Subgroups — #28374890

Order 13952.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₂²⋊D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295676	C₂	1	1	yes	maximal, characteristic
3	#20295375	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295357	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295355	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295309	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295253	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295252	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295229	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295163	C₂	1	1	yes	maximal, characteristic, complemented
11	SD₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295089	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#16812623	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#448×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182028	—	2	1	no	—
17	C₂×Q₈×D₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×#13182017	C₂²	1	1	yes	characteristic
19	C₂²×#11724784	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
21	#16812698	C₂²	1	1	yes	characteristic
22	#16812697	—	2	1	no	—
23	#16812697	C₂²	1	1	yes	characteristic
24	#16812681	—	2	1	no	—
25	#16812680	—	2	1	no	—
26	#16812678	—	2	1	no	—
27	#16812678	C₂²	1	1	yes	characteristic
28	#16812676	—	2	1	no	—
29	#16812675	—	2	1	no	—
30	#16812675	C₂²	1	1	yes	characteristic
31	#16812670	C₂²	1	1	yes	characteristic
32	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
33	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
34	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
35	C₄×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₁₀₉	C₂²	1	1	yes	characteristic
38	#16812631	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	#16812624	C₂²	1	1	yes	characteristic
43	#16812624	C₂²	1	1	yes	characteristic
44	#16812623	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	#16812622	C₂²	1	1	yes	characteristic
49	#16812620	C₂²	1	1	yes	characteristic
50	#16812611	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
52	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
55	#121×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	#13182028	—	4	4	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	—	4	1	no	—
65	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
66	C₂×C₂²⋊D₁₀₉	D₄	1	1	yes	characteristic
67	C₂²×Dic₁₀₉	—	2	1	no	—
68	C₂²×Dic₁₀₉	—	2	1	no	—
69	C₂×C₄×D₁₀₉	—	2	1	no	—
70	C₂×C₄×D₁₀₉	—	2	1	no	—
71	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
72	#13182017	—	4	1	no	—
73	C₂×#11724784	—	4	1	no	—
74	C₂×#11724784	C₂³	1	1	yes	characteristic
75	C₂×#11724784	C₂³	1	1	yes	characteristic
76	C₈×D₁₀₉	—	4	1	no	—
77	C₄⋊Dic₁₀₉	—	2	1	no	—
78	C₄⋊Dic₁₀₉	—	2	1	no	—
79	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
82	#13182008	—	2	1	no	—
83	#13182008	—	2	1	no	—
