Subgroups — #34030338

Order 18688.

558 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626346	C₂	1	2	yes	maximal, complemented
2	#22626315	C₂	1	2	yes	maximal, complemented
3	#22626311	C₂	1	4	yes	maximal, complemented
4	#22626288	C₂	1	2	yes	maximal, complemented
5	#22626288	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14556665	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×#11924031	C₂	1	1	yes	maximal, characteristic, complemented
8	#41×#10596831	C₂	1	1	yes	maximal, characteristic
9	C₂×#14556665	C₂²	1	4	yes	—
10	C₂²×#11924031	C₂²	1	2	yes	—
11	C₄⋊#11924031	C₂²	1	4	yes	—
12	C₄⋊#11924031	C₂²	1	2	yes	—
13	C₂²⋊#11924031	C₂²	1	2	yes	—
14	C₂²⋊#11924031	C₂²	1	4	yes	—
15	#17989475	C₂²	1	4	yes	—
16	C₄×#11924031	C₂²	1	2	yes	—
17	C₄×#11924031	C₂²	1	1	yes	characteristic
18	C₄×#11924031	C₂²	1	1	yes	characteristic
19	C₄×#11924031	C₂²	1	2	yes	—
20	Q₈×#10596831	C₄	1	1	yes	characteristic
21	Q₈×#10596831	C₂²	1	1	yes	characteristic
22	D₄×#10596831	C₄	1	2	yes	—
23	D₄×#10596831	C₄	1	1	yes	characteristic
24	D₄×#10596831	C₂²	1	2	yes	—
25	D₄×#10596831	C₂²	1	1	yes	characteristic
26	#17988640	C₄	1	2	yes	—
27	#17988640	C₄	1	1	yes	characteristic
28	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
29	C₂×C₄×#10596831	C₂²	1	2	yes	—
30	C₄×#41×C₇₃	C₄	1	1	yes	characteristic
31	#14556665	—	2	8	no	—
32	C₂×#11924031	—	2	4	no	—
33	C₂×#11924031	C₂³	1	2	yes	—
34	C₂×#11924031	C₂³	1	4	yes	—
35	C₂×#11924031	C₂³	1	2	yes	—
36	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
37	C₂²×#10596831	C₂×C₄	1	2	yes	—
38	C₂²×#10596831	C₂³	1	1	yes	characteristic
39	C₂²×#10596831	C₂³	1	2	yes	—
40	C₄⋊#10596831	C₂×C₄	1	2	yes	—
41	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
42	C₄⋊#10596831	C₂×C₄	1	2	yes	—
43	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
44	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
45	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
46	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
47	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
48	C₄×#10596831	C₂×C₄	1	2	yes	—
49	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
50	C₄×#10596831	C₂×C₄	1	2	yes	—
51	C₄×#10596831	C₂³	1	1	yes	characteristic
52	C₄×#10596831	C₂³	1	2	yes	—
53	C₄×#10596831	C₂³	1	1	yes	characteristic
54	C₂×#41×C₇₃	C₈	1	1	yes	characteristic
55	C₄×Q₈×C₇₃	C₈	1	1	yes	characteristic
56	C₄×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
57	C₄×D₄×C₇₃	C₈	1	2	yes	—
58	C₄×D₄×C₇₃	C₈	1	1	yes	characteristic
59	C₄×D₄×C₇₃	C₂×C₄	1	2	yes	—
60	C₄×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
61	#117×C₇₃	C₈	1	2	yes	—
62	#117×C₇₃	C₈	1	1	yes	characteristic
63	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
64	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
65	#11924031	—	2	2	no	—
66	#11924031	—	2	4	no	—
67	#11924031	#41	1	4	yes	complemented
68	C₂×#10596831	—	2	2	no	—
69	C₂×#10596831	—	2	4	no	—
70	C₂×#10596831	—	2	2	no	—
71	C₂×#10596831	—	2	4	no	—
72	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
73	C₂×#10596831	C₂²×C₄	1	2	yes	—
74	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
75	C₂×#10596831	C₂²×C₄	1	2	yes	—
76	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
77	C₂×#10596831	C₂⁴	1	1	yes	characteristic
78	#41×C₇₃	—	2	4	no	—
79	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
80	C₂×D₄×C₇₃	C₂×C₈	1	2	yes	—
81	C₂×D₄×C₇₃	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
83	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
84	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
85	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
