Subgroups — C₂³×#14524016 #31762467

Order 17152.

19998 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×C₂²⋊D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂⁴×C₂²⋊D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×C₄×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#14524016	C₂	1	56	yes	maximal, direct product factor
6	C₂³×Dic₆₇⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×(C₂×D₆₇)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×D₆₇	C₂²	1	1	yes	characteristic
10	C₂⁵×D₆₇	C₂²	1	1	yes	characteristic
11	C₂³×D₂₆₈	—	2	16	no	—
12	C₂³×D₂₆₈	C₂²	1	28	yes	complemented
13	C₂³×C₂²⋊D₆₇	—	2	16	no	—
14	C₂³×C₂²⋊D₆₇	—	2	16	no	—
15	C₂³×C₂²⋊D₆₇	C₂²	1	28	yes	complemented
16	C₂³×C₂²⋊D₆₇	C₂²	1	28	yes	complemented
17	C₂⁴×Dic₆₇	C₂²	1	1	yes	characteristic
18	C₂⁴×Dic₆₇	C₂²	1	1	yes	characteristic
19	C₂³×C₄×D₆₇	—	2	16	no	—
20	C₂³×C₄×D₆₇	C₂²	1	28	yes	complemented
21	C₂×#14524016	C₂²	1	448	yes	direct product factor
22	C₂²×Dic₆₇⋊C₄	C₂²	1	28	yes	complemented
23	C₂²×(C₂×D₆₇)⋊C₄	C₂²	1	28	yes	complemented
24	C₂⁶×C₆₇	C₂²	1	1	yes	characteristic
25	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
26	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	28	yes	complemented
28	C₂⁴×D₆₇	—	2	8	no	—
29	C₂⁴×D₆₇	—	2	8	no	—
30	C₂⁴×D₆₇	—	2	8	no	—
31	C₂⁴×D₆₇	—	2	8	no	—
32	C₂⁴×D₆₇	—	2	8	no	—
33	C₂⁴×D₆₇	D₄	1	16	yes	complemented
34	C₂⁴×D₆₇	C₂³	1	14	yes	—
35	C₂⁴×D₆₇	C₂³	1	14	yes	—
36	C₂²×D₂₆₈	—	2	224	no	—
37	C₂²×D₂₆₈	C₂³	1	112	yes	complemented
38	C₂²×C₂²⋊D₆₇	—	2	224	no	—
39	C₂²×C₂²⋊D₆₇	—	2	224	no	—
40	C₂²×C₂²⋊D₆₇	C₂³	1	112	yes	complemented
41	C₂²×C₂²⋊D₆₇	C₂³	1	112	yes	complemented
42	C₂³×Dic₆₇	—	2	8	no	—
43	C₂³×Dic₆₇	D₄	1	16	yes	complemented
44	C₂³×Dic₆₇	C₂³	1	14	yes	—
45	C₂³×Dic₆₇	C₂³	1	14	yes	—
46	C₂²×C₄×D₆₇	—	2	224	no	—
47	C₂²×C₄×D₆₇	C₂³	1	112	yes	complemented
48	#14524016	C₂³	1	512	yes	direct product factor
49	C₂×Dic₆₇⋊C₄	C₂³	1	112	yes	complemented
