Group #19160953 — subgroups

Subgroups — C₂₃×#13327 #19160953

Order 5888.

594 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×SD₃₂	C₂	1	2	yes	maximal
2	C₂×C₂₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#358	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1970	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₁₆×C₂₃	—	2	1	no	—
7	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
10	C₂×C₂₃×SD₃₂	—	2	4	no	—
11	C₂×C₂₃×SD₃₂	—	2	4	no	—
12	C₂×C₂₃×SD₃₂	—	2	4	no	—
13	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#494	—	2	4	no	—
15	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#102	—	2	1	no	—
17	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#390	—	2	1	no	—
19	C₂₃×#358	—	2	4	no	—
20	C₂²×Q₈×C₂₃	—	2	2	no	—
21	C₂²×D₄×C₂₃	—	2	1	no	—
22	C₂×Q₁₆×C₂₃	—	2	4	no	—
23	C₂×Q₁₆×C₂₃	—	2	2	no	—
24	C₂×Q₁₆×C₂₃	—	2	2	no	—
25	C₂×Q₁₆×C₂₃	—	2	2	no	—
26	C₂×Q₁₆×C₂₃	—	2	2	no	—
27	C₂×Q₁₆×C₂₃	—	2	2	no	—
28	C₂×Q₁₆×C₂₃	—	4	2	no	—
29	C₂×Q₁₆×C₂₃	—	4	2	no	—
30	C₂×Q₁₆×C₂₃	D₄	1	4	yes	—
31	C₂×D₈×C₂₃	—	2	2	no	—
32	C₂×D₈×C₂₃	—	2	2	no	—
33	C₂×D₈×C₂₃	—	2	1	no	—
34	C₂×D₈×C₂₃	D₄	1	2	yes	—
35	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
37	C₂₃×C₄⋊Q₈	—	2	4	no	—
38	C₂₃×C₄⋊Q₈	—	4	2	no	—
39	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
40	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
41	C₂×C₂²⋊C₄×C₂₃	—	4	2	no	—
42	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
43	C₂₃×SD₃₂	—	2	16	no	—
44	C₂×C₁₆×C₂₃	—	2	2	no	—
45	C₂×C₁₆×C₂₃	—	2	2	no	—
46	C₂×C₁₆×C₂₃	D₄	1	4	yes	—
47	C₂₃×#107	—	2	2	no	—
48	C₂₃×#102	—	4	2	no	—
49	C₄×C₈×C₂₃	—	2	2	no	—
50	C₂⁴×C₂₃	—	4	1	no	—
51	C₂×Q₈×C₂₃	—	2	4	no	—
52	C₂×Q₈×C₂₃	—	4	2	no	—
53	C₂×Q₈×C₂₃	—	4	2	no	—
54	C₂×Q₈×C₂₃	—	4	2	no	—
55	C₂×Q₈×C₂₃	—	4	2	no	—
56	C₂×Q₈×C₂₃	—	4	2	no	—
57	C₂×D₄×C₂₃	—	2	2	no	—
58	C₂×D₄×C₂₃	—	4	1	no	—
59	C₂×D₄×C₂₃	—	4	1	no	—
60	C₂×D₄×C₂₃	—	4	1	no	—
61	C₂×D₄×C₂₃	—	4	1	no	—
62	C₂×D₄×C₂₃	—	4	1	no	—
63	C₂²×C₄×C₂₃	—	4	2	no	—
64	C₂²×C₄×C₂₃	—	4	1	no	—
65	C₂²×C₄×C₂₃	D₈	1	2	yes	—
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	Q₁₆×C₂₃	—	2	4	no	—
68	Q₁₆×C₂₃	—	2	4	no	—
69	Q₁₆×C₂₃	—	4	8	no	—
70	Q₁₆×C₂₃	—	4	2	no	—
71	Q₁₆×C₂₃	—	4	2	no	—
72	D₈×C₂₃	—	2	4	no	—
73	D₈×C₂₃	—	4	2	no	—
74	C₂×C₈×C₂₃	—	2	1	no	—
75	C₂×C₈×C₂₃	—	2	1	no	—
76	C₂×C₈×C₂₃	D₈	1	2	yes	—
77	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
81	C₄⋊C₄×C₂₃	—	4	8	no	—
82	C₄⋊C₄×C₂₃	—	4	1	no	—
83	C₄⋊C₄×C₂₃	—	4	1	no	—
84	C₂²⋊C₄×C₂₃	—	8	4	no	—
85	C₄²×C₂₃	—	2	1	no	—
86	C₄²×C₂₃	—	2	1	no	—
87	C₁₆×C₂₃	—	2	4	no	—
88	#13327	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₂₃	—	4	2	no	—
90	C₂³×C₂₃	—	8	1	no	—
91	C₂³×C₂₃	—	8	1	no	—
92	C₂³×C₂₃	—	8	2	no	—
93	C₂³×C₂₃	—	8	2	no	—
94	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
