Subgroups — #36370321

Order 22144.

470 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012560	C₂	1	1	yes	maximal, characteristic
2	#26012559	C₂	1	1	yes	maximal, characteristic, complemented
3	#26012531	C₂	1	2	yes	maximal, complemented
4	#26012530	C₂	1	2	yes	maximal, complemented
5	Q₁₆×#10621924	C₂	1	2	yes	maximal, complemented
6	SD₁₆×#10621924	C₂	1	2	yes	maximal, complemented
7	#26012502	C₂	1	1	yes	maximal, characteristic, complemented
8	#26012481	C₂	1	1	yes	maximal, characteristic
9	C₂×#19149742	C₂	1	1	yes	maximal, characteristic, complemented
10	#26012457	C₂	1	1	yes	maximal, characteristic, complemented
11	#137×D₁₇₃	C₂	1	1	yes	maximal, characteristic
12	C₂×D₁₇₃⋊C₈	C₂²	1	1	yes	characteristic
13	Q₈×#10621924	—	2	1	no	—
14	Q₈×#10621924	—	2	1	no	—
15	Q₈×#10621924	C₂²	1	2	yes	—
16	Q₈×#10621924	C₂²	1	2	yes	—
17	Q₈×#10621924	C₂²	1	2	yes	—
18	D₄×#10621924	—	2	2	no	—
19	D₄×#10621924	C₂²	1	2	yes	—
20	#19149758	—	2	1	no	—
21	#19149758	—	2	2	no	—
22	#19149758	C₂²	1	1	yes	characteristic
23	#19149758	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊#10621924	—	2	1	no	—
25	C₂×C₄⋊#10621924	C₂²	1	1	yes	characteristic
26	#19149748	C₂²	1	2	yes	—
27	#19149747	C₂²	1	2	yes	—
28	#19149742	C₂²	1	2	yes	—
29	#19149742	C₂²	1	2	yes	—
30	#19149742	C₂²	1	1	yes	characteristic
31	#19149742	C₂²	1	1	yes	characteristic
32	#19149741	C₂²	1	1	yes	characteristic
33	#19149741	C₂²	1	1	yes	characteristic
34	#19149733	C₂²	1	2	yes	—
35	C₈×#10621924	C₂²	1	2	yes	—
36	#41×D₁₇₃	C₂²	1	1	yes	characteristic
37	C₂×Q₈×D₁₇₃	C₂²	1	1	yes	characteristic
38	#19149712	C₄	1	1	yes	characteristic, complemented
39	#19149711	C₄	1	2	yes	complemented
40	#19149710	C₄	1	1	yes	characteristic, complemented
41	#19149709	C₄	1	1	yes	characteristic, complemented
42	#19149708	C₄	1	2	yes	complemented
43	Q₁₆×D₁₇₃	C₂²	1	2	yes	—
44	SD₁₆×D₁₇₃	C₂²	1	2	yes	—
45	M₁₆×D₁₇₃	C₂²	1	1	yes	characteristic
46	#137×C₁₇₃	C₄	1	1	yes	characteristic, complemented
47	C₂²×#10621924	—	2	2	no	—
48	C₂²×#10621924	—	2	1	no	—
49	D₁₇₃⋊C₈	—	2	1	no	—
50	D₁₇₃⋊C₈	C₂³	1	1	yes	characteristic
51	D₁₇₃⋊C₈	C₂³	1	1	yes	characteristic
52	C₂×#13063443	—	2	1	no	—
53	C₄⋊#10621924	—	2	2	no	—
54	C₄⋊#10621924	—	2	2	no	—
55	C₄⋊#10621924	—	2	2	no	—
56	C₄⋊#10621924	—	2	1	no	—
57	C₄⋊#10621924	D₄	1	1	yes	characteristic
58	C₄⋊#10621924	D₄	1	1	yes	characteristic
59	C₄⋊#10621924	C₂³	1	1	yes	characteristic
60	C₄⋊#10621924	C₂³	1	1	yes	characteristic
