Subgroups — #34034864

Order 18688.

498 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627158	C₂	1	2	yes	maximal, complemented
2	#22627158	C₂	1	2	yes	maximal, complemented
3	#22627156	C₂	1	2	yes	maximal, complemented
4	#22627155	C₂	1	2	yes	maximal, complemented
5	#22627123	C₂	1	1	yes	maximal, characteristic
6	#22627123	C₂	1	1	yes	maximal, characteristic
7	#22627117	C₂	1	1	yes	maximal, characteristic
8	#22627093	C₂	1	1	yes	maximal, characteristic
9	#22627093	C₂	1	1	yes	maximal, characteristic
10	C₄×#14556688	C₂	1	1	yes	maximal, characteristic
11	#22626721	C₂	1	1	yes	maximal, characteristic
12	C₂×#14556688	C₂²	1	1	yes	characteristic
13	C₂×#14556688	C₂²	1	1	yes	characteristic
14	C₂×#14556686	C₂²	1	1	yes	characteristic
15	C₂×#14556686	C₂²	1	1	yes	characteristic
16	#17989690	C₂²	1	1	yes	characteristic
17	#17989690	C₂²	1	1	yes	characteristic
18	#17989690	C₂²	1	2	yes	—
19	#17989690	C₂²	1	2	yes	—
20	C₄⋊#11924040	C₂²	1	1	yes	characteristic
21	C₄⋊#11924040	C₂²	1	1	yes	characteristic
22	C₄⋊#11924040	C₂²	1	2	yes	—
23	C₄⋊#11924040	C₂²	1	2	yes	—
24	#17989681	C₂²	1	2	yes	—
25	#17989681	C₂²	1	2	yes	—
26	#17989681	C₂²	1	2	yes	—
27	C₄×#11924040	C₂²	1	2	yes	—
28	C₄×#11924040	C₂²	1	2	yes	—
29	C₄×#11924040	C₂²	1	2	yes	—
30	#17989639	C₄	1	2	yes	—
31	#17989639	C₄	1	2	yes	—
32	#17989636	C₂²	1	2	yes	—
33	#17989636	C₂²	1	2	yes	—
34	#17989594	C₄	1	1	yes	characteristic
35	#17989594	C₄	1	1	yes	characteristic
36	#17989592	C₄	1	1	yes	characteristic
37	#17989552	C₂²	1	1	yes	characteristic
38	#17989552	C₂²	1	1	yes	characteristic
39	C₄×#11924036	C₂²	1	1	yes	characteristic
40	#17989143	C₄	1	1	yes	characteristic
41	#14556688	—	2	2	no	—
42	#14556688	Q₈	1	4	yes	complemented
43	#14556686	—	2	2	no	—
44	#14556686	—	2	2	no	—
45	C₂×#11924040	C₂³	1	1	yes	characteristic
46	C₂×#11924040	C₂³	1	1	yes	characteristic
47	C₂×#11924040	C₂³	1	2	yes	—
48	C₂×#11924040	C₂³	1	1	yes	characteristic
49	C₂×#11924040	C₂³	1	1	yes	characteristic
50	C₂×#11924040	C₂³	1	2	yes	—
51	C₂×#11924038	C₂×C₄	1	1	yes	characteristic
52	C₂×#11924038	C₂×C₄	1	1	yes	characteristic
53	C₂×#11924036	C₂×C₄	1	1	yes	characteristic
54	C₂×#11924036	C₂³	1	1	yes	characteristic
55	#14556658	C₂×C₄	1	2	yes	—
56	#14556658	C₂×C₄	1	2	yes	—
57	#14556658	C₂×C₄	1	2	yes	—
58	#14556658	C₂×C₄	1	2	yes	—
59	C₄⋊#10596833	C₂×C₄	1	2	yes	—
60	C₄⋊#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	2	yes	—
68	C₄×#10596833	C₂³	1	1	yes	characteristic
69	#14556590	C₈	1	1	yes	characteristic
70	#14556590	C₈	1	1	yes	characteristic
71	#14556583	C₈	1	1	yes	characteristic
72	#14556569	C₈	1	2	yes	—
