Subgroups — #36362209

Order 22144.

384 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14583826	C₂	1	1	yes	maximal, characteristic
2	#26011551	C₂	1	2	yes	maximal, complemented
3	#26011549	C₂	1	2	yes	maximal, complemented
4	#26011548	C₂	1	1	yes	maximal, characteristic, complemented
5	#26011546	C₂	1	1	yes	maximal, characteristic, complemented
6	#26011546	C₂	1	1	yes	maximal, characteristic, complemented
7	#26011516	C₂	1	2	yes	maximal, complemented
8	#26011456	C₂	1	2	yes	maximal
9	#26011443	C₂	1	1	yes	maximal, characteristic, complemented
10	#97×D₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#432×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14583826	C₂²	1	1	yes	characteristic
13	C₂×#14583809	C₂²	1	2	yes	—
14	C₂×C₈×D₁₇₃	C₂²	1	2	yes	—
15	C₄×Dic₃₄₆	C₄	1	2	yes	—
16	C₄×D₆₉₂	C₄	1	2	yes	—
17	C₄×C₂²⋊D₁₇₃	C₄	1	2	yes	—
18	C₄×C₂²⋊D₁₇₃	C₄	1	2	yes	—
19	#19149638	C₂²	1	1	yes	characteristic
20	#19149636	C₂²	1	1	yes	characteristic
21	#19149636	C₂²	1	1	yes	characteristic
22	C₄²×D₁₇₃	C₂²	1	1	yes	characteristic
23	C₄²×D₁₇₃	C₂²	1	1	yes	characteristic
24	C₄⋊#13063436	C₂²	1	2	yes	—
25	#19149608	C₂²	1	2	yes	—
26	#19149608	C₂²	1	2	yes	—
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
31	#19149590	C₂²	1	1	yes	characteristic
32	#19149589	C₂²	1	2	yes	—
33	C₈×Dic₁₇₃	C₂²	1	2	yes	—
34	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₁₇₃	C₂²	1	2	yes	—
36	C₂²⋊C₈×C₁₇₃	C₂²	1	2	yes	—
37	#97×C₁₇₃	C₂²	1	1	yes	characteristic
38	#97×C₁₇₃	C₂²	1	1	yes	characteristic
39	#14583826	—	2	4	no	—
40	C₂×Dic₃₄₆	C₂×C₄	1	1	yes	characteristic
41	C₂×D₆₉₂	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊D₁₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊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	C₂×#13063436	C₂³	1	2	yes	—
50	C₂×#13063436	C₂³	1	2	yes	—
51	C₈×D₁₇₃	—	2	4	no	—
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	C₂×C₄	1	2	yes	—
63	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₁₇₃	C₂³	1	2	yes	—
65	C₂×C₈×C₁₇₃	C₂³	1	2	yes	—
66	C₄²×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	C₂²×C₄	1	1	yes	characteristic
70	C₂²×D₁₇₃	C₂²×C₄	1	1	yes	characteristic
71	Dic₃₄₆	—	2	2	no	—
72	D₆₉₂	—	2	2	no	—
73	C₂²⋊D₁₇₃	—	2	2	no	—
74	C₂²⋊D₁₇₃	—	2	2	no	—
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	C₂×Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
79	C₄×D₁₇₃	—	2	2	no	—
80	C₄×D₁₇₃	—	4	2	no	—
81	C₄×D₁₇₃	—	4	2	no	—
82	C₄×D₁₇₃	#41	1	4	yes	—
83	#13063436	—	2	2	no	—
84	#13063436	—	2	2	no	—
85	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₇₃	—	2	2	no	—
87	C₂×C₄×C₁₇₃	—	4	2	no	—
88	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
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	#117	1	2	yes	—
101	C₄×C₁₇₃	—	2	1	no	—
102	C₄×C₁₇₃	—	2	2	no	—
103	C₄×C₁₇₃	—	2	2	no	—
104	C₄×C₁₇₃	C₂×#41	1	2	yes	—
105	Dic₁₇₃	—	2	1	no	—
106	Dic₁₇₃	—	2	2	no	—
107	Dic₁₇₃	—	2	2	no	—
108	Dic₁₇₃	#117	1	2	yes	—
109	C₂×C₁₇₃	—	4	1	no	—
110	C₂×C₁₇₃	C₂×#117	1	1	yes	characteristic
111	C₂×C₁₇₃	#568	1	2	yes	—
112	D₁₇₃	—	2	2	no	—
113	D₁₇₃	—	4	1	no	—
114	C₁₇₃	#2923	1	1	yes	characteristic, complemented, Sylow
115	#2923	—	173	1	no	maximal, Sylow
116	C₄×#41	—	173	1	no	—
117	#433	—	173	2	no	—
118	#433	—	173	2	no	—
119	#432	—	173	1	no	—
120	#432	—	173	1	no	—
121	#432	—	173	1	no	—