61	C₄⋊#10621924	C₂³	1	2	yes	—
62	C₂²⋊#10621924	—	2	2	no	—
63	C₂²⋊#10621924	—	2	1	no	—
64	C₂²⋊#10621924	D₄	1	2	yes	—
65	C₄×#10621924	—	2	2	no	—
66	C₄×#10621924	—	2	2	no	—
67	C₄×#10621924	—	2	2	no	—
68	C₄×#10621924	—	2	1	no	—
69	C₄×#10621924	C₂³	1	2	yes	—
70	#14583827	C₂×C₄	1	1	yes	characteristic
71	#14583826	—	2	1	no	—
72	#14583826	C₂×C₄	1	1	yes	characteristic
73	#14583825	—	2	1	no	—
74	#14583825	C₂×C₄	1	1	yes	characteristic
75	C₂×Dic₃₄₆	—	2	1	no	—
76	C₂×Dic₃₄₆	C₂×C₄	1	1	yes	characteristic, complemented
77	Q₈×D₁₇₃	—	2	1	no	—
78	Q₈×D₁₇₃	—	2	1	no	—
79	Q₈×D₁₇₃	C₂×C₄	1	2	yes	complemented
80	Q₈×D₁₇₃	C₂³	1	2	yes	—
81	Q₈×D₁₇₃	C₂³	1	1	yes	characteristic
82	D₄×D₁₇₃	—	2	1	no	—
83	D₄×D₁₇₃	C₂³	1	1	yes	characteristic
84	C₂×C₄×D₁₇₃	—	2	1	no	—
85	C₂×C₄×D₁₇₃	—	2	1	no	—
86	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
87	Dic₆₉₂	C₂×C₄	1	2	yes	complemented
88	#14583816	C₂×C₄	1	2	yes	complemented
89	#14583816	C₂×C₄	1	2	yes	complemented
90	#14583815	C₂×C₄	1	2	yes	complemented
91	#14583814	C₂×C₄	1	2	yes	complemented
92	#14583813	C₂×C₄	1	2	yes	complemented
93	#14583810	C₂×C₄	1	1	yes	characteristic, complemented
94	#14583809	C₂×C₄	1	2	yes	complemented
95	C₈×D₁₇₃	C₂³	1	2	yes	—
96	#41×C₁₇₃	C₂×C₄	1	1	yes	characteristic
97	C₂×Q₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic, complemented
98	Q₁₆×C₁₇₃	C₂×C₄	1	2	yes	complemented
99	SD₁₆×C₁₇₃	C₂×C₄	1	2	yes	complemented
100	M₁₆×C₁₇₃	C₂×C₄	1	1	yes	characteristic, complemented
101	C₂×#10621924	—	2	1	no	—
102	C₂×#10621924	—	2	1	no	—
103	C₂×#10621924	—	2	2	no	—
104	C₂×#10621924	—	8	2	no	—
105	C₂×#10621924	—	8	1	no	—
106	C₂×#10621924	C₂×D₄	1	2	yes	—
107	C₂×#10621924	C₂×D₄	1	2	yes	—
108	#13063443	—	2	1	no	—
109	#13063443	—	2	1	no	—
110	C₂²×D₁₇₃	—	2	1	no	—
111	C₂²×D₁₇₃	C₂×D₄	1	1	yes	characteristic
112	Dic₃₄₆	—	2	1	no	—
113	Dic₃₄₆	—	2	2	no	—
114	Dic₃₄₆	—	2	2	no	—
115	Dic₃₄₆	—	2	1	no	—
116	Dic₃₄₆	C₂²×C₄	1	2	yes	—
117	Dic₃₄₆	C₂²×C₄	1	1	yes	characteristic
118	D₆₉₂	—	2	1	no	—
119	D₆₉₂	C₂²×C₄	1	1	yes	characteristic
120	C₂²⋊D₁₇₃	—	2	1	no	—
121	C₂²⋊D₁₇₃	—	2	1	no	—
122	C₂²⋊D₁₇₃	—	2	1	no	—
123	C₂×Dic₁₇₃	—	2	1	no	—
124	C₂×Dic₁₇₃	—	2	1	no	—
125	C₂×Dic₁₇₃	#41	1	1	yes	characteristic
126	C₄×D₁₇₃	—	2	2	no	—
127	C₄×D₁₇₃	—	2	1	no	—
128	C₄×D₁₇₃	—	2	2	no	—
129	C₄×D₁₇₃	—	2	1	no	—
130	C₄×D₁₇₃	—	2	1	no	—
131	C₄×D₁₇₃	C₂²×C₄	1	1	yes	characteristic
132	C₄×D₁₇₃	C₂×D₄	1	1	yes	characteristic
133	C₄×D₁₇₃	#41	1	1	yes	characteristic
