Subgroups — #36370341

Order 22144.

452 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012571	C₂	1	1	yes	maximal, characteristic, complemented
2	#26012569	C₂	1	1	yes	maximal, characteristic, complemented
3	#26012536	C₂	1	4	yes	maximal, complemented
4	#26012534	C₂	1	4	yes	maximal, complemented
5	D₁₇₃⋊#108	C₂	1	1	yes	maximal, characteristic, complemented
6	#26012490	C₂	1	1	yes	maximal, characteristic, complemented
7	#26012489	C₂	1	1	yes	maximal, characteristic, complemented
8	#26012454	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#19149700	C₂	1	1	yes	maximal, characteristic
10	#19149767	—	2	2	no	—
11	#19149767	C₂²	1	4	yes	complemented
12	#19149766	—	2	2	no	—
13	#19149766	C₂²	1	4	yes	complemented
14	D₁₇₃⋊M₁₆	—	2	2	no	—
15	D₁₇₃⋊M₁₆	C₂²	1	1	yes	characteristic
16	D₁₇₃⋊M₁₆	C₂²	1	1	yes	characteristic
17	D₁₇₃⋊M₁₆	C₂²	1	1	yes	characteristic
18	C₂×#14583835	—	2	2	no	—
19	#19149758	C₂²	1	1	yes	characteristic
20	#19149750	C₂²	1	2	yes	complemented
21	#19149749	C₂²	1	2	yes	complemented
22	#19149744	C₂²	1	4	yes	complemented
23	#19149743	C₂²	1	4	yes	complemented
24	#19149733	C₂²	1	2	yes	complemented
25	C₈×#10621924	C₂²	1	2	yes	complemented
26	C₂×#14583827	C₂²	1	1	yes	characteristic
27	C₂×#14583825	C₂²	1	1	yes	characteristic
28	#19149700	C₄	1	2	yes	complemented
29	#19149700	C₄	1	2	yes	complemented
30	#19149700	C₂²	1	2	yes	—
31	#19149700	C₂²	1	2	yes	—
32	C₂×#14583816	C₄	1	1	yes	characteristic, complemented
33	C₂×#14583815	C₄	1	1	yes	characteristic, complemented
34	C₂×#14583811	C₄	1	1	yes	characteristic, complemented
35	C₂×C₈×D₁₇₃	C₂²	1	1	yes	characteristic
36	C₂×SD₁₆×C₁₇₃	C₄	1	1	yes	characteristic, complemented
37	#14583836	—	2	4	no	—
38	#14583836	D₄	1	2	yes	—
39	#14583836	C₂³	1	2	yes	—
40	#14583836	C₂³	1	2	yes	—
41	#14583835	—	2	4	no	—
42	#14583835	—	2	1	no	—
43	#14583835	—	2	1	no	—
44	#14583835	D₄	1	2	yes	—
45	D₁₇₃⋊C₈	—	2	4	no	—
46	D₁₇₃⋊C₈	—	2	1	no	—
47	D₁₇₃⋊C₈	—	2	1	no	—
48	D₁₇₃⋊C₈	C₂³	1	2	yes	—
49	C₂×#13063443	—	2	4	no	—
50	C₄⋊#10621924	—	2	1	no	—
51	C₂²⋊#10621924	—	2	1	no	—
52	C₄×#10621924	C₂³	1	2	yes	—
53	#14583827	—	2	1	no	—
54	#14583827	—	2	1	no	—
55	#14583827	C₂×C₄	1	2	yes	—
56	#14583827	C₂³	1	2	yes	—
57	#14583825	—	2	1	no	—
58	#14583825	—	2	1	no	—
59	#14583825	C₂×C₄	1	2	yes	—
60	#14583825	C₂³	1	2	yes	—
61	C₂×Dic₃₄₆	C₂×C₄	1	1	yes	characteristic
62	C₂×D₆₉₂	—	2	1	no	—
