Subgroups — #28373326

Order 13952.

358 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295636	C₂	1	1	yes	maximal, characteristic
2	#20295504	C₂	1	1	yes	maximal, characteristic
3	#20295368	C₂	1	1	yes	maximal, characteristic
4	#20295364	C₂	1	1	yes	maximal, characteristic
5	#20295335	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295331	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295234	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295234	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295182	C₂	1	1	yes	maximal, characteristic
10	#20295133	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295130	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295118	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295084	C₂	1	1	yes	maximal, characteristic
15	#451×C₁₀₉	C₂	1	1	yes	maximal, characteristic
16	C₂×#13182019	C₂²	1	1	yes	characteristic
17	C₂×#13182019	C₂²	1	1	yes	characteristic
18	#16812697	C₂²	1	1	yes	characteristic
19	#16812692	—	2	1	no	—
20	#16812692	—	2	1	no	—
21	#16812690	—	2	1	no	—
22	#16812690	C₂²	1	1	yes	characteristic
23	#16812689	—	2	1	no	—
24	#16812678	C₂²	1	1	yes	characteristic
25	Q₈×Dic₁₀₉	—	2	1	no	—
26	Q₈×Dic₁₀₉	—	2	1	no	—
27	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
28	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
29	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
30	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
31	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
32	#16812664	C₂²	1	1	yes	characteristic
33	#16812658	—	2	1	no	—
34	#16812658	—	2	1	no	—
35	#16812658	C₂²	1	1	yes	characteristic
36	#16812658	C₂²	1	1	yes	characteristic
37	#16812633	C₂²	1	1	yes	characteristic
38	#16812632	C₂²	1	1	yes	characteristic
39	#16812631	C₂²	1	1	yes	characteristic
40	#16812630	C₂²	1	1	yes	characteristic
41	#16812628	C₂²	1	1	yes	characteristic
42	#16812628	C₂²	1	1	yes	characteristic
43	#16812624	C₂²	1	1	yes	characteristic
44	#16812624	C₂²	1	1	yes	characteristic
45	#16812623	C₂²	1	1	yes	characteristic
46	#16812623	C₂²	1	1	yes	characteristic
47	#16812623	C₂²	1	1	yes	characteristic
48	#16812623	C₂²	1	1	yes	characteristic
49	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
50	#16812609	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
52	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
56	#103×C₁₀₉	C₂²	1	1	yes	characteristic
57	#103×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
60	#13182019	—	2	2	no	—
61	#13182019	—	2	2	no	—
62	C₂×#11724784	C₂³	1	1	yes	characteristic
63	C₂×#11724784	C₂³	1	1	yes	characteristic
64	C₄⋊Dic₁₀₉	—	2	1	no	—
65	C₄⋊Dic₁₀₉	—	2	1	no	—
66	C₄⋊Dic₁₀₉	—	2	1	no	—
67	C₄⋊Dic₁₀₉	—	2	1	no	—
68	C₄⋊Dic₁₀₉	—	2	1	no	—
69	C₄⋊Dic₁₀₉	—	2	1	no	—
70	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
71	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
72	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
75	#13182008	—	2	1	no	—
76	#13182008	—	2	1	no	—
77	#13182008	—	2	1	no	—
78	#13182008	—	2	1	no	—
79	#13182008	—	2	1	no	—
80	C₂²⋊Dic₁₀₉	—	2	1	no	—
81	C₂²⋊Dic₁₀₉	—	2	1	no	—
82	C₂²⋊Dic₁₀₉	D₄	1	1	yes	characteristic
83	C₂²⋊Dic₁₀₉	D₄	1	1	yes	characteristic
84	C₄×Dic₁₀₉	—	2	1	no	—