134	C₄×D₁₇₃	C₂⁴	1	1	yes	characteristic
135	#13063436	C₂²×C₄	1	2	yes	—
136	Q₈×C₁₇₃	—	2	1	no	—
137	Q₈×C₁₇₃	C₂²×C₄	1	2	yes	—
138	Q₈×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
139	D₄×C₁₇₃	—	2	1	no	—
140	D₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₇₃	—	2	1	no	—
142	C₂×C₄×C₁₇₃	—	2	1	no	—
143	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
144	C₈×C₁₇₃	C₂²×C₄	1	2	yes	—
145	C₂²×C₁₇₃	—	2	1	no	—
146	C₂²×C₁₇₃	C₄×D₄	1	1	yes	characteristic
147	C₂×D₁₇₃	—	2	1	no	—
148	C₂×D₁₇₃	—	2	1	no	—
149	C₂×D₁₇₃	—	2	1	no	—
150	C₂×D₁₇₃	—	4	1	no	—
151	C₂×D₁₇₃	—	4	1	no	—
152	C₂×D₁₇₃	C₄×D₄	1	1	yes	characteristic
153	C₂×D₁₇₃	C₂²×D₄	1	1	yes	characteristic
154	#10621924	—	2	2	no	—
155	#10621924	—	2	2	no	—
156	#10621924	—	4	1	no	—
157	#10621924	—	4	2	no	—
158	#10621924	—	4	1	no	—
159	C₄×C₁₇₃	—	2	1	no	—
160	C₄×C₁₇₃	—	2	2	no	—
161	C₄×C₁₇₃	C₄×D₄	1	1	yes	characteristic
162	C₄×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
163	Dic₁₇₃	—	2	1	no	—
164	Dic₁₇₃	—	2	2	no	—
165	Dic₁₇₃	C₄×D₄	1	1	yes	characteristic
166	Dic₁₇₃	C₂×#41	1	1	yes	characteristic
167	C₂×C₁₇₃	—	2	1	no	—
168	C₂×C₁₇₃	—	4	1	no	—
169	C₂×C₁₇₃	C₂×C₄×D₄	1	1	yes	characteristic
170	D₁₇₃	—	2	1	no	—
171	D₁₇₃	—	4	1	no	—
172	D₁₇₃	#577	1	1	yes	characteristic
173	D₁₇₃	#578	1	1	yes	characteristic
174	C₁₇₃	#2947	1	1	yes	characteristic, complemented, Sylow
175	#2947	—	173	1	no	maximal, Sylow
176	C₂×#137	—	173	1	no	—
177	#520	—	173	1	no	—
178	#519	—	173	1	no	—
179	#441	—	173	2	no	—
180	#440	—	173	2	no	—
181	C₄×Q₁₆	—	173	2	no	—
182	C₄×SD₁₆	—	173	2	no	—
183	#428	—	173	1	no	—
184	#416	—	173	1	no	—
185	C₂×#103	—	173	1	no	—
186	#405	—	173	1	no	—
187	C₂×#41	—	173	1	no	—
188	C₂²×Q₈	—	173	1	no	—
189	#137	—	173	1	no	—
190	#137	—	173	1	no	—
191	#137	—	173	2	no	—
192	#137	—	173	2	no	—
193	#137	—	173	1	no	—
194	#137	#10621924	1	1	yes	characteristic, complemented
195	C₂×Q₁₆	—	173	2	no	—
196	C₂×SD₁₆	—	173	2	no	—
197	C₂×M₁₆	—	173	1	no	—
198	C₂²×C₈	—	173	1	no	—
199	C₄×Q₈	—	346	1	no	—
200	C₄×Q₈	—	346	1	no	—
201	C₄×Q₈	—	173	2	no	—
202	C₄×Q₈	—	173	2	no	—
203	C₄×Q₈	—	173	2	no	—
204	C₄×D₄	—	346	2	no	—
205	C₄×D₄	—	173	2	no	—
206	#117	—	346	1	no	—
207	#117	—	346	2	no	—
208	#117	—	173	1	no	—
209	#117	—	173	1	no	—
210	C₂×C₄⋊C₄	—	346	1	no	—
211	C₂×C₄⋊C₄	—	173	1	no	—
212	#107	—	173	2	no	—
213	#106	—	173	2	no	—
214	#103	—	173	2	no	—
215	#103	—	173	2	no	—
216	#103	—	173	1	no	—
217	#103	—	173	1	no	—
218	#102	—	173	1	no	—
