Subgroups — #36370378

Order 22144.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012543	C₂	1	1	yes	maximal, characteristic, complemented
2	#26012540	C₂	1	1	yes	maximal, characteristic, complemented
3	#26012524	C₂	1	1	yes	maximal, characteristic, complemented
4	#26012522	C₂	1	1	yes	maximal, characteristic, complemented
5	#26012520	C₂	1	1	yes	maximal, characteristic, complemented
6	#26012518	C₂	1	1	yes	maximal, characteristic, complemented
7	#26012518	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×#13063443	C₂	1	1	yes	maximal, characteristic, complemented
9	#26012516	C₂	1	1	yes	maximal, characteristic, complemented
10	#26012515	C₂	1	1	yes	maximal, characteristic, complemented
11	#26012511	C₂	1	1	yes	maximal, characteristic, complemented
12	#26012456	C₂	1	1	yes	maximal, characteristic, complemented
13	#26012455	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₄×#13063443	C₂	1	1	yes	maximal, characteristic, complemented
15	#26012018	C₂	1	1	yes	maximal, characteristic
16	C₂×#14583836	C₂²	1	1	yes	characteristic
17	C₂×#14583835	C₂²	1	1	yes	characteristic
18	C₂×D₁₇₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂²×#13063443	C₂²	1	1	yes	characteristic
20	C₂²×#13063443	C₂²	1	1	yes	characteristic
21	C₂²×#13063443	C₂²	1	1	yes	characteristic
22	#19149746	C₂²	1	1	yes	characteristic
23	#19149746	C₂²	1	1	yes	characteristic
24	#19149746	C₂²	1	1	yes	characteristic
25	#19149746	C₂²	1	1	yes	characteristic
26	C₄⋊#13063443	C₂²	1	1	yes	characteristic
27	C₄⋊#13063443	C₂²	1	1	yes	characteristic
28	#19149736	C₂²	1	1	yes	characteristic
29	#19149736	C₂²	1	1	yes	characteristic
30	C₂²⋊#13063443	C₂²	1	1	yes	characteristic
31	C₂²⋊#13063443	C₂²	1	1	yes	characteristic
32	C₂²⋊#13063443	C₂²	1	1	yes	characteristic
33	C₂²⋊#13063443	C₂²	1	1	yes	characteristic
34	#19149734	C₂²	1	1	yes	characteristic
35	#19149734	C₂²	1	1	yes	characteristic
36	#19149734	C₂²	1	1	yes	characteristic
37	#19149734	C₂²	1	1	yes	characteristic
38	C₄×#13063443	C₂²	1	1	yes	characteristic
39	C₄×#13063443	C₂²	1	1	yes	characteristic
40	C₄×#13063443	C₂²	1	1	yes	characteristic
41	C₄×#13063443	C₂²	1	1	yes	characteristic
42	C₄×#13063443	C₂²	1	1	yes	characteristic
43	C₄×#13063443	C₂²	1	1	yes	characteristic
44	#19149674	C₄	1	1	yes	characteristic
45	#19149672	C₄	1	1	yes	characteristic
46	#19149669	C₄	1	1	yes	characteristic
47	#19149665	C₄	1	1	yes	characteristic
48	#19149662	C₄	1	1	yes	characteristic
49	#19149656	C₄	1	1	yes	characteristic
50	#19149651	C₄	1	1	yes	characteristic
51	#19149644	C₂²	1	1	yes	characteristic
52	D₄×Dic₁₇₃	C₂²	1	1	yes	characteristic
53	#19149639	C₂²	1	1	yes	characteristic
54	#19149639	C₂²	1	1	yes	characteristic
55	#19149637	C₂²	1	1	yes	characteristic
56	#19149635	C₂²	1	1	yes	characteristic
57	C₂×C₄×Dic₁₇₃	C₂²	1	1	yes	characteristic
58	#123×C₁₇₃	C₄	1	1	yes	characteristic
59	#14583836	—	2	2	no	—
60	#14583835	—	2	2	no	—
61	D₁₇₃⋊C₈	—	2	2	no	—
62	C₂×#13063443	—	2	2	no	—
63	C₂×#13063443	—	2	2	no	—
64	C₂×#13063443	—	2	2	no	—
65	C₂×#13063443	C₂³	1	1	yes	characteristic