86	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
88	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
89	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
90	C₄⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
91	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
92	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
93	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
94	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
95	C₄²×C₇₃	C₂×C₈	1	2	yes	—
96	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
97	C₄²×C₇₃	C₂×C₈	1	2	yes	—
98	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
99	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
100	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
102	C₂³×C₇₃	C₂²×C₈	1	2	yes	—
103	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
104	Q₈×C₇₃	—	2	2	no	—
105	D₄×C₇₃	—	2	4	no	—
106	D₄×C₇₃	—	2	2	no	—
107	C₂×C₄×C₇₃	—	2	1	no	—
108	C₂×C₄×C₇₃	—	2	2	no	—
109	C₂×C₄×C₇₃	—	2	1	no	—
110	C₂×C₄×C₇₃	—	2	2	no	—
111	C₂×C₄×C₇₃	—	2	4	no	—
112	C₂×C₄×C₇₃	—	2	2	no	—
113	C₂×C₄×C₇₃	—	2	4	no	—
114	C₂×C₄×C₇₃	M₃₂	1	4	yes	complemented
115	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
118	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
120	C₂×C₄×C₇₃	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	C₂³×C₄	1	1	yes	characteristic
124	#10596831	—	2	1	no	—
125	#10596831	—	2	2	no	—
126	#10596831	—	2	1	no	—
127	#10596831	—	2	2	no	—
128	#10596831	#131	1	2	yes	—
129	#10596831	C₂×#41	1	2	yes	—
130	C₂²×C₇₃	—	2	1	no	—
131	C₂²×C₇₃	—	2	1	no	—
132	C₂²×C₇₃	—	2	2	no	—
133	C₂²×C₇₃	—	2	2	no	—
134	C₂²×C₇₃	—	2	2	no	—
135	C₂²×C₇₃	C₂×M₃₂	1	2	yes	—
136	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
137	C₄×C₇₃	—	2	2	no	—
138	C₄×C₇₃	—	2	1	no	—
139	C₄×C₇₃	—	2	2	no	—
140	C₄×C₇₃	—	2	1	no	—
141	C₄×C₇₃	—	2	2	no	—
142	C₄×C₇₃	C₂×M₃₂	1	2	yes	—
143	C₄×C₇₃	C₂×M₃₂	1	2	yes	—
144	C₄×C₇₃	#504	1	2	yes	—
145	C₄×C₇₃	C₄×#41	1	1	yes	characteristic
146	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
147	#33479	—	73	1	no	maximal, Sylow
148	C₂×C₇₃	—	2	2	no	—
149	C₂×C₇₃	—	4	2	no	—
150	C₂×C₇₃	C₈×#41	1	1	yes	characteristic
151	C₂×C₇₃	C₂²×M₃₂	1	1	yes	characteristic
152	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
153	C₈×#41	—	73	1	no	—
154	#2183	—	73	2	no	—
155	#2169	—	73	2	no	—
156	#2168	—	73	4	no	—
157	#2162	—	73	2	no	—
158	#2162	—	73	1	no	—
159	C₄×M₃₂	—	73	2	no	—
160	C₂×C₄×C₁₆	—	73	1	no	—
161	C₇₃	#33479	1	1	yes	characteristic, complemented, Sylow
162	C₄×#41	#64408	1	1	yes	characteristic
163	C₂×M₃₂	—	73	4	no	—
164	C₂²×C₁₆	—	73	2	no	—
165	C₈×Q₈	—	73	1	no	—
166	C₈×Q₈	—	73	1	no	—
167	C₈×D₄	—	73	1	no	—
168	C₈×D₄	—	73	2	no	—
169	C₈×D₄	—	73	2	no	—
170	C₈×D₄	—	73	1	no	—
171	#431	—	73	2	no	—
172	#431	—	73	1	no	—
173	C₂×C₄×C₈	—	73	1	no	—
174	C₂×C₄×C₈	—	73	2	no	—
175	C₄⋊C₁₆	—	73	4	no	—
176	C₄⋊C₁₆	—	73	2	no	—
177	C₂²⋊C₁₆	—	73	2	no	—
178	C₂²⋊C₁₆	—	73	4	no	—
179	#346	—	73	4	no	—
180	C₄×C₁₆	—	73	1	no	—
181	C₄×C₁₆	—	73	1	no	—
182	C₄×C₁₆	—	73	2	no	—
183	C₄×C₁₆	—	73	2	no	—
184	C₂×#41	#10596833	1	1	yes	characteristic
185	C₂²×C₈	—	73	1	no	—
186	C₂²×C₈	—	73	2	no	—
187	C₂²×C₈	—	73	1	no	—
188	C₂²×C₈	—	73	2	no	—
189	C₄×Q₈	#10596833	1	1	yes	characteristic
190	C₄×Q₈	C₂×#64408	1	1	yes	characteristic
191	C₄×D₄	#10596833	1	2	yes	—
192	C₄×D₄	#10596833	1	1	yes	characteristic
193	C₄×D₄	C₂×#64408	1	2	yes	—
194	C₄×D₄	C₂×#64408	1	1	yes	characteristic
195	#117	#10596833	1	2	yes	—
196	#117	#10596833	1	1	yes	characteristic
197	C₂×C₄²	C₂×#64408	1	2	yes	—
198	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
199	M₃₂	—	146	8	no	—
200	C₂×C₁₆	—	73	2	no	—
201	C₂×C₁₆	—	73	2	no	—
