Subgroups — #28371996

Order 13952.

412 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13182029	C₂	1	1	yes	maximal, characteristic
2	#20295155	C₂	1	2	yes	maximal, complemented
3	#20295101	C₂	1	2	yes	maximal, complemented
4	#20295100	C₂	1	2	yes	maximal, complemented
5	#20295096	C₂	1	2	yes	maximal, complemented
6	#20295077	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295077	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#13182013	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#13182012	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#431×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182029	C₄	1	1	yes	characteristic
13	C₂×#13182013	C₂²	1	2	yes	—
14	C₂×#13182012	C₂²	1	2	yes	—
15	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
16	C₄×Dic₂₁₈	C₄	1	1	yes	characteristic
17	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
18	C₄×D₄₃₆	C₄	1	1	yes	characteristic
19	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
20	C₄×C₂²⋊D₁₀₉	C₄	1	1	yes	characteristic
21	C₄×C₂²⋊D₁₀₉	C₄	1	1	yes	characteristic
22	C₄×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic
24	#16812658	C₄	1	1	yes	characteristic
25	#16812656	C₄	1	1	yes	characteristic
26	#16812656	C₄	1	1	yes	characteristic
27	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
28	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
29	#16812628	C₂²	1	2	yes	—
30	#16812628	C₂²	1	2	yes	—
31	#16812611	C₂²	1	2	yes	—
32	#16812611	C₂²	1	2	yes	—
33	#16812610	C₂²	1	2	yes	—
34	#16812609	C₂²	1	2	yes	—
35	C₄×#11724784	C₂²	1	1	yes	characteristic
36	C₄×#11724784	C₂²	1	1	yes	characteristic
37	C₈×Dic₁₀₉	C₂²	1	2	yes	—
38	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
40	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
41	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
42	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
43	#13182029	—	2	4	no	—
44	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
45	C₂×D₄₃₆	C₂×C₄	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
54	#13182013	—	2	4	no	—
55	#13182012	—	2	4	no	—
56	C₂×#11724784	C₂³	1	2	yes	—
57	C₂×#11724784	C₂³	1	2	yes	—
58	C₈×D₁₀₉	—	2	4	no	—
59	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
60	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
61	#13182008	C₂×C₄	1	1	yes	characteristic
62	#13182008	C₂×C₄	1	1	yes	characteristic
63	#13182008	C₂×C₄	1	1	yes	characteristic
64	#13182008	C₂×C₄	1	1	yes	characteristic
65	C₂²⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
66	C₂²⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
67	#13182006	C₂×C₄	1	1	yes	characteristic
68	#13182006	C₂×C₄	1	1	yes	characteristic
69	#13182006	C₂×C₄	1	1	yes	characteristic
70	#13182006	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
73	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
74	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
75	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
76	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
77	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
81	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
82	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
