Subgroups — #34030438

Order 18688.

452 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626348	C₂	1	2	yes	maximal, complemented
2	#22626347	C₂	1	2	yes	maximal, complemented
3	#22626347	C₂	1	1	yes	maximal, characteristic, complemented
4	#22626346	C₂	1	1	yes	maximal, characteristic, complemented
5	#22626345	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×#11924031	C₂	1	1	yes	maximal, characteristic, complemented
7	#22626299	C₂	1	2	yes	maximal
8	#22626298	C₂	1	2	yes	maximal
9	#22626295	C₂	1	1	yes	maximal, characteristic
10	#22626290	C₂	1	1	yes	maximal, characteristic
11	#22618862	C₂	1	1	yes	maximal, characteristic
12	#17989508	C₂²	1	2	yes	—
13	#17989508	C₂²	1	1	yes	characteristic
14	#17989508	C₂²	1	2	yes	—
15	#17989508	C₂²	1	1	yes	characteristic
16	C₄⋊#11924031	C₂²	1	1	yes	characteristic
17	C₄⋊#11924031	C₂²	1	2	yes	—
18	C₄⋊#11924031	C₂²	1	1	yes	characteristic
19	C₄⋊#11924031	C₂²	1	2	yes	—
20	#17989482	C₂²	1	1	yes	characteristic
21	#17989482	C₂²	1	1	yes	characteristic
22	#17989482	C₂²	1	2	yes	—
23	#17989476	C₂²	1	2	yes	—
24	#17989476	C₂²	1	2	yes	—
25	#17989475	C₂²	1	2	yes	—
26	#17989473	C₂²	1	2	yes	—
27	C₄×#11924031	C₂²	1	1	yes	characteristic
28	C₄×#11924031	C₂²	1	2	yes	—
29	C₄×#11924031	C₂²	1	1	yes	characteristic
30	#17989141	C₄	1	1	yes	characteristic
31	#17988717	C₄	1	1	yes	characteristic
32	#17988715	C₄	1	2	yes	—
33	Q₈×#10596831	C₂²	1	1	yes	characteristic
34	#17988711	C₂²	1	1	yes	characteristic
35	#17988711	C₂²	1	2	yes	—
36	#17988612	C₄	1	2	yes	—
37	#17988609	C₄	1	1	yes	characteristic
38	#17988544	C₂²	1	2	yes	—
39	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
40	#446×C₇₃	C₄	1	1	yes	characteristic
41	C₂×#11924031	C₂³	1	1	yes	characteristic
42	C₂×#11924031	C₂³	1	1	yes	characteristic
43	C₂×#11924031	C₂³	1	1	yes	characteristic
44	C₂×#11924031	C₂³	1	2	yes	—
45	C₂×#11924031	C₂³	1	1	yes	characteristic
46	C₂×#11924031	C₂³	1	2	yes	—
47	C₂×#11924011	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
50	#14556590	C₈	1	1	yes	characteristic
51	#14556589	C₈	1	1	yes	characteristic
52	#14556581	C₈	1	2	yes	—
53	#14556571	C₈	1	1	yes	characteristic
54	#14556569	C₈	1	2	yes	—
55	#14556567	C₈	1	1	yes	characteristic
56	C₄×Dic₁₄₆	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
58	#14556548	C₂×C₄	1	2	yes	—
59	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
60	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
61	C₄⋊#10596831	C₂×C₄	1	2	yes	—
62	#14556520	C₂×C₄	1	2	yes	—
63	#14556520	C₂×C₄	1	2	yes	—
64	#14556520	C₂×C₄	1	2	yes	—
65	#14556501	C₂×C₄	1	2	yes	—
66	C₄×#10596831	C₂³	1	1	yes	characteristic
67	C₄×#10596831	C₂³	1	2	yes	—
68	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₇₃	C₂³	1	2	yes	—
70	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
71	C₄×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
72	C₄⋊C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
73	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
74	#97×C₇₃	C₂×C₄	1	2	yes	—
75	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
76	#11924031	—	2	1	no	—
77	#11924031	—	2	1	no	—
78	#11924031	—	2	1	no	—
79	#11924031	—	2	2	no	—
80	#11924031	—	2	1	no	—
81	#11924031	—	2	2	no	—
82	C₂×Dic₁₄₆	C₂×C₈	1	1	yes	characteristic
83	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
84	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
87	#11924011	—	4	2	no	—
88	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