84	#13182008	—	2	1	no	—
85	#13182008	—	2	1	no	—
86	#13182008	—	2	1	no	—
87	#13182008	—	2	1	no	—
88	#13182008	D₄	1	1	yes	characteristic
89	C₂²⋊Dic₁₀₉	—	2	1	no	—
90	C₂²⋊Dic₁₀₉	—	2	1	no	—
91	C₂²⋊Dic₁₀₉	—	2	1	no	—
92	C₂²⋊Dic₁₀₉	D₄	1	1	yes	characteristic
93	#13182006	—	2	1	no	—
94	#13182006	D₄	1	1	yes	characteristic
95	C₄×Dic₁₀₉	—	2	1	no	—
96	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₁₀₉	—	2	1	no	—
98	C₂×Q₈×C₁₀₉	—	2	2	no	—
99	C₂×Q₈×C₁₀₉	—	4	1	no	—
100	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₂×D₄×C₁₀₉	—	2	1	no	—
103	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₂²×C₄×C₁₀₉	—	2	1	no	—
105	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
106	SD₁₆×C₁₀₉	—	2	2	no	—
107	SD₁₆×C₁₀₉	—	4	1	no	—
108	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
109	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
110	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
111	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
112	C₂³×C₁₀₉	—	2	1	no	—
113	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
114	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₁₀₉	—	2	1	no	—
116	Q₈×C₁₀₉	—	2	1	no	—
117	Q₈×C₁₀₉	—	2	1	no	—
118	Q₈×C₁₀₉	—	4	2	no	—
119	Q₈×C₁₀₉	#41	1	2	yes	complemented
120	D₄×C₁₀₉	—	2	1	no	—
121	D₄×C₁₀₉	—	2	1	no	—
122	D₄×C₁₀₉	—	4	2	no	—
123	C₂×C₄×C₁₀₉	—	2	1	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	—	4	2	no	—
129	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
131	C₂²⋊D₁₀₉	—	4	2	no	—
132	C₂²⋊D₁₀₉	SD₁₆	1	4	yes	complemented
133	C₂×Dic₁₀₉	—	2	1	no	—
134	C₂×Dic₁₀₉	—	2	1	no	—
135	C₂×Dic₁₀₉	—	2	1	no	—
136	C₂×Dic₁₀₉	—	2	1	no	—
137	C₂×Dic₁₀₉	—	4	2	no	—
138	C₂×Dic₁₀₉	—	4	2	no	—
139	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
140	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
141	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
142	C₄×D₁₀₉	—	2	1	no	—
143	C₄×D₁₀₉	—	2	1	no	—
144	C₄×D₁₀₉	—	4	2	no	—
145	C₄×D₁₀₉	—	4	2	no	—
146	Dic₂₁₈	—	2	1	no	—
147	Dic₂₁₈	—	2	1	no	—
148	Dic₂₁₈	—	4	2	no	—
149	Dic₂₁₈	—	4	2	no	—
150	C₈×C₁₀₉	—	2	1	no	—
151	C₈×C₁₀₉	—	2	1	no	—
152	#11724784	—	2	1	no	—
153	#11724784	—	2	1	no	—
154	C₂²×C₁₀₉	—	2	1	no	—
155	C₂²×C₁₀₉	—	2	1	no	—
156	C₂²×C₁₀₉	—	4	1	no	—
157	C₂²×C₁₀₉	—	4	1	no	—
158	C₂²×C₁₀₉	—	4	1	no	—
159	C₂²×C₁₀₉	C₂×SD₁₆	1	2	yes	—
160	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
161	C₂×D₁₀₉	—	2	1	no	—
162	C₂×D₁₀₉	—	2	1	no	—
163	C₂×D₁₀₉	C₂×SD₁₆	1	2	yes	—
164	C₄×C₁₀₉	—	2	2	no	—
165	C₄×C₁₀₉	—	2	1	no	—
166	C₄×C₁₀₉	—	4	1	no	—
167	C₄×C₁₀₉	—	4	1	no	—
168	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
169	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
170	Dic₁₀₉	—	2	1	no	—
171	Dic₁₀₉	—	2	1	no	—
172	Dic₁₀₉	—	4	1	no	—
173	Dic₁₀₉	—	4	1	no	—
174	Dic₁₀₉	—	4	1	no	—
175	Dic₁₀₉	—	4	1	no	—
176	Dic₁₀₉	C₂×SD₁₆	1	2	yes	—
177	C₂×C₁₀₉	—	2	2	no	—
178	C₂×C₁₀₉	—	8	1	no	—
179	C₂×C₁₀₉	#546	1	1	yes	characteristic
