Subgroups — #28372007

Order 13952.

380 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295560	C₂	1	1	yes	maximal, characteristic
2	#20295156	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295105	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295097	C₂	1	2	yes	maximal, complemented
6	C₈×D₄₃₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₂²⋊D₁₀₉	C₂	1	2	yes	maximal, complemented
8	#20295085	C₂	1	2	yes	maximal, complemented
9	#20295083	C₂	1	1	yes	maximal, characteristic, complemented
10	#20294980	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182012	C₂²	1	2	yes	—
13	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
14	#16812693	C₄	1	2	yes	—
15	#16812691	C₄	1	1	yes	characteristic
16	#16812687	C₄	1	1	yes	characteristic
17	#16812682	C₄	1	2	yes	—
18	#16812678	C₄	1	1	yes	characteristic
19	#16812673	C₄	1	1	yes	characteristic
20	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
21	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
22	C₄×C₂²⋊D₁₀₉	C₂²	1	2	yes	—
23	#16812656	C₂²	1	2	yes	—
24	C₄⋊#11724784	C₂²	1	1	yes	characteristic
25	C₄⋊#11724784	C₂²	1	1	yes	characteristic
26	#16812628	C₂²	1	2	yes	—
27	#16812628	C₂²	1	2	yes	—
28	C₂²⋊#11724784	C₂²	1	2	yes	—
29	#16812611	C₂²	1	2	yes	—
30	#16812611	C₂²	1	2	yes	—
31	#16812610	C₂²	1	2	yes	—
32	#16812609	C₂²	1	2	yes	—
33	C₈×Dic₁₀₉	C₂²	1	2	yes	—
34	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
36	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
37	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄₃₆	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
45	#13182012	—	2	4	no	—
46	C₂×#11724784	C₂³	1	2	yes	—
47	C₂×#11724784	C₂³	1	2	yes	—
48	C₈×D₁₀₉	—	2	4	no	—
49	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
51	#13182008	C₂×C₄	1	2	yes	—
52	#13182008	C₂×C₄	1	2	yes	—
53	#13182008	C₂×C₄	1	2	yes	—
54	#13182008	C₂×C₄	1	2	yes	—
55	C₂²⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
56	#13182006	C₂×C₄	1	2	yes	—
57	#13182006	C₂×C₄	1	2	yes	—
58	#13182006	C₂×C₄	1	2	yes	—
59	#13182006	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₀₉	C₂³	1	2	yes	—
61	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₁₀₉	—	2	2	no	—
65	C₂×C₈×C₁₀₉	—	2	2	no	—
66	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
70	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
72	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
73	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
74	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×D₁₀₉	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₁₀₉	—	2	2	no	—
77	C₂×C₄×C₁₀₉	—	2	2	no	—
78	C₂×C₄×C₁₀₉	—	2	1	no	—
79	C₂×C₄×C₁₀₉	—	2	1	no	—
80	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
86	C₂²⋊D₁₀₉	—	2	4	no	—
87	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
88	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
89	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
90	D₄₃₆	—	2	2	no	—
91	C₄×D₁₀₉	—	2	2	no	—
92	C₄×D₁₀₉	—	2	2	no	—
93	Dic₂₁₈	—	2	2	no	—
94	C₈×C₁₀₉	—	2	1	no	—
95	C₈×C₁₀₉	—	2	1	no	—
96	C₈×C₁₀₉	#41	1	2	yes	—
97	C₈×C₁₀₉	#41	1	2	yes	—
98	#11724784	—	2	2	no	—
99	#11724784	—	2	2	no	—
100	C₂²×C₁₀₉	—	2	1	no	—
101	C₂²×C₁₀₉	—	2	1	no	—
102	C₂²×C₁₀₉	#131	1	2	yes	—
103	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
104	C₂×D₁₀₉	—	2	2	no	—
105	C₂×D₁₀₉	—	2	2	no	—
106	C₂×D₁₀₉	—	2	2	no	—
107	C₄×C₁₀₉	—	2	1	no	—
108	C₄×C₁₀₉	—	2	1	no	—
109	C₄×C₁₀₉	—	2	1	no	—
110	C₄×C₁₀₉	—	2	1	no	—
111	C₄×C₁₀₉	#131	1	2	yes	—
112	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
113	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
114	Dic₁₀₉	—	2	2	no	—
115	Dic₁₀₉	—	2	2	no	—
116	Dic₁₀₉	—	2	2	no	—
117	C₂×C₁₀₉	—	2	2	no	—
118	C₂×C₁₀₉	C₄×#41	1	1	yes	characteristic
119	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
120	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
121	D₁₀₉	—	4	2	no	—
122	#2966	—	109	1	no	maximal, Sylow
123	C₁₀₉	#2966	1	1	yes	characteristic, complemented, Sylow