73	#14556569	C₈	1	2	yes	—
74	#14556556	C₂×C₄	1	2	yes	—
75	#14556556	C₂×C₄	1	2	yes	—
76	#14556549	C₂×C₄	1	1	yes	characteristic
77	#14556549	C₂×C₄	1	1	yes	characteristic
78	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
79	#125×C₇₃	C₈	1	1	yes	characteristic
80	#11924040	—	2	1	no	—
81	#11924040	—	2	1	no	—
82	#11924040	—	2	1	no	—
83	#11924040	—	2	1	no	—
84	#11924040	—	2	2	no	—
85	#11924040	C₂×Q₈	1	4	yes	—
86	#11924038	—	2	2	no	—
87	#11924038	—	2	2	no	—
88	#11924036	—	2	2	no	—
89	#11924036	C₂×Q₈	1	2	yes	—
90	#11924036	#41	1	2	yes	—
91	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
92	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
93	C₂×#10596833	C₂²×C₄	1	2	yes	—
94	C₂×#10596833	C₂²×C₄	1	2	yes	—
95	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
96	C₂×#10596833	C₂⁴	1	1	yes	characteristic
97	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
98	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
99	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
100	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
101	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
102	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
103	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
104	#11924007	C₂×C₈	1	2	yes	—
105	#11924007	C₂×C₈	1	2	yes	—
106	#11924007	C₂×C₈	1	2	yes	—
107	#11924007	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	C₂²×C₄	1	1	yes	characteristic
112	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
113	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
114	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
115	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
116	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
117	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
118	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
119	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
120	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
124	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
128	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
129	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×Dic₇₃	C₂³×C₄	1	1	yes	characteristic
131	C₄×D₇₃	—	2	2	no	—
132	C₄×D₇₃	—	4	1	no	—
133	C₄×D₇₃	C₄×Q₈	1	2	yes	—
134	C₄×D₇₃	#131	1	2	yes	—
135	Dic₁₄₆	—	2	4	no	—
136	Dic₁₄₆	—	4	2	no	—
137	#10596833	—	2	1	no	—
138	#10596833	—	2	1	no	—
139	#10596833	—	2	2	no	—
140	#10596833	—	2	2	no	—
141	#10596833	C₄×Q₈	1	2	yes	—
142	#10596833	C₂²×Q₈	1	1	yes	characteristic
143	#10596833	C₂×#41	1	1	yes	characteristic
144	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
