Subgroups — #34030344

Order 18688.

526 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626346	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#11924031	C₂	1	1	yes	maximal, characteristic, complemented
3	#22626315	C₂	1	1	yes	maximal, characteristic, complemented
4	#22626311	C₂	1	2	yes	maximal, complemented
5	D₄×#11924031	C₂	1	2	yes	maximal, complemented
6	D₄×#11924031	C₂	1	1	yes	maximal, characteristic, complemented
7	#22626296	C₂	1	2	yes	maximal, complemented
8	#22626292	C₂	1	1	yes	maximal, characteristic, complemented
9	#22626140	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×#11924031	C₂	1	1	yes	maximal, characteristic, complemented
11	#22618907	C₂	1	1	yes	maximal, characteristic
12	C₂×#14556665	C₂²	1	2	yes	—
13	C₂²×#11924031	C₂²	1	1	yes	characteristic
14	C₂²×#11924031	C₂²	1	2	yes	—
15	C₂²×#11924031	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	2	yes	—
19	C₄⋊#11924031	C₂²	1	1	yes	characteristic
20	C₂²⋊#11924031	C₂²	1	2	yes	—
21	C₂²⋊#11924031	C₂²	1	2	yes	—
22	C₂²⋊#11924031	C₂²	1	2	yes	—
23	#17989475	C₂²	1	2	yes	—
24	#17989475	C₂²	1	2	yes	—
25	C₄×#11924031	C₂²	1	1	yes	characteristic
26	C₄×#11924031	C₂²	1	2	yes	—
27	C₄×#11924031	C₂²	1	1	yes	characteristic
28	C₄×#11924031	C₂²	1	1	yes	characteristic
29	C₄×#11924031	C₂²	1	1	yes	characteristic
30	#17988718	C₄	1	1	yes	characteristic
31	Q₈×#10596831	C₂²	1	1	yes	characteristic
32	#17988665	C₄	1	1	yes	characteristic
33	#17988659	C₄	1	2	yes	—
34	D₄×#10596831	C₂²	1	2	yes	—
35	D₄×#10596831	C₂²	1	1	yes	characteristic
36	#17988648	C₄	1	2	yes	—
37	#17988644	C₄	1	1	yes	characteristic
38	#17988543	C₂²	1	2	yes	—
39	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
40	#529×C₇₃	C₄	1	1	yes	characteristic
41	#14556665	—	2	4	no	—
42	C₂×#11924031	—	2	2	no	—
43	C₂×#11924031	—	2	4	no	—
44	C₂×#11924031	—	2	2	no	—
45	C₂×#11924031	C₂³	1	1	yes	characteristic
46	C₂×#11924031	C₂³	1	1	yes	characteristic
47	C₂×#11924031	C₂³	1	2	yes	—
48	C₂×#11924031	C₂³	1	2	yes	—
49	C₂×#11924031	C₂³	1	1	yes	characteristic
50	C₂×#11924031	C₂³	1	1	yes	characteristic
51	C₂×#11924012	C₂×C₄	1	2	yes	—
52	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
53	C₂²×#10596831	C₂³	1	1	yes	characteristic
54	C₂²×#10596831	C₂³	1	2	yes	—
55	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
56	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
57	C₄⋊#10596831	C₂×C₄	1	2	yes	—
58	C₄⋊#10596831	C₂×C₄	1	2	yes	—
59	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
60	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
61	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
62	#14556502	C₂×C₄	1	2	yes	—
63	#14556502	C₂×C₄	1	2	yes	—
64	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
65	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
66	C₄×#10596831	C₂³	1	1	yes	characteristic
67	C₄×#10596831	C₂³	1	2	yes	—
68	C₄×#10596831	C₂³	1	1	yes	characteristic
69	#126×C₇₃	C₈	1	2	yes	—
70	#125×C₇₃	C₈	1	1	yes	characteristic
