Subgroups — #34030493

Order 18688.

484 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626346	C₂	1	2	yes	maximal, complemented
2	#22626317	C₂	1	2	yes	maximal, complemented
3	#22626314	C₂	1	2	yes	maximal, complemented
4	#22626314	C₂	1	2	yes	maximal, complemented
5	#22626298	C₂	1	2	yes	maximal, complemented
6	#22626294	C₂	1	1	yes	maximal, characteristic, complemented
7	#22626267	C₂	1	2	yes	maximal, complemented
8	#22626264	C₂	1	1	yes	maximal, characteristic, complemented
9	#22618905	C₂	1	1	yes	maximal, characteristic
10	C₂×#14556666	C₂²	1	1	yes	characteristic
11	C₂×#14556666	C₂²	1	2	yes	—
12	C₂×#14556666	C₂²	1	1	yes	characteristic
13	C₂×#14556664	C₂²	1	2	yes	—
14	#17989508	C₂²	1	2	yes	—
15	#17989508	C₂²	1	1	yes	characteristic
16	#17989508	C₂²	1	1	yes	characteristic
17	C₄⋊#11924031	C₂²	1	2	yes	—
18	C₄⋊#11924031	C₂²	1	2	yes	—
19	C₄⋊#11924031	C₂²	1	2	yes	—
20	#17989482	C₂²	1	2	yes	—
21	#17989482	C₂²	1	2	yes	—
22	#17989482	C₂²	1	2	yes	—
23	#17989475	C₂²	1	2	yes	—
24	#17989475	C₂²	1	2	yes	—
25	C₄×#11924031	C₂²	1	2	yes	—
26	#17989124	C₄	1	1	yes	characteristic
27	Q₈×#10596831	C₂²	1	1	yes	characteristic
28	#17988666	C₄	1	1	yes	characteristic
29	#17988660	C₄	1	2	yes	—
30	#17988652	C₂²	1	1	yes	characteristic
31	#17988652	C₂²	1	2	yes	—
32	#17988647	C₄	1	2	yes	—
33	#17988643	C₄	1	1	yes	characteristic
34	#17988544	C₂²	1	2	yes	—
35	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
36	#446×C₇₃	C₄	1	1	yes	characteristic
37	#14556666	—	4	2	no	—
38	#14556666	—	4	1	no	—
39	#14556666	—	4	1	no	—
40	#14556664	—	4	2	no	—
41	C₂×#11924031	C₂³	1	2	yes	—
42	C₂×#11924031	C₂³	1	2	yes	—
43	C₂×#11924031	C₂³	1	2	yes	—
44	C₂×#11924031	C₂³	1	2	yes	—
45	C₂×#11924011	C₂×C₄	1	2	yes	—
46	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×D₇₃	C₂³	1	2	yes	—
48	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
49	#14556586	C₈	1	2	yes	—
50	#14556582	C₈	1	1	yes	characteristic
51	#14556580	C₈	1	1	yes	characteristic
52	#14556576	C₈	1	2	yes	—
53	#14556567	C₈	1	1	yes	characteristic
54	#14556566	C₈	1	1	yes	characteristic
55	C₄×D₂₉₂	C₂×C₄	1	2	yes	—
56	C₄×D₂₉₂	C₂×C₄	1	1	yes	characteristic
57	#14556548	C₂×C₄	1	2	yes	—
58	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
59	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
60	C₄⋊#10596831	C₂×C₄	1	2	yes	—
61	#14556503	C₂×C₄	1	2	yes	—
62	#14556503	C₂×C₄	1	2	yes	—
63	#14556503	C₂×C₄	1	2	yes	—
64	#14556501	C₂×C₄	1	2	yes	—
65	C₄×#10596831	C₂³	1	2	yes	—
66	C₄×#10596831	C₂³	1	1	yes	characteristic
67	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