63	C₂×D₆₉₂	C₂×C₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₇₃	—	2	1	no	—
65	C₂²×Dic₁₇₃	—	2	1	no	—
66	C₂×C₄×D₁₇₃	—	2	1	no	—
67	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
68	#14583816	C₂×C₄	1	2	yes	—
69	#14583816	C₂×C₄	1	2	yes	—
70	#14583815	C₂×C₄	1	2	yes	—
71	#14583815	C₂×C₄	1	2	yes	—
72	#14583811	C₂×C₄	1	2	yes	—
73	#14583811	C₂×C₄	1	2	yes	—
74	C₂×#13063436	C₂×C₄	1	1	yes	characteristic
75	C₈×D₁₇₃	C₂×C₄	1	1	yes	characteristic
76	C₈×D₁₇₃	C₂×C₄	1	1	yes	characteristic
77	C₈×D₁₇₃	C₂³	1	1	yes	characteristic
78	C₈×D₁₇₃	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
80	C₂×D₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
81	SD₁₆×C₁₇₃	C₂×C₄	1	2	yes	—
82	SD₁₆×C₁₇₃	C₂×C₄	1	2	yes	—
83	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
84	C₂×#10621924	—	2	2	no	—
85	#13063443	—	2	2	no	—
86	#13063443	—	2	2	no	—
87	#13063443	C₂×D₄	1	4	yes	—
88	C₂²×D₁₇₃	—	2	1	no	—
89	C₂²×D₁₇₃	#41	1	1	yes	characteristic
90	Dic₃₄₆	—	2	1	no	—
91	Dic₃₄₆	C₂²×C₄	1	2	yes	—
92	D₆₉₂	—	2	2	no	—
93	D₆₉₂	—	2	2	no	—
94	D₆₉₂	—	2	1	no	—
95	D₆₉₂	C₂²×C₄	1	2	yes	—
96	C₂²⋊D₁₇₃	—	2	2	no	—
97	C₂²⋊D₁₇₃	—	2	2	no	—
98	C₂×Dic₁₇₃	—	2	1	no	—
99	C₂×Dic₁₇₃	—	2	2	no	—
100	C₂×Dic₁₇₃	—	2	2	no	—
101	C₂×Dic₁₇₃	C₂×D₄	1	1	yes	characteristic
102	C₄×D₁₇₃	—	2	2	no	—
103	C₄×D₁₇₃	—	2	2	no	—
104	C₄×D₁₇₃	C₂²×C₄	1	1	yes	characteristic
105	C₄×D₁₇₃	C₂×D₄	1	1	yes	characteristic
106	C₄×D₁₇₃	#41	1	1	yes	characteristic
107	C₄×D₁₇₃	C₂⁴	1	1	yes	characteristic
108	#13063436	C₂²×C₄	1	1	yes	characteristic
109	#13063436	C₂²×C₄	1	1	yes	characteristic
110	C₂³×C₁₇₃	—	2	1	no	—
111	Q₈×C₁₇₃	—	2	1	no	—
112	Q₈×C₁₇₃	C₂²×C₄	1	2	yes	—
113	D₄×C₁₇₃	—	2	1	no	—
114	D₄×C₁₇₃	C₂²×C₄	1	2	yes	—
115	C₂×C₄×C₁₇₃	—	2	1	no	—
116	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
117	C₈×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
118	C₈×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₁₇₃	—	2	2	no	—
120	C₂²×C₁₇₃	—	8	1	no	—
121	C₂²×C₁₇₃	C₄×D₄	1	1	yes	characteristic
122	C₂×D₁₇₃	—	2	2	no	—
123	C₂×D₁₇₃	—	4	1	no	—
124	C₂×D₁₇₃	—	8	1	no	—
125	C₂×D₁₇₃	C₄×D₄	1	1	yes	characteristic
126	C₂×D₁₇₃	C₂×#41	1	1	yes	characteristic
127	#10621924	—	4	2	no	—
128	C₄×C₁₇₃	—	2	2	no	—
129	C₄×C₁₇₃	C₄×D₄	1	1	yes	characteristic
130	C₄×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
131	Dic₁₇₃	—	2	2	no	—
132	Dic₁₇₃	C₄×D₄	1	1	yes	characteristic
133	Dic₁₇₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₁₇₃	—	2	1	no	—
135	C₂×C₁₇₃	—	4	2	no	—
