Subgroups — #28372835

Order 13952.

376 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295582	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#20295334	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295333	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295330	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295320	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295319	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295310	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295309	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295067	C₂	1	1	yes	maximal, characteristic
11	#20295065	C₂	1	1	yes	maximal, characteristic
12	#20295048	C₂	1	1	yes	maximal, characteristic
13	#20295033	C₂	1	1	yes	maximal, characteristic
14	#20295033	C₂	1	1	yes	maximal, characteristic
15	#479×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×#13182012	C₂²	1	1	yes	characteristic
18	C₂×#13182012	C₂²	1	1	yes	characteristic
19	#16812698	C₂²	1	1	yes	characteristic
20	#16812698	C₂²	1	1	yes	characteristic
21	#16812691	C₂²	1	1	yes	characteristic
22	#16812677	—	2	1	no	—
23	#16812677	—	2	1	no	—
24	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
25	C₄×Dic₂₁₈	—	2	1	no	—
26	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
27	#16812664	—	2	1	no	—
28	#16812664	—	2	1	no	—
29	#16812664	—	2	1	no	—
30	#16812664	C₂²	1	1	yes	characteristic
31	#16812657	C₂²	1	1	yes	characteristic
32	#16812657	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₁₀₉	—	2	1	no	—
34	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
35	C₄²×D₁₀₉	—	2	1	no	—
36	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
37	#16812633	C₂²	1	1	yes	characteristic
38	#16812632	C₂²	1	1	yes	characteristic
39	#16812632	C₂²	1	1	yes	characteristic
40	#16812631	C₂²	1	1	yes	characteristic
41	#16812630	C₂²	1	1	yes	characteristic
42	#16812630	C₂²	1	1	yes	characteristic
43	C₄⋊#11724784	C₂²	1	1	yes	characteristic
44	#16812628	C₂²	1	1	yes	characteristic
45	#16812628	C₂²	1	1	yes	characteristic
46	#16812624	C₂²	1	1	yes	characteristic
47	#16812624	C₂²	1	1	yes	characteristic
48	#16812623	C₂²	1	1	yes	characteristic
49	#16812623	C₂²	1	1	yes	characteristic
50	#16812622	C₂²	1	1	yes	characteristic
51	#16812622	C₂²	1	1	yes	characteristic
52	#125×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
54	#107×C₁₀₉	C₂²	1	1	yes	characteristic
55	#106×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
57	#103×C₁₀₉	C₂²	1	1	yes	characteristic
58	#103×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	—	2	1	no	—
60	C₂×Dic₂₁₈	—	2	1	no	—
61	C₂×Dic₂₁₈	—	2	1	no	—
62	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
63	Q₈×D₁₀₉	—	2	2	no	—
64	Q₈×D₁₀₉	Q₈	1	4	yes	complemented
65	C₂×C₄×D₁₀₉	—	2	1	no	—
66	C₂×C₄×D₁₀₉	—	2	1	no	—
67	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
68	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
70	#13182012	—	2	2	no	—
71	#13182012	—	2	2	no	—
72	C₂×#11724784	C₂³	1	1	yes	characteristic
73	C₂×#11724784	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₀₉	—	2	1	no	—
75	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
78	#13182008	—	2	1	no	—
79	#13182008	—	2	1	no	—
80	#13182008	—	2	1	no	—
81	#13182008	—	2	1	no	—
82	#13182008	—	2	1	no	—
83	#13182008	—	2	1	no	—
84	#13182008	—	2	1	no	—
85	#13182006	—	2	1	no	—
86	#13182006	—	2	1	no	—