50	C₂×(C₂×D₆₇)⋊C₄	C₂³	1	112	yes	complemented
51	C₂⁵×C₆₇	—	2	8	no	—
52	C₂⁵×C₆₇	—	2	8	no	—
53	C₂⁵×C₆₇	—	2	8	no	—
54	C₂⁵×C₆₇	C₂³	1	14	yes	—
55	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₆₇	—	2	8	no	—
57	C₂³×C₄×C₆₇	—	2	8	no	—
58	C₂³×C₄×C₆₇	C₂³	1	14	yes	—
59	C₂³×C₄×C₆₇	C₂³	1	14	yes	—
60	C₂×C₂²⋊C₄×C₆₇	C₂³	1	112	yes	complemented
61	C₂³×D₆₇	—	2	128	no	—
62	C₂³×D₆₇	—	2	56	no	—
63	C₂³×D₆₇	—	2	56	no	—
64	C₂³×D₆₇	—	2	56	no	—
65	C₂³×D₆₇	—	2	56	no	—
66	C₂³×D₆₇	—	2	56	no	—
67	C₂³×D₆₇	—	4	64	no	—
68	C₂³×D₆₇	C₂×D₄	1	112	yes	complemented
69	C₂³×D₆₇	C₂⁴	1	28	yes	—
70	C₂³×D₆₇	C₂⁴	1	28	yes	—
71	C₂×D₂₆₈	—	2	448	no	—
72	C₂×D₂₆₈	C₂⁴	1	64	yes	complemented
73	C₂×C₂²⋊D₆₇	—	2	448	no	—
74	C₂×C₂²⋊D₆₇	—	2	448	no	—
75	C₂×C₂²⋊D₆₇	C₂⁴	1	64	yes	complemented
76	C₂×C₂²⋊D₆₇	C₂⁴	1	64	yes	complemented
77	C₂²×Dic₆₇	—	2	56	no	—
78	C₂²×Dic₆₇	C₂×D₄	1	112	yes	complemented
79	C₂²×Dic₆₇	C₂⁴	1	28	yes	—
80	C₂²×Dic₆₇	C₂⁴	1	28	yes	—
81	C₂×C₄×D₆₇	—	2	448	no	—
82	C₂×C₄×D₆₇	C₂⁴	1	64	yes	complemented
83	Dic₆₇⋊C₄	C₂⁴	1	64	yes	complemented
84	(C₂×D₆₇)⋊C₄	C₂⁴	1	64	yes	complemented
85	C₂⁴×C₆₇	—	2	56	no	—
86	C₂⁴×C₆₇	—	2	56	no	—
87	C₂⁴×C₆₇	—	2	56	no	—
88	C₂⁴×C₆₇	—	4	64	no	—
89	C₂⁴×C₆₇	C₂×D₄	1	8	yes	—
90	C₂⁴×C₆₇	C₂×D₄	1	8	yes	—
91	C₂⁴×C₆₇	#41	1	8	yes	—
92	C₂⁴×C₆₇	C₂⁴	1	28	yes	—
93	C₂⁴×C₆₇	C₂⁴	1	7	yes	—
94	C₂²×C₄×C₆₇	—	2	56	no	—
95	C₂²×C₄×C₆₇	—	2	56	no	—
96	C₂²×C₄×C₆₇	C₂⁴	1	28	yes	—
97	C₂²×C₄×C₆₇	C₂⁴	1	28	yes	—
98	C₂²⋊C₄×C₆₇	C₂⁴	1	64	yes	complemented
99	C₂³×C₆₇	—	2	56	no	—
100	C₂³×C₆₇	—	2	56	no	—
101	C₂³×C₆₇	—	2	56	no	—
102	C₂³×C₆₇	—	4	224	no	—
103	C₂³×C₆₇	#121	1	64	yes	complemented
104	C₂³×C₆₇	C₂²×D₄	1	28	yes	—
105	C₂³×C₆₇	C₂²×D₄	1	28	yes	—
106	C₂³×C₆₇	C₂×#41	1	28	yes	—
107	C₂³×C₆₇	C₂⁵	1	8	yes	—
108	C₂³×C₆₇	C₂⁵	1	7	yes	—
109	C₂²×D₆₇	—	2	448	no	—
110	C₂²×D₆₇	—	2	56	no	—
111	C₂²×D₆₇	—	2	56	no	—
112	C₂²×D₆₇	—	2	56	no	—
113	C₂²×D₆₇	—	2	56	no	—
114	C₂²×D₆₇	—	2	56	no	—
