Subgroups — #36362197

Order 22144.

392 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#117×D₁₇₃	C₂	1	1	yes	maximal, characteristic
2	#26011553	C₂	1	1	yes	maximal, characteristic, complemented
3	#26011552	C₂	1	1	yes	maximal, characteristic, complemented
4	#26011551	C₂	1	2	yes	maximal, complemented
5	#26011550	C₂	1	1	yes	maximal, characteristic, complemented
6	#26011549	C₂	1	2	yes	maximal, complemented
7	#26011513	C₂	1	2	yes	maximal, complemented
8	#26011453	C₂	1	2	yes	maximal
9	#26011443	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#14583809	C₂	1	1	yes	maximal, characteristic, complemented
11	#433×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₁₇₃	C₂²	1	1	yes	characteristic
13	C₂×#14583809	C₂²	1	2	yes	—
14	C₂×#14583809	C₂²	1	2	yes	—
15	#19149638	C₂²	1	1	yes	characteristic
16	#19149637	C₄	1	2	yes	—
17	#19149636	C₂²	1	1	yes	characteristic
18	#19149636	C₂²	1	1	yes	characteristic
19	#19149635	C₄	1	2	yes	—
20	C₄⋊C₄×D₁₇₃	C₄	1	2	yes	—
21	C₂²⋊C₄×D₁₇₃	C₄	1	2	yes	—
22	C₄²×D₁₇₃	C₂²	1	1	yes	characteristic
23	C₄²×D₁₇₃	C₂²	1	1	yes	characteristic
24	C₄⋊#13063436	C₂²	1	2	yes	complemented
25	#19149608	C₂²	1	2	yes	complemented
26	#19149608	C₂²	1	2	yes	complemented
27	C₂²⋊#13063436	C₂²	1	2	yes	—
28	#19149591	C₂²	1	2	yes	—
29	#19149591	C₂²	1	2	yes	—
30	#19149590	C₂²	1	1	yes	characteristic, complemented
31	#19149589	C₂²	1	2	yes	complemented
32	C₄×#13063436	C₂²	1	1	yes	characteristic, complemented
33	C₈×Dic₁₇₃	C₂²	1	2	yes	complemented
34	#117×C₁₇₃	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₁₇₃	C₂²	1	2	yes	complemented
36	C₂²⋊C₈×C₁₇₃	C₂²	1	2	yes	—
37	#97×C₁₇₃	C₂²	1	1	yes	characteristic, complemented
38	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic, complemented
39	C₂³×D₁₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₁₇₃	—	2	4	no	—
42	C₂×C₄×D₁₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₇₃	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₁₇₃	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
48	#14583809	—	2	4	no	—
49	#14583809	—	2	4	no	—
50	C₂×#13063436	C₂³	1	2	yes	—
51	C₂×#13063436	C₂³	1	2	yes	—
52	C₄⋊Dic₁₇₃	C₂×C₄	1	2	yes	—
53	#14583805	C₂×C₄	1	2	yes	—
54	#14583805	C₂×C₄	1	2	yes	—
55	C₂²⋊Dic₁₇₃	C₂×C₄	1	2	yes	—
56	#14583803	C₂×C₄	1	2	yes	—
57	#14583803	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₇₃	C₂×C₄	1	2	yes	—
59	C₄×Dic₁₇₃	C₂×C₄	1	2	yes	—
60	C₄×Dic₁₇₃	C₂³	1	1	yes	characteristic
61	C₄×Dic₁₇₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₁₇₃	C₂³	1	2	yes	—
64	C₂×C₈×C₁₇₃	C₂³	1	2	yes	—
65	C₄⋊C₄×C₁₇₃	C₂×C₄	1	2	yes	—
66	C₂²⋊C₄×C₁₇₃	C₂×C₄	1	2	yes	—
67	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
68	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
69	C₂²×D₁₇₃	—	2	2	no	—
70	C₂²×D₁₇₃	—	4	2	no	—
71	C₂²×D₁₇₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×D₁₇₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×Dic₁₇₃	—	2	2	no	—
74	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×Dic₁₇₃	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₇₃	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
78	C₄×D₁₇₃	—	2	2	no	—
79	C₄×D₁₇₃	—	2	2	no	—
80	C₄×D₁₇₃	—	2	2	no	—
81	C₄×D₁₇₃	—	4	2	no	—
82	C₄×D₁₇₃	—	4	2	no	—
83	C₄×D₁₇₃	#41	1	4	yes	—
84	#13063436	—	2	2	no	—
85	#13063436	—	2	2	no	—
86	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₇₃	—	2	2	no	—
88	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₇₃	C₂⁴	1	1	yes	characteristic
92	C₈×C₁₇₃	—	2	2	no	—
93	C₈×C₁₇₃	—	2	2	no	—
94	C₂²×C₁₇₃	—	2	1	no	—
95	C₂²×C₁₇₃	—	2	2	no	—
96	C₂²×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
97	C₂×D₁₇₃	—	2	2	no	—
98	C₂×D₁₇₃	—	2	1	no	—
99	C₂×D₁₇₃	—	2	2	no	—
100	C₂×D₁₇₃	—	4	4	no	—
101	C₂×D₁₇₃	#117	1	2	yes	—