202	C₂×C₁₆	—	73	4	no	—
203	C₂×C₁₆	—	146	4	no	—
204	C₄⋊C₈	—	73	2	no	—
205	C₄⋊C₈	—	73	1	no	—
206	C₄⋊C₈	—	73	2	no	—
207	C₄⋊C₈	—	73	1	no	—
208	C₂²⋊C₈	—	73	2	no	—
209	C₂²⋊C₈	—	73	2	no	—
210	C₂²⋊C₈	—	73	2	no	—
211	C₄×C₈	—	73	1	no	—
212	C₄×C₈	—	73	2	no	—
213	C₄×C₈	—	73	1	no	—
214	C₄×C₈	—	73	2	no	—
215	C₄×C₈	—	73	1	no	—
216	C₄×C₈	—	73	2	no	—
217	C₄×C₈	—	73	1	no	—
218	#41	—	2	4	no	—
219	C₂×Q₈	C₂×#10596833	1	1	yes	characteristic
220	C₂×D₄	C₂×#10596833	1	2	yes	—
221	C₂×D₄	C₂×#10596833	1	1	yes	characteristic
222	C₂²×C₄	C₂×#10596833	1	2	yes	—
223	C₂²×C₄	C₂×#10596833	1	2	yes	—
224	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
225	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
226	C₂²×C₄	C₂²×#64408	1	2	yes	—
227	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
228	C₂×C₈	—	73	1	no	—
229	C₂×C₈	—	73	1	no	—
230	C₂×C₈	—	73	1	no	—
231	C₂×C₈	—	73	2	no	—
232	C₂×C₈	—	73	1	no	—
233	C₂×C₈	—	73	2	no	—
234	C₂×C₈	—	146	2	no	—
235	C₂×C₈	—	146	4	no	—
236	C₂×C₈	—	146	2	no	—
237	C₂×C₈	—	146	4	no	—
238	C₄⋊C₄	C₂×#10596833	1	2	yes	—
239	C₄⋊C₄	C₂×#10596833	1	1	yes	characteristic
240	C₄⋊C₄	C₂×#10596833	1	2	yes	—
241	C₄⋊C₄	C₂×#10596833	1	1	yes	characteristic
242	C₂²⋊C₄	C₂×#10596833	1	2	yes	—
243	C₂²⋊C₄	C₂×#10596833	1	2	yes	—
244	C₂²⋊C₄	C₂×#10596833	1	2	yes	—
245	C₄²	C₂×#10596833	1	2	yes	—
246	C₄²	C₂×#10596833	1	1	yes	characteristic
247	C₄²	C₂×#10596833	1	2	yes	—
248	C₄²	C₂×#10596833	1	1	yes	characteristic
249	C₄²	C₂²×#64408	1	2	yes	—
250	C₄²	C₂²×#64408	1	1	yes	characteristic
251	C₄²	C₂²×#64408	1	1	yes	characteristic
252	C₁₆	—	73	4	no	—
253	C₁₆	—	146	2	no	—
254	C₁₆	—	146	4	no	—
255	C₂³	C₂²×#10596833	1	2	yes	—
256	C₂³	C₂²×#10596833	1	1	yes	characteristic
257	Q₈	—	2	2	no	—
258	D₄	—	2	4	no	—
259	D₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	#14556665	1	4	yes	complemented
268	C₂×C₄	C₂²×#10596833	1	2	yes	—
269	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
270	C₂×C₄	C₂²×#10596833	1	2	yes	—
271	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
272	C₂×C₄	C₂²×#10596833	1	2	yes	—
273	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
274	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
275	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
276	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
277	C₈	—	73	2	no	—
278	C₈	—	73	2	no	—
279	C₈	—	146	1	no	—
280	C₈	—	146	2	no	—
281	C₈	—	146	1	no	—
282	C₈	—	146	2	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	C₂×#14556665	1	2	yes	—
289	C₂²	C₂³×#10596833	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	2	no	—
295	C₄	C₂×#14556665	1	2	yes	—
296	C₄	C₂×#14556665	1	2	yes	—
297	C₄	#17989645	1	2	yes	—
298	C₄	#41×#64408	1	1	yes	characteristic
299	C₄	D₇₃⋊#131	1	1	yes	characteristic
300	C₂	—	2	2	no	—
301	C₂	—	4	2	no	—
302	C₂	#41×#10596833	1	1	yes	characteristic
303	C₂	C₂²×#14556665	1	1	yes	characteristic
304	C₂	C₂×#17989645	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₄×#41×C₇₃	C₄
prime radical	C₂×#41×C₇₃	C₈
cosocle	C₂×#10596831	C₂⁴
hypercenter	C₄×#41	#64408
hypocenter	C₇₃	#33479
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #22626346	C₂	semidirect
2. #22626315	C₂	semidirect
3. #22626311	C₂	semidirect
4. #22626288	C₂	semidirect
5. #22626288	C₂	semidirect
6. C₄×#14556665	C₂	semidirect
7. C₂×C₄×#11924031	C₂	semidirect
67. #11924031	#41	semidirect
114. C₂×C₄×C₇₃	M₃₂	semidirect
161. C₇₃	#33479	semidirect
267. C₂×C₄	#14556665	semidirect

Maximal subgroups

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

subgroup	index
#22626346	2
#22626315	2
#22626311	2
#22626288	2
C₄×#14556665	2
C₂×C₄×#11924031	2
#41×#10596831	2
#33479	73