124	#529	—	109	1	no	—
125	#446	—	109	1	no	—
126	C₈×Q₈	—	109	1	no	—
127	#436	—	109	1	no	—
128	#435	—	109	2	no	—
129	C₈×D₄	—	109	2	no	—
130	C₈×D₄	—	109	1	no	—
131	#433	—	109	2	no	—
132	#432	—	109	1	no	—
133	#405	—	109	2	no	—
134	C₂×C₄×C₈	D₁₀₉	1	1	yes	characteristic, complemented
135	C₂×M₁₆	—	109	2	no	—
136	C₂²×C₈	—	109	2	no	—
137	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
138	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
139	#126	—	109	2	no	—
140	#125	—	109	1	no	—
141	#124	—	109	1	no	—
142	#123	—	109	2	no	—
143	C₂²⋊Q₈	—	109	1	no	—
144	#121	—	109	1	no	—
145	C₄×Q₈	—	109	1	no	—
146	C₄×D₄	—	109	2	no	—
147	C₄×D₄	—	109	1	no	—
148	#117	—	109	2	no	—
149	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
150	C₄⋊C₈	—	109	2	no	—
151	C₄⋊C₈	—	109	2	no	—
152	C₄⋊C₈	—	109	1	no	—
153	C₄⋊C₈	—	109	1	no	—
154	C₂²⋊C₈	—	109	2	no	—
155	C₂²⋊C₈	—	109	2	no	—
156	C₂²⋊C₈	—	109	2	no	—
157	#97	—	109	2	no	—
158	#97	—	109	2	no	—
159	C₄×C₈	—	109	2	no	—
160	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
161	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
162	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
163	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
164	C₂×Q₈	—	109	1	no	—
165	C₂×D₄	—	109	2	no	—
166	C₂×D₄	—	109	1	no	—
167	C₂²×C₄	—	109	2	no	—
168	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
169	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
171	M₁₆	—	218	4	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	109	2	no	—
175	C₂×C₈	—	109	2	no	—
176	C₂×C₈	—	218	4	no	—
177	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
179	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
180	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
181	C₄⋊C₄	—	109	2	no	—
182	C₄⋊C₄	—	109	2	no	—
183	C₄⋊C₄	—	109	2	no	—
184	C₄⋊C₄	—	109	2	no	—
185	C₄⋊C₄	—	109	1	no	—
186	C₄⋊C₄	—	109	1	no	—
187	C₂²⋊C₄	—	109	2	no	—
188	C₂²⋊C₄	—	109	2	no	—
189	C₂²⋊C₄	—	109	2	no	—
190	C₂²⋊C₄	—	109	2	no	—
191	C₂²⋊C₄	—	109	2	no	—
192	C₄²	—	109	2	no	—
193	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
194	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
195	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
196	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
197	C₂³	—	109	2	no	—
198	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
199	Q₈	—	218	2	no	—
200	D₄	—	218	4	no	—
201	D₄	—	218	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	109	2	no	—
207	C₂×C₄	—	109	2	no	—
208	C₂×C₄	—	109	2	no	—
209	C₂×C₄	—	218	2	no	—
210	C₂×C₄	—	218	2	no	—
211	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	218	2	no	—
220	C₈	—	218	2	no	—
221	C₈	#13182029	1	2	yes	—
222	C₈	#13182029	1	2	yes	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	218	2	no	—
226	C₂²	—	218	2	no	—
227	C₂²	—	218	2	no	—
228	C₂²	#16812705	1	2	yes	—
229	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	218	2	no	—
235	C₄	—	218	2	no	—
236	C₄	—	218	2	no	—
237	C₄	#16812705	1	2	yes	—
238	C₄	C₂×#13182029	1	1	yes	characteristic
239	C₄	C₂×#13182029	1	1	yes	characteristic
240	C₂	—	2	2	no	—
241	C₂	—	436	2	no	—
242	C₂	C₄×#13182029	1	1	yes	characteristic
243	C₂	C₂×#16812705	1	1	yes	characteristic
244	C₂	C₂×#16812705	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×C₄×C₈×C₁₀₉	C₂
prime radical	#16812673	C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₂×C₄×C₈	D₁₀₉
hypocenter	C₁₀₉	#2966
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20295156	C₂	semidirect
3. C₈×Dic₂₁₈	C₂	semidirect
4. #20295105	C₂	semidirect
5. #20295097	C₂	semidirect
6. C₈×D₄₃₆	C₂	semidirect
7. C₈×C₂²⋊D₁₀₉	C₂	semidirect
8. #20295085	C₂	semidirect
9. #20295083	C₂	semidirect
10. #20294980	C₂	semidirect
11. C₂×C₄×C₈×C₁₀₉	C₂	semidirect
123. C₁₀₉	#2966	semidirect
134. C₂×C₄×C₈	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295560	2
#20295156	2
C₈×Dic₂₁₈	2
#20295105	2
#20295097	2
C₈×D₄₃₆	2
C₈×C₂²⋊D₁₀₉	2
#20295085	2
#20295083	2
#20294980	2
C₂×C₄×C₈×C₁₀₉	2
#2966	109