83	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
84	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
85	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
86	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
88	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₀₉	—	2	2	no	—
90	C₂×C₄×C₁₀₉	—	2	2	no	—
91	C₂×C₄×C₁₀₉	—	2	2	no	—
92	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
98	C₂²⋊D₁₀₉	—	2	2	no	—
99	C₂²⋊D₁₀₉	—	2	2	no	—
100	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
101	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
106	D₄₃₆	—	2	2	no	—
107	C₄×D₁₀₉	—	2	2	no	—
108	C₄×D₁₀₉	—	2	2	no	—
109	C₄×D₁₀₉	—	2	2	no	—
110	C₄×D₁₀₉	—	2	2	no	—
111	C₄×D₁₀₉	—	2	2	no	—
112	C₄×D₁₀₉	M₁₆	1	4	yes	complemented
113	Dic₂₁₈	—	2	2	no	—
114	C₈×C₁₀₉	—	2	2	no	—
115	C₈×C₁₀₉	—	2	2	no	—
116	#11724784	—	2	2	no	—
117	#11724784	#41	1	4	yes	complemented
118	C₂²×C₁₀₉	—	2	1	no	—
119	C₂²×C₁₀₉	—	2	1	no	—
120	C₂²×C₁₀₉	—	2	1	no	—
121	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
122	C₂×D₁₀₉	—	2	1	no	—
123	C₂×D₁₀₉	—	2	1	no	—
124	C₂×D₁₀₉	—	2	1	no	—
125	C₂×D₁₀₉	—	2	1	no	—
126	C₂×D₁₀₉	—	2	1	no	—
127	C₂×D₁₀₉	C₂×M₁₆	1	2	yes	—
128	C₄×C₁₀₉	—	2	1	no	—
129	C₄×C₁₀₉	—	2	1	no	—
130	C₄×C₁₀₉	—	2	1	no	—
131	C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
132	C₄×C₁₀₉	#131	1	2	yes	—
133	C₄×C₁₀₉	C₂×#41	1	2	yes	—
134	Dic₁₀₉	—	2	1	no	—
135	Dic₁₀₉	—	2	1	no	—
136	Dic₁₀₉	—	2	1	no	—
137	Dic₁₀₉	—	2	1	no	—
138	Dic₁₀₉	—	2	1	no	—
139	Dic₁₀₉	C₂×M₁₆	1	2	yes	—
140	C₂×C₁₀₉	—	4	1	no	—
141	C₂×C₁₀₉	C₄×#41	1	1	yes	characteristic
142	C₂×C₁₀₉	C₂²×M₁₆	1	1	yes	characteristic
143	C₂×C₁₀₉	C₂×#131	1	1	yes	characteristic
144	D₁₀₉	—	2	2	no	—
145	D₁₀₉	—	4	1	no	—
146	#2965	—	109	1	no	maximal, Sylow
147	C₁₀₉	#2965	1	1	yes	characteristic, complemented, Sylow
148	C₄×#41	—	109	1	no	—
149	#446	—	109	2	no	—
150	#436	—	109	2	no	—
151	#435	—	109	2	no	—
152	#435	—	109	2	no	—
153	#431	—	109	1	no	—
154	#431	—	109	1	no	—
155	#431	D₁₀₉	1	1	yes	characteristic, complemented
156	C₄×M₁₆	—	109	1	no	—
157	C₄×M₁₆	—	109	1	no	—
158	C₂×C₄×C₈	—	109	1	no	—
159	C₂×#41	—	109	1	no	—
160	C₂×M₁₆	—	109	2	no	—
161	C₂×M₁₆	—	109	2	no	—
162	C₂²×C₈	—	109	2	no	—
163	C₄×Q₈	—	109	1	no	—
164	C₄×Q₈	—	109	1	no	—
165	C₄×D₄	—	109	1	no	—
166	C₄×D₄	—	109	1	no	—
167	C₄×D₄	—	109	1	no	—
168	C₄×D₄	—	109	1	no	—
169	C₄×D₄	—	109	1	no	—
170	C₄×D₄	—	109	1	no	—
171	#117	—	109	1	no	—
172	#117	—	109	1	no	—
173	#117	—	109	1	no	—
174	C₂×C₄²	—	109	1	no	—
175	C₂×C₄²	—	109	1	no	—
176	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
177	C₄⋊C₈	—	109	2	no	—
178	C₄⋊C₈	—	109	2	no	—
179	C₄⋊C₈	C₂×D₁₀₉	1	2	yes	—
180	C₂²⋊C₈	—	109	2	no	—
181	C₂²⋊C₈	—	109	2	no	—
182	C₂²⋊C₈	C₂×D₁₀₉	1	2	yes	—
183	#97	—	109	2	no	—
184	#97	—	109	2	no	—
185	C₄×C₈	—	109	2	no	—
186	C₄×C₈	—	109	1	no	—
187	C₄×C₈	—	109	1	no	—
188	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
189	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
190	#41	—	218	4	no	—
191	C₂×Q₈	—	109	1	no	—
192	C₂×D₄	—	109	1	no	—
193	C₂×D₄	—	109	1	no	—
194	C₂×D₄	—	109	1	no	—
195	C₂²×C₄	—	109	1	no	—
196	C₂²×C₄	—	109	1	no	—
197	C₂²×C₄	—	109	1	no	—
198	C₂²×C₄	—	109	1	no	—