95	Q₈×C₂₃	—	4	4	no	—
96	Q₈×C₂₃	—	4	4	no	—
97	Q₈×C₂₃	—	8	2	no	—
98	Q₈×C₂₃	—	8	2	no	—
99	D₄×C₂₃	—	4	2	no	—
100	D₄×C₂₃	—	4	2	no	—
101	D₄×C₂₃	—	8	1	no	—
102	D₄×C₂₃	—	8	1	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	4	4	no	—
106	C₂×C₄×C₂₃	—	8	2	no	—
107	C₂×C₄×C₂₃	—	8	2	no	—
108	C₂×C₄×C₂₃	—	8	1	no	—
109	C₂×C₄×C₂₃	—	8	2	no	—
110	C₂×C₄×C₂₃	SD₃₂	1	4	yes	complemented
111	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
112	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
113	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₈×C₂₃	—	2	2	no	—
118	C₈×C₂₃	#120	1	2	yes	—
119	C₈×C₂₃	#121	1	2	yes	—
120	C₂²×SD₃₂	C₂×C₂₃	1	2	yes	—
121	C₂×#494	C₂×C₂₃	1	1	yes	characteristic, complemented
122	C₂×#358	C₂×C₂₃	1	2	yes	complemented
123	#1970	C₂×C₂₃	1	1	yes	characteristic
124	#1374	C₂×C₂₃	1	1	yes	characteristic, complemented
125	C₂²×C₂₃	—	8	4	no	—
126	C₂²×C₂₃	—	8	2	no	—
127	C₂²×C₂₃	—	16	2	no	—
128	C₂²×C₂₃	—	16	1	no	—
129	C₂²×C₂₃	—	16	1	no	—
130	C₂²×C₂₃	#447	1	1	yes	characteristic
131	C₂²×C₂₃	#459	1	1	yes	characteristic
132	C₂²×C₂₃	#493	1	1	yes	characteristic
133	C₂²×C₂₃	C₂×SD₃₂	1	2	yes	—
134	C₂²×C₂₃	#510	1	2	yes	—
135	C₄×C₂₃	—	2	4	no	—
136	C₄×C₂₃	—	8	4	no	—
137	C₄×C₂₃	—	16	1	no	—
138	C₄×C₂₃	#392	1	1	yes	characteristic
139	C₄×C₂₃	#447	1	1	yes	characteristic
140	C₄×C₂₃	#459	1	1	yes	characteristic
141	C₄×C₂₃	#496	1	1	yes	characteristic
142	C₂²×Q₁₆	—	2	1	no	—
143	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
144	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
145	C₂×C₄⋊Q₈	—	2	1	no	—
146	C₂×SD₃₂	—	2	4	no	—
147	C₂×SD₃₂	—	2	4	no	—
148	C₂×SD₃₂	—	2	4	no	—
149	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
150	#494	—	2	4	no	—
151	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
152	C₂×#102	—	2	1	no	—
153	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
154	#390	—	2	1	no	—
155	#358	—	2	4	no	—
156	C₂×C₂₃	—	16	2	no	—
157	C₂×C₂₃	#2011	1	1	yes	characteristic
158	C₂×C₂₃	#2185	1	2	yes	—
159	C₂×C₂₃	#2208	1	2	yes	—
160	C₂×C₂₃	#2248	1	1	yes	characteristic
161	C₂×C₂₃	#2251	1	1	yes	characteristic
162	C₂²×Q₈	—	2	2	no	—
163	C₂²×D₄	—	2	1	no	—
164	C₂×Q₁₆	—	2	4	no	—
165	C₂×Q₁₆	—	2	2	no	—
166	C₂×Q₁₆	—	2	2	no	—
167	C₂×Q₁₆	—	2	2	no	—
168	C₂×Q₁₆	—	2	2	no	—
169	C₂×Q₁₆	—	2	2	no	—
170	C₂×Q₁₆	—	4	2	no	—
171	C₂×Q₁₆	—	4	2	no	—
172	C₂×Q₁₆	D₄×C₂₃	1	4	yes	—
173	C₂×D₈	—	2	2	no	—
174	C₂×D₈	—	2	2	no	—
175	C₂×D₈	—	2	1	no	—
176	C₂×D₈	D₄×C₂₃	1	2	yes	—
177	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
179	C₄⋊Q₈	—	2	4	no	—
180	C₄⋊Q₈	—	4	2	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	—	4	2	no	—
183	C₂×C₂²⋊C₄	—	4	2	no	—
184	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
185	SD₃₂	—	2	16	no	—
186	C₂×C₁₆	—	2	2	no	—
187	C₂×C₁₆	—	2	2	no	—
188	C₂×C₁₆	D₄×C₂₃	1	4	yes	—