85	C₄×Dic₁₀₉	—	2	1	no	—
86	C₄×Dic₁₀₉	—	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	C₂³	1	1	yes	characteristic
92	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₁₀₉	—	2	2	no	—
94	C₂×Q₈×C₁₀₉	—	2	2	no	—
95	C₂×Q₈×C₁₀₉	—	2	1	no	—
96	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₁₀₉	—	2	1	no	—
100	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
101	Q₁₆×C₁₀₉	—	2	2	no	—
102	Q₁₆×C₁₀₉	—	2	2	no	—
103	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₀₉	—	2	1	no	—
106	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
108	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
109	Q₈×C₁₀₉	—	2	4	no	—
110	Q₈×C₁₀₉	—	2	1	no	—
111	Q₈×C₁₀₉	—	2	1	no	—
112	Q₈×C₁₀₉	—	2	1	no	—
113	Q₈×C₁₀₉	—	2	1	no	—
114	Q₈×C₁₀₉	#41	1	2	yes	—
115	Q₈×C₁₀₉	#41	1	2	yes	—
116	C₂×C₄×C₁₀₉	—	2	4	no	—
117	C₂×C₄×C₁₀₉	—	2	1	no	—
118	C₂×C₄×C₁₀₉	—	2	1	no	—
119	C₂×C₄×C₁₀₉	—	2	1	no	—
120	C₂×C₄×C₁₀₉	—	2	1	no	—
121	C₂×C₄×C₁₀₉	—	2	1	no	—
122	C₂×C₄×C₁₀₉	—	2	1	no	—
123	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
125	C₂×Dic₁₀₉	—	2	1	no	—
126	C₂×Dic₁₀₉	—	2	1	no	—
127	C₂×Dic₁₀₉	—	2	1	no	—
128	C₂×Dic₁₀₉	—	2	1	no	—
129	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
133	Dic₂₁₈	—	2	1	no	—
134	Dic₂₁₈	—	2	1	no	—
135	C₈×C₁₀₉	—	2	1	no	—
136	C₈×C₁₀₉	—	2	1	no	—
137	#11724784	—	2	1	no	—
138	#11724784	—	2	1	no	—
139	C₂²×C₁₀₉	—	2	1	no	—
140	C₂²×C₁₀₉	—	2	1	no	—
141	C₂²×C₁₀₉	—	2	1	no	—
142	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
143	C₄×C₁₀₉	—	2	2	no	—
144	C₄×C₁₀₉	—	2	2	no	—
145	C₄×C₁₀₉	—	2	1	no	—
146	C₄×C₁₀₉	—	4	1	no	—
147	C₄×C₁₀₉	—	4	1	no	—
148	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
149	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
150	Dic₁₀₉	—	2	1	no	—
151	Dic₁₀₉	—	2	1	no	—
152	Dic₁₀₉	—	2	1	no	—
153	Dic₁₀₉	—	2	1	no	—
154	Dic₁₀₉	—	4	1	no	—
155	Dic₁₀₉	—	4	1	no	—
156	Dic₁₀₉	—	4	1	no	—
157	Dic₁₀₉	—	4	1	no	—
158	C₂×C₁₀₉	—	4	1	no	—
159	C₂×C₁₀₉	#525	1	1	yes	characteristic
160	C₂×C₁₀₉	#578	1	1	yes	characteristic
161	C₂×C₁₀₉	#578	1	1	yes	characteristic
162	#3127	—	109	1	no	maximal, Sylow
163	C₁₀₉	#3127	1	1	yes	characteristic, complemented, Sylow
164	#541	—	109	1	no	—
165	#519	—	109	1	no	—
166	#485	—	109	1	no	—
167	#485	—	109	1	no	—
168	#479	—	109	1	no	—
169	#479	—	109	1	no	—
170	#462	—	109	1	no	—
171	#462	—	109	1	no	—
172	#451	—	109	1	no	—
173	#451	D₁₀₉	1	1	yes	characteristic
174	#441	—	109	1	no	—
175	#441	—	109	1	no	—
176	C₄×Q₁₆	—	109	1	no	—
177	C₄×Q₁₆	—	109	1	no	—
178	#433	—	109	1	no	—
179	C₂²×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
180	C₂×Q₁₆	—	109	1	no	—
181	C₂×Q₁₆	—	109	1	no	—
182	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
183	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
184	C₄⋊Q₈	—	109	1	no	—
185	#126	—	218	1	no	—
186	#126	—	218	1	no	—
187	#125	—	109	1	no	—
188	#125	—	218	1	no	—
189	#125	—	218	1	no	—
190	C₂²⋊Q₈	—	109	1	no	—
191	C₂²⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
192	C₄×Q₈	—	109	1	no	—
193	C₄×Q₈	—	109	1	no	—
194	C₄×Q₈	—	109	1	no	—
195	C₄×Q₈	—	109	1	no	—
196	C₄×Q₈	—	109	1	no	—
197	C₄×Q₈	—	109	1	no	—
198	C₄×Q₈	—	218	1	no	—
199	C₄×Q₈	—	218	1	no	—
200	#117	—	109	1	no	—
201	#117	—	109	1	no	—
202	#117	—	218	1	no	—
203	#117	—	218	1	no	—