180	C₂×C₁₀₉	C₂²×SD₁₆	1	1	yes	characteristic
181	C₂×C₁₀₉	#578	1	1	yes	characteristic
182	D₁₀₉	—	2	2	no	—
183	#3298	—	109	1	no	maximal, Sylow
184	C₁₀₉	#3298	1	1	yes	characteristic, complemented, Sylow
185	D₄×Q₈	—	109	1	no	—
186	#547	—	109	1	no	—
187	#487	—	109	1	no	—
188	#483	—	109	1	no	—
189	#483	—	109	1	no	—
190	#475	—	109	1	no	—
191	#465	—	109	1	no	—
192	#465	—	109	1	no	—
193	#461	—	109	1	no	—
194	#448	—	109	1	no	—
195	#448	D₁₀₉	1	1	yes	characteristic, complemented
196	C₄×SD₁₆	—	109	1	no	—
197	C₈×D₄	—	109	1	no	—
198	C₂×#103	—	109	1	no	—
199	C₂×#103	—	109	1	no	—
200	C₂×#41	—	218	1	no	—
201	C₂²×Q₈	—	109	1	no	—
202	C₂²×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
203	C₂×SD₁₆	—	109	1	no	—
204	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
205	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
206	C₂²×C₈	—	109	1	no	—
207	C₂²×C₈	—	109	1	no	—
208	C₄⋊Q₈	—	109	1	no	—
209	C₄⋊Q₈	—	109	1	no	—
210	C₄⋊Q₈	—	218	1	no	—
211	#123	—	218	1	no	—
212	#123	—	218	1	no	—
213	C₂²⋊Q₈	—	109	1	no	—
214	C₂²⋊Q₈	—	109	1	no	—
215	C₂²⋊Q₈	—	218	1	no	—
216	C₂²⋊Q₈	—	218	1	no	—
217	C₂²⋊Q₈	—	218	1	no	—
218	#121	—	109	1	no	—
219	#121	C₂×D₁₀₉	1	1	yes	characteristic
220	C₄×Q₈	—	109	1	no	—
221	C₄×D₄	—	109	1	no	—
222	C₄×D₄	—	109	1	no	—
223	C₄×D₄	—	218	1	no	—
224	C₂×C₄⋊C₄	—	109	1	no	—
225	C₂×C₄⋊C₄	—	109	1	no	—
226	#106	—	109	1	no	—
227	#106	—	109	1	no	—
228	#106	—	109	1	no	—
229	C₄⋊C₈	—	109	1	no	—
230	#103	—	109	1	no	—
231	#103	—	109	1	no	—
232	#103	—	109	1	no	—
233	#103	—	109	1	no	—
234	#103	—	109	1	no	—
235	#103	—	109	1	no	—
236	#103	—	109	1	no	—
237	#103	C₂×D₁₀₉	1	1	yes	characteristic
238	#103	C₂×D₁₀₉	1	1	yes	characteristic
239	#102	—	109	1	no	—
240	C₂²⋊C₈	—	109	1	no	—
241	C₂²⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
242	C₄×C₈	—	109	1	no	—
243	#41	—	436	4	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	109	1	no	—
248	C₂×Q₈	—	436	1	no	—
249	C₂×Q₈	—	218	1	no	—
250	C₂×Q₈	—	218	1	no	—
251	C₂×Q₈	—	218	2	no	—
252	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
253	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
254	C₂×D₄	—	2	1	no	—
255	C₂×D₄	—	109	1	no	—
256	C₂×D₄	—	218	1	no	—
257	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	109	1	no	—
260	C₂²×C₄	—	218	1	no	—
261	C₂²×C₄	—	218	1	no	—
262	C₂²×C₄	—	218	1	no	—
263	C₂²×C₄	—	218	1	no	—
264	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
265	SD₁₆	—	2	2	no	—
266	SD₁₆	—	4	1	no	—
267	SD₁₆	—	436	1	no	—
268	C₂×C₈	—	109	1	no	—
269	C₂×C₈	—	109	1	no	—
270	C₂×C₈	—	436	1	no	—
271	C₂×C₈	—	436	1	no	—
272	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
273	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
274	C₄⋊C₄	—	109	1	no	—
275	C₄⋊C₄	—	109	1	no	—
276	C₄⋊C₄	—	109	1	no	—
277	C₄⋊C₄	—	109	1	no	—
278	C₄⋊C₄	—	218	1	no	—
279	C₄⋊C₄	—	218	1	no	—
280	C₄⋊C₄	—	218	1	no	—
281	C₄⋊C₄	—	218	1	no	—
282	C₄⋊C₄	—	218	1	no	—
283	C₄⋊C₄	—	218	1	no	—
