Subgroups — C₄×#10625095 #15675932

Order 2944.

656 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625095	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092323	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092315	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092301	C₂	1	2	yes	maximal, complemented
6	#13092290	C₂	1	2	yes	maximal
7	#13092277	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₂₃⋊C₄	C₂	1	2	yes	maximal, complemented
9	C₄×C₄⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₂²⋊Dic₂₃	C₂	1	2	yes	maximal
11	C₂²×C₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
12	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₂₃	—	2	4	no	—
17	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊Dic₂₃	C₂²	1	2	yes	—
22	C₂×C₂²⋊Dic₂₃	C₂²	1	2	yes	—
23	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
26	#10625095	C₄	1	8	yes	direct product factor
27	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
28	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
29	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
30	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
31	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
32	C₄×Dic₄₆	—	2	4	no	—
33	C₂₃⋊#95	C₂²	1	2	yes	—
34	C₂₃⋊#95	C₂²	1	2	yes	—
35	C₂₃⋊#95	C₂²	1	2	yes	—
36	C₂₃⋊#95	C₂²	1	2	yes	—
37	C₂₃⋊#95	C₂²	1	2	yes	—
38	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	—	2	2	no	—
40	C₂²×C₄×C₂₃	—	2	2	no	—
41	C₂²×C₄×C₂₃	—	2	2	no	—
42	C₂²×C₄×C₂₃	—	2	2	no	—
43	C₂²×C₄×C₂₃	—	2	2	no	—
44	C₂²×C₄×C₂₃	—	2	2	no	—
45	C₂²×C₄×C₂₃	—	2	2	no	—
46	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
47	C₂²×C₄×C₂₃	Q₈	1	4	yes	complemented
48	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂²×Dic₂₃	C₂³	1	2	yes	—
55	C₂²×Dic₂₃	C₂³	1	2	yes	—
56	C₂²×Dic₂₃	C₂³	1	2	yes	—
57	C₂²×Dic₂₃	C₂³	1	2	yes	—
58	C₂×Dic₄₆	—	2	2	no	—
59	C₂×Dic₄₆	—	2	2	no	—
60	C₂×Dic₄₆	C₂×C₄	1	4	yes	complemented
61	C₄²×C₂₃	—	2	2	no	—
62	C₄²×C₂₃	—	2	2	no	—
63	C₄²×C₂₃	—	2	2	no	—
64	C₄²×C₂₃	D₄	1	4	yes	—
65	C₂²⋊Dic₂₃	—	2	4	no	—
66	C₂²⋊Dic₂₃	C₂×C₄	1	8	yes	—
67	C₄⋊Dic₂₃	—	2	2	no	—
68	C₄⋊Dic₂₃	—	2	2	no	—
69	C₄⋊Dic₂₃	C₂×C₄	1	4	yes	complemented
70	Dic₂₃⋊C₄	—	2	4	no	—
71	Dic₂₃⋊C₄	—	2	4	no	—
72	Dic₂₃⋊C₄	C₂×C₄	1	8	yes	complemented
73	C₄×Dic₂₃	—	2	4	no	—
74	C₄×Dic₂₃	—	2	4	no	—
75	C₂³×C₂₃	—	2	1	no	—
76	C₂³×C₂₃	—	2	1	no	—
77	C₂³×C₂₃	—	2	1	no	—
78	C₂³×C₂₃	—	2	1	no	—
79	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
80	C₂³×C₂₃	C₂×Q₈	1	2	yes	—
81	C₂³×C₂₃	#41	1	2	yes	—
82	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
83	C₂×C₄×C₂₃	—	2	8	no	—
84	C₂×C₄×C₂₃	—	2	1	no	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	1	no	—
87	C₂×C₄×C₂₃	—	2	1	no	—
88	C₂×C₄×C₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	2	1	no	—
90	C₂×C₄×C₂₃	—	2	1	no	—
91	C₂×C₄×C₂₃	—	2	1	no	—
92	C₂×C₄×C₂₃	—	2	8	no	—
93	C₂×C₄×C₂₃	—	2	8	no	—
94	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
100	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
101	C₂×C₄×C₂₃	#41	1	2	yes	—
102	C₂×C₄×C₂₃	#41	1	2	yes	—
103	C₂×C₄×C₂₃	#41	1	2	yes	—
104	C₂×Dic₂₃	—	2	2	no	—
105	C₂×Dic₂₃	—	2	2	no	—
106	C₂×Dic₂₃	—	2	2	no	—
107	C₂×Dic₂₃	—	2	2	no	—
108	C₂×Dic₂₃	—	2	2	no	—
109	C₂×Dic₂₃	—	2	2	no	—
110	C₂×Dic₂₃	—	2	2	no	—