189	#107	—	2	2	no	—
190	#102	—	4	2	no	—
191	C₄×C₈	—	2	2	no	—
192	C₂₃	#13327	1	1	yes	characteristic, direct product factor, Sylow
193	C₂⁴	—	4	1	no	—
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	—	4	2	no	—
196	C₂×Q₈	—	4	2	no	—
197	C₂×Q₈	—	4	2	no	—
198	C₂×Q₈	—	4	2	no	—
199	C₂×Q₈	—	4	2	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	4	1	no	—
202	C₂×D₄	—	4	1	no	—
203	C₂×D₄	—	4	1	no	—
204	C₂×D₄	—	4	1	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	D₈×C₂₃	1	2	yes	—
209	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
210	Q₁₆	—	2	4	no	—
211	Q₁₆	—	2	4	no	—
212	Q₁₆	—	4	8	no	—
213	Q₁₆	—	4	2	no	—
214	Q₁₆	—	4	2	no	—
215	D₈	—	2	4	no	—
216	D₈	—	4	2	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	D₈×C₂₃	1	2	yes	—
220	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
221	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
222	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
223	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
224	C₄⋊C₄	—	4	8	no	—
225	C₄⋊C₄	—	4	1	no	—
226	C₄⋊C₄	—	4	1	no	—
227	C₂²⋊C₄	—	8	4	no	—
228	C₄²	—	2	1	no	—
229	C₄²	—	2	1	no	—
230	C₁₆	—	2	4	no	—
231	C₂³	—	4	2	no	—
232	C₂³	—	8	1	no	—
233	C₂³	—	8	1	no	—
234	C₂³	—	8	2	no	—
235	C₂³	—	8	2	no	—
236	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
237	Q₈	—	4	4	no	—
238	Q₈	—	4	4	no	—
239	Q₈	—	8	2	no	—
240	Q₈	—	8	2	no	—
241	D₄	—	4	2	no	—
242	D₄	—	4	2	no	—
243	D₄	—	8	1	no	—
244	D₄	—	8	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	8	2	no	—
249	C₂×C₄	—	8	2	no	—
250	C₂×C₄	—	8	1	no	—
251	C₂×C₄	—	8	2	no	—
252	C₂×C₄	C₂₃×SD₃₂	1	4	yes	complemented
253	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
254	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
255	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
256	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
257	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
258	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	C₂₃×#120	1	2	yes	—
261	C₈	C₂₃×#121	1	2	yes	—
262	C₂²	—	8	4	no	—
263	C₂²	—	8	2	no	—
264	C₂²	—	16	2	no	—
265	C₂²	—	16	1	no	—
266	C₂²	—	16	1	no	—
267	C₂²	C₂₃×#447	1	1	yes	characteristic
268	C₂²	C₂₃×#459	1	1	yes	characteristic
269	C₂²	C₂₃×#493	1	1	yes	characteristic
270	C₂²	C₂×C₂₃×SD₃₂	1	2	yes	—
271	C₂²	C₂₃×#510	1	2	yes	—
272	C₄	—	2	4	no	—
273	C₄	—	8	4	no	—
274	C₄	—	16	1	no	—
275	C₄	C₂₃×#392	1	1	yes	characteristic
276	C₄	C₂₃×#447	1	1	yes	characteristic
277	C₄	C₂₃×#459	1	1	yes	characteristic
278	C₄	C₂₃×#496	1	1	yes	characteristic
279	C₂	—	16	2	no	—
280	C₂	C₂₃×#2011	1	1	yes	characteristic
281	C₂	C₂₃×#2185	1	2	yes	—
282	C₂	C₂₃×#2208	1	2	yes	—
283	C₂	C₂₃×#2248	1	1	yes	characteristic
284	C₂	C₂₃×#2251	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₂₃×SD₃₂