66	C₂×#13063443	C₂³	1	1	yes	characteristic
67	C₂×#13063443	C₂³	1	1	yes	characteristic
68	C₂×#13063443	C₂³	1	1	yes	characteristic
69	C₂×#13063443	C₂³	1	1	yes	characteristic
70	C₂×#13063443	C₂³	1	1	yes	characteristic
71	C₂×#13063443	C₂³	1	1	yes	characteristic
72	C₂×#13063443	C₂³	1	1	yes	characteristic
73	C₂×Dic₃₄₆	C₂×C₄	1	1	yes	characteristic
74	C₂×C₂²⋊D₁₇₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₂²⋊D₁₇₃	C₂×C₄	1	1	yes	characteristic
76	C₂²×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
77	C₂²×Dic₁₇₃	C₂³	1	1	yes	characteristic
78	C₂²×Dic₁₇₃	C₂³	1	1	yes	characteristic
79	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
81	C₄⋊Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
82	#14583805	C₂×C₄	1	1	yes	characteristic
83	#14583805	C₂×C₄	1	1	yes	characteristic
84	#14583805	C₂×C₄	1	1	yes	characteristic
85	#14583805	C₂×C₄	1	1	yes	characteristic
86	#14583805	C₂×C₄	1	1	yes	characteristic
87	#14583805	C₂×C₄	1	1	yes	characteristic
88	C₂²⋊Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
89	C₂²⋊Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
90	C₂²⋊Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
91	#14583803	C₂×C₄	1	1	yes	characteristic
92	#14583803	C₂×C₄	1	1	yes	characteristic
93	#14583803	C₂×C₄	1	1	yes	characteristic
94	#14583803	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
96	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
97	C₄×Dic₁₇₃	C₂³	1	1	yes	characteristic
98	C₄×Dic₁₇₃	C₂³	1	1	yes	characteristic
99	C₄×Dic₁₇₃	C₂³	1	1	yes	characteristic
100	C₄×Dic₁₇₃	C₂³	1	1	yes	characteristic
101	C₂×D₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
103	C₄⋊C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
104	C₄⋊C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
105	C₂²⋊C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
106	C₂²⋊C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
108	#13063443	—	2	1	no	—
109	#13063443	—	2	1	no	—
110	#13063443	—	2	1	no	—
111	#13063443	—	2	1	no	—
112	#13063443	—	2	1	no	—
113	#13063443	—	2	1	no	—
114	#13063443	#41	1	2	yes	—
115	#13063443	#41	1	2	yes	—
116	C₂²×D₁₇₃	C₂²×C₄	1	1	yes	characteristic
117	Dic₃₄₆	—	2	2	no	—
118	C₂²⋊D₁₇₃	—	2	2	no	—
119	C₂²⋊D₁₇₃	—	2	2	no	—
120	C₂×Dic₁₇₃	—	2	2	no	—
121	C₂×Dic₁₇₃	—	2	1	no	—
122	C₂×Dic₁₇₃	—	2	1	no	—
123	C₂×Dic₁₇₃	—	2	1	no	—
124	C₂×Dic₁₇₃	—	2	1	no	—
125	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
126	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
130	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
131	C₂×Dic₁₇₃	C₂⁴	1	1	yes	characteristic
132	C₄×D₁₇₃	—	2	1	no	—
133	C₄×D₁₇₃	—	2	1	no	—
134	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
135	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
136	D₄×C₁₇₃	—	2	2	no	—
137	C₂×C₄×C₁₇₃	—	2	2	no	—
138	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
142	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₁₇₃	—	2	1	no	—
144	C₂²×C₁₇₃	—	2	1	no	—
145	C₂²×C₁₇₃	—	2	1	no	—
146	C₂²×C₁₇₃	—	2	1	no	—
147	C₂²×C₁₇₃	—	2	1	no	—
148	C₂²×C₁₇₃	#131	1	2	yes	—