102	C₄×C₁₇₃	—	2	1	no	—
103	C₄×C₁₇₃	—	2	2	no	—
104	C₄×C₁₇₃	—	2	2	no	—
105	C₄×C₁₇₃	C₂×#41	1	2	yes	—
106	Dic₁₇₃	—	2	1	no	—
107	Dic₁₇₃	—	2	2	no	—
108	Dic₁₇₃	—	2	2	no	—
109	Dic₁₇₃	#117	1	2	yes	—
110	C₂×C₁₇₃	—	4	1	no	—
111	C₂×C₁₇₃	C₂×#117	1	1	yes	characteristic
112	C₂×C₁₇₃	#568	1	2	yes	—
113	D₁₇₃	—	2	2	no	—
114	D₁₇₃	—	4	1	no	—
115	C₁₇₃	#2921	1	1	yes	characteristic, complemented, Sylow
116	#2921	—	173	1	no	maximal, Sylow
117	C₂×#117	—	173	1	no	—
118	#433	—	173	1	no	—
119	#433	—	173	2	no	—
120	#433	—	173	2	no	—
121	#433	—	173	1	no	—
122	#433	—	173	1	no	—
123	#433	D₁₇₃	1	1	yes	characteristic, complemented
124	#423	—	173	2	no	—
125	#407	—	173	2	no	—
126	C₄×M₁₆	—	173	1	no	—
127	C₄×M₁₆	—	173	1	no	—
128	C₂³×C₄	—	173	1	no	—
129	C₂×M₁₆	—	173	2	no	—
130	C₂×M₁₆	—	173	2	no	—
131	#117	—	173	1	no	—
132	#117	—	173	2	no	—
133	#117	—	173	2	no	—
134	#117	—	173	1	no	—
135	#117	—	173	1	no	—
136	#117	C₂×D₁₇₃	1	1	yes	characteristic
137	C₂×C₄⋊C₄	—	173	2	no	—
138	C₂×C₂²⋊C₄	—	173	2	no	—
139	C₂×C₄²	—	173	1	no	—
140	C₂×C₄²	—	173	1	no	—
141	C₄⋊C₈	—	173	2	no	—
142	C₄⋊C₈	—	173	2	no	—
143	C₄⋊C₈	—	173	2	no	—
144	C₄⋊C₈	C₂×D₁₇₃	1	2	yes	complemented
145	C₂²⋊C₈	—	173	2	no	—
146	C₂²⋊C₈	—	173	2	no	—
147	C₂²⋊C₈	—	173	2	no	—
148	C₂²⋊C₈	C₂×D₁₇₃	1	2	yes	—
149	#97	—	173	2	no	—
150	#97	—	173	1	no	—
151	#97	C₂×D₁₇₃	1	1	yes	characteristic, complemented
152	C₄×C₈	—	173	2	no	—
153	C₄×C₈	—	173	1	no	—
154	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic, complemented
155	C₂⁴	—	173	1	no	—
156	C₂²×C₄	—	346	4	no	—
157	C₂²×C₄	—	173	1	no	—
158	C₂²×C₄	—	173	1	no	—
159	C₂²×C₄	—	173	1	no	—
160	C₂²×C₄	—	173	1	no	—
161	C₂²×C₄	—	173	1	no	—
162	C₂²×C₄	—	173	2	no	—
163	C₂²×C₄	—	173	2	no	—
164	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
165	M₁₆	—	346	4	no	—
166	M₁₆	—	346	4	no	—
167	C₂×C₈	—	173	2	no	—
168	C₂×C₈	—	173	2	no	—
169	C₂×C₈	C₂²×D₁₇₃	1	2	yes	—
170	C₂×C₈	C₂²×D₁₇₃	1	2	yes	—
171	C₄⋊C₄	—	173	2	no	—
172	C₄⋊C₄	—	173	2	no	—
173	C₄⋊C₄	—	173	2	no	—
174	C₄⋊C₄	C₄×D₁₇₃	1	2	yes	—
175	C₂²⋊C₄	—	173	2	no	—
176	C₂²⋊C₄	—	173	2	no	—
177	C₂²⋊C₄	—	173	2	no	—
178	C₂²⋊C₄	C₄×D₁₇₃	1	2	yes	—
179	C₄²	—	173	2	no	—
180	C₄²	—	173	2	no	—
181	C₄²	—	173	1	no	—
182	C₄²	—	173	1	no	—
183	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
184	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
185	C₂³	—	346	2	no	—
186	C₂³	—	173	1	no	—
187	C₂³	—	173	1	no	—
188	C₂³	—	692	2	no	—
189	C₂³	C₂×C₄×D₁₇₃	1	1	yes	characteristic
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	346	2	no	—
192	C₂×C₄	—	346	2	no	—
193	C₂×C₄	—	346	2	no	—
194	C₂×C₄	—	346	2	no	—
195	C₂×C₄	—	173	2	no	—
196	C₂×C₄	—	173	2	no	—
197	C₂×C₄	—	173	1	no	—
198	C₂×C₄	—	173	1	no	—
199	C₂×C₄	—	173	4	no	—
200	C₂×C₄	—	692	2	no	—
201	C₂×C₄	—	692	2	no	—
202	C₂×C₄	C₂×C₄×D₁₇₃	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₁₇₃	1	2	yes	—
204	C₂×C₄	C₂×C₄×D₁₇₃	1	1	yes	characteristic
205	C₂×C₄	C₂³×D₁₇₃	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	2	2	no	—
208	C₈	—	346	2	no	—
209	C₈	—	346	2	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	346	2	no	—
213	C₂²	—	346	1	no	—
214	C₂²	—	346	2	no	—
215	C₂²	—	173	2	no	—
216	C₂²	—	692	4	no	—
217	C₂²	C₂²×C₄×D₁₇₃	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	346	1	no	—
222	C₄	—	346	2	no	—
223	C₄	—	346	2	no	—
224	C₄	—	173	2	no	—
225	C₄	#41×D₁₇₃	1	2	yes	—
226	C₂	—	4	1	no	—
227	C₂	—	346	2	no	—
228	C₂	—	692	1	no	—
229	C₂	#117×D₁₇₃	1	1	yes	characteristic
230	C₂	#26012256	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇₃
commutator, socle	C₂²×C₁₇₃	C₂³×C₄
Fitting	#433×C₁₇₃	C₂
prime radical	C₂³×D₁₇₃	C₂×C₄
cosocle	C₂×C₄×C₁₇₃	C₂⁴
hypercenter	#433	D₁₇₃
hypocenter	C₁₇₃	#2921
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #433

