Subgroups — #28371599

Order 13952.

484 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812707	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812705	C₂	1	1	yes	maximal, characteristic
3	#20295494	C₂	1	1	yes	maximal, characteristic
4	#20295053	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295051	C₂	1	2	yes	maximal, complemented
6	#20295051	C₂	1	2	yes	maximal, complemented
7	#20295049	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295048	C₂	1	2	yes	maximal, complemented
9	C₂×#16812628	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₈×D₁₀₉	C₂	1	2	yes	maximal, complemented
11	#424×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182029	C₂²	1	1	yes	characteristic
13	#16812707	—	2	4	no	—
14	#16812705	—	2	4	no	—
15	C₂×#13182013	C₂²	1	1	yes	characteristic
16	C₂×#13182012	C₂²	1	2	yes	—
17	C₂×#13182012	C₂²	1	2	yes	—
18	C₂²×#11724784	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
20	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
21	#16812664	C₄	1	1	yes	characteristic
22	#16812664	C₄	1	1	yes	characteristic
23	#16812661	C₄	1	1	yes	characteristic
24	#16812661	C₄	1	1	yes	characteristic
25	#16812659	C₄	1	2	yes	—
26	#16812659	C₄	1	2	yes	—
27	#16812656	C₂²	1	2	yes	—
28	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
29	C₄²×D₁₀₉	C₂²	1	2	yes	—
30	C₄⋊#11724784	C₂²	1	2	yes	—
31	C₄⋊#11724784	C₂²	1	2	yes	—
32	#16812628	C₂²	1	1	yes	characteristic
33	#16812628	C₂²	1	2	yes	—
34	#16812628	C₂²	1	1	yes	characteristic
35	#16812628	C₂²	1	1	yes	characteristic
36	#16812628	C₂²	1	1	yes	characteristic
37	#16812628	C₂²	1	2	yes	—
38	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
40	#117×C₁₀₉	C₂²	1	1	yes	characteristic
41	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
42	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
43	#13182029	D₄	1	4	yes	—
44	#13182029	Q₈	1	4	yes	—
45	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
46	C₂×D₄₃₆	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	2	yes	—
48	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
52	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
53	#13182013	—	2	2	no	—
54	#13182012	—	2	4	no	—
55	#13182012	—	2	4	no	—
56	C₂×#11724784	—	2	2	no	—
57	C₂×#11724784	C₂³	1	1	yes	characteristic
58	C₂×#11724784	C₂³	1	2	yes	—
59	C₂×#11724784	C₂³	1	1	yes	characteristic
60	C₈×D₁₀₉	—	2	4	no	—
61	C₈×D₁₀₉	—	2	4	no	—
62	#13182008	C₂×C₄	1	2	yes	—
63	#13182008	C₂×C₄	1	2	yes	—
64	#13182006	C₂×C₄	1	2	yes	—
65	#13182006	C₂×C₄	1	2	yes	—
66	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
67	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
68	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
71	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
72	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
73	M₁₆×C₁₀₉	—	2	2	no	—
74	C₂×C₈×C₁₀₉	—	2	2	no	—
75	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
78	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
79	C₄⋊C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
