Subgroups — #36364202

Order 22144.

362 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012112	C₂	1	2	yes	maximal, complemented
2	#26011896	C₂	1	1	yes	maximal, characteristic, complemented
3	#26011875	C₂	1	1	yes	maximal, characteristic, complemented
4	#26011851	C₂	1	2	yes	maximal, complemented
5	#26011737	C₂	1	2	yes	maximal, complemented
6	#26011732	C₂	1	1	yes	maximal, characteristic, complemented
7	#26011727	C₂	1	1	yes	maximal, characteristic
8	#26011593	C₂	1	2	yes	maximal, complemented
9	C₄×Dic₆₉₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₁₃₈₄	C₂	1	1	yes	maximal, characteristic, complemented
11	#405×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₆₉₂	C₂²	1	1	yes	characteristic
13	C₂×#14583815	C₂²	1	2	yes	—
14	C₂×D₁₃₈₄	C₂²	1	1	yes	characteristic
15	#19149679	C₂²	1	2	yes	—
16	#19149674	—	2	2	no	—
17	#19149667	C₂²	1	2	yes	—
18	#19149665	—	2	2	no	—
19	#19149661	—	2	2	no	—
20	#19149658	C₂²	1	2	yes	—
21	#19149657	—	2	2	no	—
22	#19149653	C₂²	1	2	yes	—
23	C₄×Dic₃₄₆	C₂²	1	2	yes	—
24	C₄×D₆₉₂	C₂²	1	2	yes	—
25	#19149613	C₂²	1	1	yes	characteristic
26	#19149613	C₂²	1	1	yes	characteristic
27	#19149613	C₂²	1	2	yes	—
28	#19149604	C₂²	1	2	yes	—
29	#19149604	C₂²	1	2	yes	—
30	#19149601	C₂²	1	2	yes	—
31	#19149601	C₂²	1	2	yes	—
32	C₂×M₁₆×C₁₇₃	C₂²	1	1	yes	characteristic
33	C₂²×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
34	#117×C₁₇₃	C₂²	1	1	yes	characteristic
35	#97×C₁₇₃	C₂²	1	1	yes	characteristic
36	#97×C₁₇₃	C₂²	1	1	yes	characteristic
37	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
39	C₂×Dic₃₄₆	—	2	2	no	—
40	C₂×Dic₃₄₆	C₂³	1	2	yes	—
41	C₂×D₆₉₂	C₂³	1	2	yes	—
42	C₂×C₂²⋊D₁₇₃	—	2	2	no	—
43	C₂×C₄×D₁₇₃	—	2	2	no	—
44	Dic₆₉₂	—	2	2	no	—
45	#14583815	—	2	4	no	—
46	D₁₃₈₄	—	2	2	no	—
47	C₄⋊Dic₁₇₃	—	2	2	no	—
48	C₄⋊Dic₁₇₃	C₂³	1	2	yes	—
49	C₄⋊Dic₁₇₃	C₂³	1	2	yes	—
50	#14583805	—	2	2	no	—
51	#14583805	—	2	2	no	—
52	C₂²⋊Dic₁₇₃	—	2	2	no	—
53	#14583803	—	2	2	no	—
54	#14583803	—	2	2	no	—
55	#14583803	—	2	2	no	—
56	#14583803	—	2	2	no	—
57	C₄×Dic₁₇₃	—	2	2	no	—
58	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
59	M₁₆×C₁₇₃	—	2	2	no	—
60	C₂×C₈×C₁₇₃	—	2	2	no	—
61	C₂×C₈×C₁₇₃	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₁₇₃	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₁₇₃	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₁₇₃	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₁₇₃	D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₁₇₃	D₄	1	1	yes	characteristic
67	C₂²⋊C₄×C₁₇₃	D₄	1	1	yes	characteristic
68	C₂²⋊C₄×C₁₇₃	D₄	1	1	yes	characteristic
69	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
70	C₄²×C₁₇₃	C₂³	1	1	yes	characteristic
