Subgroups — #33241508

Order 18688.

514 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620445	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988992	C₂	1	1	yes	maximal, characteristic
3	#22612217	C₂	1	4	yes	maximal, complemented
4	#22612213	C₂	1	2	yes	maximal, complemented
5	#22612206	C₂	1	2	yes	maximal, complemented
6	#22611719	C₂	1	4	yes	maximal
7	C₄×M₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic
8	#17989144	—	2	1	no	—
9	C₂×#14556591	—	2	1	no	—
10	#17988992	—	2	1	no	—
11	#17988992	—	2	1	no	—
12	#17988992	C₂²	1	2	yes	—
13	#17988992	C₂²	1	2	yes	—
14	#17988992	C₂²	1	1	yes	characteristic
15	#17988992	C₂²	1	1	yes	characteristic
16	#17988989	—	2	1	no	—
17	#17988989	C₂²	1	2	yes	—
18	#17988985	—	2	1	no	—
19	#17988625	—	2	1	no	—
20	#17988625	C₂²	1	2	yes	—
21	C₂×#14556525	—	2	1	no	—
22	C₂×#14556525	C₂²	1	2	yes	—
23	#17988415	C₂²	1	8	yes	—
24	#17988413	C₂²	1	8	yes	—
25	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
27	#346×C₇₃	C₂²	1	2	yes	—
28	C₄×C₁₆×C₇₃	C₂²	1	2	yes	—
29	C₂²×Dic₁₄₆	—	4	1	no	—
30	#14556591	—	2	2	no	—
31	#14556591	—	2	2	no	—
32	#14556591	—	2	1	no	—
33	#14556591	—	2	1	no	—
34	#14556591	—	4	1	no	—
35	#14556591	—	4	1	no	—
36	#14556583	—	2	2	no	—
37	#14556570	—	4	1	no	—
38	#14556570	—	4	1	no	—
39	C₄×Dic₁₄₆	—	4	1	no	—
40	C₄×Dic₁₄₆	—	4	1	no	—
41	#14556550	—	2	2	no	—
42	C₂×C₄⋊Dic₇₃	—	2	2	no	—
43	C₂×C₄⋊Dic₇₃	—	4	1	no	—
44	#14556525	—	2	2	no	—
45	#14556525	—	2	2	no	—
46	#14556525	—	2	2	no	—
47	#14556525	—	2	1	no	—
48	#14556525	—	2	1	no	—
49	#14556525	C₂³	1	4	yes	—
50	#14556525	C₂³	1	2	yes	—
51	#14556525	C₂³	1	2	yes	—
52	#14556523	—	2	2	no	—
53	#14556523	—	2	2	no	—
54	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
57	M₃₂×C₇₃	Q₈	1	8	yes	complemented
58	C₂×C₁₆×C₇₃	C₂³	1	4	yes	—
59	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
63	C₂×Dic₁₄₆	—	4	1	no	—
64	C₂×Dic₁₄₆	—	4	1	no	—
65	C₂×Dic₁₄₆	—	4	2	no	—
66	C₂×Dic₁₄₆	—	4	2	no	—
67	C₂×Dic₁₄₆	—	8	4	no	—
68	C₂²×Dic₇₃	—	4	2	no	—
69	C₄⋊Dic₇₃	—	2	2	no	—
70	C₄⋊Dic₇₃	—	2	2	no	—
71	C₄⋊Dic₇₃	—	2	4	no	—
72	C₄⋊Dic₇₃	—	4	2	no	—
73	C₄⋊Dic₇₃	—	4	2	no	—
74	C₄⋊Dic₇₃	—	4	2	no	—
75	C₄⋊Dic₇₃	—	4	1	no	—
76	C₄⋊Dic₇₃	—	4	1	no	—
77	#11924007	—	4	2	no	—
78	#11924007	—	4	2	no	—
79	C₂²⋊Dic₇₃	—	4	2	no	—
80	C₄×Dic₇₃	—	4	2	no	—
81	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₇₃	Q₁₆	1	4	yes	—
85	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
89	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
90	C₄²×C₇₃	D₈	1	1	yes	characteristic
91	C₄²×C₇₃	D₈	1	1	yes	characteristic
92	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
94	C₁₆×C₇₃	C₂×Q₈	1	8	yes	complemented
95	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₇₃	—	2	2	no	—
97	C₂×C₄×C₇₃	—	2	2	no	—
98	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
99	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