89	C₂×#10596831	C₂²×C₄	1	2	yes	—
90	C₂×#10596831	C₂⁴	1	1	yes	characteristic
91	C₈×D₇₃	—	2	1	no	—
92	C₈×D₇₃	—	2	1	no	—
93	C₈×D₇₃	—	4	1	no	—
94	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
95	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
96	#11924007	C₂×C₈	1	2	yes	—
97	#11924007	C₂×C₈	1	2	yes	—
98	#11924007	C₂×C₈	1	2	yes	—
99	#11924007	C₂×C₈	1	2	yes	—
100	#11924007	C₂×C₈	1	2	yes	—
101	#11924005	C₂×C₈	1	2	yes	—
102	#11924005	C₂×C₈	1	2	yes	—
103	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
104	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
105	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
106	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
107	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
108	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
111	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
112	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
113	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
115	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
116	Q₈×C₇₃	—	2	2	no	—
117	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
120	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂³×C₄	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	1	yes	characteristic
125	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
126	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
127	C₄×D₇₃	—	2	1	no	—
128	C₄×D₇₃	—	2	1	no	—
129	C₄×D₇₃	—	2	2	no	—
130	C₄×D₇₃	—	4	1	no	—
131	Dic₁₄₆	—	2	2	no	—
132	Dic₁₄₆	—	4	2	no	—
133	C₈×C₇₃	—	2	1	no	—
134	C₈×C₇₃	—	2	1	no	—
135	C₈×C₇₃	—	2	2	no	—
136	#10596831	—	2	1	no	—
137	#10596831	—	2	2	no	—
138	#10596831	2₋¹⁺⁴	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	1	no	—
144	C₄×C₇₃	—	2	2	no	—
145	C₄×C₇₃	—	2	2	no	—
146	C₄×C₇₃	—	2	1	no	—
147	C₄×C₇₃	#504	1	2	yes	—
148	C₄×C₇₃	#519	1	1	yes	characteristic
149	C₄×C₇₃	#568	1	1	yes	characteristic
150	Dic₇₃	—	2	2	no	—
151	Dic₇₃	—	2	1	no	—
152	Dic₇₃	—	2	1	no	—
153	Dic₇₃	—	2	2	no	—
154	Dic₇₃	—	2	1	no	—
155	#33499	—	73	1	no	maximal, Sylow
156	C₂×C₇₃	#2982	1	1	yes	characteristic
157	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
158	C₂×C₇₃	#3407	1	1	yes	characteristic
159	D₇₃	—	4	1	no	—
160	#2960	—	73	1	no	—
161	#2183	—	73	1	no	—
162	#2183	—	73	1	no	—
163	#2183	—	73	2	no	—
164	#2183	—	73	2	no	—
165	Q₈×C₁₆	—	73	1	no	—
166	Q₈×C₁₆	—	73	1	no	—
167	#2164	—	73	2	no	—
168	#2164	—	73	2	no	—
169	#2163	—	73	1	no	—
170	#2162	—	73	1	no	—
171	C₇₃	#33499	1	1	yes	characteristic, complemented, Sylow
172	#533	—	73	1	no	—
173	#446	—	73	2	no	—
174	#446	—	73	1	no	—
175	#446	#64408	1	1	yes	characteristic
176	C₈×Q₈	—	73	1	no	—
177	C₈×Q₈	—	73	1	no	—
178	C₈×Q₈	—	73	2	no	—
179	#424	—	73	2	no	—
180	#423	—	73	1	no	—
181	#405	—	73	2	no	—
182	C₂×C₄×C₈	—	73	1	no	—
183	C₄⋊C₁₆	—	73	1	no	—
184	C₄⋊C₁₆	—	73	1	no	—
185	C₄⋊C₁₆	—	73	1	no	—
186	C₄⋊C₁₆	—	73	1	no	—
187	C₄⋊C₁₆	—	73	2	no	—
188	C₄⋊C₁₆	—	73	2	no	—
189	C₄⋊C₁₆	—	73	2	no	—
190	C₄⋊C₁₆	—	73	2	no	—
191	C₂²⋊C₁₆	—	73	1	no	—
192	C₂²⋊C₁₆	—	73	1	no	—
193	C₂²⋊C₁₆	—	73	2	no	—
194	#346	—	73	2	no	—
195	#346	—	73	2	no	—
196	#346	—	73	2	no	—
197	C₄×C₁₆	—	73	1	no	—
198	C₄×C₁₆	—	73	1	no	—
199	C₄×C₁₆	—	73	2	no	—
200	C₄×C₁₆	—	73	2	no	—
201	C₂×M₁₆	—	73	2	no	—
202	C₂²×C₈	—	73	1	no	—
203	C₂²×C₈	—	73	1	no	—
204	C₄⋊Q₈	—	73	1	no	—
205	C₄⋊Q₈	—	73	1	no	—
206	#125	—	73	2	no	—
207	C₂²⋊Q₈	—	73	1	no	—
208	C₂²⋊Q₈	—	73	2	no	—
209	C₂²⋊Q₈	—	73	1	no	—
