Subgroups — #34030349

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	#22626317	C₂	1	1	yes	maximal, characteristic, complemented
4	#22626314	C₂	1	2	yes	maximal, complemented
5	#22626310	C₂	1	2	yes	maximal, complemented
6	#22626310	C₂	1	1	yes	maximal, characteristic, complemented
7	#22626299	C₂	1	2	yes	maximal, complemented
8	#22626295	C₂	1	1	yes	maximal, characteristic, complemented
9	#22626141	C₂	1	2	yes	maximal, complemented
10	C₄×#14556664	C₂	1	1	yes	maximal, characteristic, complemented
11	#22618905	C₂	1	1	yes	maximal, characteristic
12	C₂×#14556666	C₂²	1	2	yes	—
13	C₂×#14556664	C₂²	1	1	yes	characteristic
14	C₂×#14556664	C₂²	1	2	yes	—
15	C₂×#14556664	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	#17989482	C₂²	1	2	yes	—
21	#17989482	C₂²	1	2	yes	—
22	#17989482	C₂²	1	2	yes	—
23	#17989476	C₂²	1	2	yes	—
24	#17989475	C₂²	1	2	yes	—
25	#17989473	C₂²	1	1	yes	characteristic
26	#17989473	C₂²	1	1	yes	characteristic
27	C₄×#11924031	C₂²	1	1	yes	characteristic
28	C₄×#11924031	C₂²	1	2	yes	—
29	C₄×#11924031	C₂²	1	1	yes	characteristic
30	#17989124	C₄	1	1	yes	characteristic
31	Q₈×#10596831	C₂²	1	1	yes	characteristic
32	#17988666	C₄	1	1	yes	characteristic
33	#17988660	C₄	1	2	yes	—
34	#17988652	C₂²	1	2	yes	—
35	#17988652	C₂²	1	1	yes	characteristic
36	#17988647	C₄	1	2	yes	—
37	#17988643	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	#14556666	—	2	4	no	—
42	#14556664	—	2	2	no	—
43	#14556664	—	2	4	no	—
44	#14556664	—	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×#11924011	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
54	C₂×C₈×D₇₃	C₂³	1	2	yes	—
55	#14556586	C₈	1	2	yes	—
56	#14556582	C₈	1	1	yes	characteristic
57	#14556580	C₈	1	1	yes	characteristic
58	#14556576	C₈	1	2	yes	—
59	#14556567	C₈	1	1	yes	characteristic
60	#14556566	C₈	1	1	yes	characteristic
61	C₄×D₂₉₂	C₂×C₄	1	2	yes	—
62	C₄×D₂₉₂	C₂×C₄	1	1	yes	characteristic
63	#14556548	C₂×C₄	1	2	yes	—
64	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
65	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
66	C₄⋊#10596831	C₂×C₄	1	2	yes	—
67	#14556503	C₂×C₄	1	2	yes	—
68	#14556503	C₂×C₄	1	2	yes	—
69	#14556503	C₂×C₄	1	2	yes	—
70	#14556501	C₂×C₄	1	2	yes	—
71	C₄×#10596831	C₂³	1	1	yes	characteristic
72	C₄×#10596831	C₂³	1	2	yes	—
73	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
74	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
75	C₄×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
76	C₄⋊C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
77	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
78	#97×C₇₃	C₂×C₄	1	2	yes	—
79	C₄×C₈×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	C₂×D₂₉₂	C₂×C₈	1	2	yes	—
87	C₂×D₂₉₂	C₂×C₈	1	1	yes	characteristic
88	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
89	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
90	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
92	#11924011	—	2	4	no	—
93	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
94	C₂×#10596831	C₂²×C₄	1	2	yes	—
95	C₂×#10596831	C₂⁴	1	1	yes	characteristic
96	C₈×D₇₃	—	2	2	no	—
97	C₈×D₇₃	—	2	1	no	—
98	C₈×D₇₃	—	2	1	no	—
99	C₈×D₇₃	—	2	2	no	—
100	C₈×D₇₃	—	2	2	no	—
101	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
102	C₄⋊Dic₇₃	C₂×C₈	1	2	yes	—
103	#11924007	C₂×C₈	1	2	yes	—
104	#11924007	C₂×C₈	1	2	yes	—
105	#11924005	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	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
114	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
116	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
117	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
118	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
119	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
120	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
121	C₂²×D₇₃	C₂²×C₈	1	2	yes	—
122	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
123	Q₈×C₇₃	—	2	2	no	—
124	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
126	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
131	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
133	D₂₉₂	—	2	4	no	—
134	D₂₉₂	—	2	2	no	—
135	C₄×D₇₃	—	2	2	no	—
136	C₄×D₇₃	—	2	1	no	—
137	C₄×D₇₃	—	2	2	no	—
138	C₄×D₇₃	—	2	1	no	—
139	C₄×D₇₃	—	2	2	no	—
140	C₄×D₇₃	—	2	2	no	—
141	C₈×C₇₃	—	2	1	no	—
