Subgroups — #28374919

Order 13952.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295716	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295380	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295364	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295337	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295333	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295265	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295253	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295245	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295188	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295182	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#13182019	C₂	1	1	yes	maximal, characteristic, complemented
12	D₄×#11724784	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295022	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#16812623	C₂	1	1	yes	maximal, characteristic, complemented
15	#548×C₁₀₉	C₂	1	1	yes	maximal, characteristic
16	C₂×#13182019	C₂²	1	1	yes	characteristic
17	C₂×#13182019	C₂²	1	1	yes	characteristic
18	C₂×#13182017	C₂²	1	1	yes	characteristic
19	C₂²×#11724784	C₂²	1	1	yes	characteristic
20	C₂²×#11724784	C₂²	1	1	yes	characteristic
21	#16812692	—	2	1	no	—
22	#16812691	C₂²	1	1	yes	characteristic
23	#16812689	—	2	1	no	—
24	#16812680	—	2	1	no	—
25	#16812678	C₂²	1	1	yes	characteristic
26	#16812678	C₂²	1	1	yes	characteristic
27	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
28	#16812658	C₂²	1	1	yes	characteristic
29	#16812655	—	2	1	no	—
30	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
31	#16812632	C₂²	1	1	yes	characteristic
32	#16812632	C₂²	1	1	yes	characteristic
33	#16812630	C₂²	1	1	yes	characteristic
34	C₄⋊#11724784	C₂²	1	1	yes	characteristic
35	#16812623	C₂²	1	1	yes	characteristic
36	#16812623	C₂²	1	1	yes	characteristic
37	#16812623	C₂²	1	1	yes	characteristic
38	#16812623	C₂²	1	1	yes	characteristic
39	#16812623	C₂²	1	1	yes	characteristic
40	#16812623	C₂²	1	1	yes	characteristic
41	#16812622	C₂²	1	1	yes	characteristic
42	#16812620	C₂²	1	1	yes	characteristic
43	#16812620	C₂²	1	1	yes	characteristic
44	#16812620	C₂²	1	1	yes	characteristic
45	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
46	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
47	C₄×#11724784	C₂²	1	1	yes	characteristic
48	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
49	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
50	#124×C₁₀₉	—	2	1	no	—
51	#124×C₁₀₉	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₁₀₉	—	2	1	no	—
53	C₂²⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
54	#121×C₁₀₉	—	2	1	no	—
55	#121×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
57	C₄×D₄×C₁₀₉	—	2	1	no	—
58	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
60	C₂²×Dic₁₀₉	—	2	1	no	—
61	#13182019	—	2	2	no	—
62	#13182019	—	4	1	no	—
63	#13182017	—	4	1	no	—
64	C₂×#11724784	—	4	1	no	—
65	C₂×#11724784	—	4	1	no	—
66	C₂×#11724784	C₂³	1	1	yes	characteristic
67	C₂×#11724784	C₂³	1	1	yes	characteristic
68	C₂×#11724784	C₂³	1	1	yes	characteristic
69	C₂×#11724784	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₁₀₉	—	2	1	no	—
71	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
74	#13182008	—	2	1	no	—
75	#13182008	—	2	1	no	—
76	#13182008	—	2	1	no	—
77	#13182008	—	2	1	no	—
78	#13182008	—	2	1	no	—
79	C₂²⋊Dic₁₀₉	—	2	1	no	—
80	C₂²⋊Dic₁₀₉	—	2	1	no	—
81	C₂²⋊Dic₁₀₉	—	2	1	no	—
82	C₄×Dic₁₀₉	—	2	1	no	—
83	C₄×Dic₁₀₉	—	2	1	no	—
84	#41×C₁₀₉	—	2	2	no	—
85	#41×C₁₀₉	—	4	1	no	—
86	C₂×Q₈×C₁₀₉	—	2	1	no	—
87	C₂×Q₈×C₁₀₉	—	2	2	no	—
88	C₂×Q₈×C₁₀₉	—	4	1	no	—
89	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
90	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
91	C₂×D₄×C₁₀₉	—	2	1	no	—
92	C₂×D₄×C₁₀₉	—	2	1	no	—
93	C₂×D₄×C₁₀₉	D₄	1	1	yes	characteristic
94	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₁₀₉	—	2	1	no	—
