Subgroups — #28376093

Order 13952.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295762	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295745	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295417	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295409	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295246	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295245	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295239	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295239	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295229	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295223	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295154	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295138	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295134	C₂	1	1	yes	maximal, characteristic, complemented
14	SD₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#496×C₁₀₉	C₂	1	1	yes	maximal, characteristic
16	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
17	C₂×#13182019	C₂²	1	1	yes	characteristic
18	C₂×#13182017	C₂²	1	1	yes	characteristic
19	C₂×#13182017	C₂²	1	1	yes	characteristic
20	C₂×#13182017	C₂²	1	1	yes	characteristic
21	#16812697	C₂²	1	1	yes	characteristic
22	#16812692	—	2	1	no	—
23	#16812692	—	2	1	no	—
24	#16812686	—	2	1	no	—
25	#16812686	C₂²	1	1	yes	characteristic
26	#16812686	C₂²	1	1	yes	characteristic
27	#16812686	C₂²	1	1	yes	characteristic
28	#16812680	—	2	1	no	—
29	#16812680	—	2	1	no	—
30	#16812676	—	2	1	no	—
31	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
32	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
33	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
34	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
35	D₄×Dic₁₀₉	—	2	1	no	—
36	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
37	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
38	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
39	#16812655	—	2	1	no	—
40	#16812631	C₂²	1	1	yes	characteristic
41	C₄⋊#11724784	C₂²	1	1	yes	characteristic
42	#16812628	C₂²	1	1	yes	characteristic
43	#16812628	C₂²	1	1	yes	characteristic
44	#16812624	C₂²	1	1	yes	characteristic
45	#16812624	C₂²	1	1	yes	characteristic
46	#16812623	C₂²	1	1	yes	characteristic
47	#16812620	C₂²	1	1	yes	characteristic
48	#16812620	C₂²	1	1	yes	characteristic
49	#16812620	C₂²	1	1	yes	characteristic
50	#16812609	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₂×D₈×C₁₀₉	C₂²	1	1	yes	characteristic
56	#127×C₁₀₉	C₂²	1	1	yes	characteristic
57	#124×C₁₀₉	C₂²	1	1	yes	characteristic
58	#97×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	D₄	1	1	yes	characteristic
60	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
61	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
62	C₂²×Dic₁₀₉	—	2	1	no	—