136	C₂×C₁₇₃	C₂×C₄×D₄	1	1	yes	characteristic
137	D₁₇₃	—	2	1	no	—
138	D₁₇₃	—	2	1	no	—
139	D₁₇₃	—	4	2	no	—
140	C₁₇₃	#2957	1	1	yes	characteristic, complemented, Sylow
141	#2957	—	173	1	no	maximal, Sylow
142	C₂×#135	—	173	1	no	—
143	#568	—	173	1	no	—
144	#568	—	173	1	no	—
145	#444	—	173	4	no	—
146	#443	—	173	4	no	—
147	C₂×#108	—	173	1	no	—
148	#421	—	173	1	no	—
149	#421	—	173	1	no	—
150	#405	—	173	1	no	—
151	C₂×#41	—	173	1	no	—
152	C₂×#41	—	173	1	no	—
153	#135	—	173	2	no	—
154	#135	—	173	2	no	—
155	#135	—	173	2	no	—
156	#135	—	173	2	no	—
157	C₂×Q₁₆	—	173	1	no	—
158	C₂×SD₁₆	—	173	1	no	—
159	C₂×SD₁₆	#10621924	1	1	yes	characteristic, complemented
160	C₂×D₈	—	173	1	no	—
161	#131	—	346	2	no	—
162	#131	—	346	2	no	—
163	#131	—	173	4	no	—
164	#131	—	173	4	no	—
165	C₂×M₁₆	—	346	2	no	—
166	C₂×M₁₆	—	346	2	no	—
167	C₂×M₁₆	—	173	1	no	—
168	C₂×M₁₆	—	173	1	no	—
169	C₂×M₁₆	—	173	1	no	—
170	C₂²×C₈	—	173	1	no	—
171	#117	—	173	1	no	—
172	#108	—	173	2	no	—
173	#108	—	173	2	no	—
174	#104	—	173	4	no	—
175	#104	—	173	4	no	—
176	#97	—	173	2	no	—
177	C₄×C₈	—	173	2	no	—
178	#41	—	346	1	no	—
179	#41	—	346	1	no	—
180	#41	—	346	1	no	—
181	#41	—	346	1	no	—
182	#41	—	173	2	no	—
183	#41	—	173	2	no	—
184	#41	—	173	2	no	—
185	#41	—	173	2	no	—
186	C₂×Q₈	—	173	1	no	—
187	C₂×Q₈	C₂×#10621924	1	1	yes	characteristic
188	C₂×D₄	—	346	1	no	—
189	C₂×D₄	—	346	1	no	—
190	C₂×D₄	—	173	1	no	—
191	C₂×D₄	C₂×#10621924	1	1	yes	characteristic
192	C₂²×C₄	—	346	1	no	—
193	C₂²×C₄	—	346	1	no	—
194	C₂²×C₄	—	173	1	no	—
195	Q₁₆	—	173	2	no	—
196	Q₁₆	—	173	2	no	—
197	SD₁₆	—	173	2	no	—
198	SD₁₆	—	173	2	no	—
199	SD₁₆	C₂×#10621924	1	2	yes	—
200	SD₁₆	C₂×#10621924	1	2	yes	—
201	D₈	—	173	2	no	—
202	D₈	—	173	2	no	—
203	M₁₆	—	346	1	no	—
204	M₁₆	—	346	1	no	—
205	M₁₆	—	346	4	no	—
206	M₁₆	—	346	4	no	—
207	M₁₆	—	173	2	no	—
208	M₁₆	—	173	2	no	—
209	M₁₆	—	173	2	no	—
210	M₁₆	—	173	2	no	—
211	C₂×C₈	—	346	1	no	—
212	C₂×C₈	—	346	1	no	—
213	C₂×C₈	—	346	4	no	—
214	C₂×C₈	—	346	4	no	—
215	C₂×C₈	—	173	2	no	—
216	C₂×C₈	—	173	1	no	—
217	C₂×C₈	—	173	1	no	—
218	C₂×C₈	—	173	1	no	—
219	C₂×C₈	—	173	1	no	—
220	C₂×C₈	—	173	1	no	—
221	C₂×C₈	C₂×#10621924	1	1	yes	characteristic
222	C₄⋊C₄	—	346	1	no	—
223	C₂²⋊C₄	—	346	1	no	—
224	C₄²	—	173	2	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	346	1	no	—
227	C₂³	—	173	1	no	—
228	Q₈	—	2	1	no	—
229	Q₈	—	346	1	no	—
230	Q₈	—	173	2	no	—
231	Q₈	C₂²×#10621924	1	2	yes	—
