Subgroups — #28371895

Order 13952.

456 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#20295502	C₂	1	2	yes	maximal
3	#20295131	C₂	1	2	yes	maximal, complemented
4	#20295130	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₄₃₆	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295120	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295063	C₂	1	1	yes	maximal, characteristic
9	#20295033	C₂	1	2	yes	maximal
10	#20294980	C₂	1	1	yes	maximal, characteristic
11	C₄×Q₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₁₀₉	C₂²	1	2	yes	—
13	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
14	C₂×#13182019	C₂²	1	2	yes	—
15	Q₁₆×D₁₀₉	C₄	1	8	yes	complemented
16	C₂×#13182012	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
18	Q₈×Dic₁₀₉	C₂²	1	2	yes	—
19	C₄×Dic₂₁₈	—	2	2	no	—
20	C₄×Dic₂₁₈	C₂²	1	2	yes	—
21	#16812664	—	2	2	no	—
22	#16812664	C₂²	1	2	yes	—
23	#16812657	—	2	2	no	—
24	#16812657	C₂²	1	2	yes	—
25	#16812656	—	2	2	no	—
26	#16812656	C₂²	1	1	yes	characteristic
27	#16812633	C₂²	1	1	yes	characteristic
28	#16812630	C₂²	1	2	yes	—
29	#16812624	C₂²	1	2	yes	—
30	#16812623	C₂²	1	2	yes	—
31	#16812622	C₂²	1	2	yes	—
32	#16812610	C₂²	1	1	yes	characteristic
33	#16812609	C₂²	1	1	yes	characteristic
34	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
37	#107×C₁₀₉	C₂²	1	1	yes	characteristic
38	#103×C₁₀₉	C₂²	1	2	yes	—
39	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
40	C₂×Dic₂₁₈	—	2	2	no	—
41	C₂×Dic₂₁₈	C₂³	1	2	yes	—
42	Q₈×D₁₀₉	—	2	4	no	—
43	Q₈×D₁₀₉	C₂×C₄	1	8	yes	—
44	C₂×C₄×D₁₀₉	—	2	2	no	—
45	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
46	Dic₄₃₆	C₂×C₄	1	4	yes	—
47	#13182019	C₂×C₄	1	8	yes	—
48	#13182012	—	2	2	no	—
49	C₂×#11724784	C₂³	1	1	yes	characteristic
50	C₂×#11724784	C₂³	1	1	yes	characteristic
51	C₈×D₁₀₉	C₂×C₄	1	4	yes	—
52	C₄⋊Dic₁₀₉	—	2	2	no	—
53	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
54	#13182008	—	2	2	no	—
55	#13182008	—	2	2	no	—
56	#13182008	D₄	1	1	yes	characteristic
57	#13182008	D₄	1	1	yes	characteristic
58	#13182006	—	2	2	no	—
59	#13182006	D₄	1	1	yes	characteristic
60	#13182006	D₄	1	1	yes	characteristic
61	C₄×Dic₁₀₉	—	2	2	no	—
62	C₄×Dic₁₀₉	—	2	2	no	—
63	C₄×Dic₁₀₉	—	2	2	no	—
64	C₄×Dic₁₀₉	—	2	2	no	—
65	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
66	C₂×Q₈×C₁₀₉	C₂³	1	2	yes	—
67	Q₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
68	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₀₉	—	2	2	no	—
71	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
72	C₄²×C₁₀₉	—	2	2	no	—
73	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
74	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
75	Q₈×C₁₀₉	—	2	2	no	—
76	Q₈×C₁₀₉	C₂²×C₄	1	4	yes	—
77	C₂×C₄×C₁₀₉	—	2	2	no	—
78	C₂×C₄×C₁₀₉	—	2	2	no	—
79	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₁₀₉	—	2	2	no	—
83	C₂×Dic₁₀₉	—	2	2	no	—
84	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
85	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
86	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
87	C₄×D₁₀₉	—	2	8	no	—
88	C₄×D₁₀₉	C₂²×C₄	1	2	yes	—
89	C₄×D₁₀₉	#41	1	2	yes	—
90	Dic₂₁₈	—	2	2	no	—
91	Dic₂₁₈	—	2	8	no	—
92	Dic₂₁₈	C₂²×C₄	1	4	yes	—
93	C₈×C₁₀₉	—	2	1	no	—
94	C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
95	#11724784	—	2	1	no	—
96	#11724784	C₂²×C₄	1	2	yes	—
97	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
98	C₂×D₁₀₉	—	2	1	no	—
99	C₂×D₁₀₉	—	2	1	no	—
100	C₂×D₁₀₉	C₄×D₄	1	2	yes	—
101	C₄×C₁₀₉	—	2	4	no	—
102	C₄×C₁₀₉	—	2	1	no	—
103	C₄×C₁₀₉	—	2	1	no	—
104	C₄×C₁₀₉	—	4	2	no	—
105	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
106	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
107	Dic₁₀₉	—	2	4	no	—
108	Dic₁₀₉	—	2	1	no	—
109	Dic₁₀₉	—	2	1	no	—
110	Dic₁₀₉	—	4	2	no	—
111	Dic₁₀₉	C₄×D₄	1	2	yes	—
112	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
113	C₂×C₁₀₉	#578	1	1	yes	characteristic
