Subgroups — #15676484

Order 2944.

496 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625059	C₂	1	1	yes	maximal, characteristic
2	#13092323	C₂	1	2	yes	maximal
3	#13092321	C₂	1	2	yes	maximal
4	#13092316	C₂	1	1	yes	maximal, characteristic
5	#13092315	C₂	1	1	yes	maximal, characteristic
6	#13092314	C₂	1	2	yes	maximal
7	C₄⋊C₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic
8	C₄⋊C₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic
9	C₄×Dic₂₃⋊C₄	C₂	1	2	yes	maximal
10	C₄×Dic₂₃⋊C₄	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₄₆	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
22	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
23	C₂×Dic₂₃⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
25	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
27	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
28	#10625059	C₄	1	8	yes	complemented
29	C₂₃×#95	C₂²	1	2	yes	—
30	C₂₃⋊#95	C₂²	1	2	yes	—
31	C₂₃⋊#95	C₂²	1	2	yes	—
32	C₂₃⋊#95	C₂²	1	1	yes	characteristic
33	C₂₃⋊#95	C₂²	1	1	yes	characteristic
34	C₂₃⋊#95	C₂²	1	2	yes	—
35	C₂₃⋊#95	C₂²	1	2	yes	—
36	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
41	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
42	C₂²×Dic₂₃	C₂³	1	2	yes	—
43	C₂²×Dic₂₃	C₂³	1	2	yes	—
44	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
45	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
46	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
47	C₂×Dic₄₆	—	4	2	no	—
48	C₂×Dic₄₆	—	4	2	no	—
49	C₂×Dic₄₆	C₂×C₄	1	8	yes	—
50	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
51	C₄⋊C₄×C₂₃	Q₈	1	4	yes	complemented
52	C₄²×C₂₃	—	2	2	no	—
53	C₄²×C₂₃	—	4	2	no	—
54	C₄⋊Dic₂₃	—	2	2	no	—
55	C₄⋊Dic₂₃	—	4	2	no	—
56	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	—
57	Dic₂₃⋊C₄	—	2	2	no	—
58	Dic₂₃⋊C₄	—	4	2	no	—
59	Dic₂₃⋊C₄	C₂×C₄	1	8	yes	—
60	Dic₂₃⋊C₄	Q₈	1	4	yes	complemented
61	C₄×Dic₂₃	—	2	2	no	—
62	C₄×Dic₂₃	—	2	4	no	—
63	C₄×Dic₂₃	—	4	2	no	—
64	C₄×Dic₂₃	C₂×C₄	1	4	yes	—
65	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
66	C₂×C₄×C₂₃	—	2	2	no	—
67	C₂×C₄×C₂₃	—	2	1	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	2	1	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	1	no	—
73	C₂×C₄×C₂₃	—	2	1	no	—
74	C₂×C₄×C₂₃	—	2	1	no	—
75	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
76	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
78	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
79	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
80	C₂×C₄×C₂₃	#41	1	2	yes	—
81	C₂×Dic₂₃	—	2	1	no	—
82	C₂×Dic₂₃	—	2	1	no	—
83	C₂×Dic₂₃	—	2	2	no	—
84	C₂×Dic₂₃	—	2	2	no	—
85	C₂×Dic₂₃	—	2	1	no	—
86	C₂×Dic₂₃	—	2	1	no	—