69	C₄×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
70	C₄⋊C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
71	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
72	#97×C₇₃	C₂×C₄	1	2	yes	—
73	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
74	#11924031	—	2	2	no	—
75	#11924031	—	2	2	no	—
76	#11924031	—	2	2	no	—
77	#11924031	—	2	2	no	—
78	C₂×D₂₉₂	C₂×C₈	1	2	yes	—
79	C₂×D₂₉₂	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
81	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
84	#11924011	—	4	2	no	—
85	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
86	C₂×#10596831	C₂²×C₄	1	2	yes	—
87	C₂×#10596831	C₂⁴	1	1	yes	characteristic
88	C₈×D₇₃	—	2	2	no	—
89	C₈×D₇₃	—	2	1	no	—
90	C₈×D₇₃	—	2	2	no	—
91	C₈×D₇₃	—	2	1	no	—
92	C₈×D₇₃	—	4	1	no	—
93	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
94	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
95	#11924007	C₂×C₈	1	2	yes	—
96	#11924007	C₂×C₈	1	2	yes	—
97	#11924005	C₂×C₈	1	2	yes	—
98	#11924005	C₂×C₈	1	2	yes	—
99	#11924005	C₂×C₈	1	2	yes	—
100	#11924005	C₂×C₈	1	2	yes	—
101	#11924005	C₂×C₈	1	2	yes	—
102	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
103	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
104	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
106	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
109	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
110	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
111	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
113	C₂²×D₇₃	C₂²×C₈	1	2	yes	—
114	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
115	Q₈×C₇₃	—	4	1	no	—
116	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
120	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
123	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
124	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
125	D₂₉₂	—	4	1	no	—
126	D₂₉₂	—	4	2	no	—
127	C₄×D₇₃	—	2	2	no	—
128	C₄×D₇₃	—	2	1	no	—
129	C₄×D₇₃	—	2	1	no	—
130	C₄×D₇₃	—	2	2	no	—
131	C₄×D₇₃	—	4	1	no	—
132	C₄×D₇₃	—	4	1	no	—
133	C₈×C₇₃	—	2	1	no	—
134	C₈×C₇₃	—	2	2	no	—
135	C₈×C₇₃	—	2	1	no	—
136	#10596831	—	2	1	no	—
137	#10596831	—	2	2	no	—
138	#10596831	H(5,𝔽₂)	1	1	yes	characteristic
139	#10596831	2₋¹⁺⁴	1	1	yes	characteristic
140	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
141	C₂×D₇₃	—	2	1	no	—
142	C₂×D₇₃	—	2	1	no	—
143	C₂×D₇₃	—	2	2	no	—
144	C₂×D₇₃	—	2	2	no	—
145	C₂×D₇₃	—	2	1	no	—
146	C₂×D₇₃	—	2	2	no	—
147	C₄×C₇₃	—	2	1	no	—
148	C₄×C₇₃	—	2	1	no	—
149	C₄×C₇₃	—	2	2	no	—
150	C₄×C₇₃	—	2	2	no	—
