Subgroups — #34035017

Order 18688.

502 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627157	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#11924040	C₂	1	1	yes	maximal, characteristic, complemented
3	#22627146	C₂	1	1	yes	maximal, characteristic, complemented
4	#22627141	C₂	1	2	yes	maximal, complemented
5	#22627140	C₂	1	1	yes	maximal, characteristic, complemented
6	#22627139	C₂	1	2	yes	maximal, complemented
7	#22627137	C₂	1	2	yes	maximal, complemented
8	#22627129	C₂	1	1	yes	maximal, characteristic, complemented
9	#22627123	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊#11924040	C₂	1	1	yes	maximal, characteristic, complemented
11	#22626747	C₂	1	1	yes	maximal, characteristic
12	C₂×#14556688	C₂²	1	2	yes	—
13	C₂×#14556686	C₂²	1	2	yes	—
14	C₂²×#11924040	C₂²	1	2	yes	—
15	#17989690	C₂²	1	1	yes	characteristic
16	#17989690	C₂²	1	2	yes	—
17	#17989690	C₂²	1	1	yes	characteristic
18	C₄⋊#11924040	C₂²	1	2	yes	—
19	C₄⋊#11924040	C₂²	1	1	yes	characteristic
20	C₄⋊#11924040	C₂²	1	1	yes	characteristic
21	C₄⋊#11924040	C₂²	1	2	yes	—
22	#17989684	C₂²	1	2	yes	—
23	#17989684	C₂²	1	2	yes	—
24	C₂²⋊#11924040	C₂²	1	1	yes	characteristic
25	C₂²⋊#11924040	C₂²	1	1	yes	characteristic
26	#17989681	C₂²	1	2	yes	—
27	C₄×#11924040	C₂²	1	1	yes	characteristic
28	C₄×#11924040	C₂²	1	1	yes	characteristic
29	C₄×#11924040	C₂²	1	2	yes	—
30	#17989638	C₄	1	1	yes	characteristic
31	Q₈×#10596833	C₂²	1	1	yes	characteristic
32	#17989621	C₄	1	1	yes	characteristic
33	#17989616	C₄	1	2	yes	—
34	#17989615	C₂²	1	1	yes	characteristic
35	#17989614	C₂²	1	2	yes	—
36	#17989612	C₄	1	2	yes	—
37	#17989608	C₄	1	1	yes	characteristic
38	#17989552	C₂²	1	2	yes	—
39	C₂×C₄×#10596833	C₂²	1	1	yes	characteristic
40	#17989123	C₄	1	1	yes	characteristic
41	#14556688	—	4	2	no	—
42	#14556686	—	4	2	no	—
43	C₂×#11924040	—	4	2	no	—
44	C₂×#11924040	C₂³	1	1	yes	characteristic
45	C₂×#11924040	C₂³	1	1	yes	characteristic
46	C₂×#11924040	C₂³	1	1	yes	characteristic
47	C₂×#11924040	C₂³	1	1	yes	characteristic
48	C₂×#11924040	C₂³	1	2	yes	—
49	C₂×#11924040	C₂³	1	2	yes	—
50	C₂×#11924038	C₂×C₄	1	2	yes	—
51	C₂×#11924036	C₂³	1	2	yes	—
52	C₂²×#10596833	C₂×C₄	1	1	yes	characteristic
53	C₂²×#10596833	C₂³	1	1	yes	characteristic
54	#14556658	C₂×C₄	1	2	yes	—
55	C₄⋊#10596833	C₂×C₄	1	1	yes	characteristic
56	C₄⋊#10596833	C₂×C₄	1	2	yes	—
57	C₄⋊#10596833	C₂×C₄	1	1	yes	characteristic
58	#14556648	C₂×C₄	1	2	yes	—
59	#14556648	C₂×C₄	1	2	yes	—
60	C₂²⋊#10596833	C₂×C₄	1	2	yes	—
61	#14556646	C₂×C₄	1	2	yes	—
62	#14556646	C₂×C₄	1	2	yes	—
63	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
64	C₄×#10596833	C₂×C₄	1	1	yes	characteristic
65	C₄×#10596833	C₂³	1	1	yes	characteristic
66	C₄×#10596833	C₂³	1	2	yes	—
67	C₄×#10596833	C₂³	1	1	yes	characteristic
68	#14556586	C₈	1	2	yes	—
69	#14556582	C₈	1	1	yes	characteristic