115	C₂²×D₆₇	—	4	224	no	—
116	C₂²×D₆₇	C₂²×D₄	1	112	yes	complemented
117	C₂²×D₆₇	C₂⁵	1	8	yes	—
118	C₂²×D₆₇	C₂⁵	1	8	yes	—
119	C₂×C₄×C₆₇	—	2	56	no	—
120	C₂×C₄×C₆₇	—	2	56	no	—
121	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
122	C₂×C₄×C₆₇	C₂⁵	1	8	yes	—
123	C₂²⋊D₆₇	—	2	128	no	—
124	C₂²⋊D₆₇	—	2	128	no	—
125	C₂×Dic₆₇	—	2	56	no	—
126	C₂×Dic₆₇	C₂²×D₄	1	112	yes	complemented
127	C₂×Dic₆₇	C₂⁵	1	8	yes	—
128	C₂×Dic₆₇	C₂⁵	1	8	yes	—
129	D₂₆₈	—	2	128	no	—
130	C₄×D₆₇	—	2	128	no	—
131	C₂²×C₆₇	—	2	8	no	—
132	C₂²×C₆₇	—	2	8	no	—
133	C₂²×C₆₇	—	2	8	no	—
134	C₂²×C₆₇	—	4	112	no	—
135	C₂²×C₆₇	C₂×#121	1	112	yes	complemented
136	C₂²×C₆₇	C₂³×D₄	1	14	yes	—
137	C₂²×C₆₇	C₂³×D₄	1	14	yes	—
138	C₂²×C₆₇	C₂²×#41	1	14	yes	—
139	C₂²×C₆₇	C₂⁶	1	1	yes	characteristic
140	C₂×D₆₇	—	2	8	no	—
141	C₂×D₆₇	—	2	8	no	—
142	C₂×D₆₇	—	2	224	no	—
143	C₂×D₆₇	—	2	8	no	—
144	C₂×D₆₇	—	2	8	no	—
145	C₂×D₆₇	—	2	8	no	—
146	C₂×D₆₇	—	4	112	no	—
147	C₂×D₆₇	C₂³×D₄	1	16	yes	complemented
148	C₄×C₆₇	—	2	8	no	—
149	C₄×C₆₇	—	2	8	no	—
150	Dic₆₇	—	2	8	no	—
151	Dic₆₇	C₂³×D₄	1	16	yes	complemented
152	C₂³×#121	—	67	1	no	maximal, Sylow
153	C₂×C₆₇	—	4	8	no	—
154	C₂×C₆₇	C₂²×#121	1	28	yes	complemented
155	C₂×C₆₇	C₂⁴×D₄	1	1	yes	characteristic
156	C₂×C₆₇	C₂⁴×D₄	1	1	yes	characteristic
157	C₂×C₆₇	C₂³×#41	1	1	yes	characteristic
158	D₆₇	—	2	16	no	—
159	D₆₇	—	4	8	no	—
160	C₂⁴×D₄	—	67	1	no	—
161	C₂⁴×D₄	—	67	1	no	—
162	C₂⁴×D₄	—	67	1	no	—
163	C₂⁵×C₄	—	67	1	no	—
164	C₂²×#121	—	67	56	no	—
165	C₂³×C₄⋊C₄	—	67	1	no	—
166	C₂³×C₂²⋊C₄	—	67	1	no	—
167	C₂³×C₂²⋊C₄	D₆₇	1	1	yes	characteristic, complemented
168	C₆₇	C₂³×#121	1	1	yes	characteristic, complemented, Sylow
169	C₂⁶	—	67	1	no	—
170	C₂⁶	—	67	1	no	—
171	C₂⁶	C₂×D₆₇	1	1	yes	characteristic
172	C₂³×D₄	—	67	28	no	—
173	C₂³×D₄	—	67	28	no	—
174	C₂³×D₄	—	67	28	no	—
175	C₂³×D₄	—	134	16	no	—
176	C₂³×D₄	—	134	16	no	—
177	C₂³×D₄	—	134	16	no	—
178	C₂⁴×C₄	—	67	1	no	—
179	C₂⁴×C₄	—	67	1	no	—
