Subgroups — #28371621

Order 13952.

492 conjugacy classes of subgroups.

The subgroup lattice has 198 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	#20295069	C₂	1	4	yes	maximal, complemented
4	#20295066	C₂	1	4	yes	maximal, complemented
5	#20295062	C₂	1	4	yes	maximal, complemented
6	C₂×#117×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#13182013	C₄	1	8	yes	complemented
8	C₂×#13182013	C₂²	1	2	yes	—
9	C₂×#13182013	C₂²	1	2	yes	—
10	C₂²×#11724784	C₂²	1	2	yes	—
11	#16812658	—	2	1	no	—
12	#16812658	—	2	1	no	—
13	#16812658	C₂²	1	2	yes	—
14	#16812658	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
16	C₂×C₂²⋊Dic₁₀₉	—	2	1	no	—
17	C₂×C₄×Dic₁₀₉	—	2	1	no	—
18	#16812632	C₂²	1	8	yes	complemented
19	#16812630	C₂²	1	8	yes	complemented
20	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
21	#117×C₁₀₉	—	2	1	no	—
22	#117×C₁₀₉	—	2	1	no	—
23	#117×C₁₀₉	C₂²	1	2	yes	—
24	#117×C₁₀₉	C₂²	1	2	yes	—
25	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₁₀₉	—	2	1	no	—
27	C₂×C₄²×C₁₀₉	—	2	1	no	—
28	C₂²×Dic₁₀₉	—	2	2	no	—
29	#13182013	C₂×C₄	1	8	yes	complemented
30	#13182013	C₂×C₄	1	8	yes	complemented
31	C₂×#11724784	C₂×C₄	1	8	yes	complemented
32	C₂×#11724784	C₂³	1	4	yes	—
33	C₄⋊Dic₁₀₉	—	2	2	no	—
34	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
35	C₂²⋊Dic₁₀₉	—	2	2	no	—
36	C₂²⋊Dic₁₀₉	—	2	2	no	—
37	C₄×Dic₁₀₉	—	2	2	no	—
38	C₄×Dic₁₀₉	—	2	2	no	—
39	C₂⁴×C₁₀₉	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₀₉	—	2	2	no	—
41	C₂²×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
42	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₀₉	Q₈	1	4	yes	—
46	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₁₀₉	—	2	2	no	—
50	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
51	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
52	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
53	C₄²×C₁₀₉	—	2	2	no	—
54	C₄²×C₁₀₉	—	2	2	no	—
55	C₂³×C₁₀₉	—	4	2	no	—
56	C₂³×C₁₀₉	C₄⋊C₄	1	4	yes	—
57	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
58	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
59	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₀₉	—	2	4	no	—
61	C₂×C₄×C₁₀₉	—	4	4	no	—
62	C₂×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
63	C₂×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
64	C₂×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
65	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
66	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
72	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
73	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
74	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
75	C₂×Dic₁₀₉	—	2	4	no	—
76	C₂×Dic₁₀₉	—	4	4	no	—
77	#11724784	C₂²×C₄	1	8	yes	—
78	C₂²×C₁₀₉	—	2	2	no	—
79	C₂²×C₁₀₉	—	2	2	no	—
80	C₂²×C₁₀₉	—	2	2	no	—
81	C₂²×C₁₀₉	—	4	4	no	—
82	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
83	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
84	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
85	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
86	C₄×C₁₀₉	—	4	4	no	—
87	C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
88	C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
89	C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
90	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
91	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
92	Dic₁₀₉	—	4	4	no	—
93	C₂×C₁₀₉	—	2	2	no	—
94	C₂×C₁₀₉	—	2	2	no	—
95	C₂×C₁₀₉	—	2	2	no	—
96	C₂×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₁₀₉	#575	1	2	yes	—
98	#2911	—	109	1	no	maximal, Sylow
99	C₁₀₉	#2911	1	1	yes	characteristic, complemented, Sylow
100	C₂²×M₁₆	—	109	1	no	—
101	C₂×#117	—	109	1	no	—
102	C₂×#117	D₁₀₉	1	1	yes	characteristic