C₂

yes

maximal

C₂×C₂₃×#494

C₂

yes

maximal, characteristic, complemented

C₂×C₂₃×#358

C₂

yes

maximal, complemented

C₂₃×#1970

C₂

yes

maximal, characteristic

C₂₃×#1374

C₂

yes

maximal, characteristic, complemented

C₂²×Q₁₆×C₂₃

—

C₂²×Q₁₆×C₂₃

C₂²

yes

—

C₂²×D₈×C₂₃

C₂²

yes

characteristic

C₂×C₂₃×C₄⋊Q₈

—

C₂×C₂₃×SD₃₂

—

C₂×C₂₃×SD₃₂

—

C₂×C₂₃×SD₃₂

—

C₂²×C₁₆×C₂₃

C₂²

yes

—

C₂₃×#494

—

C₂×C₂₃×#107

C₂²

yes

characteristic

C₂×C₂₃×#102

—

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#390

—

C₂₃×#358

—

C₂²×Q₈×C₂₃

—

C₂²×D₄×C₂₃

—

C₂×Q₁₆×C₂₃

—

C₂×Q₁₆×C₂₃

—

C₂×Q₁₆×C₂₃

—

C₂×Q₁₆×C₂₃

—

C₂×Q₁₆×C₂₃

—

C₂×Q₁₆×C₂₃

—

C₂×Q₁₆×C₂₃

—

C₂×Q₁₆×C₂₃

—

C₂×Q₁₆×C₂₃

D₄

yes

—

C₂×D₈×C₂₃

—

C₂×D₈×C₂₃

—

C₂×D₈×C₂₃

—

C₂×D₈×C₂₃

D₄

yes

—

C₂²×C₈×C₂₃

D₄

yes

characteristic

C₂²×C₈×C₂₃

C₂³

yes

characteristic

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₂²⋊C₄×C₂₃

—

C₂×C₄²×C₂₃

D₄

yes

characteristic

C₂₃×SD₃₂

—

C₂×C₁₆×C₂₃

—

C₂×C₁₆×C₂₃

—

C₂×C₁₆×C₂₃

D₄

yes

—

C₂₃×#107

—

C₂₃×#102

—

C₄×C₈×C₂₃

—

C₂⁴×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₈

yes

—

C₂²×C₄×C₂₃

C₂×D₄

yes

characteristic

Q₁₆×C₂₃

—

Q₁₆×C₂₃

—

Q₁₆×C₂₃

—

Q₁₆×C₂₃

—

Q₁₆×C₂₃

—

D₈×C₂₃

—

D₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

D₈

yes

—

C₂×C₈×C₂₃

C₂×D₄

yes

characteristic

C₂×C₈×C₂₃

C₂×D₄

yes

—

C₂×C₈×C₂₃

C₂×D₄

yes

—

C₂×C₈×C₂₃

#41

yes

characteristic

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₁₆×C₂₃

—

#13327

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂×D₈

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

100

D₄×C₂₃

—

101

D₄×C₂₃

—

102

D₄×C₂₃

—

103

C₂×C₄×C₂₃

—

104

C₂×C₄×C₂₃

—

105

C₂×C₄×C₂₃

—

106

C₂×C₄×C₂₃

—

107

C₂×C₄×C₂₃

—

108

C₂×C₄×C₂₃

—

109

C₂×C₄×C₂₃

—

110

C₂×C₄×C₂₃

SD₃₂

yes

complemented

111

C₂×C₄×C₂₃

#120

yes

characteristic

112

C₂×C₄×C₂₃

#121

yes

characteristic

113

C₂×C₄×C₂₃

#127

yes

characteristic

114

C₂×C₄×C₂₃

C₂×D₈

yes

characteristic

115

C₂×C₄×C₂₃

AGL(1,ℤ/8)

yes

characteristic

116

C₂×C₄×C₂₃

AGL(1,ℤ/8)