63	C₂²×Dic₁₀₉	—	2	1	no	—
64	C₂²×Dic₁₀₉	—	2	1	no	—
65	Dic₄₃₆	—	2	2	no	—
66	#13182019	—	4	1	no	—
67	#13182017	—	4	1	no	—
68	#13182017	—	4	1	no	—
69	#13182017	—	4	1	no	—
70	C₂×#11724784	C₂³	1	1	yes	characteristic
71	C₂×#11724784	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₁₀₉	—	2	1	no	—
73	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
74	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
76	#13182008	—	2	1	no	—
77	#13182008	—	2	1	no	—
78	#13182008	—	2	1	no	—
79	#13182008	—	2	1	no	—
80	#13182008	D₄	1	1	yes	characteristic
81	#13182008	D₄	1	1	yes	characteristic
82	C₂²⋊Dic₁₀₉	—	2	1	no	—
83	C₂²⋊Dic₁₀₉	—	2	1	no	—
84	C₂²⋊Dic₁₀₉	—	2	1	no	—
85	C₂²⋊Dic₁₀₉	—	2	1	no	—
86	C₂²⋊Dic₁₀₉	—	2	1	no	—
87	C₂²⋊Dic₁₀₉	—	2	1	no	—
88	C₂²⋊Dic₁₀₉	—	2	1	no	—
89	C₂²⋊Dic₁₀₉	—	2	1	no	—
90	C₂²⋊Dic₁₀₉	—	2	1	no	—
91	C₄×Dic₁₀₉	—	2	1	no	—
92	C₄×Dic₁₀₉	—	2	1	no	—
93	C₄×Dic₁₀₉	—	2	1	no	—
94	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
95	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₁₀₉	—	2	1	no	—
98	C₂×D₄×C₁₀₉	—	2	1	no	—
99	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
102	SD₁₆×C₁₀₉	—	2	2	no	—
103	SD₁₆×C₁₀₉	—	4	1	no	—
104	D₈×C₁₀₉	—	2	2	no	—
105	D₈×C₁₀₉	—	4	1	no	—
106	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
107	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
109	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
110	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
111	C₂³×C₁₀₉	—	2	1	no	—
112	C₂³×C₁₀₉	—	2	1	no	—
113	C₂³×C₁₀₉	—	2	1	no	—
114	Q₈×C₁₀₉	—	2	1	no	—
115	Q₈×C₁₀₉	—	2	1	no	—
116	D₄×C₁₀₉	—	2	1	no	—
117	D₄×C₁₀₉	—	2	1	no	—
118	D₄×C₁₀₉	—	2	1	no	—
119	D₄×C₁₀₉	—	2	1	no	—
120	D₄×C₁₀₉	—	2	1	no	—
121	D₄×C₁₀₉	—	2	1	no	—
122	D₄×C₁₀₉	—	4	2	no	—
123	D₄×C₁₀₉	—	8	1	no	—
124	C₂×C₄×C₁₀₉	—	2	1	no	—
125	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
128	C₂×Dic₁₀₉	—	2	1	no	—
129	C₂×Dic₁₀₉	—	2	1	no	—
130	C₂×Dic₁₀₉	—	2	1	no	—
131	C₂×Dic₁₀₉	—	2	1	no	—
132	C₂×Dic₁₀₉	—	4	2	no	—
133	C₂×Dic₁₀₉	—	8	1	no	—
134	C₂×Dic₁₀₉	—	8	1	no	—
135	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
138	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
139	Dic₂₁₈	—	2	1	no	—
140	Dic₂₁₈	—	2	1	no	—
141	Dic₂₁₈	—	2	1	no	—
142	Dic₂₁₈	—	2	1	no	—
143	Dic₂₁₈	—	4	1	no	—
144	C₈×C₁₀₉	—	2	1	no	—
145	C₈×C₁₀₉	#41	1	2	yes	—
146	#11724784	—	2	1	no	—
147	#11724784	—	2	1	no	—
148	C₂²×C₁₀₉	—	4	1	no	—
149	C₂²×C₁₀₉	—	4	1	no	—
150	C₂²×C₁₀₉	—	4	1	no	—
151	C₂²×C₁₀₉	—	4	1	no	—
152	C₂²×C₁₀₉	—	4	1	no	—
153	C₂²×C₁₀₉	—	4	1	no	—
154	C₂²×C₁₀₉	—	4	1	no	—
155	C₂²×C₁₀₉	—	4	1	no	—
156	C₂²×C₁₀₉	—	4	1	no	—
157	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
158	C₄×C₁₀₉	—	2	1	no	—
159	C₄×C₁₀₉	—	2	1	no	—
160	C₄×C₁₀₉	—	4	1	no	—
161	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
162	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
163	Dic₁₀₉	—	2	1	no	—