103	#428	—	109	4	no	—
104	#428	—	109	4	no	—
105	#427	—	109	4	no	—
106	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
107	C₂×M₁₆	—	109	8	no	—
108	C₂×M₁₆	—	109	2	no	—
109	C₂×M₁₆	—	109	2	no	—
110	C₂²×C₈	—	109	2	no	—
111	#117	—	2	1	no	—
112	#117	—	2	1	no	—
113	#117	—	109	2	no	—
114	#117	—	109	2	no	—
115	#117	—	218	1	no	—
116	#117	—	218	1	no	—
117	#117	C₂×D₁₀₉	1	2	yes	—
118	#117	C₂×D₁₀₉	1	2	yes	—
119	C₂×C₄⋊C₄	—	109	2	no	—
120	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
121	C₂×C₂²⋊C₄	—	2	1	no	—
122	C₂×C₂²⋊C₄	—	218	1	no	—
123	C₂×C₄²	—	2	1	no	—
124	C₂×C₄²	—	218	1	no	—
125	#107	—	109	8	no	—
126	#106	—	109	8	no	—
127	C₂⁴	C₂²⋊D₁₀₉	1	1	yes	characteristic
128	C₂²×C₄	—	2	2	no	—
129	C₂²×C₄	—	218	2	no	—
130	C₂²×C₄	Dic₂₁₈	1	4	yes	—
131	C₂²×C₄	C₄×D₁₀₉	1	4	yes	—
132	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
133	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
134	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
135	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
137	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
138	M₁₆	—	109	8	no	—
139	M₁₆	—	109	8	no	—
140	C₂×C₈	—	109	8	no	—
141	C₂×C₈	—	109	4	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	109	4	no	—
144	C₄⋊C₄	—	218	2	no	—
145	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
146	C₂²⋊C₄	—	2	2	no	—
147	C₂²⋊C₄	—	2	2	no	—
148	C₂²⋊C₄	—	218	2	no	—
149	C₂²⋊C₄	—	218	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	2	no	—
152	C₄²	—	218	2	no	—
153	C₄²	—	218	2	no	—
154	C₂³	—	4	2	no	—
155	C₂³	#13182008	1	4	yes	—
156	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
157	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
158	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	436	4	no	—
162	C₂×C₄	—	218	4	no	—
163	C₂×C₄	#13182008	1	4	yes	—
164	C₂×C₄	#13182008	1	4	yes	—
165	C₂×C₄	#13182008	1	4	yes	—
166	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
167	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
168	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
169	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
170	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
171	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
172	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
173	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
174	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
175	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
176	C₈	—	109	8	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	2	2	no	—
180	C₂²	—	4	4	no	—
181	C₂²	C₂×#13182008	1	2	yes	—
182	C₂²	C₂×#13182008	1	2	yes	—
183	C₂²	C₂×#13182008	1	2	yes	—
184	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
185	C₄	—	4	4	no	—
186	C₄	—	436	4	no	—
187	C₄	C₂×#13182008	1	2	yes	—
188	C₄	C₂×#13182008	1	2	yes	—
189	C₄	C₂×#13182008	1	2	yes	—
190	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
191	C₄	C₂²×Dic₂₁₈	1	1	yes	characteristic
192	C₂	—	2	2	no	—
193	C₂	—	2	2	no	—
194	C₂	—	2	2	no	—
195	C₂	C₂²×#13182008	1	1	yes	characteristic
196	C₂	#20295824	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×#117×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂⁴×C₁₀₉	D₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₂×#117	D₁₀₉
hypocenter	C₁₀₉	#2911
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. #20295069	C₂	semidirect
4. #20295066	C₂	semidirect
5. #20295062	C₂	semidirect
7. C₂×#13182013	C₄	semidirect
18. #16812632	C₂²	semidirect
19. #16812630	C₂²	semidirect
29. #13182013	C₂×C₄	semidirect
30. #13182013	C₂×C₄	semidirect
31. C₂×#11724784	C₂×C₄	semidirect
99. C₁₀₉	#2911	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
#20295069	2
#20295066	2
#20295062	2
C₂×#117×C₁₀₉	2
#2911	109