no	—
51	D₄×C₁₂₇	—	2	2	no	—
52	C₂×C₄×C₁₂₇	—	2	4	no	—
53	C₂×C₄×C₁₂₇	—	2	8	no	—
54	C₂×C₄×C₁₂₇	—	2	4	no	—
55	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
58	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₁₂₇	#41	1	4	yes	—
61	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
62	C₂²×C₁₂₇	—	2	2	no	—
63	C₂²×C₁₂₇	—	2	2	no	—
64	C₂²×C₁₂₇	—	2	4	no	—
65	C₂²×C₁₂₇	#117	1	2	yes	—
66	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
67	C₄×C₁₂₇	—	2	2	no	—
68	C₄×C₁₂₇	—	2	2	no	—
69	C₄×C₁₂₇	—	2	4	no	—
70	C₄×C₁₂₇	#117	1	2	yes	—
71	C₄×C₁₂₇	#131	1	4	yes	—
72	C₄×C₁₂₇	C₂×#41	1	2	yes	—
73	C₂×C₁₂₇	—	2	2	no	—
74	C₂×C₁₂₇	—	4	2	no	—
75	C₂×C₁₂₇	C₂×#117	1	1	yes	characteristic
76	C₂×C₁₂₇	C₂×#131	1	2	yes	—
77	#2922	—	127	1	no	maximal, Sylow
78	C₁₂₇	#2922	1	1	yes	characteristic, complemented, Sylow
79	C₄×#41	D₁₂₇	1	1	yes	characteristic
80	#433	—	127	4	no	—
81	#431	—	127	4	no	—
82	#424	—	127	2	no	—
83	#408	—	127	2	no	—
84	C₄×M₁₆	—	127	1	no	—
85	C₂×C₄×C₈	—	127	1	no	—
86	C₂×#41	C₂×D₁₂₇	1	1	yes	characteristic
87	C₂×M₁₆	—	127	2	no	—
88	C₂²×C₈	—	127	2	no	—
89	C₄×Q₈	Dic₁₂₇	1	2	yes	—
90	C₄×D₄	Dic₁₂₇	1	4	yes	—
91	C₄×D₄	Dic₁₂₇	1	2	yes	—
92	#117	C₂×D₁₂₇	1	1	yes	characteristic
93	#117	C₂×D₁₂₇	1	2	yes	—
94	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
95	C₂×C₄²	C₂×D₁₂₇	1	2	yes	—
96	C₄⋊C₈	—	127	8	no	—
97	C₂²⋊C₈	—	127	8	no	—
98	#97	—	127	2	no	—
99	C₄×C₈	—	127	2	no	—
100	C₄×C₈	—	127	2	no	—
101	C₄×C₈	—	127	2	no	—
102	#41	—	2	4	no	—
103	C₂×Q₈	C₂×Dic₁₂₇	1	1	yes	characteristic
104	C₂×D₄	C₂×Dic₁₂₇	1	2	yes	—
105	C₂×D₄	C₂×Dic₁₂₇	1	1	yes	characteristic
106	C₂²×C₄	C₂×Dic₁₂₇	1	4	yes	—
107	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
108	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
109	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
110	M₁₆	—	254	4	no	—
111	C₂×C₈	—	127	4	no	—
112	C₂×C₈	—	127	4	no	—
113	C₂×C₈	—	254	4	no	—
114	C₄⋊C₄	C₂×Dic₁₂₇	1	4	yes	—
115	C₄⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
116	C₂²⋊C₄	C₂×Dic₁₂₇	1	4	yes	—
117	C₂²⋊C₄	C₂×Dic₁₂₇	1	2	yes	—
118	C₄²	C₂×Dic₁₂₇	1	4	yes	—
119	C₄²	C₂×Dic₁₂₇	1	2	yes	—
120	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
121	C₄²	C₂²×D₁₂₇	1	2	yes	—
122	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
123	C₂³	C₂²×Dic₁₂₇	1	2	yes	—
124	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	D₄	—	2	4	no	—
127	D₄	—	2	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	8	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
132	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
133	C₂×C₄	C₂²×Dic₁₂₇	1	4	yes	—
134	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
135	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
136	C₂×C₄	#14523248	1	4	yes	—
137	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
138	C₈	—	254	4	no	—
139	C₈	—	254	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	2	4	no	—
143	C₂²	#16861168	1	2	yes	—
144	C₂²	C₂³×Dic₁₂₇	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	2	4	no	—
148	C₄	#16861168	1	2	yes	—
149	C₄	#16861216	1	4	yes	—
150	C₄	C₂×#14523248	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	4	2	no	—
153	C₂	C₂×#16861168	1	1	yes	characteristic
154	C₂	C₂×#16861216	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₄×#41×C₁₂₇	C₂
prime radical	C₂×#41×C₁₂₇	C₂²
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₄×#41	D₁₂₇
hypocenter	C₁₂₇	#2922
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #22549937	C₂	semidirect
2. #22549929	C₂	semidirect
3. #22549903	C₂	semidirect
5. C₄×#14523232	C₂	semidirect
6. C₂×C₄×C₁₂₇⋊C₈	C₂	semidirect
78. C₁₂₇	#2922	semidirect

Maximal subgroups

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

subgroup	index
#22549937	2
#22549929	2
#22549903	2
#22549843	2
C₄×#14523232	2
C₂×C₄×C₁₂₇⋊C₈	2
C₄×#41×C₁₂₇	2
#2922	127