Subgroups — #28373246

Order 13952.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295638	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295361	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295356	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295333	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295326	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295236	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295231	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295188	C₂	1	1	yes	maximal, characteristic
9	#20295188	C₂	1	1	yes	maximal, characteristic
10	#20295131	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295126	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#13182019	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#13182017	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295087	C₂	1	1	yes	maximal, characteristic, complemented
15	#520×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	C₂×#13182017	C₂²	1	1	yes	characteristic
20	#16812699	C₂²	1	1	yes	characteristic
21	#16812692	—	2	1	no	—
22	#16812692	—	2	1	no	—
23	#16812691	C₂²	1	1	yes	characteristic
24	#16812689	—	2	1	no	—
25	#16812689	—	2	1	no	—
26	#16812678	C₂²	1	1	yes	characteristic
27	#16812678	C₂²	1	1	yes	characteristic
28	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
29	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
30	#16812632	C₂²	1	1	yes	characteristic
31	#16812632	C₂²	1	1	yes	characteristic
32	#16812630	C₂²	1	1	yes	characteristic
33	#16812630	C₂²	1	1	yes	characteristic
34	C₄⋊#11724784	C₂²	1	1	yes	characteristic
35	C₄⋊#11724784	C₂²	1	1	yes	characteristic
36	#16812623	C₂²	1	1	yes	characteristic
37	#16812623	C₂²	1	1	yes	characteristic
38	#16812622	C₂²	1	1	yes	characteristic
39	#16812622	C₂²	1	1	yes	characteristic
40	#16812622	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	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
45	C₂²⋊#11724784	C₂²	1	1	yes	characteristic
46	#16812610	C₂²	1	1	yes	characteristic
47	C₄×#11724784	C₂²	1	1	yes	characteristic
48	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
49	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
50	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
51	C₄×D₄×C₁₀₉	—	2	1	no	—
52	C₄×D₄×C₁₀₉	—	2	1	no	—
53	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
55	#117×C₁₀₉	—	2	1	no	—
56	#117×C₁₀₉	C₂²	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
58	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
60	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
61	#13182019	—	2	2	no	—
62	#13182019	—	2	2	no	—
63	#13182017	—	2	2	no	—
64	#13182017	—	2	2	no	—
65	C₂×#11724784	C₂³	1	1	yes	characteristic
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	C₄⋊Dic₁₀₉	—	2	1	no	—
70	C₄⋊Dic₁₀₉	—	2	1	no	—
71	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
73	#13182008	—	2	1	no	—
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	C₄×Dic₁₀₉	—	2	1	no	—
82	C₄×Dic₁₀₉	—	2	1	no	—
83	#41×C₁₀₉	—	2	2	no	—
84	#41×C₁₀₉	—	2	2	no	—
85	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
86	C₂×D₄×C₁₀₉	—	2	1	no	—
87	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
88	C₂²×C₄×C₁₀₉	—	2	1	no	—
89	C₂²×C₄×C₁₀₉	—	2	1	no	—
90	C₂²×C₄×C₁₀₉	—	2	1	no	—
91	C₂²×C₄×C₁₀₉	—	2	1	no	—
92	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₁₀₉	—	2	1	no	—
94	C₄⋊C₄×C₁₀₉	—	2	1	no	—
95	C₄⋊C₄×C₁₀₉	—	2	1	no	—