71	#124×C₇₃	C₈	1	1	yes	characteristic
72	#123×C₇₃	C₈	1	2	yes	—
73	C₂²⋊Q₈×C₇₃	C₈	1	1	yes	characteristic
74	#121×C₇₃	C₈	1	1	yes	characteristic
75	C₄×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
76	C₄×D₄×C₇₃	C₂×C₄	1	2	yes	—
77	C₄×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
78	#117×C₇₃	C₂×C₄	1	2	yes	—
79	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
80	#11924031	—	2	1	no	—
81	#11924031	—	2	1	no	—
82	#11924031	—	2	2	no	—
83	#11924031	—	2	2	no	—
84	#11924031	#41	1	2	yes	—
85	#11924031	#41	1	2	yes	—
86	#11924012	—	2	4	no	—
87	C₂×#10596831	—	2	2	no	—
88	C₂×#10596831	—	2	1	no	—
89	C₂×#10596831	—	2	1	no	—
90	C₂×#10596831	—	2	2	no	—
91	C₂×#10596831	—	2	2	no	—
92	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
93	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
94	C₂×#10596831	C₂²×C₄	1	2	yes	—
95	C₂×#10596831	C₂²×C₄	1	2	yes	—
96	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
97	C₂×#10596831	C₂⁴	1	1	yes	characteristic
98	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
99	C₂×D₄×C₇₃	C₂×C₈	1	2	yes	—
100	C₂×D₄×C₇₃	C₂×C₈	1	1	yes	characteristic
101	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
102	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
103	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
105	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
106	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
107	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
108	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
109	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
110	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
111	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
112	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
113	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
114	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
115	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
116	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
117	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
118	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
121	C₂³×C₇₃	C₂²×C₈	1	2	yes	—
122	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
123	Q₈×C₇₃	—	2	2	no	—
124	D₄×C₇₃	—	2	4	no	—
125	D₄×C₇₃	—	2	2	no	—
126	C₂×C₄×C₇₃	—	2	2	no	—
127	C₂×C₄×C₇₃	—	2	1	no	—
128	C₂×C₄×C₇₃	—	2	2	no	—
129	C₂×C₄×C₇₃	—	2	1	no	—
130	C₂×C₄×C₇₃	—	2	2	no	—
131	C₂×C₄×C₇₃	—	2	2	no	—
132	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
137	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
138	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
139	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
141	#10596831	—	2	1	no	—
142	#10596831	—	2	2	no	—
143	#10596831	—	2	1	no	—
144	#10596831	—	2	2	no	—
145	#10596831	#131	1	2	yes	—
146	#10596831	C₂×#41	1	1	yes	characteristic
147	#10596831	C₂×#41	1	1	yes	characteristic
148	C₂²×C₇₃	—	2	1	no	—
149	C₂²×C₇₃	—	2	1	no	—
150	C₂²×C₇₃	—	2	2	no	—
151	C₂²×C₇₃	—	2	2	no	—