Complements

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

N	H ≅ G/N	type
2. #26011553	C₂	semidirect
3. #26011552	C₂	semidirect
4. #26011551	C₂	semidirect
5. #26011550	C₂	semidirect
6. #26011549	C₂	semidirect
7. #26011513	C₂	semidirect
9. #26011443	C₂	semidirect
10. C₄×#14583809	C₂	semidirect
11. #433×C₁₇₃	C₂	semidirect
24. C₄⋊#13063436	C₂²	semidirect
25. #19149608	C₂²	semidirect
26. #19149608	C₂²	semidirect
30. #19149590	C₂²	semidirect
31. #19149589	C₂²	semidirect
32. C₄×#13063436	C₂²	semidirect
33. C₈×Dic₁₇₃	C₂²	semidirect
35. C₄⋊C₈×C₁₇₃	C₂²	semidirect
37. #97×C₁₇₃	C₂²	semidirect
38. C₄×C₈×C₁₇₃	C₂²	semidirect
115. C₁₇₃	#2921	semidirect
123. #433	D₁₇₃	semidirect
144. C₄⋊C₈	C₂×D₁₇₃	semidirect
151. #97	C₂×D₁₇₃	semidirect
154. C₄×C₈	C₂×D₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#117×D₁₇₃	2
#26011553	2
#26011552	2
#26011551	2
#26011550	2
#26011549	2
#26011513	2
#26011453	2
#26011443	2
C₄×#14583809	2
#433×C₁₇₃	2
#2921	173