111	C₂×Dic₂₃	—	2	2	no	—
112	C₂×Dic₂₃	—	2	2	no	—
113	C₂×Dic₂₃	—	2	2	no	—
114	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
115	C₂×Dic₂₃	C₂²×C₄	1	4	yes	—
116	Dic₄₆	—	2	8	no	—
117	C₄×C₂²⋊Q₈	—	23	1	no	maximal, Sylow
118	C₂²×C₂₃	—	2	2	no	—
119	C₂²×C₂₃	—	2	2	no	—
120	C₂²×C₂₃	—	2	2	no	—
121	C₂²×C₂₃	—	2	2	no	—
122	C₂²×C₂₃	—	2	2	no	—
123	C₂²×C₂₃	—	2	2	no	—
124	C₂²×C₂₃	C₄×Q₈	1	4	yes	complemented
125	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
126	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
127	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
128	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
129	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
130	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
131	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
132	C₄×C₂₃	—	2	2	no	—
133	C₄×C₂₃	—	2	2	no	—
134	C₄×C₂₃	—	2	2	no	—
135	C₄×C₂₃	—	2	2	no	—
136	C₄×C₂₃	C₄×D₄	1	4	yes	—
137	C₄×C₂₃	C₂²⋊Q₈	1	4	yes	complemented
138	Dic₂₃	—	2	4	no	—
139	Dic₂₃	—	2	4	no	—
140	Dic₂₃	—	4	2	no	—
141	Dic₂₃	—	4	2	no	—
142	C₂×C₂²⋊Q₈	—	23	1	no	—
143	C₂×C₄×Q₈	—	23	1	no	—
144	C₂²×C₄²	D₂₃	1	1	yes	characteristic
145	#391	—	23	1	no	—
146	#389	—	23	1	no	—
147	#387	—	23	2	no	—
148	#385	—	23	2	no	—
149	#380	—	23	1	no	—
150	C₄×C₄⋊C₄	—	23	1	no	—
151	C₄×C₄⋊C₄	—	23	2	no	—
152	C₄×C₂²⋊C₄	—	23	2	no	—
153	C₂×C₂₃	—	2	4	no	—
154	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
155	C₂×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
156	C₂×C₂₃	C₄×#41	1	1	yes	characteristic
157	C₂×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₂×C₂₃	#525	1	1	yes	characteristic
159	C₂×C₂₃	#529	1	1	yes	characteristic
160	C₂×C₂₃	#533	1	1	yes	characteristic
161	C₂²×Q₈	—	23	1	no	—
162	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
163	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
164	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
165	C₂²⋊Q₈	—	23	8	no	—
166	C₄×Q₈	—	46	4	no	—
167	C₂×C₄⋊C₄	—	23	2	no	—
168	C₂×C₄⋊C₄	—	23	1	no	—
169	C₂×C₄⋊C₄	—	23	2	no	—
170	C₂×C₄⋊C₄	—	23	1	no	—
171	C₂×C₄⋊C₄	—	23	2	no	—
172	C₂×C₄⋊C₄	—	23	1	no	—
173	C₂×C₂²⋊C₄	—	23	2	no	—
174	C₂×C₂²⋊C₄	—	23	2	no	—
175	C₂×C₄²	—	2	4	no	—
176	C₂×C₄²	—	23	2	no	—
177	C₂×C₄²	—	23	2	no	—
178	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
179	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
180	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
181	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
182	#95	—	23	2	no	—
183	#95	—	23	2	no	—
184	#95	—	23	2	no	—
185	#95	—	23	2	no	—
186	#95	—	23	2	no	—
187	C₂₃	C₄×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
188	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
189	C₂×Q₈	—	23	4	no	—
190	C₂×Q₈	—	46	2	no	—
191	C₂×Q₈	—	46	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	23	2	no	—
200	C₂²×C₄	—	23	2	no	—
201	C₂²×C₄	—	23	2	no	—
202	C₂²×C₄	—	23	2	no	—
203	C₂²×C₄	Dic₄₆	1	4	yes	complemented
204	C₂²×C₄	C₄×D₂₃	1	4	yes	—
205	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
206	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
207	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
208	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
209	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