152	C₂²×C₇₃	—	2	2	no	—
153	C₂²×C₇₃	#504	1	2	yes	—
154	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
155	C₄×C₇₃	—	2	1	no	—
156	C₄×C₇₃	—	2	1	no	—
157	C₄×C₇₃	—	2	1	no	—
158	C₄×C₇₃	—	2	2	no	—
159	C₄×C₇₃	—	2	2	no	—
160	C₄×C₇₃	—	2	1	no	—
161	C₄×C₇₃	—	2	2	no	—
162	C₄×C₇₃	#504	1	2	yes	—
163	C₄×C₇₃	C₄×#41	1	1	yes	characteristic
164	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
165	#33480	—	73	1	no	maximal, Sylow
166	C₂×C₇₃	—	2	2	no	—
167	C₂×C₇₃	—	4	2	no	—
168	C₂×C₇₃	C₈×#41	1	1	yes	characteristic
169	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
170	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
171	#2966	—	73	1	no	—
172	#2183	—	73	1	no	—
173	Q₈×C₁₆	—	73	1	no	—
174	#2169	—	73	1	no	—
175	#2168	—	73	2	no	—
176	D₄×C₁₆	—	73	1	no	—
177	D₄×C₁₆	—	73	2	no	—
178	#2164	—	73	2	no	—
179	#2163	—	73	1	no	—
180	#2108	—	73	2	no	—
181	C₂×C₄×C₁₆	—	73	1	no	—
182	C₇₃	#33480	1	1	yes	characteristic, complemented, Sylow
183	#529	#64408	1	1	yes	characteristic
184	C₂×M₃₂	—	73	2	no	—
185	C₂²×C₁₆	—	73	1	no	—
186	C₂²×C₁₆	—	73	1	no	—
187	C₂²×C₁₆	—	73	2	no	—
188	#446	—	73	1	no	—
189	C₈×Q₈	—	73	1	no	—
190	#436	—	73	1	no	—
191	#435	—	73	2	no	—
192	C₈×D₄	—	73	2	no	—
193	C₈×D₄	—	73	1	no	—
194	#433	—	73	2	no	—
195	#432	—	73	1	no	—
196	#405	—	73	2	no	—
197	C₂×C₄×C₈	—	73	1	no	—
198	C₄⋊C₁₆	—	73	1	no	—
199	C₄⋊C₁₆	—	73	1	no	—
200	C₄⋊C₁₆	—	73	2	no	—
201	C₄⋊C₁₆	—	73	2	no	—
202	C₂²⋊C₁₆	—	73	2	no	—
203	C₂²⋊C₁₆	—	73	2	no	—
204	C₂²⋊C₁₆	—	73	2	no	—
205	#346	—	73	2	no	—
206	#346	—	73	2	no	—
207	C₄×C₁₆	—	73	1	no	—
208	C₄×C₁₆	—	73	1	no	—
209	C₄×C₁₆	—	73	1	no	—
210	C₄×C₁₆	—	73	1	no	—
211	C₄×C₁₆	—	73	2	no	—
212	C₂×M₁₆	—	73	2	no	—
213	C₂²×C₈	—	73	1	no	—
214	C₂²×C₈	—	73	2	no	—
215	C₂²×C₈	—	73	1	no	—
216	#126	#10596833	1	2	yes	—
217	#125	#10596833	1	1	yes	characteristic
218	#124	#10596833	1	1	yes	characteristic
219	#123	#10596833	1	2	yes	—
220	C₂²⋊Q₈	#10596833	1	1	yes	characteristic
221	#121	#10596833	1	1	yes	characteristic
222	C₄×Q₈	C₂×#64408	1	1	yes	characteristic
223	C₄×D₄	C₂×#64408	1	2	yes	—
224	C₄×D₄	C₂×#64408	1	1	yes	characteristic
225	#117	C₂×#64408	1	2	yes	—
226	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
227	M₃₂	—	146	4	no	—
228	C₂×C₁₆	—	73	1	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	2	no	—
234	C₂×C₁₆	—	146	2	no	—
235	C₂×C₁₆	—	146	4	no	—
236	C₂×C₁₆	—	146	2	no	—
237	C₄⋊C₈	—	73	1	no	—
238	C₄⋊C₈	—	73	1	no	—
239	C₄⋊C₈	—	73	2	no	—
240	C₄⋊C₈	—	73	2	no	—
241	C₂²⋊C₈	—	73	2	no	—
242	C₂²⋊C₈	—	73	2	no	—
243	C₂²⋊C₈	—	73	2	no	—
244	#97	—	73	2	no	—
245	#97	—	73	2	no	—
246	C₄×C₈	—	73	1	no	—
247	C₄×C₈	—	73	1	no	—
248	C₄×C₈	—	73	1	no	—
249	C₄×C₈	—	73	1	no	—
250	C₄×C₈	—	73	2	no	—
251	C₂×Q₈	C₂×#10596833	1	1	yes	characteristic
252	C₂×D₄	C₂×#10596833	1	2	yes	—
253	C₂×D₄	C₂×#10596833	1	1	yes	characteristic
254	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
255	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
256	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