204	#107	—	109	1	no	—
205	#107	—	109	1	no	—
206	#106	—	109	1	no	—
207	#106	—	109	1	no	—
208	C₄⋊C₈	—	109	1	no	—
209	C₄⋊C₈	—	109	1	no	—
210	#103	—	109	1	no	—
211	#103	—	109	1	no	—
212	#103	—	109	1	no	—
213	#103	—	109	1	no	—
214	#103	—	109	1	no	—
215	#103	—	109	1	no	—
216	#103	C₂×D₁₀₉	1	1	yes	characteristic
217	#103	C₂×D₁₀₉	1	1	yes	characteristic
218	C₂²⋊C₈	—	109	1	no	—
219	C₂²⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
220	#97	—	109	1	no	—
221	C₄×C₈	—	109	1	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	1	no	—
225	C₂×Q₈	—	109	1	no	—
226	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
227	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
228	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
231	Q₁₆	—	2	2	no	—
232	Q₁₆	—	2	2	no	—
233	Q₁₆	—	218	2	no	—
234	Q₁₆	—	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	—	109	1	no	—
241	C₄⋊C₄	—	109	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	—	218	1	no	—
254	C₄⋊C₄	—	218	1	no	—
255	C₄⋊C₄	—	218	1	no	—
256	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	109	1	no	—
259	C₂²⋊C₄	—	109	1	no	—
260	C₂²⋊C₄	—	218	1	no	—
261	C₂²⋊C₄	—	218	1	no	—
262	C₄²	—	109	1	no	—
263	C₄²	—	109	1	no	—
264	C₄²	—	218	1	no	—
265	C₄²	—	218	1	no	—
266	C₄²	—	218	1	no	—
267	C₄²	—	218	1	no	—
268	C₄²	—	218	1	no	—
269	C₄²	—	218	1	no	—
270	C₄²	—	218	1	no	—
271	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
272	Q₈	—	2	4	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	2	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	218	1	no	—
278	Q₈	—	218	1	no	—
279	Q₈	#13182028	1	2	yes	—
280	Q₈	#13182028	1	2	yes	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	109	1	no	—
289	C₂×C₄	—	109	1	no	—
290	C₂×C₄	—	109	1	no	—
291	C₂×C₄	—	109	1	no	—
292	C₂×C₄	—	218	1	no	—
293	C₂×C₄	—	218	1	no	—
294	C₂×C₄	—	218	1	no	—
295	C₂×C₄	—	218	1	no	—
296	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	218	1	no	—
301	C₈	—	218	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	436	1	no	—
312	C₄	—	436	1	no	—
313	C₄	—	436	1	no	—
314	C₄	—	436	1	no	—
315	C₄	—	218	1	no	—
316	C₄	—	218	1	no	—
317	C₄	—	218	1	no	—
318	C₄	—	218	1	no	—
319	C₄	C₂×#13182028	1	1	yes	characteristic
320	C₄	C₂×#13182028	1	1	yes	characteristic
321	C₂	—	4	1	no	—
322	C₂	#20295532	1	1	yes	characteristic
323	C₂	#20295839	1	1	yes	characteristic
324	C₂	#20295842	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#451×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂³×C₁₀₉	C₂×D₄
hypercenter	#451	D₁₀₉
hypocenter	C₁₀₉	#3127
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #451

Complements

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

N	H ≅ G/N	type
5. #20295335	C₂	semidirect
6. #20295331	C₂	semidirect
7. #20295234	C₂	semidirect
8. #20295234	C₂	semidirect
10. #20295133	C₂	semidirect
11. #20295130	C₂	semidirect
12. Q₁₆×Dic₁₀₉	C₂	semidirect
13. #20295118	C₂	semidirect
163. C₁₀₉	#3127	semidirect

Maximal subgroups

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

subgroup	index
#20295636	2
#20295504	2
#20295368	2
#20295364	2
#20295335	2
#20295331	2
#20295234	2
#20295182	2
#20295133	2
#20295130	2
Q₁₆×Dic₁₀₉	2
#20295118	2
#20295084	2
#451×C₁₀₉	2
#3127	109