232	D₄	—	2	1	no	—
233	D₄	—	346	2	no	—
234	D₄	—	346	2	no	—
235	D₄	—	346	2	no	—
236	D₄	—	346	2	no	—
237	D₄	—	346	1	no	—
238	D₄	—	173	2	no	—
239	D₄	C₂²×#10621924	1	2	yes	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	346	2	no	—
242	C₂×C₄	—	346	2	no	—
243	C₂×C₄	—	346	1	no	—
244	C₂×C₄	—	346	2	no	—
245	C₂×C₄	—	346	2	no	—
246	C₂×C₄	—	346	2	no	—
247	C₂×C₄	—	173	1	no	—
248	C₂×C₄	—	173	1	no	—
249	C₂×C₄	—	173	1	no	—
250	C₂×C₄	—	173	1	no	—
251	C₂×C₄	—	173	1	no	—
252	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
253	C₈	—	346	2	no	—
254	C₈	—	346	2	no	—
255	C₈	—	173	4	no	—
256	C₈	—	173	1	no	—
257	C₈	—	173	1	no	—
258	C₈	C₂²×#10621924	1	1	yes	characteristic
259	C₈	C₂²×#10621924	1	1	yes	characteristic
260	C₂²	—	2	2	no	—
261	C₂²	—	8	1	no	—
262	C₂²	—	346	2	no	—
263	C₂²	—	1384	1	no	—
264	C₂²	—	173	1	no	—
265	C₂²	—	173	1	no	—
266	C₂²	—	692	1	no	—
267	C₂²	D₄×#10621924	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	346	2	no	—
270	C₄	—	173	1	no	—
271	C₄	—	173	1	no	—
272	C₄	—	692	2	no	—
273	C₄	D₄×#10621924	1	1	yes	characteristic
274	C₄	C₂³×#10621924	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	4	2	no	—
277	C₂	—	346	1	no	—
278	C₂	—	346	1	no	—
279	C₂	—	692	2	no	—
280	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	C₂×SD₁₆×C₁₇₃	C₄
socle	C₂×C₁₇₃	C₂×C₄×D₄
prime radical	C₂×#19149700	C₂
cosocle	C₄×D₁₇₃	C₂⁴
hypercenter	C₂×SD₁₆	#10621924
hypocenter	C₁₇₃	#2957
LCS	G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #26012571	C₂	semidirect
2. #26012569	C₂	semidirect
3. #26012536	C₂	semidirect
4. #26012534	C₂	semidirect
5. D₁₇₃⋊#108	C₂	semidirect
6. #26012490	C₂	semidirect
7. #26012489	C₂	semidirect
8. #26012454	C₂	semidirect
11. #19149767	C₂²	semidirect
13. #19149766	C₂²	semidirect
20. #19149750	C₂²	semidirect
21. #19149749	C₂²	semidirect
22. #19149744	C₂²	semidirect
23. #19149743	C₂²	semidirect
24. #19149733	C₂²	semidirect
25. C₈×#10621924	C₂²	semidirect
28. #19149700	C₄	semidirect
29. #19149700	C₄	semidirect
32. C₂×#14583816	C₄	semidirect
33. C₂×#14583815	C₄	semidirect
34. C₂×#14583811	C₄	semidirect
36. C₂×SD₁₆×C₁₇₃	C₄	semidirect
140. C₁₇₃	#2957	semidirect
159. C₂×SD₁₆	#10621924	semidirect

Maximal subgroups

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

subgroup	index
#26012571	2
#26012569	2
#26012536	2
#26012534	2
D₁₇₃⋊#108	2
#26012490	2
#26012489	2
#26012454	2
C₂×#19149700	2
#2957	173