149	C₂²×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
150	C₂×D₁₇₃	—	2	1	no	—
151	C₂×D₁₇₃	—	2	1	no	—
152	C₂×D₁₇₃	—	2	1	no	—
153	C₄×C₁₇₃	—	2	1	no	—
154	C₄×C₁₇₃	—	2	1	no	—
155	C₄×C₁₇₃	—	2	1	no	—
156	C₄×C₁₇₃	—	2	1	no	—
157	C₄×C₁₇₃	—	2	1	no	—
158	Dic₁₇₃	—	2	1	no	—
159	Dic₁₇₃	—	2	1	no	—
160	Dic₁₇₃	—	2	1	no	—
161	Dic₁₇₃	—	2	1	no	—
162	Dic₁₇₃	—	2	1	no	—
163	Dic₁₇₃	#131	1	2	yes	—
164	Dic₁₇₃	C₂×#41	1	1	yes	characteristic
165	Dic₁₇₃	C₂×#41	1	1	yes	characteristic
166	C₂×C₁₇₃	—	2	2	no	—
167	C₂×C₁₇₃	—	4	1	no	—
168	C₂×C₁₇₃	C₄×#41	1	1	yes	characteristic
169	C₂×C₁₇₃	C₂×#131	1	1	yes	characteristic
170	C₂×C₁₇₃	C₂×#131	1	1	yes	characteristic
171	D₁₇₃	—	4	1	no	—
172	C₁₇₃	#2966	1	1	yes	characteristic, complemented, Sylow
173	#2966	—	173	1	no	maximal, Sylow
174	#529	—	173	1	no	—
175	#446	—	173	1	no	—
176	C₈×Q₈	—	173	1	no	—
177	#436	—	173	1	no	—
178	#435	—	173	1	no	—
179	#435	—	173	1	no	—
180	C₈×D₄	—	173	1	no	—
181	C₈×D₄	—	173	1	no	—
182	C₈×D₄	—	173	1	no	—
183	#433	—	173	1	no	—
184	#433	—	173	1	no	—
185	#432	—	173	1	no	—
186	#405	—	173	1	no	—
187	#405	—	173	1	no	—
188	C₂×C₄×C₈	—	173	1	no	—
189	C₂×M₁₆	—	173	1	no	—
190	C₂×M₁₆	—	173	1	no	—
191	C₂²×C₈	—	173	1	no	—
192	C₂²×C₈	—	173	1	no	—
193	C₂²×C₈	—	173	1	no	—
194	C₂²×C₈	—	173	1	no	—
195	#126	—	173	1	no	—
196	#126	—	173	1	no	—
197	#125	—	173	1	no	—
198	#124	—	173	1	no	—
199	#123	—	173	1	no	—
200	#123	#10621924	1	1	yes	characteristic
201	C₂²⋊Q₈	—	173	1	no	—
202	#121	—	173	1	no	—
203	C₄×Q₈	—	173	1	no	—
204	C₄×D₄	—	173	1	no	—
205	C₄×D₄	—	173	1	no	—
206	C₄×D₄	—	173	1	no	—
207	#117	—	173	1	no	—
208	#117	—	173	1	no	—
209	C₂×C₄²	—	173	1	no	—
210	C₄⋊C₈	—	173	1	no	—
211	C₄⋊C₈	—	173	1	no	—
212	C₄⋊C₈	—	173	1	no	—
213	C₄⋊C₈	—	173	1	no	—
214	C₄⋊C₈	—	173	1	no	—
215	C₄⋊C₈	—	173	1	no	—
216	C₂²⋊C₈	—	173	1	no	—
217	C₂²⋊C₈	—	173	1	no	—
218	C₂²⋊C₈	—	173	1	no	—
219	C₂²⋊C₈	—	173	1	no	—
220	C₂²⋊C₈	—	173	1	no	—
221	C₂²⋊C₈	—	173	1	no	—
222	#97	—	173	1	no	—
223	#97	—	173	1	no	—
224	#97	—	173	1	no	—
225	#97	—	173	1	no	—
226	C₄×C₈	—	173	1	no	—
227	C₄×C₈	—	173	1	no	—
228	C₄×C₈	—	173	1	no	—
229	C₄×C₈	—	173	1	no	—
230	C₄×C₈	—	173	1	no	—
231	C₄×C₈	—	173	1	no	—
232	C₂×Q₈	—	173	1	no	—
233	C₂×D₄	—	173	1	no	—
234	C₂×D₄	—	173	1	no	—
235	C₂×D₄	C₂×#10621924	1	1	yes	characteristic
236	C₂²×C₄	—	173	1	no	—
237	C₂²×C₄	—	173	1	no	—
238	C₂²×C₄	—	173	1	no	—
239	C₂²×C₄	—	173	1	no	—
240	C₂²×C₄	C₂×#10621924	1	1	yes	characteristic
241	M₁₆	—	346	2	no	—
242	M₁₆	—	346	2	no	—
243	C₂×C₈	—	346	2	no	—
244	C₂×C₈	—	346	2	no	—
245	C₂×C₈	—	346	2	no	—
246	C₂×C₈	—	346	2	no	—
247	C₂×C₈	—	173	1	no	—
248	C₂×C₈	—	173	1	no	—
249	C₂×C₈	—	173	1	no	—
250	C₂×C₈	—	173	1	no	—
251	C₂×C₈	—	173	1	no	—
252	C₂×C₈	—	173	1	no	—
253	C₂×C₈	—	173	1	no	—
254	C₂×C₈	—	173	1	no	—
255	C₄⋊C₄	—	173	1	no	—
256	C₄⋊C₄	—	173	1	no	—
257	C₄⋊C₄	—	173	1	no	—
258	C₄⋊C₄	—	173	1	no	—
259	C₄⋊C₄	—	173	1	no	—
260	C₄⋊C₄	—	173	1	no	—
261	C₄⋊C₄	—	173	1	no	—
262	C₄⋊C₄	—	173	1	no	—
263	C₄⋊C₄	C₂×#10621924	1	1	yes	characteristic
264	C₄⋊C₄	C₂×#10621924	1	1	yes	characteristic
265	C₂²⋊C₄	—	173	1	no	—