180	C₂⁴×C₄	—	67	28	no	—
181	C₂⁴×C₄	—	134	16	no	—
182	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
183	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
184	C₂×#121	—	67	448	no	—
185	C₂²×C₄⋊C₄	—	67	28	no	—
186	C₂²×C₂²⋊C₄	—	67	28	no	—
187	C₂²×C₂²⋊C₄	C₂×D₆₇	1	28	yes	complemented
188	C₂⁵	—	2	8	no	—
189	C₂⁵	—	2	8	no	—
190	C₂⁵	—	2	8	no	—
191	C₂⁵	—	67	16	no	—
192	C₂⁵	—	67	14	no	—
193	C₂⁵	—	67	14	no	—
194	C₂⁵	—	134	8	no	—
195	C₂⁵	—	134	8	no	—
196	C₂⁵	—	134	8	no	—
197	C₂⁵	—	134	8	no	—
198	C₂⁵	—	134	8	no	—
199	C₂⁵	C₂²×D₆₇	1	14	yes	—
200	C₂⁵	C₂²×D₆₇	1	1	yes	characteristic
201	C₂²×D₄	—	67	112	no	—
202	C₂²×D₄	—	67	112	no	—
203	C₂²×D₄	—	67	112	no	—
204	C₂²×D₄	—	134	224	no	—
205	C₂²×D₄	—	134	224	no	—
206	C₂²×D₄	—	134	224	no	—
207	C₂³×C₄	—	2	8	no	—
208	C₂³×C₄	—	2	8	no	—
209	C₂³×C₄	—	67	16	no	—
210	C₂³×C₄	—	67	14	no	—
211	C₂³×C₄	—	67	14	no	—
212	C₂³×C₄	—	67	112	no	—
213	C₂³×C₄	—	134	224	no	—
214	C₂³×C₄	—	134	8	no	—
215	C₂³×C₄	C₂²×D₆₇	1	14	yes	—
216	C₂³×C₄	C₂²×D₆₇	1	14	yes	—
217	#121	—	67	512	no	—
218	C₂×C₄⋊C₄	—	67	112	no	—
219	C₂×C₂²⋊C₄	—	67	112	no	—
220	C₂×C₂²⋊C₄	C₂²×D₆₇	1	112	yes	complemented
221	C₂⁴	—	2	56	no	—
222	C₂⁴	—	2	56	no	—
223	C₂⁴	—	2	56	no	—
224	C₂⁴	—	4	64	no	—
225	C₂⁴	—	268	64	no	—
226	C₂⁴	—	67	112	no	—
227	C₂⁴	—	67	28	no	—
228	C₂⁴	—	67	28	no	—
229	C₂⁴	—	134	56	no	—
230	C₂⁴	—	134	56	no	—
231	C₂⁴	—	134	56	no	—
232	C₂⁴	—	134	128	no	—
233	C₂⁴	—	134	56	no	—
234	C₂⁴	—	134	56	no	—
235	C₂⁴	D₄×D₆₇	1	8	yes	—
236	C₂⁴	D₄×D₆₇	1	8	yes	—
237	C₂⁴	#11761628	1	8	yes	—
238	C₂⁴	C₂³×D₆₇	1	28	yes	—
239	C₂⁴	C₂³×D₆₇	1	7	yes	—
240	C₂×D₄	—	67	64	no	—
241	C₂×D₄	—	67	64	no	—
242	C₂×D₄	—	67	64	no	—
243	C₂×D₄	—	134	448	no	—
244	C₂×D₄	—	134	448	no	—
245	C₂×D₄	—	134	448	no	—
246	C₂²×C₄	—	2	56	no	—
247	C₂²×C₄	—	2	56	no	—
248	C₂²×C₄	—	67	112	no	—
249	C₂²×C₄	—	67	28	no	—
250	C₂²×C₄	—	67	28	no	—
251	C₂²×C₄	—	67	64	no	—
252	C₂²×C₄	—	134	448	no	—