257	C₂²×C₄	C₂²×#64408	1	2	yes	—
258	M₁₆	—	146	4	no	—
259	C₂×C₈	—	73	1	no	—
260	C₂×C₈	—	73	1	no	—
261	C₂×C₈	—	73	1	no	—
262	C₂×C₈	—	73	1	no	—
263	C₂×C₈	—	73	2	no	—
264	C₂×C₈	—	73	2	no	—
265	C₂×C₈	—	146	1	no	—
266	C₂×C₈	—	146	1	no	—
267	C₂×C₈	—	146	2	no	—
268	C₂×C₈	—	146	2	no	—
269	C₂×C₈	—	146	2	no	—
270	C₄⋊C₄	C₂×#10596833	1	2	yes	—
271	C₄⋊C₄	C₂×#10596833	1	2	yes	—
272	C₄⋊C₄	C₂×#10596833	1	1	yes	characteristic
273	C₄⋊C₄	C₂×#10596833	1	2	yes	—
274	C₄⋊C₄	C₂×#10596833	1	1	yes	characteristic
275	C₄⋊C₄	C₂×#10596833	1	2	yes	—
276	C₂²⋊C₄	C₂×#10596833	1	2	yes	—
277	C₂²⋊C₄	C₂×#10596833	1	2	yes	—
278	C₂²⋊C₄	C₂×#10596833	1	2	yes	—
279	C₂²⋊C₄	C₂×#10596833	1	2	yes	—
280	C₂²⋊C₄	C₂×#10596833	1	2	yes	—
281	C₄²	C₂×#10596833	1	1	yes	characteristic
282	C₄²	C₂×#10596833	1	1	yes	characteristic
283	C₄²	C₂²×#64408	1	1	yes	characteristic
284	C₄²	C₂²×#64408	1	1	yes	characteristic
285	C₄²	C₂²×#64408	1	2	yes	—
286	C₁₆	—	73	2	no	—
287	C₁₆	—	73	2	no	—
288	C₁₆	—	146	1	no	—
289	C₁₆	—	146	1	no	—
290	C₁₆	—	146	2	no	—
291	C₁₆	—	146	2	no	—
292	C₂³	C₂²×#10596833	1	2	yes	—
293	C₂³	C₂²×#10596833	1	1	yes	characteristic
294	Q₈	—	2	2	no	—
295	D₄	—	2	4	no	—
296	D₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
304	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
305	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
306	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
307	C₂×C₄	C₂²×#10596833	1	2	yes	—
308	C₂×C₄	C₂²×#10596833	1	2	yes	—
309	C₂×C₄	C₂²×#10596833	1	2	yes	—
310	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
311	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
312	C₈	—	73	1	no	—
313	C₈	—	73	1	no	—
314	C₈	—	73	2	no	—
315	C₈	—	146	1	no	—
316	C₈	—	146	2	no	—
317	C₈	—	146	1	no	—
318	C₈	—	146	2	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	#17989645	1	2	yes	—
325	C₂²	C₂³×#10596833	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	2	no	—
333	C₄	#17989645	1	2	yes	—
334	C₄	#41×#64408	1	1	yes	characteristic
335	C₄	D₇₃⋊#131	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	4	2	no	—
338	C₂	#41×#10596833	1	1	yes	characteristic
339	C₂	C₂×#17989645	1	1	yes	characteristic
340	C₂	C₂×#17989645	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	#529×C₇₃	C₄
prime radical	#121×C₇₃	C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	#529	#64408
hypocenter	C₇₃	#33480
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
1. #22626346	C₂	semidirect
2. Q₈×#11924031	C₂	semidirect
3. #22626315	C₂	semidirect
4. #22626311	C₂	semidirect
5. D₄×#11924031	C₂	semidirect
6. D₄×#11924031	C₂	semidirect
7. #22626296	C₂	semidirect
8. #22626292	C₂	semidirect
9. #22626140	C₂	semidirect
10. C₂×C₄×#11924031	C₂	semidirect
182. C₇₃	#33480	semidirect

Maximal subgroups

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

subgroup	index
#22626346	2
Q₈×#11924031	2
#22626315	2
#22626311	2
D₄×#11924031	2
#22626296	2
#22626292	2
#22626140	2
C₂×C₄×#11924031	2
#22618907	2
#33480	73