71	C₂²×D₁₇₃	—	2	2	no	—
72	Dic₃₄₆	—	2	2	no	—
73	Dic₃₄₆	—	2	2	no	—
74	Dic₃₄₆	—	8	2	no	—
75	D₆₉₂	—	2	2	no	—
76	D₆₉₂	—	2	2	no	—
77	C₂²⋊D₁₇₃	—	8	2	no	—
78	C₂×Dic₁₇₃	—	2	2	no	—
79	C₂×Dic₁₇₃	—	2	2	no	—
80	C₂×Dic₁₇₃	—	2	2	no	—
81	C₄×D₁₇₃	—	8	2	no	—
82	C₂³×C₁₇₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₇₃	—	2	1	no	—
84	C₂×C₄×C₁₇₃	—	2	1	no	—
85	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₁₇₃	C₂⁴	1	1	yes	characteristic
91	C₈×C₁₇₃	—	2	1	no	—
92	C₈×C₁₇₃	—	2	1	no	—
93	C₈×C₁₇₃	#41	1	2	yes	—
94	C₈×C₁₇₃	#41	1	2	yes	—
95	C₂²×C₁₇₃	—	2	1	no	—
96	C₂²×C₁₇₃	—	2	1	no	—
97	C₂²×C₁₇₃	—	2	1	no	—
98	C₂²×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×D₁₇₃	—	4	2	no	—
100	C₂×D₁₇₃	—	4	2	no	—
101	C₂×D₁₇₃	—	4	2	no	—
102	C₄×C₁₇₃	—	2	1	no	—
103	C₄×C₁₇₃	—	2	1	no	—
104	C₄×C₁₇₃	—	2	1	no	—
105	C₄×C₁₇₃	—	2	1	no	—
106	C₄×C₁₇₃	—	2	1	no	—
107	C₄×C₁₇₃	C₂×#41	1	1	yes	characteristic
108	C₄×C₁₇₃	C₂×#41	1	1	yes	characteristic
109	Dic₁₇₃	—	4	2	no	—
110	Dic₁₇₃	—	4	2	no	—
111	Dic₁₇₃	—	4	2	no	—
112	C₂×C₁₇₃	—	4	1	no	—
113	C₂×C₁₇₃	#532	1	1	yes	characteristic
114	C₂×C₁₇₃	#576	1	1	yes	characteristic
115	C₂×C₁₇₃	#578	1	1	yes	characteristic
116	D₁₇₃	—	8	2	no	—
117	C₁₇₃	#3175	1	1	yes	characteristic, complemented, Sylow
118	#3175	—	173	1	no	maximal, Sylow
119	#542	—	173	2	no	—
120	#500	—	173	1	no	—
121	#495	—	173	1	no	—
122	#489	—	173	2	no	—
123	#468	—	173	2	no	—
124	#467	—	173	1	no	—
125	#466	—	173	1	no	—
126	#440	—	173	2	no	—
127	C₄×Q₁₆	—	173	1	no	—
128	C₄×D₈	—	173	1	no	—
129	#405	D₁₇₃	1	1	yes	characteristic, complemented
130	C₂×Q₁₆	—	173	1	no	—
131	C₂×SD₁₆	—	173	2	no	—
132	C₂×D₈	—	173	1	no	—
133	C₂×M₁₆	C₂×D₁₇₃	1	1	yes	characteristic
134	C₂²×C₈	C₂×D₁₇₃	1	1	yes	characteristic
135	C₄⋊Q₈	—	173	2	no	—
136	#126	—	346	2	no	—
137	#124	—	346	2	no	—
138	#124	—	173	2	no	—
139	#123	—	346	2	no	—
140	C₂²⋊Q₈	—	346	2	no	—
141	C₂²⋊Q₈	—	173	2	no	—
142	#121	—	173	2	no	—
143	C₄×Q₈	—	173	2	no	—
144	C₄×D₄	—	173	2	no	—
145	#117	C₂×D₁₇₃	1	1	yes	characteristic
146	#107	—	173	1	no	—
147	#107	—	173	1	no	—
148	#107	—	173	2	no	—
149	#103	—	173	2	no	—
150	#103	—	173	2	no	—
151	#102	—	173	2	no	—
152	#102	—	173	2	no	—
153	#97	C₂×D₁₇₃	1	1	yes	characteristic
154	#97	C₂×D₁₇₃	1	1	yes	characteristic
155	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
156	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
157	C₂×Q₈	—	346	2	no	—
158	C₂×Q₈	—	173	2	no	—
159	C₂×D₄	—	346	2	no	—
160	C₂×D₄	—	173	2	no	—
161	C₂²×C₄	—	346	2	no	—
162	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
163	Q₁₆	—	346	2	no	—
164	SD₁₆	—	346	4	no	—
165	D₈	—	346	2	no	—
166	M₁₆	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	C₂²×D₁₇₃	1	1	yes	characteristic