122	#432	D₁₇₃	1	1	yes	characteristic, complemented
123	#424	—	173	2	no	—
124	#408	—	173	2	no	—
125	C₄×M₁₆	—	173	1	no	—
126	C₂×#97	—	173	1	no	—
127	C₂×#41	—	173	1	no	—
128	C₂×M₁₆	—	173	2	no	—
129	C₂²×C₈	—	173	2	no	—
130	C₄×Q₈	—	173	2	no	—
131	C₄×D₄	—	173	2	no	—
132	C₄×D₄	—	173	2	no	—
133	C₄×D₄	—	173	2	no	—
134	#117	—	173	1	no	—
135	#117	—	173	1	no	—
136	#117	—	173	1	no	—
137	C₂×C₄²	—	173	1	no	—
138	C₂×C₄²	—	173	1	no	—
139	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
140	C₄⋊C₈	—	173	2	no	—
141	C₄⋊C₈	—	173	2	no	—
142	C₄⋊C₈	—	173	2	no	—
143	C₄⋊C₈	C₂×D₁₇₃	1	2	yes	—
144	C₂²⋊C₈	—	173	2	no	—
145	C₂²⋊C₈	—	173	2	no	—
146	C₂²⋊C₈	—	173	2	no	—
147	C₂²⋊C₈	C₂×D₁₇₃	1	2	yes	—
148	#97	—	173	2	no	—
149	#97	—	173	1	no	—
150	#97	—	173	1	no	—
151	#97	C₂×D₁₇₃	1	1	yes	characteristic
152	#97	C₂×D₁₇₃	1	1	yes	characteristic
153	C₄×C₈	—	173	2	no	—
154	#41	—	346	4	no	—
155	C₂×Q₈	—	173	1	no	—
156	C₂×D₄	—	173	1	no	—
157	C₂×D₄	—	173	1	no	—
158	C₂×D₄	—	173	1	no	—
159	C₂²×C₄	—	173	1	no	—
160	C₂²×C₄	—	173	1	no	—
161	C₂²×C₄	—	173	2	no	—
162	C₂²×C₄	—	173	2	no	—
163	C₂²×C₄	C₄×D₁₇₃	1	2	yes	—
164	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
165	M₁₆	—	346	4	no	—
166	C₂×C₈	—	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	C₂²⋊C₄	—	173	2	no	—
175	C₂²⋊C₄	—	173	2	no	—
176	C₂²⋊C₄	—	173	2	no	—
177	C₄²	—	173	2	no	—
178	C₄²	—	173	2	no	—
179	C₄²	—	173	1	no	—
180	C₄²	—	173	1	no	—
181	C₄²	C₄×D₁₇₃	1	2	yes	—
182	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
183	C₄²	C₂²×D₁₇₃	1	1	yes	characteristic
184	C₂³	—	173	1	no	—
185	C₂³	—	173	1	no	—
186	C₂³	C₂×C₄×D₁₇₃	1	1	yes	characteristic
187	Q₈	—	346	2	no	—
188	D₄	—	346	2	no	—
189	D₄	—	346	2	no	—
190	D₄	—	346	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	346	2	no	—
194	C₂×C₄	—	173	2	no	—
195	C₂×C₄	—	173	2	no	—
196	C₂×C₄	—	173	1	no	—
197	C₂×C₄	—	173	1	no	—
198	C₂×C₄	—	173	4	no	—
199	C₂×C₄	—	692	2	no	—
200	C₂×C₄	—	692	2	no	—
201	C₂×C₄	C₂×C₄×D₁₇₃	1	2	yes	—
202	C₂×C₄	C₂×C₄×D₁₇₃	1	1	yes	characteristic
203	C₂×C₄	C₂×C₄×D₁₇₃	1	2	yes	—
204	C₂×C₄	C₂³×D₁₇₃	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₈	—	346	2	no	—
208	C₈	—	346	2	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	346	2	no	—
212	C₂²	—	346	1	no	—
213	C₂²	—	346	2	no	—
214	C₂²	—	173	2	no	—
215	C₂²	C₂²×C₄×D₁₇₃	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	346	1	no	—
220	C₄	—	346	2	no	—
221	C₄	—	346	2	no	—
222	C₄	—	173	2	no	—
223	C₄	#41×D₁₇₃	1	2	yes	—
224	C₂	—	4	1	no	—
225	C₂	—	346	2	no	—
226	C₂	—	692	1	no	—
227	C₂	#117×D₁₇₃	1	1	yes	characteristic
228	C₂	#26012257	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇₃
commutator, socle	C₂²×C₁₇₃	C₂³×C₄
Fitting	#432×C₁₇₃	C₂
prime radical	C₂×#14583826	C₂²
cosocle	C₂×C₄×C₁₇₃	C₂⁴
hypercenter	#432	D₁₇₃
hypocenter	C₁₇₃	#2923
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #26011551	C₂	semidirect
3. #26011549	C₂	semidirect
4. #26011548	C₂	semidirect
5. #26011546	C₂	semidirect
6. #26011546	C₂	semidirect
7. #26011516	C₂	semidirect
9. #26011443	C₂	semidirect
10. #97×D₁₇₃	C₂	semidirect
11. #432×C₁₇₃	C₂	semidirect
114. C₁₇₃	#2923	semidirect
122. #432	D₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14583826	2
#26011551	2
#26011549	2
#26011548	2
#26011546	2
#26011516	2
#26011456	2
#26011443	2
#97×D₁₇₃	2
#432×C₁₇₃	2
#2923	173