164	Dic₁₀₉	—	2	1	no	—
165	Dic₁₀₉	—	2	1	no	—
166	Dic₁₀₉	—	2	1	no	—
167	Dic₁₀₉	—	4	1	no	—
168	Dic₁₀₉	—	4	1	no	—
169	Dic₁₀₉	—	4	1	no	—
170	Dic₁₀₉	—	4	1	no	—
171	C₂×C₁₀₉	—	8	1	no	—
172	C₂×C₁₀₉	—	8	1	no	—
173	C₂×C₁₀₉	—	8	1	no	—
174	C₂×C₁₀₉	#550	1	1	yes	characteristic
175	C₂×C₁₀₉	#575	1	1	yes	characteristic
176	C₂×C₁₀₉	#577	1	1	yes	characteristic
177	#3401	—	109	1	no	maximal, Sylow
178	C₁₀₉	#3401	1	1	yes	characteristic, complemented, Sylow
179	#559	—	109	1	no	—
180	#556	—	109	1	no	—
181	#497	—	109	1	no	—
182	#496	D₁₀₉	1	1	yes	characteristic
183	#495	—	109	1	no	—
184	#464	—	109	1	no	—
185	#464	—	109	1	no	—
186	#463	—	109	1	no	—
187	#463	—	109	1	no	—
188	#461	—	109	1	no	—
189	#460	—	109	1	no	—
190	#446	—	109	1	no	—
191	#442	—	109	1	no	—
192	#441	—	109	1	no	—
193	C₄×SD₁₆	—	109	1	no	—
194	C₂×Q₁₆	—	109	1	no	—
195	C₂×Q₁₆	—	109	1	no	—
196	C₂×SD₁₆	—	109	1	no	—
197	C₂×SD₁₆	—	109	1	no	—
198	C₂×SD₁₆	—	109	1	no	—
199	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
200	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
201	C₂×D₈	C₂×D₁₀₉	1	1	yes	characteristic
202	C₂×D₈	C₂×D₁₀₉	1	1	yes	characteristic
203	C₄⋊Q₈	—	109	1	no	—
204	#127	C₂×D₁₀₉	1	1	yes	characteristic
205	#126	—	218	1	no	—
206	#126	—	218	1	no	—
207	#124	—	109	1	no	—
208	#124	—	109	1	no	—
209	#124	—	109	1	no	—
210	#124	—	218	1	no	—
211	#124	C₂×D₁₀₉	1	1	yes	characteristic
212	#123	—	218	1	no	—
213	#123	—	218	1	no	—
214	C₂²⋊Q₈	—	218	1	no	—
215	C₄×Q₈	—	109	1	no	—
216	C₄×Q₈	—	109	1	no	—
217	C₄×Q₈	—	109	1	no	—
218	C₄×Q₈	—	109	1	no	—
219	C₄×D₄	—	109	1	no	—
220	C₄×D₄	—	109	1	no	—
221	C₄×D₄	—	109	1	no	—
222	C₄×D₄	—	218	1	no	—
223	#117	—	218	1	no	—
224	#106	—	109	1	no	—
225	C₄⋊C₈	—	109	1	no	—
226	C₄⋊C₈	—	109	1	no	—
227	C₄⋊C₈	—	109	1	no	—
228	#103	—	109	1	no	—
229	#103	—	109	1	no	—
230	#103	—	109	1	no	—
231	#102	—	109	1	no	—
232	#102	—	109	1	no	—
233	#102	—	109	1	no	—
234	#97	—	109	1	no	—
235	#97	C₂×D₁₀₉	1	1	yes	characteristic
236	C₄×C₈	—	109	1	no	—
237	C₂×Q₈	—	109	1	no	—
238	C₂×Q₈	—	109	1	no	—
239	C₂×Q₈	—	109	1	no	—
240	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
244	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
245	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
246	C₂²×C₄	—	218	1	no	—
247	C₂²×C₄	—	218	1	no	—
248	C₂²×C₄	—	218	1	no	—
249	Q₁₆	—	436	1	no	—
250	Q₁₆	—	218	2	no	—
251	SD₁₆	—	2	2	no	—
252	SD₁₆	—	4	1	no	—
253	SD₁₆	—	436	1	no	—
254	SD₁₆	—	436	1	no	—
255	SD₁₆	—	436	1	no	—
256	D₈	—	2	2	no	—
257	D₈	—	4	1	no	—
258	C₂×C₈	—	109	1	no	—
259	C₂×C₈	—	109	1	no	—
260	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
262	C₄⋊C₄	—	109	1	no	—
263	C₄⋊C₄	—	109	1	no	—
264	C₄⋊C₄	—	109	1	no	—
265	C₄⋊C₄	—	109	1	no	—
266	C₄⋊C₄	—	109	1	no	—
267	C₄⋊C₄	—	218	1	no	—
268	C₄⋊C₄	—	218	1	no	—
269	C₄⋊C₄	—	218	1	no	—
270	C₄⋊C₄	—	218	1	no	—
271	C₄⋊C₄	—	218	1	no	—