169	C₂×C₈	C₂²×D₁₇₃	1	1	yes	characteristic
170	C₂×C₈	C₂²×D₁₇₃	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₁₇₃	1	1	yes	characteristic
172	C₄⋊C₄	—	346	2	no	—
173	C₄⋊C₄	—	346	2	no	—
174	C₄⋊C₄	—	346	2	no	—
175	C₄⋊C₄	—	173	2	no	—
176	C₄⋊C₄	—	173	2	no	—
177	C₄⋊C₄	D₆₉₂	1	1	yes	characteristic
178	C₄⋊C₄	D₆₉₂	1	1	yes	characteristic
179	C₂²⋊C₄	—	346	2	no	—
180	C₂²⋊C₄	—	346	2	no	—
181	C₂²⋊C₄	—	346	2	no	—
182	C₂²⋊C₄	—	346	2	no	—
183	C₂²⋊C₄	—	346	2	no	—
184	C₂²⋊C₄	D₆₉₂	1	1	yes	characteristic
185	C₂²⋊C₄	D₆₉₂	1	1	yes	characteristic
186	C₄²	—	346	2	no	—
187	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
188	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
189	C₂³	—	346	2	no	—
190	C₂³	C₂×D₆₉₂	1	1	yes	characteristic
191	Q₈	—	346	2	no	—
192	Q₈	—	346	2	no	—
193	Q₈	—	1384	2	no	—
194	D₄	—	346	2	no	—
195	D₄	—	346	2	no	—
196	D₄	—	1384	2	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	346	2	no	—
200	C₂×C₄	—	346	2	no	—
201	C₂×C₄	—	346	2	no	—
202	C₂×C₄	—	1384	2	no	—
203	C₂×C₄	C₂×D₆₉₂	1	1	yes	characteristic
204	C₂×C₄	C₂×D₆₉₂	1	1	yes	characteristic
205	C₂×C₄	C₂×D₆₉₂	1	1	yes	characteristic
206	C₂×C₄	C₂×D₆₉₂	1	1	yes	characteristic
207	C₂×C₄	C₂×D₆₉₂	1	1	yes	characteristic
208	C₂×C₄	C₂³×D₁₇₃	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	#14583826	1	2	yes	—
212	C₈	#14583826	1	2	yes	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	—	692	2	no	—
217	C₂²	—	692	2	no	—
218	C₂²	—	692	2	no	—
219	C₂²	C₂²×D₆₉₂	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	692	2	no	—
226	C₄	—	692	2	no	—
227	C₄	—	692	2	no	—
228	C₄	C₂×#14583826	1	1	yes	characteristic
229	C₄	C₂×#14583826	1	1	yes	characteristic
230	C₂	—	4	1	no	—
231	C₂	—	1384	2	no	—
232	C₂	#26012041	1	1	yes	characteristic
233	C₂	#26012296	1	1	yes	characteristic
234	C₂	#26012308	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₆₉₂
commutator, cosocle	C₂×C₄×C₁₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇₃
Fitting	#405×C₁₇₃	C₂
socle	C₂²×C₁₇₃	C₂²×D₄
prime radical	#26011727	C₂
hypercenter	#405	D₁₇₃
hypocenter	C₁₇₃	#3175
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
1. #26012112	C₂	semidirect
2. #26011896	C₂	semidirect
3. #26011875	C₂	semidirect
4. #26011851	C₂	semidirect
5. #26011737	C₂	semidirect
6. #26011732	C₂	semidirect
8. #26011593	C₂	semidirect
9. C₄×Dic₆₉₂	C₂	semidirect
10. C₄×D₁₃₈₄	C₂	semidirect
11. #405×C₁₇₃	C₂	semidirect
117. C₁₇₃	#3175	semidirect
129. #405	D₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#26012112	2
#26011896	2
#26011875	2
#26011851	2
#26011737	2
#26011732	2
#26011727	2
#26011593	2
C₄×Dic₆₉₂	2
C₄×D₁₃₈₄	2
#405×C₁₇₃	2
#3175	173