210	C₄×Q₈	—	73	2	no	—
211	C₄×Q₈	—	73	1	no	—
212	C₄×Q₈	C₂×#64408	1	1	yes	characteristic
213	#117	—	73	2	no	—
214	C₂×C₄²	—	73	1	no	—
215	C₂×C₁₆	—	73	1	no	—
216	C₂×C₁₆	—	73	1	no	—
217	C₂×C₁₆	—	73	1	no	—
218	C₂×C₁₆	—	73	1	no	—
219	C₂×C₁₆	—	73	2	no	—
220	C₂×C₁₆	—	73	2	no	—
221	C₄⋊C₈	—	73	2	no	—
222	C₄⋊C₈	—	73	2	no	—
223	C₄⋊C₈	—	73	2	no	—
224	C₄⋊C₈	—	73	1	no	—
225	C₄⋊C₈	—	73	2	no	—
226	C₄⋊C₈	C₂×#64408	1	1	yes	characteristic
227	C₄⋊C₈	C₂×#64408	1	2	yes	—
228	#97	—	73	2	no	—
229	#97	C₂×#64408	1	2	yes	—
230	C₄×C₈	—	73	1	no	—
231	C₄×C₈	—	73	2	no	—
232	C₄×C₈	—	73	1	no	—
233	C₄×C₈	—	73	1	no	—
234	C₄×C₈	—	73	2	no	—
235	C₄×C₈	C₂×#64408	1	1	yes	characteristic
236	C₂×Q₈	—	73	2	no	—
237	C₂×Q₈	—	73	1	no	—
238	C₂×Q₈	#11924036	1	1	yes	characteristic
239	C₂²×C₄	—	73	1	no	—
240	C₂²×C₄	—	73	1	no	—
241	C₂²×C₄	—	73	1	no	—
242	M₁₆	—	292	2	no	—
243	C₂×C₈	—	292	1	no	—
244	C₂×C₈	—	73	1	no	—
245	C₂×C₈	—	73	1	no	—
246	C₂×C₈	—	73	2	no	—
247	C₂×C₈	—	146	1	no	—
248	C₂×C₈	—	146	1	no	—
249	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
250	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
251	C₂×C₈	C₂²×#64408	1	2	yes	—
252	C₄⋊C₄	—	73	2	no	—
253	C₄⋊C₄	—	73	2	no	—
254	C₄⋊C₄	—	73	2	no	—
255	C₄⋊C₄	—	73	2	no	—
256	C₄⋊C₄	—	73	2	no	—
257	C₄⋊C₄	—	73	2	no	—
258	C₄⋊C₄	—	73	1	no	—
259	C₄⋊C₄	#11924036	1	2	yes	—
260	C₄⋊C₄	#11924036	1	1	yes	characteristic
261	C₂²⋊C₄	—	73	2	no	—
262	C₂²⋊C₄	—	73	2	no	—
263	C₄²	—	73	1	no	—
264	C₄²	—	73	1	no	—
265	C₄²	—	73	1	no	—
266	C₄²	—	73	2	no	—
267	C₄²	C₂²×#64408	1	1	yes	characteristic
268	C₄²	C₂²×#64408	1	2	yes	—
269	C₁₆	—	146	1	no	—
270	C₁₆	—	146	1	no	—
271	C₁₆	—	146	1	no	—
272	C₁₆	—	146	1	no	—
273	C₁₆	—	146	2	no	—
274	C₁₆	—	146	2	no	—
275	C₂³	—	73	1	no	—
276	Q₈	—	2	2	no	—
277	Q₈	—	292	2	no	—
278	Q₈	—	146	2	no	—
279	C₂×C₄	—	292	1	no	—
280	C₂×C₄	—	73	1	no	—
281	C₂×C₄	—	73	1	no	—
282	C₂×C₄	—	73	1	no	—
283	C₂×C₄	—	73	2	no	—
284	C₂×C₄	—	73	2	no	—
285	C₂×C₄	—	146	2	no	—
286	C₂×C₄	—	146	1	no	—
287	C₂×C₄	—	146	1	no	—
288	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
289	C₂×C₄	C₂×#11924036	1	2	yes	—
290	C₂×C₄	C₂×#11924036	1	2	yes	—
291	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
292	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	—	2	2	no	—
296	C₈	—	73	1	no	—
297	C₈	—	73	1	no	—
298	C₈	—	146	1	no	—
299	C₈	—	146	2	no	—
300	C₂²	—	146	1	no	—
301	C₂²	—	146	1	no	—
302	C₂²	—	146	1	no	—
303	C₂²	C₂²×#11924036	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	2	no	—
307	C₄	—	146	1	no	—
308	C₄	—	146	1	no	—
309	C₄	—	146	2	no	—
310	C₄	—	146	1	no	—
311	C₄	—	146	2	no	—
312	C₄	#17989646	1	2	yes	—
313	C₄	#17989655	1	1	yes	characteristic
314	C₄	#17989666	1	1	yes	characteristic
315	C₂	—	292	1	no	—
316	C₂	#22626825	1	1	yes	characteristic
317	C₂	C₂×#17989646	1	1	yes	characteristic
318	C₂	#22626899	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	C₂²×D₇₃	C₂²×C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	#446	#64408
hypocenter	C₇₃	#33499
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. #22626348	C₂	semidirect
2. #22626347	C₂	semidirect
3. #22626347	C₂	semidirect
4. #22626346	C₂	semidirect
5. #22626345	C₂	semidirect
6. Q₈×#11924031	C₂	semidirect
171. C₇₃	#33499	semidirect

Maximal subgroups

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

subgroup	index
#22626348	2
#22626347	2
#22626346	2
#22626345	2
Q₈×#11924031	2
#22626299	2
#22626298	2
#22626295	2
#22626290	2
#22618862	2
#33499	73