yes

characteristic

117

C₈×C₂₃

—

118

C₈×C₂₃

#120

yes

—

119

C₈×C₂₃

#121

yes

—

120

C₂²×SD₃₂

C₂×C₂₃

yes

—

121

C₂×#494

C₂×C₂₃

yes

characteristic, complemented

122

C₂×#358

C₂×C₂₃

yes

complemented

123

#1970

C₂×C₂₃

yes

characteristic

124

#1374

C₂×C₂₃

yes

characteristic, complemented

125

C₂²×C₂₃

—

126

C₂²×C₂₃

—

127

C₂²×C₂₃

—

128

C₂²×C₂₃

—

129

C₂²×C₂₃

—

130

C₂²×C₂₃

#447

yes

characteristic

131

C₂²×C₂₃

#459

yes

characteristic

132

C₂²×C₂₃

#493

yes

characteristic

133

C₂²×C₂₃

C₂×SD₃₂

yes

—

134

C₂²×C₂₃

#510

yes

—

135

C₄×C₂₃

—

136

C₄×C₂₃

—

137

C₄×C₂₃

—

138

C₄×C₂₃

#392

yes

characteristic

139

C₄×C₂₃

#447

yes

characteristic

140

C₄×C₂₃

#459

yes

characteristic

141

C₄×C₂₃

#496

yes

characteristic

142

C₂²×Q₁₆

—

143

C₂²×Q₁₆

C₂²×C₂₃

yes

—

144

C₂²×D₈

C₂²×C₂₃

yes

characteristic

145

C₂×C₄⋊Q₈

—

146

C₂×SD₃₂

—

147

C₂×SD₃₂

—

148

C₂×SD₃₂

—

149

C₂²×C₁₆

C₂²×C₂₃

yes

—

150

#494

—

151

C₂×#107

C₂²×C₂₃

yes

characteristic

152

C₂×#102

—

153

C₂×C₄×C₈

C₂²×C₂₃

yes

characteristic

154

#390

—

155

#358

—

156

C₂×C₂₃

—

157

C₂×C₂₃

#2011

yes

characteristic

158

C₂×C₂₃

#2185

yes

—

159

C₂×C₂₃

#2208

yes

—

160

C₂×C₂₃

#2248

yes

characteristic

161

C₂×C₂₃

#2251

yes

characteristic

162

C₂²×Q₈

—

163

C₂²×D₄

—

164

C₂×Q₁₆

—

165

C₂×Q₁₆

—

166

C₂×Q₁₆

—

167

C₂×Q₁₆

—

168

C₂×Q₁₆

—

169

C₂×Q₁₆

—

170

C₂×Q₁₆

—

171

C₂×Q₁₆

—

172

C₂×Q₁₆

D₄×C₂₃

yes

—

173

C₂×D₈

—

174

C₂×D₈

—

175

C₂×D₈

—

176

C₂×D₈

D₄×C₂₃

yes

—

177

C₂²×C₈

D₄×C₂₃

yes

characteristic

178

C₂²×C₈

C₂³×C₂₃

yes

characteristic

179

C₄⋊Q₈

—

180

C₄⋊Q₈

—

181

C₂×C₄⋊C₄

—

182

C₂×C₄⋊C₄

—

183

C₂×C₂²⋊C₄

—

184

C₂×C₄²

D₄×C₂₃

yes

characteristic

185

SD₃₂

—

186

C₂×C₁₆

—

187

C₂×C₁₆

—

188

C₂×C₁₆

D₄×C₂₃

yes

—

189

#107

—

190

#102

—

191

C₄×C₈

—

192

C₂₃

#13327

yes

characteristic, direct product factor, Sylow

193

C₂⁴

—

194

C₂×Q₈

—

195

C₂×Q₈

—

196

C₂×Q₈

—

197

C₂×Q₈

—

198

C₂×Q₈

—

199

C₂×Q₈

—

200

C₂×D₄

—

201

C₂×D₄

—

202

C₂×D₄

—

203

C₂×D₄

—

204

C₂×D₄

—

205

C₂×D₄

—

206

C₂²×C₄

—

207

C₂²×C₄

—

208

C₂²×C₄

D₈×C₂₃

yes

—

209

C₂²×C₄

C₂×D₄×C₂₃

yes

characteristic

210

Q₁₆

—

211

Q₁₆

—

212

Q₁₆

—

213

Q₁₆

—

214

Q₁₆

—

215

D₈

—

216

D₈

—

217

C₂×C₈

—

218

C₂×C₈

—

219

C₂×C₈

D₈×C₂₃

yes

—

220

C₂×C₈

C₂×D₄×C₂₃

yes

characteristic

221

C₂×C₈

C₂×D₄×C₂₃

yes

—

222

C₂×C₈

C₂×D₄×C₂₃

yes

—

223

C₂×C₈

#41×C₂₃

yes

characteristic

224

C₄⋊C₄

—

225

C₄⋊C₄

—

226

C₄⋊C₄

—

227

C₂²⋊C₄

—

228

C₄²

—

229

C₄²

—