87	C₄×Dic₁₀₉	—	2	1	no	—
88	C₄×Dic₁₀₉	—	2	1	no	—
89	C₄×Dic₁₀₉	—	2	1	no	—
90	C₄×Dic₁₀₉	—	2	1	no	—
91	C₄×Dic₁₀₉	—	2	1	no	—
92	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
93	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
94	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₁₀₉	—	2	1	no	—
99	C₄⋊C₄×C₁₀₉	—	2	1	no	—
100	C₄⋊C₄×C₁₀₉	—	2	1	no	—
101	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₄²×C₁₀₉	—	2	1	no	—
105	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
107	Q₈×C₁₀₉	—	2	1	no	—
108	Q₈×C₁₀₉	C₂×Q₈	1	2	yes	—
109	C₂×C₄×C₁₀₉	—	2	1	no	—
110	C₂×C₄×C₁₀₉	—	2	1	no	—
111	C₂×C₄×C₁₀₉	—	2	1	no	—
112	C₂×C₄×C₁₀₉	—	2	1	no	—
113	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₁₀₉	—	2	1	no	—
117	C₂×Dic₁₀₉	—	2	1	no	—
118	C₂×Dic₁₀₉	—	2	1	no	—
119	C₂×Dic₁₀₉	—	2	1	no	—
120	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
123	C₄×D₁₀₉	—	2	4	no	—
124	C₄×D₁₀₉	—	2	2	no	—
125	C₄×D₁₀₉	—	4	1	no	—
126	C₄×D₁₀₉	—	4	2	no	—
127	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
128	C₄×D₁₀₉	#41	1	2	yes	—
129	Dic₂₁₈	—	2	1	no	—
130	Dic₂₁₈	—	2	4	no	—
131	Dic₂₁₈	—	4	2	no	—
132	Dic₂₁₈	—	4	2	no	—
133	Dic₂₁₈	C₂×Q₈	1	2	yes	—
134	C₈×C₁₀₉	—	2	1	no	—
135	C₈×C₁₀₉	—	2	1	no	—
136	#11724784	—	2	1	no	—
137	#11724784	—	2	1	no	—
138	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₁₀₉	—	2	1	no	—
140	C₂×D₁₀₉	—	2	1	no	—
141	C₂×D₁₀₉	C₂²⋊Q₈	1	2	yes	—
142	C₄×C₁₀₉	—	2	2	no	—
143	C₄×C₁₀₉	—	2	1	no	—
144	C₄×C₁₀₉	—	2	1	no	—
145	C₄×C₁₀₉	—	4	1	no	—
146	C₄×C₁₀₉	—	4	1	no	—
147	C₄×C₁₀₉	—	4	1	no	—
148	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
149	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
150	Dic₁₀₉	—	2	2	no	—
151	Dic₁₀₉	—	2	1	no	—
152	Dic₁₀₉	—	4	1	no	—
153	Dic₁₀₉	—	4	1	no	—
154	Dic₁₀₉	—	4	1	no	—
155	Dic₁₀₉	C₂²⋊Q₈	1	2	yes	—
156	Dic₁₀₉	#137	1	2	yes	—
157	C₂×C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₂×C₁₀₉	C₂×#137	1	1	yes	characteristic
159	C₂×C₁₀₉	#575	1	1	yes	characteristic
160	D₁₀₉	—	2	2	no	—
161	#3078	—	109	1	no	maximal, Sylow
162	C₁₀₉	#3078	1	1	yes	characteristic, complemented, Sylow
163	#533	—	109	1	no	—
164	C₂×C₄×Q₈	—	109	1	no	—
165	#479	—	109	1	no	—
166	#479	—	109	1	no	—
167	#479	—	109	1	no	—
168	#479	D₁₀₉	1	1	yes	characteristic, complemented
169	#477	—	109	1	no	—
170	#477	—	109	1	no	—
171	#475	—	109	1	no	—
172	#475	—	109	1	no	—
173	#428	—	109	1	no	—
174	#428	—	109	1	no	—
175	#423	—	109	1	no	—
176	#419	—	109	1	no	—
177	#419	—	109	1	no	—
178	C₂²×Q₈	—	109	1	no	—
179	C₂×M₁₆	—	109	1	no	—
180	C₂×M₁₆	—	109	1	no	—
181	C₄⋊Q₈	—	109	1	no	—
182	C₄⋊Q₈	—	109	1	no	—
183	#125	—	109	1	no	—
184	#125	C₂×D₁₀₉	1	1	yes	characteristic
185	C₂²⋊Q₈	—	218	1	no	—
186	C₂²⋊Q₈	—	218	1	no	—
187	C₄×Q₈	—	109	1	no	—
188	C₄×Q₈	—	109	1	no	—
189	C₄×Q₈	—	109	1	no	—
190	C₄×Q₈	—	218	1	no	—
191	C₄×Q₈	—	218	1	no	—
192	C₄×Q₈	—	218	1	no	—
193	C₄×Q₈	—	218	1	no	—
194	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
195	#117	—	109	1	no	—
196	#117	—	109	1	no	—
197	C₂×C₄⋊C₄	—	109	1	no	—
198	C₂×C₄⋊C₄	—	218	1	no	—
199	C₂×C₄²	—	109	1	no	—
200	C₂×C₄²	—	218	1	no	—
201	#107	—	109	1	no	—
202	#107	—	109	1	no	—
203	#107	—	109	1	no	—
204	#107	C₂×D₁₀₉	1	1	yes	characteristic
205	#106	—	109	1	no	—
206	#106	—	109	1	no	—
207	#106	—	109	1	no	—
208	#106	C₂×D₁₀₉	1	1	yes	characteristic
209	C₄⋊C₈	—	109	1	no	—
210	C₄⋊C₈	—	109	1	no	—
211	C₄⋊C₈	—	109	1	no	—
212	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