145	C₂×D₇₃	—	2	1	no	—
146	C₂×D₇₃	—	2	1	no	—
147	C₂×D₇₃	C₈×Q₈	1	2	yes	—
148	C₄×C₇₃	—	2	2	no	—
149	C₄×C₇₃	—	2	1	no	—
150	C₄×C₇₃	—	2	2	no	—
151	C₄×C₇₃	—	2	1	no	—
152	C₄×C₇₃	C₈×Q₈	1	2	yes	—
153	Dic₇₃	—	2	2	no	—
154	Dic₇₃	—	2	1	no	—
155	Dic₇₃	—	2	2	no	—
156	Dic₇₃	#504	1	2	yes	—
157	Dic₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
158	Dic₇₃	C₂×#131	1	1	yes	characteristic
159	#33476	—	73	1	no	maximal, Sylow
160	C₂×C₇₃	C₂×C₈×Q₈	1	1	yes	characteristic
161	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
162	C₂×C₇₃	#3407	1	1	yes	characteristic
163	D₇₃	—	2	2	no	—
164	#2960	—	73	1	no	—
165	#2183	—	73	2	no	—
166	#2183	—	73	2	no	—
167	#2183	—	73	2	no	—
168	Q₈×C₁₆	—	73	2	no	—
169	#2151	—	73	1	no	—
170	#2151	—	73	1	no	—
171	C₂×C₄⋊C₁₆	—	73	1	no	—
172	#2108	—	73	1	no	—
173	#2108	—	73	1	no	—
174	C₄×M₃₂	—	73	1	no	—
175	C₇₃	#33476	1	1	yes	characteristic, complemented, Sylow
176	#533	—	73	1	no	—
177	C₂×M₃₂	—	73	1	no	—
178	C₂×M₃₂	—	73	1	no	—
179	C₂²×C₁₆	—	73	1	no	—
180	C₂²×C₁₆	—	73	1	no	—
181	#446	—	73	2	no	—
182	#446	—	73	2	no	—
183	C₈×Q₈	—	73	2	no	—
184	C₈×Q₈	—	73	2	no	—
185	#424	—	73	1	no	—
186	#424	—	73	1	no	—
187	#423	—	73	1	no	—
188	#405	—	73	1	no	—
189	#405	—	73	1	no	—
190	C₂×C₄×C₈	—	73	1	no	—
191	C₄⋊C₁₆	—	73	1	no	—
192	C₄⋊C₁₆	—	73	1	no	—
193	C₄⋊C₁₆	—	73	2	no	—
194	C₄⋊C₁₆	—	73	2	no	—
195	C₄⋊C₁₆	—	73	1	no	—
196	C₄⋊C₁₆	—	73	1	no	—
197	C₄⋊C₁₆	—	73	2	no	—
198	C₄⋊C₁₆	—	73	2	no	—
199	#346	—	73	2	no	—
200	#346	—	73	2	no	—
201	#346	—	73	2	no	—
202	C₄×C₁₆	—	73	2	no	—
203	C₄×C₁₆	—	73	2	no	—
204	C₄×C₁₆	—	73	2	no	—
205	C₂×M₁₆	—	73	1	no	—
206	C₂×M₁₆	—	73	1	no	—
207	C₂²×C₈	—	73	1	no	—
208	C₂²×C₈	—	73	1	no	—
209	C₄⋊Q₈	—	73	1	no	—
210	C₄⋊Q₈	—	73	1	no	—
211	#125	—	73	1	no	—
212	#125	#10596842	1	1	yes	characteristic
213	C₂²⋊Q₈	—	73	2	no	—
214	C₂²⋊Q₈	—	73	2	no	—
215	C₄×Q₈	—	73	2	no	—
216	C₄×Q₈	—	73	2	no	—
217	#117	—	73	1	no	—
218	#117	—	73	1	no	—
219	C₂×C₄²	—	73	1	no	—
220	M₃₂	—	73	4	no	—
221	M₃₂	—	146	2	no	—
222	C₂×C₁₆	—	73	2	no	—
223	C₂×C₁₆	—	73	1	no	—
224	C₂×C₁₆	—	73	1	no	—
225	C₂×C₁₆	—	73	2	no	—
226	C₂×C₁₆	—	73	1	no	—
227	C₂×C₁₆	—	73	1	no	—
228	C₂×C₁₆	—	146	2	no	—
229	C₂×C₁₆	—	146	2	no	—
230	C₄⋊C₈	—	73	2	no	—
231	C₄⋊C₈	—	73	2	no	—
232	C₄⋊C₈	—	73	2	no	—
233	C₄⋊C₈	—	73	2	no	—
234	C₄⋊C₈	—	73	2	no	—
235	C₄⋊C₈	—	73	2	no	—
236	#97	—	73	2	no	—
237	#97	—	73	2	no	—
238	C₄×C₈	—	73	1	no	—
239	C₄×C₈	—	73	2	no	—
240	C₄×C₈	—	73	2	no	—
241	C₄×C₈	—	73	1	no	—
242	C₄×C₈	—	73	1	no	—
243	C₄×C₈	—	73	1	no	—
244	C₂×Q₈	—	73	2	no	—
245	C₂×Q₈	—	73	2	no	—
246	C₂²×C₄	—	73	1	no	—
247	C₂²×C₄	—	73	1	no	—
248	C₂²×C₄	—	73	1	no	—