96	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
97	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
98	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
101	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
102	C₂²⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
103	C₂²⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
104	C₄²×C₁₀₉	—	2	1	no	—
105	C₄²×C₁₀₉	—	2	1	no	—
106	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
107	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₂³×C₁₀₉	—	2	1	no	—
109	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₁₀₉	—	2	1	no	—
111	Q₈×C₁₀₉	#41	1	2	yes	—
112	D₄×C₁₀₉	—	2	4	no	—
113	D₄×C₁₀₉	—	2	1	no	—
114	D₄×C₁₀₉	#41	1	2	yes	—
115	C₂×C₄×C₁₀₉	—	2	4	no	—
116	C₂×C₄×C₁₀₉	—	2	1	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	—	8	1	no	—
122	C₂×C₄×C₁₀₉	—	8	1	no	—
123	C₂×C₄×C₁₀₉	—	8	1	no	—
124	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
130	C₂×Dic₁₀₉	—	2	1	no	—
131	C₂×Dic₁₀₉	—	2	1	no	—
132	C₂×Dic₁₀₉	—	2	1	no	—
133	C₂×Dic₁₀₉	—	2	1	no	—
134	Dic₂₁₈	—	2	1	no	—
135	Dic₂₁₈	—	2	1	no	—
136	Dic₂₁₈	—	2	1	no	—
137	Dic₂₁₈	—	2	1	no	—
138	#11724784	—	2	1	no	—
139	#11724784	—	2	1	no	—
140	#11724784	—	2	1	no	—
141	#11724784	—	2	1	no	—
142	C₂²×C₁₀₉	—	2	1	no	—
143	C₂²×C₁₀₉	—	2	2	no	—
144	C₂²×C₁₀₉	—	2	1	no	—
145	C₂²×C₁₀₉	—	2	1	no	—
146	C₂²×C₁₀₉	—	4	1	no	—
147	C₂²×C₁₀₉	—	4	1	no	—
148	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
149	C₄×C₁₀₉	—	2	1	no	—
150	C₄×C₁₀₉	—	2	2	no	—
151	C₄×C₁₀₉	—	2	1	no	—
152	C₄×C₁₀₉	—	2	1	no	—
153	C₄×C₁₀₉	—	2	1	no	—
154	C₄×C₁₀₉	—	2	1	no	—
155	C₄×C₁₀₉	—	4	1	no	—
156	C₄×C₁₀₉	—	4	1	no	—
157	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
158	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
159	Dic₁₀₉	—	4	1	no	—
160	Dic₁₀₉	—	4	1	no	—
161	Dic₁₀₉	—	4	1	no	—
162	Dic₁₀₉	—	4	1	no	—
163	C₂×C₁₀₉	—	4	1	no	—
164	C₂×C₁₀₉	—	4	2	no	—
165	C₂×C₁₀₉	#525	1	1	yes	characteristic
166	C₂×C₁₀₉	#577	1	1	yes	characteristic
167	C₂×C₁₀₉	#578	1	1	yes	characteristic
168	#3120	—	109	1	no	maximal, Sylow
169	C₁₀₉	#3120	1	1	yes	characteristic, complemented, Sylow
170	#541	—	109	1	no	—
171	#520	D₁₀₉	1	1	yes	characteristic
172	#484	—	109	1	no	—
173	#483	—	109	1	no	—
174	#479	—	109	1	no	—
175	#478	—	109	1	no	—
176	#462	—	109	1	no	—
177	#461	—	109	1	no	—
178	#452	—	109	1	no	—
179	#452	—	109	1	no	—
180	#441	—	109	1	no	—
181	#440	—	109	1	no	—
182	C₄×Q₁₆	—	109	1	no	—
183	C₄×SD₁₆	—	109	1	no	—
184	#433	—	109	1	no	—
185	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
186	C₂×Q₁₆	—	109	1	no	—
187	C₂×Q₁₆	—	109	1	no	—
188	C₂×SD₁₆	—	109	1	no	—
189	C₂×SD₁₆	—	109	1	no	—
190	C₄⋊Q₈	—	109	1	no	—
191	#126	—	218	1	no	—
192	#126	—	218	1	no	—
193	#125	—	109	1	no	—
194	#125	—	218	1	no	—
195	#125	—	218	1	no	—
196	C₂²⋊Q₈	—	109	1	no	—
197	C₂²⋊Q₈	—	109	1	no	—
198	C₄×Q₈	—	109	1	no	—
199	C₄×Q₈	—	109	1	no	—
200	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
201	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
202	C₄×D₄	—	2	1	no	—
203	C₄×D₄	—	2	1	no	—
204	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
205	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
206	#117	—	2	1	no	—
207	#117	C₂×D₁₀₉	1	1	yes	characteristic
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
210	#107	—	109	1	no	—
211	#107	—	109	1	no	—
212	#106	—	109	1	no	—
213	#106	—	109	1	no	—
214	C₄⋊C₈	—	109	1	no	—
215	C₄⋊C₈	—	109	1	no	—
216	#103	—	109	1	no	—