151	C₄×C₇₃	#520	1	1	yes	characteristic
152	C₄×C₇₃	#568	1	1	yes	characteristic
153	Dic₇₃	—	2	1	no	—
154	Dic₇₃	—	2	1	no	—
155	Dic₇₃	—	2	1	no	—
156	Dic₇₃	—	2	2	no	—
157	#33509	—	73	1	no	maximal, Sylow
158	C₂×C₇₃	#2988	1	1	yes	characteristic
159	C₂×C₇₃	#3407	1	1	yes	characteristic
160	C₂×C₇₃	#3407	1	1	yes	characteristic
161	D₇₃	—	4	1	no	—
162	D₇₃	—	4	2	no	—
163	#2966	—	73	1	no	—
164	#2183	—	73	2	no	—
165	#2169	—	73	2	no	—
166	#2168	—	73	2	no	—
167	#2168	—	73	2	no	—
168	#2164	—	73	2	no	—
169	#2163	—	73	1	no	—
170	#2151	—	73	2	no	—
171	#2150	—	73	1	no	—
172	C₇₃	#33509	1	1	yes	characteristic, complemented, Sylow
173	#529	—	73	1	no	—
174	C₂×M₃₂	—	73	1	no	—
175	C₂×M₃₂	—	73	1	no	—
176	C₂×M₃₂	—	73	2	no	—
177	C₂²×C₁₆	—	73	2	no	—
178	#446	#64408	1	1	yes	characteristic
179	C₈×Q₈	—	73	1	no	—
180	#436	—	73	1	no	—
181	#435	—	73	2	no	—
182	C₈×D₄	—	73	1	no	—
183	C₈×D₄	—	73	2	no	—
184	#433	—	73	2	no	—
185	#432	—	73	1	no	—
186	#405	—	73	2	no	—
187	C₂×C₄×C₈	—	73	1	no	—
188	C₄⋊C₁₆	—	73	2	no	—
189	C₄⋊C₁₆	—	73	2	no	—
190	C₄⋊C₁₆	—	73	1	no	—
191	C₄⋊C₁₆	—	73	2	no	—
192	C₄⋊C₁₆	—	73	2	no	—
193	C₄⋊C₁₆	—	73	1	no	—
194	C₂²⋊C₁₆	—	73	2	no	—
195	C₂²⋊C₁₆	—	73	2	no	—
196	C₂²⋊C₁₆	—	73	2	no	—
197	#346	—	73	2	no	—
198	#346	—	73	2	no	—
199	C₄×C₁₆	—	73	2	no	—
200	C₂×M₁₆	—	73	2	no	—
201	C₂²×C₈	—	73	1	no	—
202	C₂²×C₈	—	73	1	no	—
203	C₂²×C₈	—	73	2	no	—
204	#126	—	73	2	no	—
205	#125	—	73	1	no	—
206	#124	—	73	1	no	—
207	#123	—	73	2	no	—
208	C₂²⋊Q₈	—	73	1	no	—
209	#121	—	73	1	no	—
210	C₄×Q₈	C₂×#64408	1	1	yes	characteristic
211	C₄×D₄	—	73	2	no	—
212	C₄×D₄	—	73	1	no	—
213	#117	—	73	2	no	—
214	C₂×C₄²	—	73	1	no	—
215	M₃₂	—	292	1	no	—
216	M₃₂	—	292	2	no	—
217	M₃₂	—	292	1	no	—
218	C₂×C₁₆	—	292	2	no	—
219	C₂×C₁₆	—	73	2	no	—
220	C₂×C₁₆	—	73	2	no	—
221	C₂×C₁₆	—	73	2	no	—
222	C₂×C₁₆	—	73	2	no	—
223	C₄⋊C₈	—	73	1	no	—
224	C₄⋊C₈	—	73	2	no	—
225	C₄⋊C₈	C₂×#64408	1	1	yes	characteristic
226	C₄⋊C₈	C₂×#64408	1	2	yes	—
227	C₂²⋊C₈	—	73	2	no	—
228	C₂²⋊C₈	—	73	2	no	—
229	C₂²⋊C₈	—	73	2	no	—
230	#97	—	73	2	no	—
231	#97	C₂×#64408	1	2	yes	—
232	C₄×C₈	—	73	1	no	—
233	C₄×C₈	—	73	1	no	—
234	C₄×C₈	—	73	2	no	—
235	C₄×C₈	—	73	1	no	—
236	C₄×C₈	C₂×#64408	1	1	yes	characteristic
237	C₂×Q₈	#11924036	1	1	yes	characteristic
238	C₂×D₄	—	73	2	no	—
239	C₂×D₄	—	73	1	no	—
240	C₂²×C₄	—	73	1	no	—
241	C₂²×C₄	—	73	1	no	—
242	C₂²×C₄	—	73	1	no	—
243	C₂²×C₄	—	73	2	no	—
244	M₁₆	—	292	2	no	—
245	C₂×C₈	—	292	1	no	—
246	C₂×C₈	—	73	1	no	—
247	C₂×C₈	—	73	1	no	—
248	C₂×C₈	—	73	2	no	—
249	C₂×C₈	—	146	2	no	—
250	C₂×C₈	—	146	1	no	—
251	C₂×C₈	—	146	1	no	—
252	C₂×C₈	—	146	2	no	—
