Subgroups — #34012375

Order 18688.

490 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989486	C₂	1	2	yes	maximal
2	#22626137	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17989473	C₂	1	1	yes	maximal, characteristic, complemented
4	#22624985	C₂	1	2	yes	maximal, complemented
5	#22618353	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556665	C₂²	1	2	yes	—
7	C₂²×#11924031	C₂²	1	2	yes	—
8	#17989486	C₄	1	8	yes	complemented
9	#17989476	—	2	1	no	—
10	#17989473	—	2	1	no	—
11	#17989473	—	2	1	no	—
12	#17989473	—	2	1	no	—
13	C₂×#14556599	C₂²	1	1	yes	characteristic
14	C₂×#14556597	C₄	1	1	yes	characteristic
15	#17989055	C₄	1	1	yes	characteristic
16	#17988544	—	2	1	no	—
17	#17988543	C₂²	1	1	yes	characteristic
18	#17988541	—	2	1	no	—
19	#17988539	—	2	1	no	—
20	M₁₆×Dic₇₃	C₄	1	1	yes	characteristic
21	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
22	C₄×C₈×D₇₃	—	2	1	no	—
23	#405×C₇₃	C₄	1	1	yes	characteristic
24	#14556665	—	2	2	no	—
25	#14556665	C₂×C₄	1	4	yes	complemented
26	C₂×#11924031	—	2	2	no	—
27	C₂×#11924031	—	2	2	no	—
28	C₂×#11924031	—	2	2	no	—
29	C₂×#11924031	C₂×C₄	1	4	yes	complemented
30	C₂×#11924028	C₂×C₄	1	1	yes	characteristic
31	#14556599	—	2	2	no	—
32	#14556599	C₂×C₄	1	4	yes	—
33	#14556597	—	2	2	no	—
34	#14556597	C₈	1	4	yes	—
35	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
36	C₂×#11924011	—	2	1	no	—
37	C₂×#11924011	—	2	1	no	—
38	C₂²×#10596831	C₂³	1	1	yes	characteristic
39	C₂×C₈×D₇₃	—	2	1	no	—
40	C₂×C₈×D₇₃	—	2	1	no	—
41	#14556556	C₈	1	2	yes	—
42	#14556553	C₈	1	2	yes	—
43	#14556551	—	2	2	no	—
44	#14556548	—	2	1	no	—
45	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
46	C₄²×D₇₃	—	2	1	no	—
47	#14556502	—	2	1	no	—
48	#14556501	—	2	1	no	—
49	C₄×#10596831	—	2	1	no	—
50	C₈×Dic₇₃	—	2	1	no	—
51	C₈×Dic₇₃	D₄	1	1	yes	characteristic
52	C₈×Dic₇₃	D₄	1	1	yes	characteristic
53	C₈×Dic₇₃	D₄	1	1	yes	characteristic
54	C₈×Dic₇₃	D₄	1	1	yes	characteristic
55	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
57	#117×C₇₃	C₂×C₄	1	1	yes	characteristic
58	#97×C₇₃	—	2	1	no	—
59	C₄×C₈×C₇₃	—	2	1	no	—
60	#11924031	—	2	4	no	—
61	#11924031	—	2	4	no	—
62	#11924028	—	2	1	no	—
63	#11924028	—	2	1	no	—
64	#11924028	C₄²	1	2	yes	—
65	#11924028	C₂×C₈	1	2	yes	—
66	C₂×Dic₁₄₆	C₂×C₈	1	1	yes	characteristic
67	C₂×D₂₉₂	C₂×C₈	1	1	yes	characteristic
68	C₂×C₂²⋊D₇₃	—	2	1	no	—
69	C₂²×Dic₇₃	M₁₆	1	2	yes	—
70	C₂×C₄×D₇₃	—	2	1	no	—
71	C₂×C₄×D₇₃	—	2	2	no	—
72	#11924012	—	2	1	no	—
73	#11924012	C₂×C₈	1	2	yes	—
74	#11924011	—	2	4	no	—
75	#11924011	—	2	4	no	—
76	C₂×#10596831	—	2	1	no	—
77	C₂×#10596831	C₂²×C₄	1	2	yes	—
78	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
79	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
80	C₂×#10596831	#41	1	2	yes	—
81	C₈×D₇₃	—	2	4	no	—
82	C₈×D₇₃	—	2	4	no	—
83	#11924007	—	2	2	no	—
84	#11924005	—	2	2	no	—
85	C₄×Dic₇₃	—	2	2	no	—
86	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₄×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
89	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
90	M₁₆×C₇₃	—	2	1	no	—
91	M₁₆×C₇₃	C₄²	1	2	yes	—
92	C₂×C₈×C₇₃	—	2	1	no	—
93	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