219	#102	—	173	1	no	—
220	#97	—	173	2	no	—
221	C₄×C₈	—	173	2	no	—
222	#41	—	346	1	no	—
223	#41	—	346	1	no	—
224	#41	—	173	1	no	—
225	#41	—	173	1	no	—
226	#41	—	173	1	no	—
227	#41	C₂×#10621924	1	1	yes	characteristic
228	C₂×Q₈	—	346	1	no	—
229	C₂×Q₈	—	346	1	no	—
230	C₂×Q₈	—	346	1	no	—
231	C₂×Q₈	—	173	2	no	—
232	C₂×Q₈	—	173	2	no	—
233	C₂×Q₈	—	173	1	no	—
234	C₂×Q₈	—	173	1	no	—
235	C₂×Q₈	C₂×#10621924	1	1	yes	characteristic, complemented
236	C₂×D₄	—	346	1	no	—
237	C₂×D₄	—	173	1	no	—
238	C₂²×C₄	—	346	1	no	—
239	C₂²×C₄	—	346	1	no	—
240	C₂²×C₄	—	346	1	no	—
241	C₂²×C₄	—	346	2	no	—
242	C₂²×C₄	—	173	1	no	—
243	Q₁₆	—	173	2	no	—
244	Q₁₆	—	173	2	no	—
245	Q₁₆	—	173	2	no	—
246	Q₁₆	C₂×#10621924	1	2	yes	complemented
247	SD₁₆	—	173	2	no	—
248	SD₁₆	—	173	2	no	—
249	SD₁₆	—	173	2	no	—
250	SD₁₆	C₂×#10621924	1	2	yes	complemented
251	M₁₆	—	173	1	no	—
252	M₁₆	—	173	2	no	—
253	M₁₆	C₂×#10621924	1	1	yes	characteristic, complemented
254	C₂×C₈	—	346	1	no	—
255	C₂×C₈	—	346	1	no	—
256	C₂×C₈	—	173	1	no	—
257	C₂×C₈	—	173	1	no	—
258	C₂×C₈	—	173	2	no	—
259	C₄⋊C₄	—	346	1	no	—
260	C₄⋊C₄	—	346	2	no	—
261	C₄⋊C₄	—	346	2	no	—
262	C₄⋊C₄	—	346	2	no	—
263	C₄⋊C₄	—	173	1	no	—
264	C₄⋊C₄	—	173	1	no	—
265	C₄⋊C₄	—	173	2	no	—
266	C₄⋊C₄	—	173	1	no	—
267	C₄⋊C₄	—	173	1	no	—
268	C₂²⋊C₄	—	346	1	no	—
269	C₂²⋊C₄	—	346	2	no	—
270	C₂²⋊C₄	—	173	2	no	—
271	C₄²	—	346	2	no	—
272	C₄²	—	346	2	no	—
273	C₄²	—	346	2	no	—
274	C₄²	—	346	1	no	—
275	C₄²	—	173	2	no	—
276	C₂³	—	346	1	no	—
277	C₂³	—	173	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	346	2	no	—
280	Q₈	—	346	2	no	—
281	Q₈	—	346	1	no	—
282	Q₈	—	346	1	no	—
283	Q₈	—	173	2	no	—
284	Q₈	—	173	1	no	—
285	Q₈	C₂²×#10621924	1	2	yes	—
286	Q₈	C₂²×#10621924	1	1	yes	characteristic
287	D₄	—	2	1	no	—
288	D₄	—	346	1	no	—
289	D₄	—	346	1	no	—
290	D₄	—	346	1	no	—
291	D₄	—	346	1	no	—
292	D₄	—	173	1	no	—
293	D₄	C₂²×#10621924	1	1	yes	characteristic
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	346	1	no	—
297	C₂×C₄	—	346	2	no	—
298	C₂×C₄	—	346	2	no	—
299	C₂×C₄	—	346	1	no	—
300	C₂×C₄	—	346	1	no	—
301	C₂×C₄	—	346	1	no	—
302	C₂×C₄	—	346	1	no	—
303	C₂×C₄	—	346	1	no	—
304	C₂×C₄	—	346	1	no	—
305	C₂×C₄	—	346	2	no	—
306	C₂×C₄	—	1384	2	no	—
307	C₂×C₄	—	1384	1	no	—
308	C₂×C₄	—	173	2	no	—
309	C₂×C₄	—	173	2	no	—
310	C₂×C₄	—	173	1	no	—
311	C₂×C₄	—	173	1	no	—
312	C₂×C₄	—	173	1	no	—
313	C₂×C₄	—	173	1	no	—
314	C₂×C₄	—	173	1	no	—