87	C₂×Dic₂₃	—	2	1	no	—
88	C₂×Dic₂₃	—	2	2	no	—
89	C₂×Dic₂₃	—	2	1	no	—
90	C₂×Dic₂₃	—	2	2	no	—
91	C₂×Dic₂₃	—	2	2	no	—
92	C₂×Dic₂₃	—	2	1	no	—
93	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
94	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
95	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
96	C₂×Dic₂₃	C₂×Q₈	1	2	yes	—
97	C₂×Dic₂₃	C₂×Q₈	1	2	yes	—
98	C₂×Dic₂₃	#41	1	2	yes	—
99	Dic₄₆	—	4	8	no	—
100	#2369	—	23	1	no	maximal, Sylow
101	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
102	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
103	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
104	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
105	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
106	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
107	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
108	C₄×C₂₃	—	2	4	no	—
109	C₄×C₂₃	—	2	2	no	—
110	C₄×C₂₃	—	2	4	no	—
111	C₄×C₂₃	—	4	1	no	—
112	C₄×C₂₃	C₄×Q₈	1	4	yes	—
113	Dic₂₃	—	2	2	no	—
114	Dic₂₃	—	2	4	no	—
115	Dic₂₃	—	2	2	no	—
116	Dic₂₃	—	4	1	no	—
117	Dic₂₃	—	4	2	no	—
118	Dic₂₃	C₄×Q₈	1	4	yes	—
119	C₂×C₄⋊Q₈	—	23	1	no	—
120	#391	—	23	2	no	—
121	#391	—	23	2	no	—
122	#389	—	23	1	no	—
123	#389	—	23	1	no	—
124	#389	—	23	2	no	—
125	C₄×C₄⋊C₄	—	23	1	no	—
126	C₄×C₄⋊C₄	—	23	1	no	—
127	C₄×C₄⋊C₄	—	23	2	no	—
128	C₄×C₄⋊C₄	—	23	1	no	—
129	C₄×C₄⋊C₄	D₂₃	1	1	yes	characteristic
130	C₂×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
131	C₂×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
132	C₂×C₂₃	#518	1	1	yes	characteristic
133	C₂×C₂₃	#554	1	1	yes	characteristic
134	C₂×C₂₃	#554	1	1	yes	characteristic
135	C₂×C₂₃	#555	1	1	yes	characteristic
136	C₂×C₂₃	Q₈²	1	1	yes	characteristic
137	C₂²×Q₈	—	23	2	no	—
138	C₄⋊Q₈	—	23	8	no	—
139	C₂×C₄⋊C₄	—	23	2	no	—
140	C₂×C₄⋊C₄	—	23	1	no	—
141	C₂×C₄⋊C₄	—	23	1	no	—
142	C₂×C₄⋊C₄	—	23	1	no	—
143	C₂×C₄⋊C₄	—	23	1	no	—
144	C₂×C₄⋊C₄	—	23	2	no	—
145	C₂×C₄⋊C₄	—	23	2	no	—
146	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
147	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
148	C₂×C₄²	—	23	1	no	—
149	C₂×C₄²	—	23	2	no	—
150	C₂×C₄²	—	23	1	no	—
151	C₂×C₄²	—	23	2	no	—
152	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
153	C₂×C₄²	C₂×D₂₃	1	2	yes	—
154	#95	—	23	2	no	—
155	#95	—	23	2	no	—
156	#95	—	23	2	no	—
157	#95	—	23	1	no	—
158	#95	—	23	2	no	—
159	#95	—	23	1	no	—
160	#95	C₂×D₂₃	1	2	yes	—
161	C₂₃	#2369	1	1	yes	characteristic, complemented, Sylow
162	C₂×Q₈	—	23	8	no	—
163	C₂×Q₈	—	92	2	no	—
164	C₂×Q₈	—	92	2	no	—
165	C₂²×C₄	—	23	1	no	—
166	C₂²×C₄	—	23	1	no	—
167	C₂²×C₄	—	23	1	no	—
168	C₂²×C₄	—	23	2	no	—
169	C₂²×C₄	—	23	1	no	—
170	C₂²×C₄	—	23	2	no	—