230

C₁₆

—

231

C₂³

—

232

C₂³

—

233

C₂³

—

234

C₂³

—

235

C₂³

—

236

C₂³

C₂×D₈×C₂₃

yes

characteristic

237

Q₈

—

238

Q₈

—

239

Q₈

—

240

Q₈

—

241

D₄

—

242

D₄

—

243

D₄

—

244

D₄

—

245

C₂×C₄

—

246

C₂×C₄

—

247

C₂×C₄

—

248

C₂×C₄

—

249

C₂×C₄

—

250

C₂×C₄

—

251

C₂×C₄

—

252

C₂×C₄

C₂₃×SD₃₂

yes

complemented

253

C₂×C₄

C₂₃×#120

yes

characteristic

254

C₂×C₄

C₂₃×#121

yes

characteristic

255

C₂×C₄

C₂₃×#127

yes

characteristic

256

C₂×C₄

C₂×D₈×C₂₃

yes

characteristic

257

C₂×C₄

C₂₃×AGL(1,ℤ/8)

yes

characteristic

258

C₂×C₄

C₂₃×AGL(1,ℤ/8)

yes

characteristic

259

C₈

—

260

C₈

C₂₃×#120

yes

—

261

C₈

C₂₃×#121

yes

—

262

C₂²

—

263

C₂²

—

264

C₂²

—

265

C₂²

—

266

C₂²

—

267

C₂²

C₂₃×#447

yes

characteristic

268

C₂²

C₂₃×#459

yes

characteristic

269

C₂²

C₂₃×#493

yes

characteristic

270

C₂²

C₂×C₂₃×SD₃₂

yes

—

271

C₂²

C₂₃×#510

yes

—

272

C₄

—

273

C₄

—

274

C₄

—

275

C₄

C₂₃×#392

yes

characteristic

276

C₄

C₂₃×#447

yes

characteristic

277

C₄

C₂₃×#459

yes

characteristic

278

C₄

C₂₃×#496

yes

characteristic

279

C₂

—

280

C₂

C₂₃×#2011

yes

characteristic

281

C₂

C₂₃×#2185

yes

—

282

C₂

C₂₃×#2208

yes

—

283

C₂

C₂₃×#2248

yes

characteristic

284

C₂

C₂₃×#2251

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
prime radical	C₂²×D₈×C₂₃	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₂₃

C₂×D₈

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₂₃

prime radical

C₂²×D₈×C₂₃

C₂²

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

N	H ≅ G/N	type
2. C₂×C₂₃×#494	C₂	semidirect
3. C₂×C₂₃×#358	C₂	semidirect
5. C₂₃×#1374	C₂	semidirect
88. #13327	C₂₃	direct (N × H)
110. C₂×C₄×C₂₃	SD₃₂	semidirect
121. C₂×#494	C₂×C₂₃	semidirect
122. C₂×#358	C₂×C₂₃	semidirect
124. #1374	C₂×C₂₃	semidirect
192. C₂₃	#13327	direct (N × H)
252. C₂×C₄	C₂₃×SD₃₂	semidirect

H ≅ G/N

type

2. C₂×C₂₃×#494

C₂

semidirect

3. C₂×C₂₃×#358

C₂

semidirect

5. C₂₃×#1374

C₂

semidirect

88. #13327

C₂₃

direct (N × H)

110. C₂×C₄×C₂₃

SD₃₂

semidirect

121. C₂×#494

C₂×C₂₃

semidirect

122. C₂×#358

C₂×C₂₃

semidirect

124. #1374

C₂×C₂₃

semidirect

192. C₂₃

#13327

direct (N × H)

252. C₂×C₄

C₂₃×SD₃₂

semidirect

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

subgroup	index
C₂²×C₂₃×SD₃₂	2
C₂×C₂₃×#494	2
C₂×C₂₃×#358	2
C₂₃×#1970	2
C₂₃×#1374	2
#13327	23

subgroup

index

C₂²×C₂₃×SD₃₂

C₂×C₂₃×#494

C₂×C₂₃×#358

C₂₃×#1970

C₂₃×#1374

#13327

Subgroups — C₂₃×#13327 #19160953

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#13327 #19160953

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13327 #19160953