315	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
316	C₈	—	346	1	no	—
317	C₈	—	346	1	no	—
318	C₈	—	173	2	no	—
319	C₈	C₂²×#10621924	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	346	1	no	—
322	C₂²	—	346	1	no	—
323	C₂²	—	346	1	no	—
324	C₂²	—	173	1	no	—
325	C₂²	—	173	1	no	—
326	C₂²	—	692	1	no	—
327	C₂²	—	692	1	no	—
328	C₂²	D₄×#10621924	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	2	2	no	—
331	C₄	—	346	2	no	—
332	C₄	—	346	2	no	—
333	C₄	—	346	1	no	—
334	C₄	—	346	2	no	—
335	C₄	—	173	1	no	—
336	C₄	—	173	1	no	—
337	C₄	—	692	2	no	—
338	C₄	—	692	1	no	—
339	C₄	—	692	1	no	—
340	C₄	D₄×#10621924	1	1	yes	characteristic
341	C₄	C₂³×#10621924	1	1	yes	characteristic
342	C₂	—	2	1	no	—
343	C₂	—	4	1	no	—
344	C₂	—	346	1	no	—
345	C₂	—	173	1	no	—
346	C₂	—	173	1	no	—
347	C₂	—	692	1	no	—
348	C₂	C₂×D₄×#10621924	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#10621924
commutator	C₄×C₁₇₃	C₂³×C₄
Frattini	C₄	C₂³×#10621924
Fitting	#137×C₁₇₃	C₄
socle	C₂×C₁₇₃	C₂×C₄×D₄
prime radical	#41×D₁₇₃	C₂²
cosocle	C₄×D₁₇₃	C₂⁴
hypercenter	#137	#10621924
hypocenter	C₁₇₃	#2947
LCS	G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
2. #26012559	C₂	semidirect
3. #26012531	C₂	semidirect
4. #26012530	C₂	semidirect
5. Q₁₆×#10621924	C₂	semidirect
6. SD₁₆×#10621924	C₂	semidirect
7. #26012502	C₂	semidirect
9. C₂×#19149742	C₂	semidirect
10. #26012457	C₂	semidirect
38. #19149712	C₄	semidirect
39. #19149711	C₄	semidirect
40. #19149710	C₄	semidirect
41. #19149709	C₄	semidirect
42. #19149708	C₄	semidirect
46. #137×C₁₇₃	C₄	semidirect
76. C₂×Dic₃₄₆	C₂×C₄	semidirect
79. Q₈×D₁₇₃	C₂×C₄	semidirect
87. Dic₆₉₂	C₂×C₄	semidirect
88. #14583816	C₂×C₄	semidirect
89. #14583816	C₂×C₄	semidirect
90. #14583815	C₂×C₄	semidirect
91. #14583814	C₂×C₄	semidirect
92. #14583813	C₂×C₄	semidirect
93. #14583810	C₂×C₄	semidirect
94. #14583809	C₂×C₄	semidirect
97. C₂×Q₈×C₁₇₃	C₂×C₄	semidirect
98. Q₁₆×C₁₇₃	C₂×C₄	semidirect
99. SD₁₆×C₁₇₃	C₂×C₄	semidirect
100. M₁₆×C₁₇₃	C₂×C₄	semidirect
174. C₁₇₃	#2947	semidirect
194. #137	#10621924	semidirect
235. C₂×Q₈	C₂×#10621924	semidirect
246. Q₁₆	C₂×#10621924	semidirect
250. SD₁₆	C₂×#10621924	semidirect
253. M₁₆	C₂×#10621924	semidirect

Maximal subgroups

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

subgroup	index
#26012560	2
#26012559	2
#26012531	2
#26012530	2
Q₁₆×#10621924	2
SD₁₆×#10621924	2
#26012502	2
#26012481	2
C₂×#19149742	2
#26012457	2
#137×D₁₇₃	2
#2947	173