105	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×Dic₇₃	—	4	2	no	—
107	C₂×Dic₇₃	—	4	2	no	—
108	C₂×Dic₇₃	—	4	4	no	—
109	C₂×Dic₇₃	—	8	4	no	—
110	Dic₁₄₆	—	8	4	no	—
111	Dic₁₄₆	—	8	2	no	—
112	Dic₁₄₆	—	8	4	no	—
113	Dic₁₄₆	—	8	2	no	—
114	C₈×C₇₃	C₄⋊Q₈	1	2	yes	—
115	C₈×C₇₃	C₂×Q₁₆	1	2	yes	—
116	C₈×C₇₃	C₂×Q₁₆	1	2	yes	—
117	C₈×C₇₃	C₂²×Q₈	1	2	yes	—
118	C₂²×C₇₃	—	2	1	no	—
119	C₂²×C₇₃	—	2	1	no	—
120	C₂²×C₇₃	#500	1	2	yes	—
121	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
122	C₄×C₇₃	—	2	1	no	—
123	C₄×C₇₃	—	2	1	no	—
124	C₄×C₇₃	#500	1	2	yes	—
125	C₄×C₇₃	#509	1	2	yes	—
126	C₄×C₇₃	#510	1	2	yes	—
127	C₄×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
128	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
129	Dic₇₃	—	8	4	no	—
130	Dic₇₃	—	8	2	no	—
131	Dic₇₃	—	8	2	no	—
132	#33731	—	73	1	no	maximal, Sylow
133	C₂×C₇₃	—	2	2	no	—
134	C₂×C₇₃	C₂×#500	1	1	yes	characteristic
135	C₂×C₇₃	C₂×#509	1	1	yes	characteristic
136	C₂×C₇₃	C₂×#510	1	1	yes	characteristic
137	#3160	—	73	1	no	—
138	C₂×#500	—	73	1	no	—
139	#2255	—	73	4	no	—
140	#2254	—	73	2	no	—
141	#2252	—	73	2	no	—
142	#2159	—	73	4	no	—
143	C₄×M₃₂	D₇₃	1	1	yes	characteristic
144	C₇₃	#33731	1	1	yes	characteristic, complemented, Sylow
145	#533	—	146	1	no	—
146	C₂×C₄⋊Q₈	—	146	1	no	—
147	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
148	#500	—	73	2	no	—
149	#500	—	73	2	no	—
150	#500	—	73	1	no	—
151	#500	—	73	1	no	—
152	#500	—	146	1	no	—
153	#500	—	146	1	no	—
154	#499	—	73	2	no	—
155	#499	—	146	1	no	—
156	#498	—	146	1	no	—
157	#427	—	73	2	no	—
158	#427	—	146	1	no	—
159	C₂×#107	—	73	2	no	—
160	C₂×#107	—	146	1	no	—
161	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
162	#367	—	73	8	no	—
163	#366	—	73	8	no	—
164	#346	C₂×D₇₃	1	2	yes	—
165	C₄×C₁₆	C₂×D₇₃	1	2	yes	—
166	C₂²×Q₈	—	292	1	no	—
167	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
169	C₄⋊Q₈	—	292	1	no	—
170	C₄⋊Q₈	—	292	1	no	—
171	C₄⋊Q₈	—	146	2	no	—
172	C₄⋊Q₈	—	146	2	no	—
173	C₄⋊Q₈	—	146	1	no	—
174	C₄⋊Q₈	—	146	1	no	—
175	#125	—	146	2	no	—
176	C₂²⋊Q₈	—	292	1	no	—
177	C₂²⋊Q₈	—	292	1	no	—
178	C₄×Q₈	—	292	1	no	—
179	C₄×Q₈	—	292	1	no	—
180	#117	—	146	2	no	—
181	C₂×C₄⋊C₄	—	292	1	no	—
182	C₂×C₄⋊C₄	—	146	2	no	—
183	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
184	M₃₂	Dic₁₄₆	1	8	yes	complemented
185	C₂×C₁₆	C₂²×D₇₃	1	4	yes	—
186	#107	—	73	2	no	—
187	#107	—	73	4	no	—
188	#107	—	73	2	no	—
189	#107	—	146	2	no	—
190	#107	—	146	2	no	—
191	#107	—	146	2	no	—
192	#107	—	146	1	no	—
193	#107	—	146	1	no	—
194	#106	—	146	2	no	—
195	#106	—	146	2	no	—
196	C₄×C₈	D₂₉₂	1	1	yes	characteristic
197	C₄×C₈	D₂₉₂	1	1	yes	characteristic
198	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
199	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
200	C₂×Q₈	—	292	1	no	—
201	C₂×Q₈	—	292	1	no	—
202	C₂×Q₈	—	292	2	no	—
203	C₂×Q₈	—	292	2	no	—
204	C₂×Q₈	—	584	4	no	—
205	C₂²×C₄	—	292	2	no	—
206	C₂²×C₄	D₅₈₄	1	1	yes	characteristic