253	C₂²×C₄	—	134	56	no	—
254	C₂²×C₄	C₂³×D₆₇	1	28	yes	—
255	C₂²×C₄	C₂³×D₆₇	1	28	yes	—
256	C₄⋊C₄	—	67	64	no	—
257	C₂²⋊C₄	—	67	64	no	—
258	C₂²⋊C₄	C₂³×D₆₇	1	64	yes	complemented
259	C₂³	—	2	56	no	—
260	C₂³	—	2	56	no	—
261	C₂³	—	2	56	no	—
262	C₂³	—	4	224	no	—
263	C₂³	—	268	224	no	—
264	C₂³	—	67	112	no	—
265	C₂³	—	67	8	no	—
266	C₂³	—	67	8	no	—
267	C₂³	—	134	56	no	—
268	C₂³	—	134	56	no	—
269	C₂³	—	134	56	no	—
270	C₂³	—	134	448	no	—
271	C₂³	—	134	56	no	—
272	C₂³	—	134	56	no	—
273	C₂³	#14524016	1	64	yes	direct product factor
274	C₂³	C₂×D₄×D₆₇	1	28	yes	—
275	C₂³	C₂×D₄×D₆₇	1	28	yes	—
276	C₂³	C₂×#11761628	1	28	yes	—
277	C₂³	C₂⁴×D₆₇	1	8	yes	—
278	C₂³	C₂⁴×D₆₇	1	7	yes	—
279	D₄	—	134	128	no	—
280	D₄	—	134	128	no	—
281	D₄	—	134	128	no	—
282	C₂×C₄	—	2	56	no	—
283	C₂×C₄	—	2	56	no	—
284	C₂×C₄	—	67	112	no	—
285	C₂×C₄	—	67	8	no	—
286	C₂×C₄	—	67	8	no	—
287	C₂×C₄	—	134	128	no	—
288	C₂×C₄	—	134	56	no	—
289	C₂×C₄	C₂⁴×D₆₇	1	8	yes	—
290	C₂×C₄	C₂⁴×D₆₇	1	8	yes	—
291	C₂²	—	2	8	no	—
292	C₂²	—	2	8	no	—
293	C₂²	—	2	8	no	—
294	C₂²	—	4	112	no	—
295	C₂²	—	268	112	no	—
296	C₂²	—	67	16	no	—
297	C₂²	—	134	8	no	—
298	C₂²	—	134	8	no	—
299	C₂²	—	134	8	no	—
300	C₂²	—	134	224	no	—
301	C₂²	—	134	8	no	—
302	C₂²	—	134	8	no	—
303	C₂²	C₂×#14524016	1	112	yes	direct product factor
304	C₂²	C₂²×D₄×D₆₇	1	14	yes	—
305	C₂²	C₂²×D₄×D₆₇	1	14	yes	—
306	C₂²	C₂²×#11761628	1	14	yes	—
307	C₂²	C₂⁵×D₆₇	1	1	yes	characteristic
308	C₄	—	2	8	no	—
309	C₄	—	2	8	no	—
310	C₄	—	67	16	no	—
311	C₄	—	134	8	no	—
312	C₂	—	4	8	no	—
313	C₂	—	268	8	no	—
314	C₂	—	134	16	no	—
315	C₂	C₂²×#14524016	1	28	yes	direct product factor
316	C₂	C₂³×D₄×D₆₇	1	1	yes	characteristic
317	C₂	C₂³×D₄×D₆₇	1	1	yes	characteristic
318	C₂	C₂³×#11761628	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₆₇
commutator, cosocle	C₂²×C₆₇	C₂⁶
Frattini	C₂²	C₂⁵×D₆₇
Fitting	C₂³×C₂²⋊C₄×C₆₇	C₂
socle	C₂⁵×C₆₇	C₂³
hypercenter	C₂³×C₂²⋊C₄	D₆₇
hypocenter	C₆₇	C₂³×#121