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	218	1	no	—
275	C₂²⋊C₄	—	218	1	no	—
276	C₂²⋊C₄	—	218	1	no	—
277	C₂²⋊C₄	—	218	1	no	—
278	C₂²⋊C₄	—	218	1	no	—
279	C₂²⋊C₄	—	218	1	no	—
280	C₂²⋊C₄	—	218	1	no	—
281	C₂²⋊C₄	—	218	1	no	—
282	C₂²⋊C₄	—	218	1	no	—
283	C₄²	—	109	1	no	—
284	C₄²	—	109	1	no	—
285	C₄²	—	218	1	no	—
286	C₄²	—	218	1	no	—
287	C₄²	—	218	1	no	—
288	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
289	C₂³	—	2	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	2	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	2	1	no	—
294	Q₈	—	436	1	no	—
295	Q₈	—	218	1	no	—
296	Q₈	—	218	1	no	—
297	Q₈	—	218	1	no	—
298	Q₈	—	218	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	2	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	2	1	no	—
303	D₄	—	2	1	no	—
304	D₄	—	2	1	no	—
305	D₄	—	4	2	no	—
306	D₄	—	8	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	872	1	no	—
309	C₂×C₄	—	872	1	no	—
310	C₂×C₄	—	109	1	no	—
311	C₂×C₄	—	109	1	no	—
312	C₂×C₄	—	109	1	no	—
313	C₂×C₄	—	109	1	no	—
314	C₂×C₄	—	436	2	no	—
315	C₂×C₄	—	218	1	no	—
316	C₂×C₄	—	218	1	no	—
317	C₂×C₄	—	218	1	no	—
318	C₂×C₄	—	218	1	no	—
319	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
320	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
321	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	218	1	no	—
324	C₈	—	218	1	no	—
325	C₈	#13182028	1	2	yes	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	436	1	no	—
340	C₄	—	436	1	no	—
341	C₄	—	436	1	no	—
342	C₄	—	436	1	no	—
343	C₄	—	218	1	no	—
344	C₄	—	218	1	no	—
345	C₄	—	218	1	no	—
346	C₄	—	218	1	no	—
347	C₄	C₂×#13182028	1	1	yes	characteristic
348	C₄	#16812744	1	1	yes	characteristic
349	C₂	—	8	1	no	—
350	C₂	—	8	1	no	—
351	C₂	—	8	1	no	—
352	C₂	#20295696	1	1	yes	characteristic
353	C₂	#20295822	1	1	yes	characteristic
354	C₂	#20295836	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting, prime radical	#496×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
hypercenter	#496	D₁₀₉
hypocenter	C₁₀₉	#3401
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #496

Complements

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

N	H ≅ G/N	type
1. #20295762	C₂	semidirect
2. #20295745	C₂	semidirect
3. #20295417	C₂	semidirect
4. #20295409	C₂	semidirect
5. #20295246	C₂	semidirect
6. #20295245	C₂	semidirect
7. #20295239	C₂	semidirect
8. #20295239	C₂	semidirect
9. #20295229	C₂	semidirect
10. #20295223	C₂	semidirect
11. #20295154	C₂	semidirect
12. #20295138	C₂	semidirect
13. #20295134	C₂	semidirect
14. SD₁₆×Dic₁₀₉	C₂	semidirect
178. C₁₀₉	#3401	semidirect

Maximal subgroups

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

subgroup	index
#20295762	2
#20295745	2
#20295417	2
#20295409	2
#20295246	2
#20295245	2
#20295239	2
#20295229	2
#20295223	2
#20295154	2
#20295138	2
#20295134	2
SD₁₆×Dic₁₀₉	2
#496×C₁₀₉	2
#3401	109