80	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
81	C₄²×C₁₀₉	C₂³	1	2	yes	—
82	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
83	C₂²×D₁₀₉	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
86	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
89	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
90	C₂²⋊D₁₀₉	C₄⋊C₄	1	8	yes	—
91	C₂×Dic₁₀₉	—	2	2	no	—
92	C₂×Dic₁₀₉	—	4	1	no	—
93	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
97	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
98	D₄₃₆	C₄⋊C₄	1	4	yes	—
99	C₄×D₁₀₉	—	2	4	no	—
100	C₄×D₁₀₉	—	4	2	no	—
101	C₄×D₁₀₉	C₂×D₄	1	4	yes	—
102	C₄×D₁₀₉	C₂×Q₈	1	4	yes	—
103	Dic₂₁₈	C₄⋊C₄	1	4	yes	—
104	C₈×C₁₀₉	—	2	1	no	—
105	C₈×C₁₀₉	—	2	1	no	—
106	C₈×C₁₀₉	—	2	2	no	—
107	#11724784	—	2	1	no	—
108	#11724784	—	2	2	no	—
109	#11724784	—	2	1	no	—
110	C₂²×C₁₀₉	—	2	1	no	—
111	C₂²×C₁₀₉	—	2	1	no	—
112	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
113	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
114	C₂×D₁₀₉	—	2	2	no	—
115	C₂×D₁₀₉	—	2	2	no	—
116	C₂×D₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
117	C₄×C₁₀₉	—	2	2	no	—
118	C₄×C₁₀₉	—	2	2	no	—
119	C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
120	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
121	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
122	Dic₁₀₉	—	2	1	no	—
123	Dic₁₀₉	—	2	1	no	—
124	Dic₁₀₉	—	2	1	no	—
125	Dic₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
126	Dic₁₀₉	#131	1	2	yes	complemented
127	C₂×C₁₀₉	—	2	2	no	—
128	C₂×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
130	C₂×C₁₀₉	#568	1	1	yes	characteristic
131	D₁₀₉	—	2	4	no	—
132	#2906	—	109	1	no	maximal, Sylow
133	C₁₀₉	#2906	1	1	yes	characteristic, complemented, Sylow
134	C₂×#131	—	109	1	no	—
135	C₂×#131	—	109	1	no	—
136	C₄×#41	—	109	1	no	—
137	#424	—	109	1	no	—
138	#424	—	109	2	no	—
139	#424	—	109	2	no	—
140	#424	D₁₀₉	1	1	yes	characteristic, complemented
141	#423	—	109	1	no	—
142	#423	—	109	2	no	—
143	C₂×C₄⋊C₈	—	109	1	no	—
144	C₂×C₄⋊C₈	—	109	2	no	—
145	C₂×#41	—	109	1	no	—
146	#131	—	218	4	no	—
147	#131	—	218	4	no	—
148	C₂×M₁₆	—	109	1	no	—
149	C₂×M₁₆	—	109	2	no	—
150	C₂×M₁₆	—	109	2	no	—
151	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
152	C₂²×C₈	—	109	1	no	—
153	C₂²×C₈	—	109	2	no	—
154	C₂²×C₈	—	109	2	no	—
155	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
156	C₄×Q₈	—	109	1	no	—
157	C₄×Q₈	—	109	1	no	—
158	C₄×D₄	—	109	1	no	—
159	C₄×D₄	—	109	1	no	—
160	C₄×D₄	—	109	2	no	—
161	C₄×D₄	—	109	2	no	—
162	#117	—	109	2	no	—
163	#117	C₂×D₁₀₉	1	1	yes	characteristic
164	C₂×C₄²	—	109	1	no	—
165	C₂×C₄²	—	109	2	no	—
166	C₄⋊C₈	—	109	1	no	—
167	C₄⋊C₈	—	109	2	no	—
168	C₄⋊C₈	—	109	1	no	—
169	C₄⋊C₈	—	109	1	no	—
170	C₄⋊C₈	—	109	1	no	—
171	C₄⋊C₈	—	109	2	no	—
172	C₄⋊C₈	—	109	2	no	—
173	C₄⋊C₈	—	109	2	no	—
174	C₄⋊C₈	C₂×D₁₀₉	1	2	yes	—
175	C₄⋊C₈	C₂×D₁₀₉	1	2	yes	—
176	#41	—	109	4	no	—
177	#41	—	109	4	no	—
178	C₂×Q₈	—	109	1	no	—
179	C₂×D₄	—	109	2	no	—
180	C₂×D₄	—	109	1	no	—
181	C₂²×C₄	—	109	2	no	—
182	C₂²×C₄	—	109	1	no	—
183	C₂²×C₄	—	109	1	no	—