96	C₂²×C₄×C₁₀₉	—	2	1	no	—
97	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₀₉	—	2	1	no	—
100	C₄⋊C₄×C₁₀₉	—	2	1	no	—
101	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
102	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
103	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
104	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
105	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
106	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
107	C₂²⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
108	C₂²⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
109	C₄²×C₁₀₉	—	2	1	no	—
110	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
111	C₂³×C₁₀₉	—	2	1	no	—
112	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
113	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₁₀₉	—	2	1	no	—
115	Q₈×C₁₀₉	—	2	1	no	—
116	Q₈×C₁₀₉	—	2	1	no	—
117	Q₈×C₁₀₉	—	4	2	no	—
118	Q₈×C₁₀₉	#41	1	2	yes	—
119	D₄×C₁₀₉	—	2	2	no	—
120	D₄×C₁₀₉	—	2	1	no	—
121	D₄×C₁₀₉	—	2	1	no	—
122	D₄×C₁₀₉	—	4	2	no	—
123	D₄×C₁₀₉	—	4	2	no	—
124	C₂×C₄×C₁₀₉	—	2	1	no	—
125	C₂×C₄×C₁₀₉	—	2	1	no	—
126	C₂×C₄×C₁₀₉	—	2	1	no	—
127	C₂×C₄×C₁₀₉	—	2	1	no	—
128	C₂×C₄×C₁₀₉	—	2	1	no	—
129	C₂×C₄×C₁₀₉	—	2	1	no	—
130	C₂×C₄×C₁₀₉	—	2	1	no	—
131	C₂×C₄×C₁₀₉	—	4	2	no	—
132	C₂×C₄×C₁₀₉	—	4	2	no	—
133	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
138	C₂×Dic₁₀₉	—	2	1	no	—
139	C₂×Dic₁₀₉	—	2	1	no	—
140	C₂×Dic₁₀₉	—	2	1	no	—
141	C₂×Dic₁₀₉	—	2	1	no	—
142	C₂×Dic₁₀₉	—	4	2	no	—
143	Dic₂₁₈	—	2	1	no	—
144	Dic₂₁₈	—	2	1	no	—
145	#11724784	—	2	1	no	—
146	#11724784	—	2	1	no	—
147	#11724784	—	2	1	no	—
148	#11724784	—	2	1	no	—
149	C₂²×C₁₀₉	—	2	1	no	—
150	C₂²×C₁₀₉	—	2	1	no	—
151	C₂²×C₁₀₉	—	2	1	no	—
152	C₂²×C₁₀₉	—	2	1	no	—
153	C₂²×C₁₀₉	—	2	1	no	—
154	C₂²×C₁₀₉	—	4	1	no	—
155	C₂²×C₁₀₉	—	4	1	no	—
156	C₂²×C₁₀₉	—	4	1	no	—
157	C₂²×C₁₀₉	#135	1	2	yes	—
158	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
159	C₄×C₁₀₉	—	2	1	no	—
160	C₄×C₁₀₉	—	2	1	no	—
161	C₄×C₁₀₉	—	2	2	no	—
162	C₄×C₁₀₉	—	2	1	no	—
163	C₄×C₁₀₉	—	2	1	no	—
164	C₄×C₁₀₉	—	4	1	no	—
165	C₄×C₁₀₉	—	4	1	no	—
166	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
167	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
168	Dic₁₀₉	—	4	1	no	—
169	Dic₁₀₉	—	4	1	no	—
170	Dic₁₀₉	—	4	1	no	—
171	Dic₁₀₉	—	4	1	no	—
172	C₂×C₁₀₉	—	2	2	no	—
173	C₂×C₁₀₉	—	4	1	no	—
174	C₂×C₁₀₉	—	8	1	no	—
175	C₂×C₁₀₉	#546	1	1	yes	characteristic
176	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
177	C₂×C₁₀₉	#578	1	1	yes	characteristic
178	#3300	—	109	1	no	maximal, Sylow
179	C₁₀₉	#3300	1	1	yes	characteristic, complemented, Sylow
180	#552	—	109	1	no	—
181	#548	D₁₀₉	1	1	yes	characteristic
182	#488	—	109	1	no	—
183	#485	—	109	1	no	—
184	#480	—	109	1	no	—
185	#479	—	109	1	no	—
186	#467	—	109	1	no	—
187	#465	—	109	1	no	—
188	#464	—	109	1	no	—
189	#452	—	109	1	no	—
190	#451	—	109	1	no	—
191	C₄×Q₁₆	—	109	1	no	—
192	C₈×D₄	—	109	1	no	—
193	#416	—	109	1	no	—
194	C₂×#103	—	109	1	no	—
195	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
196	C₂²×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
197	C₂×Q₁₆	—	109	1	no	—
198	C₂×Q₁₆	—	109	1	no	—
199	C₂×SD₁₆	—	109	1	no	—
200	C₂²×C₈	—	109	1	no	—
201	C₂²×C₈	—	109	1	no	—
202	#126	—	218	1	no	—
203	#125	—	109	1	no	—
204	#125	—	218	1	no	—
205	#124	—	2	1	no	—
206	#124	C₂×D₁₀₉	1	1	yes	characteristic
207	#123	—	218	1	no	—
208	C₂²⋊Q₈	—	2	1	no	—
209	C₂²⋊Q₈	—	109	1	no	—
210	C₂²⋊Q₈	—	109	1	no	—
211	C₂²⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
212	#121	—	2	1	no	—
213	#121	C₂×D₁₀₉	1	1	yes	characteristic
214	C₄×Q₈	—	109	1	no	—
215	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
216	C₄×D₄	—	2	1	no	—
217	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
218	#117	—	109	1	no	—
219	#117	—	218	1	no	—
220	C₂×C₄⋊C₄	—	109	1	no	—
221	#107	—	109	1	no	—
222	#107	—	109	1	no	—
223	#106	—	109	1	no	—
224	C₄⋊C₈	—	109	1	no	—
225	#103	—	109	1	no	—