96	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
97	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
98	C₂²⋊C₄×C₇₃	—	2	1	no	—
99	C₄²×C₇₃	—	2	1	no	—
100	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
101	C₂²×D₇₃	—	2	1	no	—
102	C₂×C₄×C₇₃	—	2	2	no	—
103	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
106	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
109	C₂²⋊D₇₃	—	2	2	no	—
110	C₂²⋊D₇₃	—	2	2	no	—
111	C₂×Dic₇₃	—	2	1	no	—
112	C₂×Dic₇₃	—	2	4	no	—
113	C₂×Dic₇₃	C₂²⋊C₈	1	2	yes	—
114	C₂×Dic₇₃	C₂²⋊C₈	1	2	yes	—
115	D₂₉₂	—	2	1	no	—
116	D₂₉₂	C₄×C₈	1	2	yes	—
117	C₄×D₇₃	—	2	2	no	—
118	C₄×D₇₃	—	2	2	no	—
119	C₄×D₇₃	—	4	4	no	—
120	Dic₁₄₆	—	2	1	no	—
121	Dic₁₄₆	C₄×C₈	1	2	yes	—
122	C₈×C₇₃	—	2	2	no	—
123	C₈×C₇₃	C₂²⋊C₈	1	2	yes	—
124	C₈×C₇₃	C₂²⋊C₈	1	2	yes	—
125	#10596831	—	2	2	no	—
126	#10596831	C₄×D₄	1	2	yes	—
127	#10596831	C₄×D₄	1	2	yes	—
128	C₂²×C₇₃	—	2	2	no	—
129	C₂²×C₇₃	C₄×M₁₆	1	1	yes	characteristic
130	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
131	C₂²×C₇₃	#431	1	1	yes	characteristic
132	C₂×D₇₃	—	2	2	no	—
133	C₂×D₇₃	—	4	2	no	—
134	C₄×C₇₃	—	4	2	no	—
135	C₄×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
136	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
137	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
138	C₄×C₇₃	#431	1	1	yes	characteristic
139	Dic₇₃	—	2	2	no	—
140	Dic₇₃	—	2	2	no	—
141	Dic₇₃	#350	1	4	yes	complemented
142	C₄×#350	—	73	1	no	maximal, Sylow
143	C₂×C₇₃	—	2	1	no	—
144	C₂×C₇₃	—	2	1	no	—
145	C₂×C₇₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
146	C₂×C₇₃	C₂×#350	1	2	yes	—
147	D₇₃	—	4	2	no	—
148	C₄×#131	—	73	1	no	—
149	C₂×#350	—	73	2	no	—
150	C₄×M₃₂	—	73	1	no	—
151	C₂×C₄×C₁₆	—	73	1	no	—
152	#1379	—	73	2	no	—
153	C₇₃	C₄×#350	1	1	yes	characteristic, complemented, Sylow
154	C₂×#131	—	73	1	no	—
155	C₂×#131	—	73	1	no	—
156	C₄×#41	—	73	1	no	—
157	C₂×M₃₂	—	73	2	no	—
158	C₂²×C₁₆	—	73	2	no	—
159	#405	—	73	1	no	—
160	#405	—	146	1	no	—
161	#405	—	146	1	no	—
162	#405	#64408	1	1	yes	characteristic
163	C₄×M₁₆	—	73	1	no	—
164	C₄×M₁₆	—	146	1	no	—
165	C₂×C₄×C₈	—	73	1	no	—
166	C₂×C₄×C₈	—	146	1	no	—
167	#350	—	73	8	no	—
168	#346	—	146	1	no	—
169	C₄×C₁₆	—	146	1	no	—
170	C₄×C₁₆	—	146	1	no	—
171	C₄×C₁₆	—	146	1	no	—
172	C₂×#41	—	73	1	no	—
173	#131	—	73	4	no	—
174	#131	—	73	4	no	—
175	#131	—	146	2	no	—
176	#131	—	146	2	no	—
177	C₂×M₁₆	—	73	1	no	—
178	C₂×M₁₆	—	146	1	no	—
179	C₂×M₁₆	—	146	1	no	—
180	C₂×M₁₆	C₂×#64408	1	1	yes	characteristic
181	C₂²×C₈	—	73	1	no	—
182	C₂²×C₈	—	146	1	no	—
183	C₂²×C₈	—	146	1	no	—
184	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
185	C₄×Q₈	—	73	2	no	—
186	C₄×D₄	—	73	2	no	—
187	C₄×D₄	—	146	2	no	—
188	#117	—	146	1	no	—
189	#117	C₂×#64408	1	1	yes	characteristic
190	C₂×C₄²	—	73	1	no	—
191	C₂×C₄²	—	146	1	no	—
192	M₃₂	—	73	4	no	—
193	M₃₂	—	146	2	no	—
194	C₂×C₁₆	—	73	4	no	—
195	C₂×C₁₆	—	146	2	no	—
196	C₂×C₁₆	—	146	2	no	—
197	C₂×C₁₆	—	146	2	no	—
198	#97	—	2	1	no	—
199	#97	—	146	1	no	—
200	#97	—	146	1	no	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	73	1	no	—
203	C₄×C₈	—	73	1	no	—
204	C₄×C₈	—	73	1	no	—