114	C₂×C₁₀₉	#578	1	1	yes	characteristic
115	D₁₀₉	—	2	2	no	—
116	#2950	—	109	1	no	maximal, Sylow
117	C₁₀₉	#2950	1	1	yes	characteristic, complemented, Sylow
118	C₂²×Q₁₆	—	109	1	no	—
119	#519	—	109	2	no	—
120	#441	—	109	2	no	—
121	#441	—	109	2	no	—
122	C₄×Q₁₆	—	109	1	no	—
123	C₄×Q₁₆	—	109	1	no	—
124	C₄×Q₁₆	—	109	1	no	—
125	C₄×Q₁₆	D₁₀₉	1	1	yes	characteristic, complemented
126	#427	—	109	1	no	—
127	#419	—	109	2	no	—
128	#405	—	109	1	no	—
129	C₂²×Q₈	—	109	2	no	—
130	C₂×Q₁₆	—	109	2	no	—
131	C₂×Q₁₆	—	109	8	no	—
132	C₂×Q₁₆	—	109	1	no	—
133	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
134	C₂×M₁₆	—	109	1	no	—
135	C₂²×C₈	—	109	1	no	—
136	C₄×Q₈	—	109	2	no	—
137	C₄×Q₈	—	109	2	no	—
138	C₄×Q₈	—	109	2	no	—
139	C₄×Q₈	—	218	2	no	—
140	C₄×Q₈	—	218	2	no	—
141	C₄×Q₈	C₂×D₁₀₉	1	2	yes	—
142	#117	—	109	1	no	—
143	#117	—	109	2	no	—
144	#117	—	218	2	no	—
145	#117	—	218	2	no	—
146	#107	—	109	1	no	—
147	#107	C₂×D₁₀₉	1	1	yes	characteristic
148	#106	—	109	2	no	—
149	#103	—	109	2	no	—
150	#103	—	109	2	no	—
151	#103	—	109	2	no	—
152	#103	C₂×D₁₀₉	1	2	yes	—
153	#97	—	109	1	no	—
154	#97	—	109	1	no	—
155	C₄×C₈	—	109	1	no	—
156	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
157	C₂×Q₈	—	109	8	no	—
158	C₂×Q₈	—	109	2	no	—
159	C₂×Q₈	—	218	2	no	—
160	C₂×Q₈	—	218	4	no	—
161	C₂×Q₈	C₂²×D₁₀₉	1	2	yes	—
162	C₂²×C₄	—	109	1	no	—
163	C₂²×C₄	—	218	2	no	—
164	Q₁₆	—	109	8	no	—
165	Q₁₆	—	109	4	no	—
166	Q₁₆	C₄×D₁₀₉	1	4	yes	—
167	M₁₆	—	218	2	no	—
168	C₂×C₈	—	109	1	no	—
169	C₂×C₈	—	109	4	no	—
170	C₂×C₈	—	109	1	no	—
171	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	109	1	no	—
175	C₄⋊C₄	—	109	1	no	—
176	C₄⋊C₄	—	109	2	no	—
177	C₄⋊C₄	—	218	2	no	—
178	C₄⋊C₄	—	218	2	no	—
179	C₄⋊C₄	—	218	2	no	—
180	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
181	C₂²⋊C₄	—	109	1	no	—
182	C₂²⋊C₄	—	109	1	no	—
183	C₂²⋊C₄	—	218	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	109	1	no	—
186	C₄²	—	218	2	no	—
187	C₄²	—	218	2	no	—
188	C₄²	—	218	2	no	—
189	C₄²	—	218	2	no	—
190	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
191	C₂³	—	109	1	no	—
192	Q₈	—	2	2	no	—
193	Q₈	—	109	4	no	—
194	Q₈	—	218	8	no	—
195	Q₈	—	218	2	no	—
196	Q₈	C₂×C₄×D₁₀₉	1	4	yes	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	109	1	no	—
200	C₂×C₄	—	109	2	no	—
201	C₂×C₄	—	109	2	no	—
202	C₂×C₄	—	109	1	no	—
203	C₂×C₄	—	109	1	no	—
204	C₂×C₄	—	218	8	no	—
205	C₂×C₄	—	218	2	no	—
206	C₂×C₄	—	218	2	no	—
207	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
208	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
209	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	109	2	no	—
212	C₈	—	218	1	no	—
213	C₈	C₂×C₄×D₁₀₉	1	2	yes	—
214	C₂²	—	109	2	no	—
215	C₂²	—	218	1	no	—
216	C₂²	—	218	1	no	—
217	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
218	C₄	—	2	4	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	4	2	no	—
222	C₄	—	109	2	no	—
223	C₄	—	436	2	no	—
224	C₄	—	218	4	no	—
225	C₄	—	218	1	no	—
226	C₄	—	218	1	no	—
227	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
228	C₄	#41×D₁₀₉	1	1	yes	characteristic
229	C₂	—	218	2	no	—
230	C₂	C₄×D₄×D₁₀₉	1	1	yes	characteristic
231	C₂	#20295839	1	1	yes	characteristic
232	C₂	#20295840	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₄×Q₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂²×D₁₀₉	C₂×D₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₄×Q₁₆	D₁₀₉
hypocenter	C₁₀₉	#2950
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #20295131	C₂	semidirect
4. #20295130	C₂	semidirect
5. C₄×Dic₄₃₆	C₂	semidirect
6. Q₁₆×Dic₁₀₉	C₂	semidirect
7. #20295120	C₂	semidirect
11. C₄×Q₁₆×C₁₀₉	C₂	semidirect
15. Q₁₆×D₁₀₉	C₄	semidirect
117. C₁₀₉	#2950	semidirect
125. C₄×Q₁₆	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×D₁₀₉	2
#20295502	2
#20295131	2
#20295130	2
C₄×Dic₄₃₆	2
Q₁₆×Dic₁₀₉	2
#20295120	2
#20295063	2
#20295033	2
#20294980	2
C₄×Q₁₆×C₁₀₉	2
#2950	109