284	C₄⋊C₄	—	218	1	no	—
285	C₄⋊C₄	—	218	1	no	—
286	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
287	C₂²⋊C₄	—	2	1	no	—
288	C₂²⋊C₄	—	109	1	no	—
289	C₂²⋊C₄	—	109	1	no	—
290	C₂²⋊C₄	—	218	1	no	—
291	C₂²⋊C₄	—	218	1	no	—
292	C₂²⋊C₄	—	218	1	no	—
293	C₂²⋊C₄	—	218	1	no	—
294	C₄²	—	109	1	no	—
295	C₄²	—	218	1	no	—
296	C₂³	—	2	1	no	—
297	C₂³	—	109	1	no	—
298	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
299	Q₈	—	2	1	no	—
300	Q₈	—	2	1	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	436	2	no	—
304	Q₈	—	436	2	no	—
305	Q₈	—	218	1	no	—
306	Q₈	—	218	1	no	—
307	Q₈	#13182028	1	2	yes	complemented
308	D₄	—	2	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	4	2	no	—
311	D₄	—	109	4	no	—
312	D₄	—	436	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	109	1	no	—
320	C₂×C₄	—	109	1	no	—
321	C₂×C₄	—	109	1	no	—
322	C₂×C₄	—	436	2	no	—
323	C₂×C₄	—	436	2	no	—
324	C₂×C₄	—	436	2	no	—
325	C₂×C₄	—	436	2	no	—
326	C₂×C₄	—	218	1	no	—
327	C₂×C₄	—	218	1	no	—
328	C₂×C₄	—	218	1	no	—
329	C₂×C₄	—	218	1	no	—
330	C₂×C₄	—	218	1	no	—
331	C₂×C₄	—	218	1	no	—
332	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
333	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	218	1	no	—
337	C₈	—	218	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	109	2	no	—
344	C₂²	—	218	1	no	—
345	C₂²	—	218	1	no	—
346	C₂²	SD₁₆×D₁₀₉	1	2	yes	—
347	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	109	2	no	—
353	C₄	—	436	1	no	—
354	C₄	—	436	1	no	—
355	C₄	—	436	1	no	—
356	C₄	—	436	1	no	—
357	C₄	—	218	1	no	—
358	C₄	—	218	1	no	—
359	C₄	C₂×#13182028	1	1	yes	characteristic
360	C₄	#16812744	1	1	yes	characteristic
361	C₂	—	2	2	no	—
362	C₂	—	8	1	no	—
363	C₂	—	218	2	no	—
364	C₂	#20295663	1	1	yes	characteristic
365	C₂	C₂×SD₁₆×D₁₀₉	1	1	yes	characteristic
366	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	#448×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#20295676	C₂
hypercenter	#448	D₁₀₉
hypocenter	C₁₀₉	#3298
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #448

Complements

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

N	H ≅ G/N	type
1. Q₈×C₂²⋊D₁₀₉	C₂	semidirect
3. #20295375	C₂	semidirect
4. #20295357	C₂	semidirect
5. #20295355	C₂	semidirect
6. #20295309	C₂	semidirect
7. #20295253	C₂	semidirect
8. #20295252	C₂	semidirect
9. #20295229	C₂	semidirect
10. #20295163	C₂	semidirect
11. SD₁₆×Dic₁₀₉	C₂	semidirect
12. #20295089	C₂	semidirect
13. C₂×#16812623	C₂	semidirect
14. #103×D₁₀₉	C₂	semidirect
15. #448×C₁₀₉	C₂	semidirect
119. Q₈×C₁₀₉	#41	semidirect
132. C₂²⋊D₁₀₉	SD₁₆	semidirect
184. C₁₀₉	#3298	semidirect
195. #448	D₁₀₉	semidirect
307. Q₈	#13182028	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₂²⋊D₁₀₉	2
#20295676	2
#20295375	2
#20295357	2
#20295355	2
#20295309	2
#20295253	2
#20295252	2
#20295229	2
#20295163	2
SD₁₆×Dic₁₀₉	2
#20295089	2
C₂×#16812623	2
#103×D₁₀₉	2
#448×C₁₀₉	2
#3298	109