249	M₁₆	—	146	2	no	—
250	M₁₆	—	146	2	no	—
251	C₂×C₈	—	73	1	no	—
252	C₂×C₈	—	73	1	no	—
253	C₂×C₈	—	73	2	no	—
254	C₂×C₈	—	73	2	no	—
255	C₂×C₈	—	73	1	no	—
256	C₂×C₈	—	73	1	no	—
257	C₂×C₈	—	73	2	no	—
258	C₂×C₈	—	73	2	no	—
259	C₂×C₈	—	146	2	no	—
260	C₄⋊C₄	—	73	2	no	—
261	C₄⋊C₄	—	73	2	no	—
262	C₄⋊C₄	—	73	2	no	—
263	C₄⋊C₄	—	73	2	no	—
264	C₄⋊C₄	—	73	1	no	—
265	C₄⋊C₄	—	73	1	no	—
266	C₄⋊C₄	C₂×#10596842	1	1	yes	characteristic
267	C₄⋊C₄	C₂×#10596842	1	1	yes	characteristic
268	C₄⋊C₄	C₂×#10596842	1	2	yes	—
269	C₄⋊C₄	C₂×#10596842	1	2	yes	—
270	C₂²⋊C₄	—	73	2	no	—
271	C₂²⋊C₄	—	73	2	no	—
272	C₄²	—	73	1	no	—
273	C₄²	—	73	1	no	—
274	C₄²	—	73	1	no	—
275	C₄²	—	73	2	no	—
276	C₄²	—	73	2	no	—
277	C₄²	C₂×#10596842	1	1	yes	characteristic
278	C₁₆	—	73	4	no	—
279	C₁₆	—	146	1	no	—
280	C₁₆	—	146	1	no	—
281	C₁₆	—	146	1	no	—
282	C₁₆	—	146	1	no	—
283	C₁₆	—	146	2	no	—
284	C₂³	—	73	1	no	—
285	Q₈	—	292	2	no	—
286	Q₈	—	146	4	no	—
287	C₂×C₄	—	292	1	no	—
288	C₂×C₄	—	73	1	no	—
289	C₂×C₄	—	73	1	no	—
290	C₂×C₄	—	73	2	no	—
291	C₂×C₄	—	73	2	no	—
292	C₂×C₄	—	73	1	no	—
293	C₂×C₄	—	73	2	no	—
294	C₂×C₄	—	73	2	no	—
295	C₂×C₄	—	146	2	no	—
296	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
297	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
298	C₂×C₄	C₂²×#10596842	1	2	yes	—
299	C₂×C₄	C₂²×#10596842	1	2	yes	—
300	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
301	C₈	—	73	1	no	—
302	C₈	—	73	1	no	—
303	C₈	—	73	2	no	—
304	C₈	—	146	1	no	—
305	C₈	—	146	1	no	—
306	C₈	—	146	2	no	—
307	C₈	—	146	2	no	—
308	C₂²	—	73	2	no	—
309	C₂²	—	146	1	no	—
310	C₂²	—	146	1	no	—
311	C₂²	C₂³×#10596842	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	2	no	—
316	C₄	—	73	1	no	—
317	C₄	—	73	1	no	—
318	C₄	—	73	2	no	—
319	C₄	—	146	2	no	—
320	C₄	—	146	1	no	—
321	C₄	—	146	2	no	—
322	C₄	Q₈×#10596842	1	2	yes	—
323	C₂	—	146	2	no	—
324	C₂	C₂×Q₈×#10596842	1	1	yes	characteristic
325	C₂	#22627182	1	1	yes	characteristic
326	C₂	#22627185	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10596842
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	#125×C₇₃	C₈
prime radical	C₂²×D₇₃	C₂²×C₈
cosocle	C₂×#10596833	C₂⁴
hypercenter	#125	#10596842
hypocenter	C₇₃	#33476
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #22627158	C₂	semidirect
2. #22627158	C₂	semidirect
3. #22627156	C₂	semidirect
4. #22627155	C₂	semidirect
42. #14556688	Q₈	semidirect
175. C₇₃	#33476	semidirect

Maximal subgroups

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

subgroup	index
#22627158	2
#22627156	2
#22627155	2
#22627123	2
#22627117	2
#22627093	2
C₄×#14556688	2
#22626721	2
#33476	73