199	C₂²×C₄	—	109	1	no	—
200	C₂²×C₄	—	109	1	no	—
201	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
202	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
203	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
204	M₁₆	—	218	4	no	—
205	M₁₆	—	218	4	no	—
206	C₂×C₈	—	109	2	no	—
207	C₂×C₈	—	109	2	no	—
208	C₂×C₈	—	218	4	no	—
209	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
210	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
211	C₄⋊C₄	—	109	1	no	—
212	C₄⋊C₄	—	109	1	no	—
213	C₄⋊C₄	—	109	1	no	—
214	C₄⋊C₄	—	109	1	no	—
215	C₄⋊C₄	—	109	1	no	—
216	C₄⋊C₄	—	109	1	no	—
217	C₂²⋊C₄	—	109	1	no	—
218	C₂²⋊C₄	—	109	1	no	—
219	C₂²⋊C₄	—	109	1	no	—
220	C₂²⋊C₄	—	109	1	no	—
221	C₂²⋊C₄	—	109	1	no	—
222	C₂²⋊C₄	—	109	1	no	—
223	C₄²	—	109	1	no	—
224	C₄²	—	109	1	no	—
225	C₄²	—	109	1	no	—
226	C₄²	—	109	1	no	—
227	C₄²	—	109	1	no	—
228	C₄²	—	109	1	no	—
229	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
230	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
231	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
232	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
233	C₂³	—	109	1	no	—
234	C₂³	—	109	1	no	—
235	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
236	Q₈	—	218	2	no	—
237	D₄	—	218	2	no	—
238	D₄	—	218	2	no	—
239	D₄	—	218	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	109	1	no	—
244	C₂×C₄	—	109	4	no	—
245	C₂×C₄	—	109	1	no	—
246	C₂×C₄	—	109	1	no	—
247	C₂×C₄	—	109	1	no	—
248	C₂×C₄	—	109	1	no	—
249	C₂×C₄	—	109	1	no	—
250	C₂×C₄	—	218	2	no	—
251	C₂×C₄	—	218	2	no	—
252	C₂×C₄	—	218	2	no	—
253	C₂×C₄	—	218	2	no	—
254	C₂×C₄	—	218	2	no	—
255	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
260	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	2	2	no	—
263	C₈	—	109	4	no	—
264	C₈	—	218	2	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	109	2	no	—
269	C₂²	—	218	1	no	—
270	C₂²	—	218	1	no	—
271	C₂²	—	218	1	no	—
272	C₂²	—	218	1	no	—
273	C₂²	—	218	1	no	—
274	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	109	2	no	—
279	C₄	—	218	1	no	—
280	C₄	—	218	1	no	—
281	C₄	—	218	1	no	—
282	C₄	—	218	1	no	—
283	C₄	—	218	1	no	—
284	C₄	M₁₆×D₁₀₉	1	2	yes	—
285	C₄	#16812705	1	2	yes	—
286	C₄	#41×D₁₀₉	1	2	yes	—
287	C₂	—	4	1	no	—
288	C₂	—	436	1	no	—
289	C₂	—	218	2	no	—
290	C₂	#20295494	1	1	yes	characteristic
291	C₂	C₂×M₁₆×D₁₀₉	1	1	yes	characteristic
292	C₂	C₂×#16812705	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#431×C₁₀₉	C₂
prime radical	C₂×#13182029	C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#431	D₁₀₉
hypocenter	C₁₀₉	#2965
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
2. #20295155	C₂	semidirect
3. #20295101	C₂	semidirect
4. #20295100	C₂	semidirect
5. #20295096	C₂	semidirect
6. #20295077	C₂	semidirect
7. #20295077	C₂	semidirect
8. C₄×#13182013	C₂	semidirect
9. C₄×#13182012	C₂	semidirect
10. C₄×C₈×D₁₀₉	C₂	semidirect
11. #431×C₁₀₉	C₂	semidirect
112. C₄×D₁₀₉	M₁₆	semidirect
117. #11724784	#41	semidirect
147. C₁₀₉	#2965	semidirect
155. #431	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13182029	2
#20295155	2
#20295101	2
#20295100	2
#20295096	2
#20295077	2
C₄×#13182013	2
C₄×#13182012	2
C₄×C₈×D₁₀₉	2
#431×C₁₀₉	2
#2965	109