70	#14556580	C₈	1	1	yes	characteristic
71	#14556575	C₈	1	2	yes	—
72	#14556565	C₈	1	1	yes	characteristic
73	Q₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
74	#14556553	C₂×C₄	1	1	yes	characteristic
75	#14556551	C₂×C₄	1	2	yes	—
76	#14556549	C₂×C₄	1	2	yes	—
77	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
78	C₂²⋊Q₈×C₇₃	C₈	1	1	yes	characteristic
79	#11924040	—	2	2	no	—
80	#11924040	—	2	2	no	—
81	#11924040	—	2	1	no	—
82	#11924040	—	2	1	no	—
83	#11924040	—	2	1	no	—
84	#11924040	—	2	1	no	—
85	#11924038	—	4	2	no	—
86	#11924036	—	2	2	no	—
87	#11924036	—	2	2	no	—
88	C₂×#10596833	—	2	1	no	—
89	C₂×#10596833	—	2	1	no	—
90	C₂×#10596833	—	4	1	no	—
91	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
92	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
93	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
94	C₂×#10596833	C₂²×C₄	1	2	yes	—
95	C₂×#10596833	C₂²×C₄	1	2	yes	—
96	C₂×#10596833	C₂⁴	1	1	yes	characteristic
97	C₂×D₂₉₂	C₂×C₈	1	1	yes	characteristic
98	C₂×C₂²⋊D₇₃	C₂×C₈	1	2	yes	—
99	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
100	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
102	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
103	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
104	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
105	#11924007	C₂×C₈	1	2	yes	—
106	#11924005	C₂×C₈	1	2	yes	—
107	#11924005	C₂×C₈	1	2	yes	—
108	#11924005	C₂×C₈	1	2	yes	—
109	#11924005	C₂×C₈	1	2	yes	—
110	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
111	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
112	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
113	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
115	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
116	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
117	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
118	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
119	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
120	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
121	C₂²×D₇₃	C₂²×C₈	1	2	yes	—
122	Q₈×C₇₃	—	2	2	no	—
123	C₂×C₄×C₇₃	—	2	2	no	—
124	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
128	C₂²⋊D₇₃	—	2	4	no	—
129	C₂×Dic₇₃	—	2	2	no	—
130	C₂×Dic₇₃	—	2	1	no	—
131	C₂×Dic₇₃	—	2	1	no	—
132	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
133	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
135	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
136	C₂×Dic₇₃	C₂³×C₄	1	1	yes	characteristic
137	D₂₉₂	—	2	2	no	—
138	C₄×D₇₃	—	2	2	no	—
139	C₄×D₇₃	—	2	2	no	—
140	#10596833	—	2	1	no	—
141	#10596833	—	2	2	no	—
142	#10596833	—	2	1	no	—
143	#10596833	—	2	1	no	—
144	#10596833	—	2	2	no	—
145	#10596833	H(5,𝔽₂)	1	1	yes	characteristic
146	#10596833	2₋¹⁺⁴	1	1	yes	characteristic