266	C₂²⋊C₄	—	173	1	no	—
267	C₂²⋊C₄	—	173	1	no	—
268	C₂²⋊C₄	—	173	1	no	—
269	C₂²⋊C₄	—	173	1	no	—
270	C₂²⋊C₄	—	173	1	no	—
271	C₂²⋊C₄	—	173	1	no	—
272	C₂²⋊C₄	C₂×#10621924	1	1	yes	characteristic
273	C₂²⋊C₄	C₂×#10621924	1	1	yes	characteristic
274	C₂²⋊C₄	C₂×#10621924	1	1	yes	characteristic
275	C₄²	—	173	1	no	—
276	C₄²	—	173	1	no	—
277	C₄²	—	173	1	no	—
278	C₄²	—	173	1	no	—
279	C₄²	—	173	1	no	—
280	C₄²	—	173	1	no	—
281	C₂³	—	173	1	no	—
282	C₂³	C₂²×#10621924	1	1	yes	characteristic
283	C₂³	C₂²×#10621924	1	1	yes	characteristic
284	Q₈	—	346	2	no	—
285	D₄	—	2	2	no	—
286	D₄	—	346	2	no	—
287	D₄	—	346	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	346	1	no	—
290	C₂×C₄	—	346	1	no	—
291	C₂×C₄	—	346	1	no	—
292	C₂×C₄	—	346	1	no	—
293	C₂×C₄	—	346	1	no	—
294	C₂×C₄	—	346	1	no	—
295	C₂×C₄	—	346	2	no	—
296	C₂×C₄	—	173	1	no	—
297	C₂×C₄	—	173	1	no	—
298	C₂×C₄	—	173	1	no	—
299	C₂×C₄	—	173	1	no	—
300	C₂×C₄	—	173	1	no	—
301	C₂×C₄	—	173	1	no	—
302	C₂×C₄	—	173	1	no	—
303	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
304	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
305	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
306	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
307	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
308	C₈	—	346	1	no	—
309	C₈	—	346	1	no	—
310	C₈	—	346	1	no	—
311	C₈	—	346	1	no	—
312	C₈	—	346	1	no	—
313	C₈	—	346	1	no	—
314	C₈	—	173	2	no	—
315	C₈	—	173	2	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	346	1	no	—
322	C₂²	—	346	1	no	—
323	C₂²	—	346	1	no	—
324	C₂²	#19149766	1	2	yes	—
325	C₂²	C₂³×#10621924	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	346	1	no	—
332	C₄	—	346	1	no	—
333	C₄	—	346	1	no	—
334	C₄	—	346	1	no	—
335	C₄	—	346	1	no	—
336	C₄	—	173	1	no	—
337	C₄	—	173	1	no	—
338	C₄	—	173	2	no	—
339	C₂	—	2	2	no	—
340	C₂	—	4	1	no	—
341	C₂	—	692	1	no	—
342	C₂	#41×#10621924	1	1	yes	characteristic
343	C₂	C₂×#19149766	1	1	yes	characteristic
344	C₂	D₁₇₃⋊#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10621924
commutator, socle	C₂²×C₁₇₃	C₂³×C₄
Fitting	#123×C₁₇₃	C₄
prime radical	#19149651	C₄
cosocle	C₂×Dic₁₇₃	C₂⁴
hypercenter	#123	#10621924
hypocenter	C₁₇₃	#2966
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ #123

Complements

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

N	H ≅ G/N	type
1. #26012543	C₂	semidirect
2. #26012540	C₂	semidirect
3. #26012524	C₂	semidirect
4. #26012522	C₂	semidirect
5. #26012520	C₂	semidirect
6. #26012518	C₂	semidirect
7. #26012518	C₂	semidirect
8. D₄×#13063443	C₂	semidirect
9. #26012516	C₂	semidirect
10. #26012515	C₂	semidirect
11. #26012511	C₂	semidirect
12. #26012456	C₂	semidirect
13. #26012455	C₂	semidirect
14. C₂×C₄×#13063443	C₂	semidirect
172. C₁₇₃	#2966	semidirect

Maximal subgroups

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

subgroup	index
#26012543	2
#26012540	2
#26012524	2
#26012522	2
#26012520	2
#26012518	2
D₄×#13063443	2
#26012516	2
#26012515	2
#26012511	2
#26012456	2
#26012455	2
C₂×C₄×#13063443	2
#26012018	2
#2966	173