184	C₂²×C₄	—	109	2	no	—
185	C₂²×C₄	—	109	2	no	—
186	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
187	M₁₆	—	2	2	no	—
188	M₁₆	—	218	4	no	—
189	M₁₆	—	218	4	no	—
190	M₁₆	—	218	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	109	1	no	—
193	C₂×C₈	—	109	2	no	—
194	C₂×C₈	—	109	1	no	—
195	C₂×C₈	—	218	4	no	—
196	C₂×C₈	—	218	4	no	—
197	C₂×C₈	—	218	2	no	—
198	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
199	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
200	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
201	C₄⋊C₄	—	109	2	no	—
202	C₄⋊C₄	—	109	2	no	—
203	C₄⋊C₄	C₄×D₁₀₉	1	1	yes	characteristic
204	C₄⋊C₄	C₄×D₁₀₉	1	1	yes	characteristic
205	C₂²⋊C₄	—	109	2	no	—
206	C₂²⋊C₄	—	109	2	no	—
207	C₂²⋊C₄	C₄×D₁₀₉	1	2	yes	—
208	C₄²	—	109	2	no	—
209	C₄²	—	109	1	no	—
210	C₄²	—	109	2	no	—
211	C₄²	—	109	1	no	—
212	C₄²	—	109	1	no	—
213	C₄²	—	109	1	no	—
214	C₄²	C₂²×D₁₀₉	1	2	yes	—
215	C₂³	—	109	2	no	—
216	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
217	Q₈	—	109	4	no	—
218	D₄	—	109	8	no	—
219	D₄	—	109	4	no	—
220	C₂×C₄	—	109	1	no	—
221	C₂×C₄	—	109	2	no	—
222	C₂×C₄	—	109	4	no	—
223	C₂×C₄	—	109	4	no	—
224	C₂×C₄	—	109	1	no	—
225	C₂×C₄	—	109	1	no	—
226	C₂×C₄	—	109	1	no	—
227	C₂×C₄	—	436	2	no	—
228	C₂×C₄	—	436	1	no	—
229	C₂×C₄	—	218	4	no	—
230	C₂×C₄	—	218	2	no	—
231	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
232	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
233	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
234	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
235	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
236	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	2	2	no	—
240	C₈	—	218	1	no	—
241	C₈	—	218	2	no	—
242	C₈	—	218	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	109	4	no	—
246	C₂²	—	218	2	no	—
247	C₂²	—	218	2	no	—
248	C₂²	C₄⋊C₄×D₁₀₉	1	2	yes	—
249	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	109	4	no	—
253	C₄	—	109	2	no	—
254	C₄	—	218	1	no	—
255	C₄	—	218	1	no	—
256	C₄	—	218	1	no	—
257	C₄	C₄⋊C₄×D₁₀₉	1	2	yes	—
258	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
259	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
260	C₂	—	2	2	no	—
261	C₂	—	218	4	no	—
262	C₂	C₂×C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
263	C₂	#131×D₁₀₉	1	1	yes	characteristic
264	C₂	#20295791	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	#424×C₁₀₉	C₂
prime radical	C₂×#13182029	C₂²
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#424	D₁₀₉
hypocenter	C₁₀₉	#2906
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
4. #20295053	C₂	semidirect
5. #20295051	C₂	semidirect
6. #20295051	C₂	semidirect
7. #20295049	C₂	semidirect
8. #20295048	C₂	semidirect
9. C₂×#16812628	C₂	semidirect
10. C₄⋊C₈×D₁₀₉	C₂	semidirect
11. #424×C₁₀₉	C₂	semidirect
126. Dic₁₀₉	#131	semidirect
133. C₁₀₉	#2906	semidirect
140. #424	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812707	2
C₂×#16812705	2
#20295494	2
#20295053	2
#20295051	2
#20295049	2
#20295048	2
C₂×#16812628	2
C₄⋊C₈×D₁₀₉	2
#424×C₁₀₉	2
#2906	109