213	#103	—	109	1	no	—
214	#103	—	109	1	no	—
215	#103	—	109	1	no	—
216	#103	—	109	1	no	—
217	#103	—	109	1	no	—
218	#103	—	109	1	no	—
219	#103	C₂×D₁₀₉	1	1	yes	characteristic
220	#103	C₂×D₁₀₉	1	1	yes	characteristic
221	C₂×Q₈	—	109	4	no	—
222	C₂×Q₈	—	109	1	no	—
223	C₂×Q₈	—	218	1	no	—
224	C₂×Q₈	—	218	1	no	—
225	C₂×Q₈	—	218	1	no	—
226	C₂×Q₈	—	218	2	no	—
227	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
228	C₂²×C₄	—	109	1	no	—
229	C₂²×C₄	—	109	1	no	—
230	C₂²×C₄	—	109	1	no	—
231	C₂²×C₄	—	218	1	no	—
232	C₂²×C₄	—	218	1	no	—
233	M₁₆	—	218	2	no	—
234	M₁₆	—	218	2	no	—
235	C₂×C₈	—	109	1	no	—
236	C₂×C₈	—	109	1	no	—
237	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	109	1	no	—
243	C₄⋊C₄	—	109	1	no	—
244	C₄⋊C₄	—	109	1	no	—
245	C₄⋊C₄	—	218	1	no	—
246	C₄⋊C₄	—	218	1	no	—
247	C₄⋊C₄	—	218	1	no	—
248	C₄⋊C₄	—	218	1	no	—
249	C₄⋊C₄	—	218	1	no	—
250	C₄⋊C₄	—	218	1	no	—
251	C₄⋊C₄	—	218	1	no	—
252	C₄⋊C₄	—	218	1	no	—
253	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
254	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
256	C₂²⋊C₄	—	218	1	no	—
257	C₂²⋊C₄	—	218	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	109	1	no	—
260	C₄²	—	109	1	no	—
261	C₄²	—	109	1	no	—
262	C₄²	—	218	1	no	—
263	C₄²	—	218	1	no	—
264	C₄²	—	218	1	no	—
265	C₄²	—	218	1	no	—
266	C₄²	—	218	1	no	—
267	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
268	C₂³	—	109	1	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	109	2	no	—
271	Q₈	—	436	2	no	—
272	Q₈	—	436	2	no	—
273	Q₈	—	218	4	no	—
274	Q₈	—	218	1	no	—
275	Q₈	Q₈×D₁₀₉	1	2	yes	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	109	1	no	—
281	C₂×C₄	—	109	1	no	—
282	C₂×C₄	—	109	2	no	—
283	C₂×C₄	—	109	2	no	—
284	C₂×C₄	—	109	1	no	—
285	C₂×C₄	—	436	1	no	—
286	C₂×C₄	—	436	2	no	—
287	C₂×C₄	—	218	2	no	—
288	C₂×C₄	—	218	4	no	—
289	C₂×C₄	—	218	1	no	—
290	C₂×C₄	—	218	1	no	—
291	C₂×C₄	—	218	1	no	—
292	C₂×C₄	—	218	1	no	—
293	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
294	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
295	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	218	1	no	—
299	C₈	—	218	1	no	—
300	C₂²	—	109	2	no	—
301	C₂²	—	218	1	no	—
302	C₂²	—	218	1	no	—
303	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	109	2	no	—
311	C₄	—	109	2	no	—
312	C₄	—	436	1	no	—
313	C₄	—	436	1	no	—
314	C₄	—	436	1	no	—
315	C₄	—	218	2	no	—
316	C₄	—	218	1	no	—
317	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
318	C₄	#41×D₁₀₉	1	1	yes	characteristic
319	C₂	—	218	2	no	—
320	C₂	C₂²⋊Q₈×D₁₀₉	1	1	yes	characteristic
321	C₂	#137×D₁₀₉	1	1	yes	characteristic
322	C₂	#20295823	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#479×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂²×D₁₀₉	C₂×D₄
hypercenter	#479	D₁₀₉
hypocenter	C₁₀₉	#3078
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
3. #20295334	C₂	semidirect
4. #20295333	C₂	semidirect
5. #20295330	C₂	semidirect
6. #20295320	C₂	semidirect
7. #20295319	C₂	semidirect
8. #20295310	C₂	semidirect
9. #20295309	C₂	semidirect
15. #479×C₁₀₉	C₂	semidirect
64. Q₈×D₁₀₉	Q₈	semidirect
162. C₁₀₉	#3078	semidirect
168. #479	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295582	2
C₄×Q₈×D₁₀₉	2
#20295334	2
#20295333	2
#20295330	2
#20295320	2
#20295319	2
#20295310	2
#20295309	2
#20295067	2
#20295065	2
#20295048	2
#20295033	2
#479×C₁₀₉	2
#3078	109