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₂₆₈	C₂	semidirect
2. C₂⁴×C₂²⋊D₆₇	C₂	semidirect
3. C₂⁴×C₂²⋊D₆₇	C₂	semidirect
4. C₂⁴×C₄×D₆₇	C₂	semidirect
5. C₂²×#14524016	C₂	direct (N × H)
6. C₂³×Dic₆₇⋊C₄	C₂	semidirect
7. C₂³×(C₂×D₆₇)⋊C₄	C₂	semidirect
8. C₂³×C₂²⋊C₄×C₆₇	C₂	semidirect
12. C₂³×D₂₆₈	C₂²	semidirect
15. C₂³×C₂²⋊D₆₇	C₂²	semidirect
16. C₂³×C₂²⋊D₆₇	C₂²	semidirect
20. C₂³×C₄×D₆₇	C₂²	semidirect
21. C₂×#14524016	C₂²	direct (N × H)
22. C₂²×Dic₆₇⋊C₄	C₂²	semidirect
23. C₂²×(C₂×D₆₇)⋊C₄	C₂²	semidirect
27. C₂²×C₂²⋊C₄×C₆₇	C₂²	semidirect
33. C₂⁴×D₆₇	D₄	semidirect
37. C₂²×D₂₆₈	C₂³	semidirect
40. C₂²×C₂²⋊D₆₇	C₂³	semidirect
41. C₂²×C₂²⋊D₆₇	C₂³	semidirect
43. C₂³×Dic₆₇	D₄	semidirect
47. C₂²×C₄×D₆₇	C₂³	semidirect
48. #14524016	C₂³	direct (N × H)
49. C₂×Dic₆₇⋊C₄	C₂³	semidirect
50. C₂×(C₂×D₆₇)⋊C₄	C₂³	semidirect
60. C₂×C₂²⋊C₄×C₆₇	C₂³	semidirect
68. C₂³×D₆₇	C₂×D₄	semidirect
72. C₂×D₂₆₈	C₂⁴	semidirect
75. C₂×C₂²⋊D₆₇	C₂⁴	semidirect
76. C₂×C₂²⋊D₆₇	C₂⁴	semidirect
78. C₂²×Dic₆₇	C₂×D₄	semidirect
82. C₂×C₄×D₆₇	C₂⁴	semidirect
83. Dic₆₇⋊C₄	C₂⁴	semidirect
84. (C₂×D₆₇)⋊C₄	C₂⁴	semidirect
98. C₂²⋊C₄×C₆₇	C₂⁴	semidirect
103. C₂³×C₆₇	#121	semidirect
116. C₂²×D₆₇	C₂²×D₄	semidirect
126. C₂×Dic₆₇	C₂²×D₄	semidirect
135. C₂²×C₆₇	C₂×#121	semidirect
147. C₂×D₆₇	C₂³×D₄	semidirect
151. Dic₆₇	C₂³×D₄	semidirect
154. C₂×C₆₇	C₂²×#121	semidirect
167. C₂³×C₂²⋊C₄	D₆₇	semidirect
168. C₆₇	C₂³×#121	semidirect
187. C₂²×C₂²⋊C₄	C₂×D₆₇	semidirect
220. C₂×C₂²⋊C₄	C₂²×D₆₇	semidirect
258. C₂²⋊C₄	C₂³×D₆₇	semidirect
273. C₂³	#14524016	direct (N × H)
303. C₂²	C₂×#14524016	direct (N × H)
315. C₂	C₂²×#14524016	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₂₆₈	2
C₂⁴×C₂²⋊D₆₇	2
C₂⁴×C₄×D₆₇	2
C₂²×#14524016	2
C₂³×Dic₆₇⋊C₄	2
C₂³×(C₂×D₆₇)⋊C₄	2
C₂³×C₂²⋊C₄×C₆₇	2
C₂³×#121	67

Subgroups — C23×#14524016 #31762467

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14524016 #31762467