Subgroups — #28371983

Order 13952.

388 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic
2	#20295155	C₂	1	4	yes	maximal, complemented
3	#20295153	C₂	1	2	yes	maximal, complemented
4	#20295150	C₂	1	2	yes	maximal, complemented
5	#20295049	C₂	1	2	yes	maximal
6	C₄×#13182013	C₂	1	1	yes	maximal, characteristic
7	M₁₆×Dic₁₀₉	C₂	1	2	yes	maximal
8	#423×C₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₁₈	C₄	1	1	yes	characteristic
10	C₂×#13182013	C₂²	1	1	yes	characteristic
11	C₂×#13182013	C₂²	1	1	yes	characteristic
12	C₄×Dic₂₁₈	C₄	1	2	yes	—
13	C₄×Dic₂₁₈	C₄	1	1	yes	characteristic
14	C₄×Dic₂₁₈	C₄	1	1	yes	characteristic
15	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
16	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
17	C₄×Dic₂₁₈	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₁₀₉	C₄	1	1	yes	characteristic
19	C₂×#13182008	C₄	1	2	yes	—
20	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
21	#16812628	C₂²	1	4	yes	—
22	#16812628	C₂²	1	4	yes	—
23	#16812610	C₂²	1	2	yes	—
24	#16812609	C₂²	1	4	yes	—
25	C₄×#11724784	C₂²	1	2	yes	—
26	C₈×Dic₁₀₉	C₂²	1	4	yes	—
27	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
28	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
30	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
31	C₂×Dic₂₁₈	—	2	4	no	—
32	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
33	C₂×Dic₂₁₈	C₂×C₄	1	2	yes	—
34	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
36	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
37	#13182013	—	2	2	no	—
38	#13182013	Q₈	1	4	yes	complemented
39	C₂×#11724784	C₂³	1	2	yes	—
40	C₂×#11724784	C₂³	1	2	yes	—
41	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
42	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
43	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
44	#13182008	C₂×C₄	1	2	yes	—
45	#13182008	C₂×C₄	1	2	yes	—
46	#13182008	C₂×C₄	1	2	yes	—
47	#13182008	C₂×C₄	1	2	yes	—
48	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
49	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
50	C₄×Dic₁₀₉	C₂³	1	2	yes	—
51	C₄×Dic₁₀₉	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
55	M₁₆×C₁₀₉	—	2	4	no	—
56	C₂×C₈×C₁₀₉	C₂³	1	4	yes	—
57	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
59	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
60	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
61	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₀₉	—	2	2	no	—
63	C₂×C₄×C₁₀₉	—	4	1	no	—
64	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
70	C₂×C₄×C₁₀₉	#41	1	2	yes	—
71	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
72	C₂×Dic₁₀₉	—	2	4	no	—
73	C₂×Dic₁₀₉	—	4	2	no	—
74	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
75	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
78	Dic₂₁₈	—	2	2	no	—
79	Dic₂₁₈	—	2	4	no	—
80	Dic₂₁₈	M₁₆	1	4	yes	complemented
81	C₈×C₁₀₉	—	2	4	no	—
82	#11724784	—	2	2	no	—
83	#11724784	C₂×Q₈	1	4	yes	complemented
84	C₂²×C₁₀₉	—	2	1	no	—
85	C₂²×C₁₀₉	—	2	1	no	—
86	C₂²×C₁₀₉	C₄×Q₈	1	2	yes	—
87	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
88	C₄×C₁₀₉	—	2	1	no	—
89	C₄×C₁₀₉	—	2	1	no	—
90	C₄×C₁₀₉	—	2	1	no	—
91	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
92	C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
93	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
94	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
95	Dic₁₀₉	—	2	2	no	—
96	Dic₁₀₉	—	2	2	no	—
97	Dic₁₀₉	—	2	2	no	—
98	Dic₁₀₉	C₂×M₁₆	1	4	yes	—
99	C₂×C₁₀₉	—	2	2	no	—
100	C₂×C₁₀₉	C₂×C₄×Q₈	1	1	yes	characteristic