147	C₂²×C₇₃	—	2	1	no	—
148	C₂²×C₇₃	—	2	1	no	—
149	C₂²×C₇₃	#504	1	2	yes	—
150	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
151	C₂×D₇₃	—	2	2	no	—
152	C₂×D₇₃	—	2	2	no	—
153	C₂×D₇₃	—	2	2	no	—
154	C₄×C₇₃	—	2	1	no	—
155	C₄×C₇₃	—	2	2	no	—
156	C₄×C₇₃	—	2	2	no	—
157	C₄×C₇₃	#504	1	2	yes	—
158	Dic₇₃	—	2	1	no	—
159	Dic₇₃	—	2	1	no	—
160	Dic₇₃	—	2	2	no	—
161	Dic₇₃	—	2	1	no	—
162	Dic₇₃	#520	1	1	yes	characteristic
163	Dic₇₃	#568	1	1	yes	characteristic
164	#33508	—	73	1	no	maximal, Sylow
165	C₂×C₇₃	—	2	2	no	—
166	C₂×C₇₃	#2988	1	1	yes	characteristic
167	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
168	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
169	D₇₃	—	4	2	no	—
170	#2966	—	73	1	no	—
171	#2183	—	73	1	no	—
172	Q₈×C₁₆	—	73	1	no	—
173	#2169	—	73	1	no	—
174	#2168	—	73	2	no	—
175	D₄×C₁₆	—	73	2	no	—
176	D₄×C₁₆	—	73	1	no	—
177	#2164	—	73	2	no	—
178	#2162	—	73	1	no	—
179	#2151	—	73	2	no	—
180	C₂×C₄⋊C₁₆	—	73	1	no	—
181	C₇₃	#33508	1	1	yes	characteristic, complemented, Sylow
182	#529	—	73	1	no	—
183	C₂×M₃₂	—	73	2	no	—
184	C₂²×C₁₆	—	73	2	no	—
185	C₂²×C₁₆	—	73	2	no	—
186	#446	—	73	1	no	—
187	C₈×Q₈	—	73	1	no	—
188	#436	—	73	1	no	—
189	#435	—	73	2	no	—
190	C₈×D₄	—	73	2	no	—
191	C₈×D₄	—	73	1	no	—
192	#433	—	73	2	no	—
193	#432	—	73	1	no	—
194	#405	—	73	2	no	—
195	C₂×C₄×C₈	—	73	1	no	—
196	C₄⋊C₁₆	—	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	1	no	—
201	C₄⋊C₁₆	—	73	2	no	—
202	C₄⋊C₁₆	—	73	1	no	—
203	C₂²⋊C₁₆	—	73	1	no	—
204	C₂²⋊C₁₆	—	73	1	no	—
205	C₂²⋊C₁₆	—	73	2	no	—
206	C₂²⋊C₁₆	—	73	2	no	—
207	#346	—	73	2	no	—
208	C₄×C₁₆	—	73	1	no	—
209	C₄×C₁₆	—	73	1	no	—
210	C₄×C₁₆	—	73	2	no	—
211	C₂×M₁₆	—	73	2	no	—
212	C₂²×C₈	—	73	1	no	—
213	C₂²×C₈	—	73	1	no	—
214	C₂²×C₈	—	73	2	no	—
215	#126	—	73	2	no	—
216	#125	—	73	1	no	—
217	#124	—	73	1	no	—
218	#123	—	73	2	no	—
219	C₂²⋊Q₈	#10596842	1	1	yes	characteristic
220	#121	—	73	1	no	—
221	C₄×Q₈	—	73	1	no	—
222	C₄×D₄	—	73	1	no	—
223	C₄×D₄	—	73	2	no	—
224	#117	—	73	2	no	—
225	C₂×C₄²	—	73	1	no	—
226	M₃₂	—	292	2	no	—
227	C₂×C₁₆	—	292	2	no	—
228	C₂×C₁₆	—	292	2	no	—
229	C₂×C₁₆	—	73	1	no	—
230	C₂×C₁₆	—	73	1	no	—
231	C₂×C₁₆	—	73	1	no	—
232	C₂×C₁₆	—	73	2	no	—
233	C₂×C₁₆	—	73	1	no	—
234	C₂×C₁₆	—	73	2	no	—
235	C₄⋊C₈	—	73	1	no	—
236	C₄⋊C₈	—	73	1	no	—
237	C₄⋊C₈	—	73	2	no	—
238	C₄⋊C₈	—	73	2	no	—
239	C₂²⋊C₈	—	73	2	no	—
240	C₂²⋊C₈	—	73	2	no	—
241	C₂²⋊C₈	—	73	2	no	—
242	#97	—	73	2	no	—
243	#97	—	73	2	no	—
244	C₄×C₈	—	73	1	no	—
245	C₄×C₈	—	73	2	no	—
246	C₄×C₈	—	73	1	no	—
247	C₄×C₈	—	73	1	no	—
248	C₄×C₈	—	73	1	no	—
249	C₂×Q₈	C₂×#10596842	1	1	yes	characteristic
250	C₂×D₄	—	73	2	no	—
251	C₂×D₄	—	73	1	no	—
252	C₂²×C₄	—	73	1	no	—
253	C₂²×C₄	—	73	1	no	—
254	C₂²×C₄	—	73	2	no	—
255	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
256	M₁₆	—	292	2	no	—
257	C₂×C₈	—	292	1	no	—
258	C₂×C₈	—	73	1	no	—