205	C₄×C₈	—	73	1	no	—
206	C₄×C₈	—	146	1	no	—
207	C₄×C₈	—	146	1	no	—
208	#41	—	73	2	no	—
209	#41	—	73	2	no	—
210	#41	—	146	1	no	—
211	#41	—	146	1	no	—
212	C₂×Q₈	—	73	1	no	—
213	C₂×D₄	—	73	1	no	—
214	C₂×D₄	—	146	1	no	—
215	C₂²×C₄	—	73	2	no	—
216	C₂²×C₄	—	146	2	no	—
217	C₂²×C₄	—	146	1	no	—
218	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
219	M₁₆	—	2	1	no	—
220	M₁₆	—	73	2	no	—
221	M₁₆	—	146	4	no	—
222	M₁₆	—	146	1	no	—
223	M₁₆	—	146	4	no	—
224	M₁₆	C₄×#64408	1	2	yes	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	73	2	no	—
227	C₂×C₈	—	73	1	no	—
228	C₂×C₈	—	73	1	no	—
229	C₂×C₈	—	73	2	no	—
230	C₂×C₈	—	146	4	no	—
231	C₂×C₈	—	146	1	no	—
232	C₂×C₈	—	146	4	no	—
233	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
234	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
235	C₂×C₈	#11924036	1	2	yes	—
236	C₂×C₈	#11924038	1	2	yes	—
237	C₄⋊C₄	—	146	2	no	—
238	C₄⋊C₄	#11924036	1	2	yes	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₂²⋊C₄	—	146	2	no	—
241	C₄²	—	2	1	no	—
242	C₄²	—	73	2	no	—
243	C₄²	—	73	1	no	—
244	C₄²	—	73	1	no	—
245	C₄²	—	146	2	no	—
246	C₁₆	—	146	4	no	—
247	C₁₆	—	146	4	no	—
248	C₂³	—	146	1	no	—
249	C₂³	D₇₃⋊M₁₆	1	1	yes	characteristic
250	Q₈	—	73	2	no	—
251	Q₈	—	146	1	no	—
252	D₄	—	73	2	no	—
253	D₄	—	146	2	no	—
254	D₄	—	146	2	no	—
255	D₄	—	146	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	292	4	no	—
258	C₂×C₄	—	73	2	no	—
259	C₂×C₄	—	73	2	no	—
260	C₂×C₄	—	146	4	no	—
261	C₂×C₄	—	146	1	no	—
262	C₂×C₄	—	146	2	no	—
263	C₂×C₄	—	146	2	no	—
264	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
266	C₂×C₄	#14556670	1	1	yes	characteristic
267	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
268	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
269	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	73	2	no	—
272	C₈	—	73	2	no	—
273	C₈	—	146	2	no	—
274	C₈	#14556648	1	2	yes	—
275	C₈	#14556648	1	2	yes	—
276	C₂²	—	2	2	no	—
277	C₂²	—	292	2	no	—
278	C₂²	—	146	2	no	—
279	C₂²	C₄×#11924038	1	1	yes	characteristic
280	C₂²	#17989555	1	1	yes	characteristic
281	C₂²	#17989605	1	1	yes	characteristic
282	C₄	—	4	2	no	—
283	C₄	—	73	4	no	—
284	C₄	—	146	2	no	—
285	C₄	—	146	2	no	—
286	C₄	C₄×C₂²⋊#64408	1	1	yes	characteristic
287	C₄	C₄×#11924036	1	1	yes	characteristic
288	C₄	C₂×#14556648	1	1	yes	characteristic
289	C₄	#17989606	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	292	2	no	—
293	C₂	C₄×#14556648	1	1	yes	characteristic
294	C₂	#22626171	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²⋊#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	#405×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₂²⋊C₈
prime radical	C₂×#11924028	C₂×C₄
cosocle	C₂²×#10596831	C₂³
hypercenter	#405	#64408
hypocenter	C₇₃	C₄×#350
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #22626137	C₂	semidirect
3. C₂×#17989473	C₂	semidirect
4. #22624985	C₂	semidirect
8. #17989486	C₄	semidirect
25. #14556665	C₂×C₄	semidirect
29. C₂×#11924031	C₂×C₄	semidirect
141. Dic₇₃	#350	semidirect
153. C₇₃	C₄×#350	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989486	2
#22626137	2
C₂×#17989473	2
#22624985	2
#22618353	2
C₄×#350	73