226	#103	—	109	1	no	—
227	#103	—	109	1	no	—
228	#103	—	109	1	no	—
229	#103	—	109	1	no	—
230	#103	—	109	1	no	—
231	#103	—	109	1	no	—
232	#102	—	109	1	no	—
233	#102	—	109	1	no	—
234	#102	—	109	1	no	—
235	C₂²⋊C₈	—	109	1	no	—
236	C₂²⋊C₈	—	109	1	no	—
237	C₄×C₈	—	109	1	no	—
238	#41	—	2	2	no	—
239	#41	—	4	1	no	—
240	C₂×Q₈	—	2	1	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	109	1	no	—
244	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
245	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
249	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	218	1	no	—
253	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
254	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
255	Q₁₆	—	436	1	no	—
256	Q₁₆	—	218	2	no	—
257	SD₁₆	—	436	1	no	—
258	C₂×C₈	—	109	1	no	—
259	C₂×C₈	—	109	1	no	—
260	C₂×C₈	—	109	1	no	—
261	C₂×C₈	—	109	1	no	—
262	C₂×C₈	—	436	1	no	—
263	C₂×C₈	—	436	1	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	109	1	no	—
267	C₄⋊C₄	—	109	1	no	—
268	C₄⋊C₄	—	109	1	no	—
269	C₄⋊C₄	—	218	1	no	—
270	C₄⋊C₄	—	218	1	no	—
271	C₄⋊C₄	—	218	1	no	—
272	C₄⋊C₄	—	218	1	no	—
273	C₄⋊C₄	—	218	1	no	—
274	C₄⋊C₄	—	218	1	no	—
275	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	218	1	no	—
282	C₂²⋊C₄	—	218	1	no	—
283	C₂²⋊C₄	—	218	1	no	—
284	C₂²⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
285	C₂²⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
286	C₄²	—	2	1	no	—
287	C₄²	—	218	1	no	—
288	C₄²	—	218	1	no	—
289	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
290	C₂³	—	2	1	no	—
291	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
292	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	2	1	no	—
296	Q₈	—	4	2	no	—
297	Q₈	—	218	1	no	—
298	Q₈	—	218	1	no	—
299	Q₈	#13182028	1	2	yes	—
300	D₄	—	2	2	no	—
301	D₄	—	2	1	no	—
302	D₄	—	2	1	no	—
303	D₄	—	4	2	no	—
304	D₄	—	4	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	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	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
320	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
321	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
322	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
323	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
324	C₈	—	218	1	no	—
325	C₈	—	218	1	no	—
326	C₈	—	218	1	no	—
327	C₈	—	218	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	#16812719	1	2	yes	—
337	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	2	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	436	1	no	—
346	C₄	—	436	1	no	—
347	C₄	—	436	1	no	—
348	C₄	—	436	1	no	—
349	C₄	C₂×#13182028	1	1	yes	characteristic
350	C₄	#16812744	1	1	yes	characteristic
351	C₂	—	2	2	no	—
352	C₂	—	4	1	no	—
353	C₂	—	8	1	no	—
354	C₂	#20295670	1	1	yes	characteristic
355	C₂	C₂×#16812719	1	1	yes	characteristic
356	C₂	#20295841	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting, prime radical	#548×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
hypercenter	#548	D₁₀₉
hypocenter	C₁₀₉	#3300
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #548

Complements

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

N	H ≅ G/N	type
1. #20295716	C₂	semidirect
2. #20295380	C₂	semidirect
3. #20295364	C₂	semidirect
4. #20295337	C₂	semidirect
5. #20295333	C₂	semidirect
6. #20295265	C₂	semidirect
7. #20295253	C₂	semidirect
8. #20295245	C₂	semidirect
9. #20295188	C₂	semidirect
10. #20295182	C₂	semidirect
11. C₄×#13182019	C₂	semidirect
12. D₄×#11724784	C₂	semidirect
13. #20295022	C₂	semidirect
14. C₂×#16812623	C₂	semidirect
179. C₁₀₉	#3300	semidirect

Maximal subgroups

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

subgroup	index
#20295716	2
#20295380	2
#20295364	2
#20295337	2
#20295333	2
#20295265	2
#20295253	2
#20295245	2
#20295188	2
#20295182	2
C₄×#13182019	2
D₄×#11724784	2
#20295022	2
C₂×#16812623	2
#548×C₁₀₉	2
#3300	109