210	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
211	C₄⋊C₄	—	23	8	no	—
212	C₄⋊C₄	—	23	4	no	—
213	C₄⋊C₄	—	46	4	no	—
214	C₄⋊C₄	—	46	2	no	—
215	C₄⋊C₄	—	46	4	no	—
216	C₄⋊C₄	—	46	2	no	—
217	C₂²⋊C₄	—	23	8	no	—
218	C₂²⋊C₄	—	46	4	no	—
219	C₄²	—	2	2	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	2	no	—
222	C₄²	—	46	4	no	—
223	C₄²	—	46	4	no	—
224	C₄²	C₂²⋊D₂₃	1	4	yes	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	C₂×Dic₄₆	1	2	yes	—
230	C₂³	C₂×C₄×D₂₃	1	2	yes	—
231	C₂³	#67288	1	2	yes	—
232	C₂³	C₂³×D₂₃	1	1	yes	characteristic
233	Q₈	—	46	8	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	23	4	no	—
246	C₂×C₄	—	23	4	no	—
247	C₂×C₄	—	46	2	no	—
248	C₂×C₄	—	46	2	no	—
249	C₂×C₄	—	46	2	no	—
250	C₂×C₄	—	46	2	no	—
251	C₂×C₄	—	46	2	no	—
252	C₂×C₄	—	46	2	no	—
253	C₂×C₄	—	46	2	no	—
254	C₂×C₄	—	46	2	no	—
255	C₂×C₄	—	46	2	no	—
256	C₂×C₄	—	46	2	no	—
257	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
258	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
259	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
260	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
261	C₂×C₄	#67288	1	2	yes	—
262	C₂×C₄	#67288	1	2	yes	—
263	C₂×C₄	#67288	1	2	yes	—
264	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
265	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
266	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	C₄×Dic₄₆	1	4	yes	complemented
274	C₂²	C₂²×Dic₄₆	1	1	yes	characteristic
275	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
276	C₂²	C₂×#67288	1	1	yes	characteristic
277	C₂²	C₂×#67288	1	1	yes	characteristic
278	C₂²	C₂×#67288	1	1	yes	characteristic
279	C₂²	C₂×#67288	1	1	yes	characteristic
280	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	46	4	no	—
286	C₄	—	46	4	no	—
287	C₄	—	92	2	no	—
288	C₄	—	92	2	no	—
289	C₄	#10625095	1	4	yes	direct product factor
290	C₄	C₄×C₂²⋊D₂₃	1	4	yes	—
291	C₂	—	2	4	no	—
292	C₂	C₂×C₄×C₂²⋊D₂₃	1	1	yes	characteristic
293	C₂	C₂×C₄×Dic₄₆	1	1	yes	characteristic
294	C₂	C₄×#67288	1	1	yes	characteristic
295	C₂	C₂×#10625095	1	1	yes	characteristic
296	C₂	#13092928	1	1	yes	characteristic
297	C₂	#13092951	1	1	yes	characteristic
298	C₂	#13092974	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Frattini	C₂³	C₂³×D₂₃
Fitting	C₂²×C₄²×C₂₃	C₂
socle, cosocle	C₂³×C₂₃	C₂⁴
prime radical	C₂⁴×C₂₃	C₂³
hypercenter	C₂²×C₄²	D₂₃
hypocenter	C₂₃	C₄×C₂²⋊Q₈
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₄₆	C₂	semidirect
3. #13092323	C₂	semidirect
4. #13092315	C₂	semidirect
5. #13092301	C₂	semidirect
8. C₄×Dic₂₃⋊C₄	C₂	semidirect
9. C₄×C₄⋊Dic₂₃	C₂	semidirect
26. #10625095	C₄	direct (N × H)
47. C₂²×C₄×C₂₃	Q₈	semidirect
60. C₂×Dic₄₆	C₂×C₄	semidirect
69. C₄⋊Dic₂₃	C₂×C₄	semidirect
72. Dic₂₃⋊C₄	C₂×C₄	semidirect
124. C₂²×C₂₃	C₄×Q₈	semidirect
137. C₄×C₂₃	C₂²⋊Q₈	semidirect
187. C₂₃	C₄×C₂²⋊Q₈	semidirect
203. C₂²×C₄	Dic₄₆	semidirect
273. C₂²	C₄×Dic₄₆	semidirect
289. C₄	#10625095	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#10625095	2
C₂×C₄×Dic₄₆	2
#13092323	2
#13092315	2
#13092301	2
#13092290	2
#13092277	2
C₄×Dic₂₃⋊C₄	2
C₄×C₄⋊Dic₂₃	2
C₄×C₂²⋊Dic₂₃	2
C₂²×C₄²×C₂₃	2
C₄×C₂²⋊Q₈	23

Subgroups — C4×#10625095 #15675932

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#10625095 #15675932