Subgroups — #28371595

Order 13952.

484 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13182013	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812658	C₂	1	1	yes	maximal, characteristic
3	#20295054	C₂	1	4	yes	maximal, complemented
4	#20295052	C₂	1	4	yes	maximal, complemented
5	#20295049	C₂	1	4	yes	maximal, complemented
6	C₂²×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#13182013	—	2	4	no	—
8	C₂×#13182013	C₂²	1	2	yes	—
9	C₂×#13182013	C₂²	1	2	yes	—
10	C₂²×#11724784	C₂²	1	2	yes	—
11	#16812658	C₄	1	2	yes	—
12	#16812658	C₄	1	2	yes	—
13	#16812658	C₂²	1	2	yes	—
14	#16812658	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₁₀₉	C₄	1	2	yes	—
16	C₂×C₂²⋊Dic₁₀₉	C₄	1	2	yes	—
17	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
18	#16812628	C₂²	1	16	yes	complemented
19	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₁₀₉	—	2	4	no	—
21	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
22	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
23	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
24	C₂²×Dic₁₀₉	C₂×C₄	1	4	yes	—
25	#13182013	—	2	4	no	—
26	#13182013	—	2	4	no	—
27	C₂×#11724784	—	2	4	no	—
28	C₂×#11724784	C₂³	1	4	yes	—
29	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
30	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
31	C₂²⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
32	C₂²⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
33	C₄×Dic₁₀₉	C₂×C₄	1	4	yes	—
34	C₄×Dic₁₀₉	C₂³	1	4	yes	—
35	C₂⁴×C₁₀₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₀₉	D₄	1	4	yes	—
40	C₂²×C₄×C₁₀₉	Q₈	1	4	yes	—
41	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
44	M₁₆×C₁₀₉	—	2	4	no	—
45	M₁₆×C₁₀₉	—	2	4	no	—
46	C₂×C₈×C₁₀₉	—	2	4	no	—
47	C₂×C₈×C₁₀₉	C₂³	1	4	yes	—
48	C₂³×C₁₀₉	—	4	2	no	—
49	C₂³×C₁₀₉	C₄⋊C₄	1	4	yes	—
50	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
51	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
52	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
54	C₂×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
55	C₂×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
56	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
63	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
64	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
65	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
66	C₂×Dic₁₀₉	—	4	4	no	—
67	C₂×Dic₁₀₉	C₂²×C₄	1	8	yes	—
68	C₈×C₁₀₉	—	2	4	no	—
69	#11724784	—	2	4	no	—
70	C₂²×C₁₀₉	—	2	2	no	—
71	C₂²×C₁₀₉	—	2	2	no	—
72	C₂²×C₁₀₉	—	2	2	no	—
73	C₂²×C₁₀₉	—	4	4	no	—
74	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
75	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
76	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
77	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
78	C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
79	C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
80	C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
81	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
82	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
83	Dic₁₀₉	—	2	8	no	—
84	C₂×C₁₀₉	—	2	2	no	—
85	C₂×C₁₀₉	—	2	2	no	—
86	C₂×C₁₀₉	—	2	2	no	—
87	C₂×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₁₀₉	#568	1	2	yes	—
89	#2905	—	109	1	no	maximal, Sylow
90	C₁₀₉	#2905	1	1	yes	characteristic, complemented, Sylow
91	C₂²×M₁₆	—	109	1	no	—
92	C₂²×M₁₆	D₁₀₉	1	1	yes	characteristic
93	C₂×#117	—	109	1	no	—
94	#424	—	109	4	no	—
95	#424	—	109	4	no	—
96	#423	—	109	4	no	—
97	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
98	C₂×M₁₆	—	2	4	no	—
99	C₂×M₁₆	—	109	2	no	—
100	C₂×M₁₆	—	109	2	no	—
101	C₂×M₁₆	—	218	4	no	—
102	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
103	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
104	C₂²×C₈	—	109	2	no	—
105	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
106	#117	—	109	2	no	—
107	#117	—	109	2	no	—
108	#117	—	109	2	no	—
109	#117	—	109	2	no	—
110	C₂×C₄⋊C₄	—	109	2	no	—
111	C₂×C₂²⋊C₄	—	109	2	no	—
112	C₂×C₄²	—	109	2	no	—
113	C₄⋊C₈	—	109	16	no	—
114	C₂⁴	C₄×D₁₀₉	1	1	yes	characteristic
115	C₂²×C₄	—	109	4	no	—
116	C₂²×C₄	Dic₂₁₈	1	4	yes	—
117	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
118	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
119	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
120	C₂²×C₄	C₂²⋊D₁₀₉	1	4	yes	—
121	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
122	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
124	M₁₆	—	2	4	no	—
125	M₁₆	—	2	4	no	—
126	M₁₆	—	218	4	no	—
127	M₁₆	—	218	4	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	109	4	no	—
130	C₂×C₈	—	218	4	no	—
131	C₂×C₈	C₂²×D₁₀₉	1	4	yes	—
132	C₄⋊C₄	—	109	4	no	—
133	C₄⋊C₄	—	109	4	no	—
134	C₂²⋊C₄	—	109	4	no	—
135	C₂²⋊C₄	—	109	4	no	—
136	C₄²	—	109	4	no	—
137	C₄²	—	109	4	no	—
138	C₂³	—	4	2	no	—
139	C₂³	#13182008	1	4	yes	—
140	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
141	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
142	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
143	C₂×C₄	—	109	8	no	—
144	C₂×C₄	—	436	4	no	—
145	C₂×C₄	#13182008	1	4	yes	—
146	C₂×C₄	#13182008	1	4	yes	—
147	C₂×C₄	#13182008	1	4	yes	—
148	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
151	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
152	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
153	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
154	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
155	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
156	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
157	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	218	4	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	4	4	no	—
164	C₂²	C₂×#13182008	1	2	yes	—
165	C₂²	C₂×#13182008	1	2	yes	—
166	C₂²	C₂×#13182008	1	2	yes	—
167	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
168	C₄	—	218	8	no	—
169	C₄	C₂×#13182008	1	2	yes	—
170	C₄	C₂×#13182008	1	2	yes	—
171	C₄	C₂×#13182008	1	2	yes	—
172	C₄	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
173	C₄	C₂²×Dic₂₁₈	1	1	yes	characteristic
174	C₂	—	2	2	no	—
175	C₂	—	2	2	no	—
176	C₂	—	2	2	no	—
177	C₂	C₂²×#13182008	1	1	yes	characteristic
178	C₂	#20295792	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	C₂²×M₁₆×C₁₀₉	C₂
prime radical	C₂⁴×C₁₀₉	C₂×C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₂²×M₁₆	D₁₀₉
hypocenter	C₁₀₉	#2905
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20295054	C₂	semidirect
4. #20295052	C₂	semidirect
5. #20295049	C₂	semidirect
18. #16812628	C₂²	semidirect
90. C₁₀₉	#2905	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13182013	2
C₂×#16812658	2
#20295054	2
#20295052	2
#20295049	2
C₂²×M₁₆×C₁₀₉	2
#2905	109