101	C₂×C₁₀₉	C₂²×M₁₆	1	1	yes	characteristic
102	C₂×C₁₀₉	#568	1	1	yes	characteristic
103	Q₈×M₁₆	—	109	1	no	maximal, Sylow
104	C₁₀₉	Q₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
105	C₂×C₄×Q₈	—	109	1	no	—
106	#446	—	109	2	no	—
107	#446	—	109	4	no	—
108	C₈×Q₈	—	109	2	no	—
109	#423	—	109	2	no	—
110	#423	D₁₀₉	1	1	yes	characteristic
111	C₄×M₁₆	—	109	2	no	—
112	C₄×M₁₆	—	109	1	no	—
113	C₂²×Q₈	—	109	1	no	—
114	C₂×M₁₆	—	109	1	no	—
115	C₂×M₁₆	—	109	1	no	—
116	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
117	C₄×Q₈	—	109	1	no	—
118	C₄×Q₈	—	109	2	no	—
119	C₄×Q₈	—	109	1	no	—
120	C₄×Q₈	—	109	1	no	—
121	C₄×Q₈	—	109	2	no	—
122	C₄×Q₈	—	109	1	no	—
123	C₂×C₄⋊C₄	—	109	2	no	—
124	C₂×C₄⋊C₄	—	109	1	no	—
125	C₂×C₄²	—	109	2	no	—
126	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
127	C₄⋊C₈	—	109	4	no	—
128	C₄⋊C₈	—	109	4	no	—
129	C₄⋊C₈	C₂×D₁₀₉	1	2	yes	—
130	C₄⋊C₈	C₂×D₁₀₉	1	2	yes	—
131	#97	—	109	4	no	—
132	#97	—	109	2	no	—
133	C₄×C₈	—	109	4	no	—
134	C₄×C₈	—	109	2	no	—
135	C₂×Q₈	—	109	1	no	—
136	C₂×Q₈	—	109	2	no	—
137	C₂×Q₈	—	109	1	no	—
138	C₂×Q₈	—	218	4	no	—
139	C₂²×C₄	—	109	2	no	—
140	C₂²×C₄	—	109	2	no	—
141	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
142	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
143	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
144	M₁₆	—	2	4	no	—
145	M₁₆	—	109	4	no	—
146	M₁₆	—	218	2	no	—
147	C₂×C₈	—	109	2	no	—
148	C₂×C₈	—	109	2	no	—
149	C₂×C₈	C₂²×D₁₀₉	1	4	yes	—
150	C₄⋊C₄	—	109	2	no	—
151	C₄⋊C₄	—	109	2	no	—
152	C₄⋊C₄	—	109	2	no	—
153	C₄⋊C₄	—	109	2	no	—
154	C₄⋊C₄	—	109	2	no	—
155	C₄⋊C₄	—	109	1	no	—
156	C₄⋊C₄	—	109	1	no	—
157	C₄²	—	109	2	no	—
158	C₄²	—	109	2	no	—
159	C₄²	—	109	2	no	—
160	C₄²	—	109	2	no	—
161	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
162	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
163	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
164	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
165	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
166	Q₈	—	109	4	no	—
167	Q₈	—	218	4	no	—
168	Q₈	—	218	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	109	2	no	—
172	C₂×C₄	—	109	2	no	—
173	C₂×C₄	—	109	2	no	—
174	C₂×C₄	—	109	2	no	—
175	C₂×C₄	—	436	2	no	—
176	C₂×C₄	—	218	4	no	—
177	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
182	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
183	C₂×C₄	#13182028	1	2	yes	—
184	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	—	109	4	no	—
187	C₈	—	218	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	#16812664	1	2	yes	—
191	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	109	4	no	—
196	C₄	—	218	2	no	—
197	C₄	—	218	2	no	—
198	C₄	—	218	2	no	—
199	C₄	#16812664	1	2	yes	—
200	C₄	M₁₆×D₁₀₉	1	2	yes	—
201	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
202	C₄	C₂×#13182028	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	C₂×#16812664	1	1	yes	characteristic
205	C₂	C₂×M₁₆×D₁₀₉	1	1	yes	characteristic
206	C₂	#20295792	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	#423×C₁₀₉	C₂
prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#423	D₁₀₉
hypocenter	C₁₀₉	Q₈×M₁₆
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
2. #20295155	C₂	semidirect
3. #20295153	C₂	semidirect
4. #20295150	C₂	semidirect
38. #13182013	Q₈	semidirect
80. Dic₂₁₈	M₁₆	semidirect
83. #11724784	C₂×Q₈	semidirect
104. C₁₀₉	Q₈×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₁₈	2
#20295155	2
#20295153	2
#20295150	2
#20295049	2
C₄×#13182013	2
M₁₆×Dic₁₀₉	2
#423×C₁₀₉	2
Q₈×M₁₆	109