259	C₂×C₈	—	73	1	no	—
260	C₂×C₈	—	73	1	no	—
261	C₂×C₈	—	73	1	no	—
262	C₂×C₈	—	73	2	no	—
263	C₂×C₈	—	73	2	no	—
264	C₂×C₈	—	146	1	no	—
265	C₂×C₈	—	146	1	no	—
266	C₂×C₈	—	146	2	no	—
267	C₂×C₈	—	146	2	no	—
268	C₄⋊C₄	—	73	2	no	—
269	C₄⋊C₄	—	73	2	no	—
270	C₄⋊C₄	—	73	1	no	—
271	C₄⋊C₄	—	73	2	no	—
272	C₄⋊C₄	C₂×#10596842	1	2	yes	—
273	C₄⋊C₄	C₂×#10596842	1	1	yes	characteristic
274	C₂²⋊C₄	—	73	2	no	—
275	C₂²⋊C₄	—	73	2	no	—
276	C₂²⋊C₄	—	73	2	no	—
277	C₂²⋊C₄	—	73	2	no	—
278	C₂²⋊C₄	C₂×#10596842	1	2	yes	—
279	C₄²	—	73	1	no	—
280	C₄²	—	73	1	no	—
281	C₄²	—	73	1	no	—
282	C₄²	—	73	1	no	—
283	C₄²	—	73	2	no	—
284	C₁₆	—	146	2	no	—
285	C₁₆	—	146	1	no	—
286	C₁₆	—	146	2	no	—
287	C₁₆	—	146	1	no	—
288	C₁₆	—	146	1	no	—
289	C₁₆	—	146	1	no	—
290	C₂³	—	73	2	no	—
291	C₂³	C₂²×#10596842	1	1	yes	characteristic
292	Q₈	—	2	2	no	—
293	D₄	—	146	4	no	—
294	D₄	—	146	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	73	1	no	—
297	C₂×C₄	—	73	1	no	—
298	C₂×C₄	—	73	1	no	—
299	C₂×C₄	—	73	1	no	—
300	C₂×C₄	—	73	2	no	—
301	C₂×C₄	—	146	2	no	—
302	C₂×C₄	—	146	2	no	—
303	C₂×C₄	—	146	1	no	—
304	C₂×C₄	—	146	1	no	—
305	C₂×C₄	—	146	2	no	—
306	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
307	C₂×C₄	C₂²×#10596842	1	2	yes	—
308	C₂×C₄	C₂²×#10596842	1	2	yes	—
309	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
310	C₈	—	73	1	no	—
311	C₈	—	73	1	no	—
312	C₈	—	146	1	no	—
313	C₈	—	146	1	no	—
314	C₈	—	146	1	no	—
315	C₈	—	146	2	no	—
316	C₈	—	146	2	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	146	2	no	—
320	C₂²	—	146	2	no	—
321	C₂²	—	146	2	no	—
322	C₂²	#17989709	1	2	yes	—
323	C₂²	C₂³×#10596842	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	2	no	—
327	C₄	—	73	1	no	—
328	C₄	—	73	1	no	—
329	C₄	—	146	1	no	—
330	C₄	—	146	1	no	—
331	C₄	—	146	1	no	—
332	C₄	—	146	2	no	—
333	C₄	#17989708	1	2	yes	—
334	C₂	—	2	2	no	—
335	C₂	—	292	2	no	—
336	C₂	#22627175	1	1	yes	characteristic
337	C₂	C₂×#17989708	1	1	yes	characteristic
338	C₂	C₂×#17989709	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10596842
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	C₂²⋊Q₈×C₇₃	C₈
prime radical	#14556565	C₈
cosocle	C₂×#10596833	C₂⁴
hypercenter	C₂²⋊Q₈	#10596842
hypocenter	C₇₃	#33508
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #22627157	C₂	semidirect
2. Q₈×#11924040	C₂	semidirect
3. #22627146	C₂	semidirect
4. #22627141	C₂	semidirect
5. #22627140	C₂	semidirect
6. #22627139	C₂	semidirect
7. #22627137	C₂	semidirect
8. #22627129	C₂	semidirect
9. #22627123	C₂	semidirect
10. C₂×C₄⋊#11924040	C₂	semidirect
181. C₇₃	#33508	semidirect

Maximal subgroups

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

subgroup	index
#22627157	2
Q₈×#11924040	2
#22627146	2
#22627141	2
#22627140	2
#22627139	2
#22627137	2
#22627129	2
#22627123	2
C₂×C₄⋊#11924040	2
#22626747	2
#33508	73