217	#103	—	109	1	no	—
218	#103	—	109	1	no	—
219	#103	—	109	1	no	—
220	#103	—	109	1	no	—
221	#103	—	109	1	no	—
222	#102	—	109	1	no	—
223	#102	—	109	1	no	—
224	C₂²⋊C₈	—	109	1	no	—
225	C₂²⋊C₈	—	109	1	no	—
226	#97	—	109	1	no	—
227	C₄×C₈	—	109	1	no	—
228	#41	—	2	2	no	—
229	#41	—	2	2	no	—
230	C₂×Q₈	—	109	1	no	—
231	C₂×Q₈	—	109	1	no	—
232	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
240	Q₁₆	—	218	2	no	—
241	Q₁₆	—	218	2	no	—
242	SD₁₆	—	218	2	no	—
243	SD₁₆	—	218	2	no	—
244	C₂×C₈	—	109	1	no	—
245	C₂×C₈	—	109	1	no	—
246	C₂×C₈	—	109	1	no	—
247	C₂×C₈	—	109	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	109	1	no	—
252	C₄⋊C₄	—	109	1	no	—
253	C₄⋊C₄	—	218	1	no	—
254	C₄⋊C₄	—	218	1	no	—
255	C₄⋊C₄	—	218	1	no	—
256	C₄⋊C₄	—	218	1	no	—
257	C₄⋊C₄	—	218	1	no	—
258	C₄⋊C₄	—	218	1	no	—
259	C₄⋊C₄	—	218	1	no	—
260	C₄⋊C₄	—	218	1	no	—
261	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
262	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	218	1	no	—
268	C₂²⋊C₄	—	218	1	no	—
269	C₂²⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
270	C₂²⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
271	C₄²	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	218	1	no	—
274	C₄²	—	218	1	no	—
275	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
276	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
277	C₂³	—	2	1	no	—
278	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
279	Q₈	—	2	1	no	—
280	Q₈	—	218	1	no	—
281	Q₈	—	218	1	no	—
282	Q₈	—	218	1	no	—
283	Q₈	—	218	1	no	—
284	Q₈	#13182029	1	2	yes	—
285	D₄	—	2	4	no	—
286	D₄	—	2	1	no	—
287	D₄	#13182029	1	2	yes	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	8	1	no	—
295	C₂×C₄	—	8	1	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	—	218	1	no	—
298	C₂×C₄	—	218	1	no	—
299	C₂×C₄	—	218	1	no	—
300	C₂×C₄	—	218	1	no	—
301	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
302	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
303	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
304	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
305	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
307	C₈	—	218	1	no	—
308	C₈	—	218	1	no	—
309	C₈	—	218	1	no	—
310	C₈	—	218	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	436	1	no	—
327	C₄	—	436	1	no	—
328	C₄	—	436	1	no	—
329	C₄	—	436	1	no	—
330	C₄	C₂×#13182029	1	1	yes	characteristic
331	C₄	C₂×#13182029	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	4	2	no	—
334	C₂	#20295537	1	1	yes	characteristic
335	C₂	#20295837	1	1	yes	characteristic
336	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	#520×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#41×C₁₀₉	C₂²
hypercenter	#520	D₁₀₉
hypocenter	C₁₀₉	#3120
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
1. #20295638	C₂	semidirect
2. #20295361	C₂	semidirect
3. #20295356	C₂	semidirect
4. #20295333	C₂	semidirect
5. #20295326	C₂	semidirect
6. #20295236	C₂	semidirect
7. #20295231	C₂	semidirect
10. #20295131	C₂	semidirect
11. #20295126	C₂	semidirect
12. C₄×#13182019	C₂	semidirect
13. C₄×#13182017	C₂	semidirect
14. #20295087	C₂	semidirect
169. C₁₀₉	#3120	semidirect

Maximal subgroups

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

subgroup	index
#20295638	2
#20295361	2
#20295356	2
#20295333	2
#20295326	2
#20295236	2
#20295231	2
#20295188	2
#20295131	2
#20295126	2
C₄×#13182019	2
C₄×#13182017	2
#20295087	2
#520×C₁₀₉	2
#3120	109