171	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
172	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
173	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
174	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
175	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
176	C₄⋊C₄	—	23	8	no	—
177	C₄⋊C₄	—	23	4	no	—
178	C₄⋊C₄	—	23	4	no	—
179	C₄⋊C₄	—	46	2	no	—
180	C₄⋊C₄	—	46	2	no	—
181	C₄⋊C₄	—	92	2	no	—
182	C₄⋊C₄	—	92	2	no	—
183	C₄⋊C₄	Dic₄₆	1	4	yes	complemented
184	C₄⋊C₄	C₄×D₂₃	1	4	yes	—
185	C₄²	—	2	2	no	—
186	C₄²	—	4	2	no	—
187	C₄²	—	23	4	no	—
188	C₄²	—	46	4	no	—
189	C₄²	—	46	2	no	—
190	C₄²	—	92	2	no	—
191	C₂³	C₂³×D₂₃	1	1	yes	characteristic
192	Q₈	—	92	8	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	23	2	no	—
203	C₂×C₄	—	23	2	no	—
204	C₂×C₄	—	23	2	no	—
205	C₂×C₄	—	23	2	no	—
206	C₂×C₄	—	23	4	no	—
207	C₂×C₄	—	23	2	no	—
208	C₂×C₄	—	46	1	no	—
209	C₂×C₄	—	46	1	no	—
210	C₂×C₄	—	46	2	no	—
211	C₂×C₄	—	46	1	no	—
212	C₂×C₄	—	46	2	no	—
213	C₂×C₄	—	46	1	no	—
214	C₂×C₄	—	46	1	no	—
215	C₂×C₄	—	46	2	no	—
216	C₂×C₄	—	46	2	no	—
217	C₂×C₄	—	46	1	no	—
218	C₂×C₄	—	46	2	no	—
219	C₂×C₄	—	46	1	no	—
220	C₂×C₄	C₂×Dic₄₆	1	4	yes	—
221	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
222	C₂×C₄	C₂×C₄×D₂₃	1	4	yes	—
223	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
224	C₂×C₄	#67288	1	2	yes	—
225	C₂×C₄	Q₈×D₂₃	1	2	yes	—
226	C₂²	C₂²×Dic₄₆	1	1	yes	characteristic
227	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
228	C₂²	C₂×#67288	1	1	yes	characteristic
229	C₂²	#10625155	1	1	yes	characteristic
230	C₂²	C₂×Q₈×D₂₃	1	1	yes	characteristic
231	C₂²	#41×D₂₃	1	1	yes	characteristic
232	C₂²	#10625160	1	1	yes	characteristic
233	C₄	—	2	2	no	—
234	C₄	—	2	4	no	—
235	C₄	—	2	4	no	—
236	C₄	—	4	1	no	—
237	C₄	—	23	4	no	—
238	C₄	—	46	2	no	—
239	C₄	—	46	2	no	—
240	C₄	—	46	4	no	—
241	C₄	—	92	1	no	—
242	C₄	—	92	2	no	—
243	C₄	C₄×Dic₄₆	1	4	yes	—
244	C₂	C₄×Q₈×D₂₃	1	1	yes	characteristic
245	C₂	C₂×C₄×Dic₄₆	1	1	yes	characteristic
246	C₂	#13092889	1	1	yes	characteristic
247	C₂	#13093122	1	1	yes	characteristic
248	C₂	#13093126	1	1	yes	characteristic
249	C₂	#13093130	1	1	yes	characteristic
250	C₂	Q₈×Dic₄₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Fitting	C₄×C₄⋊C₄×C₂₃	C₂
socle, prime radical, cosocle	C₂³×C₂₃	C₂⁴
hypercenter	C₄×C₄⋊C₄	D₂₃
hypocenter	C₂₃	#2369
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
28. #10625059	C₄	semidirect
51. C₄⋊C₄×C₂₃	Q₈	semidirect
60. Dic₂₃⋊C₄	Q₈	semidirect
161. C₂₃	#2369	semidirect
183. C₄⋊C₄	Dic₄₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625059	2
#13092323	2
#13092321	2
#13092316	2
#13092315	2
#13092314	2
C₄⋊C₄×Dic₂₃	2
C₄×Dic₂₃⋊C₄	2
C₄×C₄⋊C₄×C₂₃	2
#2369	23