253	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
254	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
255	C₂×C₈	C₂²×#64408	1	2	yes	—
256	C₄⋊C₄	—	73	2	no	—
257	C₄⋊C₄	—	73	2	no	—
258	C₄⋊C₄	—	73	1	no	—
259	C₄⋊C₄	—	73	2	no	—
260	C₄⋊C₄	#11924036	1	1	yes	characteristic
261	C₄⋊C₄	#11924036	1	2	yes	—
262	C₂²⋊C₄	—	73	2	no	—
263	C₂²⋊C₄	—	73	2	no	—
264	C₂²⋊C₄	—	73	2	no	—
265	C₂²⋊C₄	—	73	2	no	—
266	C₂²⋊C₄	—	73	2	no	—
267	C₄²	—	73	1	no	—
268	C₄²	—	73	1	no	—
269	C₄²	—	73	1	no	—
270	C₄²	C₂²×#64408	1	1	yes	characteristic
271	C₄²	C₂²×#64408	1	2	yes	—
272	C₁₆	—	146	2	no	—
273	C₁₆	—	146	2	no	—
274	C₁₆	—	146	2	no	—
275	C₁₆	—	146	2	no	—
276	C₂³	—	73	2	no	—
277	C₂³	—	73	1	no	—
278	Q₈	—	4	1	no	—
279	D₄	—	292	2	no	—
280	D₄	—	292	1	no	—
281	C₂×C₄	—	292	1	no	—
282	C₂×C₄	—	292	1	no	—
283	C₂×C₄	—	73	1	no	—
284	C₂×C₄	—	73	1	no	—
285	C₂×C₄	—	73	1	no	—
286	C₂×C₄	—	73	2	no	—
287	C₂×C₄	—	146	2	no	—
288	C₂×C₄	—	146	1	no	—
289	C₂×C₄	—	146	1	no	—
290	C₂×C₄	—	146	2	no	—
291	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
292	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
293	C₂×C₄	C₂×#11924036	1	2	yes	—
294	C₂×C₄	C₂×#11924036	1	2	yes	—
295	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
296	C₈	—	2	2	no	—
297	C₈	—	2	1	no	—
298	C₈	—	2	1	no	—
299	C₈	—	73	1	no	—
300	C₈	—	73	1	no	—
301	C₈	—	146	1	no	—
302	C₈	—	146	2	no	—
303	C₂²	—	146	1	no	—
304	C₂²	—	146	2	no	—
305	C₂²	—	146	1	no	—
306	C₂²	—	146	2	no	—
307	C₂²	—	146	1	no	—
308	C₂²	—	146	2	no	—
309	C₂²	C₂²×#11924036	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	2	no	—
314	C₄	—	146	1	no	—
315	C₄	—	146	2	no	—
316	C₄	—	146	1	no	—
317	C₄	—	146	1	no	—
318	C₄	#17989656	1	1	yes	characteristic
319	C₄	#17989667	1	1	yes	characteristic
320	C₂	—	292	1	no	—
321	C₂	—	292	2	no	—
322	C₂	#22626849	1	1	yes	characteristic
323	C₂	#22626899	1	1	yes	characteristic
324	C₂	#22626900	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	#446×C₇₃	C₄
prime radical	#14556566	C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	#446	#64408
hypocenter	C₇₃	#33509
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
1. #22626346	C₂	semidirect
2. #22626317	C₂	semidirect
3. #22626314	C₂	semidirect
4. #22626314	C₂	semidirect
5. #22626298	C₂	semidirect
6. #22626294	C₂	semidirect
7. #22626267	C₂	semidirect
8. #22626264	C₂	semidirect
172. C₇₃	#33509	semidirect

Maximal subgroups

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

subgroup	index
#22626346	2
#22626317	2
#22626314	2
#22626298	2
#22626294	2
#22626267	2
#22626264	2
#22618905	2
#33509	73