207	C₂²×C₄	D₅₈₄	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
209	C₂×C₈	Dic₂₉₂	1	4	yes	—
210	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
211	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
212	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
213	C₂×C₈	C₂×Dic₁₄₆	1	4	yes	—
214	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
215	C₄⋊C₄	—	292	2	no	—
216	C₄⋊C₄	—	292	2	no	—
217	C₄⋊C₄	—	292	2	no	—
218	C₄⋊C₄	—	292	2	no	—
219	C₄⋊C₄	—	292	2	no	—
220	C₄⋊C₄	—	292	1	no	—
221	C₄⋊C₄	—	292	1	no	—
222	C₄⋊C₄	—	146	2	no	—
223	C₄⋊C₄	—	146	2	no	—
224	C₄⋊C₄	—	146	4	no	—
225	C₂²⋊C₄	—	292	2	no	—
226	C₄²	—	292	2	no	—
227	C₄²	D₅₈₄	1	1	yes	characteristic
228	C₄²	D₅₈₄	1	1	yes	characteristic
229	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
230	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
231	C₁₆	C₂×Dic₁₄₆	1	8	yes	complemented
232	C₂³	C₂×D₅₈₄	1	1	yes	characteristic
233	Q₈	—	584	4	no	—
234	Q₈	—	584	4	no	—
235	Q₈	—	584	2	no	—
236	Q₈	—	584	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	292	2	no	—
240	C₂×C₄	—	292	2	no	—
241	C₂×C₄	—	292	4	no	—
242	C₂×C₄	—	584	4	no	—
243	C₂×C₄	#14556591	1	2	yes	—
244	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
245	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
246	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
247	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
248	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
249	C₂×C₄	C₂×Dic₂₉₂	1	2	yes	—
250	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
251	C₈	#14556591	1	2	yes	—
252	C₈	C₂×Dic₂₉₂	1	2	yes	—
253	C₈	C₂×Dic₂₉₂	1	2	yes	—
254	C₈	C₂²×Dic₁₄₆	1	2	yes	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	#17988992	1	2	yes	—
258	C₂²	C₂²×D₅₈₄	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	584	4	no	—
262	C₄	—	584	2	no	—
263	C₄	—	584	2	no	—
264	C₄	#17988992	1	2	yes	—
265	C₄	#17989025	1	2	yes	—
266	C₄	#17989030	1	2	yes	—
267	C₄	C₂×#14556591	1	1	yes	characteristic
268	C₄	C₂²×Dic₂₉₂	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	C₂×#17988992	1	1	yes	characteristic
271	C₂	C₂×#17989025	1	1	yes	characteristic
272	C₂	C₂×#17989030	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₈₄
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₄×M₃₂×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂³×C₇₃	C₂×D₈
hypercenter	C₄×M₃₂	D₇₃
hypocenter	C₇₃	#33731
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #22612217	C₂	semidirect
4. #22612213	C₂	semidirect
5. #22612206	C₂	semidirect
57. M₃₂×C₇₃	Q₈	semidirect
94. C₁₆×C₇₃	C₂×Q₈	semidirect
144. C₇₃	#33731	semidirect
184. M₃₂	Dic₁₄₆	semidirect
231. C₁₆	C₂×Dic₁₄₆	semidirect

Maximal subgroups

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

subgroup	index
#22620445	2
C₂×#17988992	2
#22612217	2
#22612213	2
#22612206	2
#22611719	2
C₄×M₃₂×C₇₃	2
#33731	73