142	C₈×C₇₃	—	2	2	no	—
143	C₈×C₇₃	#131	1	2	yes	—
144	#10596831	—	2	1	no	—
145	#10596831	—	2	2	no	—
146	#10596831	C₂×#41	1	1	yes	characteristic
147	#10596831	C₂×#41	1	1	yes	characteristic
148	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
149	C₂×D₇₃	—	2	1	no	—
150	C₂×D₇₃	—	2	1	no	—
151	C₂×D₇₃	—	2	2	no	—
152	C₂×D₇₃	—	2	2	no	—
153	C₂×D₇₃	—	2	2	no	—
154	C₂×D₇₃	#504	1	2	yes	—
155	C₄×C₇₃	—	2	1	no	—
156	C₄×C₇₃	—	2	2	no	—
157	C₄×C₇₃	—	2	2	no	—
158	C₄×C₇₃	#504	1	2	yes	—
159	C₄×C₇₃	C₄×#41	1	1	yes	characteristic
160	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
161	Dic₇₃	—	2	1	no	—
162	Dic₇₃	—	2	1	no	—
163	Dic₇₃	—	2	1	no	—
164	Dic₇₃	—	2	2	no	—
165	#33480	—	73	1	no	maximal, Sylow
166	C₂×C₇₃	C₈×#41	1	1	yes	characteristic
167	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
168	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
169	D₇₃	—	2	2	no	—
170	D₇₃	—	4	2	no	—
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	—	73	1	no	—
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	#64408	1	1	yes	characteristic
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	—	73	2	no	—
217	#125	—	73	1	no	—
218	#124	—	73	1	no	—
219	#123	—	73	2	no	—
220	C₂²⋊Q₈	—	73	1	no	—
221	#121	—	73	1	no	—
222	C₄×Q₈	C₂×#64408	1	1	yes	characteristic
223	C₄×D₄	—	73	2	no	—
224	C₄×D₄	—	73	1	no	—
225	#117	—	73	2	no	—
226	C₂×C₄²	—	73	1	no	—
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	2	no	—
239	C₄⋊C₈	C₂×#64408	1	1	yes	characteristic
240	C₄⋊C₈	C₂×#64408	1	2	yes	—
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	C₂×#64408	1	2	yes	—
246	C₄×C₈	—	73	1	no	—
247	C₄×C₈	—	73	1	no	—
248	C₄×C₈	—	73	1	no	—
249	C₄×C₈	—	73	2	no	—
250	C₄×C₈	C₂×#64408	1	1	yes	characteristic
251	C₂×Q₈	#11924036	1	1	yes	characteristic
252	C₂×D₄	—	73	2	no	—
253	C₂×D₄	—	73	1	no	—
254	C₂²×C₄	—	73	1	no	—
255	C₂²×C₄	—	73	1	no	—
256	C₂²×C₄	—	73	1	no	—
257	C₂²×C₄	—	73	2	no	—
258	M₁₆	—	146	4	no	—
259	C₂×C₈	—	73	1	no	—
260	C₂×C₈	—	73	1	no	—
261	C₂×C₈	—	73	2	no	—
262	C₂×C₈	—	146	1	no	—
263	C₂×C₈	—	146	1	no	—
264	C₂×C₈	—	146	2	no	—
265	C₂×C₈	—	146	2	no	—
266	C₂×C₈	—	146	2	no	—
267	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
268	C₂×C₈	C₂²×#64408	1	2	yes	—
269	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
270	C₄⋊C₄	—	73	2	no	—
271	C₄⋊C₄	—	73	2	no	—
272	C₄⋊C₄	—	73	1	no	—
273	C₄⋊C₄	—	73	2	no	—
274	C₄⋊C₄	#11924036	1	1	yes	characteristic
275	C₄⋊C₄	#11924036	1	2	yes	—
276	C₂²⋊C₄	—	73	2	no	—
277	C₂²⋊C₄	—	73	2	no	—
278	C₂²⋊C₄	—	73	2	no	—
279	C₂²⋊C₄	—	73	2	no	—
280	C₂²⋊C₄	—	73	2	no	—
281	C₄²	—	73	1	no	—
282	C₄²	—	73	1	no	—
283	C₄²	—	73	1	no	—
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	—	73	1	no	—
293	C₂³	—	73	2	no	—
294	Q₈	—	2	2	no	—
295	D₄	—	146	4	no	—
296	D₄	—	146	2	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	2	no	—
304	C₂×C₄	—	146	1	no	—
305	C₂×C₄	—	146	1	no	—
306	C₂×C₄	—	146	2	no	—
307	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
308	C₂×C₄	C₂×#11924036	1	2	yes	—
309	C₂×C₄	C₂×#11924036	1	2	yes	—
310	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
311	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	2	no	—
314	C₈	—	73	1	no	—
315	C₈	—	73	1	no	—
316	C₈	—	146	1	no	—
317	C₈	—	146	2	no	—
318	C₈	#14556679	1	2	yes	—
319	C₂²	—	73	2	no	—
320	C₂²	—	146	2	no	—
321	C₂²	—	146	2	no	—
322	C₂²	—	146	2	no	—
323	C₂²	—	146	1	no	—
324	C₂²	—	146	1	no	—
325	C₂²	C₂²×#11924036	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	146	1	no	—
330	C₄	—	146	1	no	—
331	C₄	—	146	1	no	—
332	C₄	—	146	2	no	—
333	C₄	#17989646	1	2	yes	—
334	C₄	#41×#64408	1	1	yes	characteristic
335	C₄	C₂×#14556679	1	1	yes	characteristic
336	C₂	—	292	2	no	—
337	C₂	—	146	2	no	—
338	C₂	#22626737	1	1	yes	characteristic
339	C₂	#22626896	1	1	yes	characteristic
340	C₂	C₂×#17989646	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	#33480
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. Q₈×#11924031	C₂	semidirect
3. #22626317	C₂	semidirect
4. #22626314	C₂	semidirect
5. #22626310	C₂	semidirect
6. #22626310	C₂	semidirect
7. #22626299	C₂	semidirect
8. #22626295	C₂	semidirect
9. #22626141	C₂	semidirect
10. C₄×#14556664	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
#22626317	2
#22626314	2
#22626310	2
#22626299	2
#22626295	2
#22626141	2
C₄×#14556664	2
#22618905	2
#33480	73