Subgroups — #30772132

Order 17152.

404 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#22560247	C₂	1	1	yes	maximal, characteristic
3	#22558304	C₂	1	4	yes	maximal, complemented
4	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14523979	C₂²	1	2	yes	—
7	#16863360	—	2	1	no	—
8	C₂²⋊(C₆₇⋊C₁₆)	—	2	1	no	—
9	C₂²⋊(C₆₇⋊C₁₆)	—	2	1	no	—
10	#16863161	—	2	1	no	—
11	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#117×C₆₇	—	2	1	no	—
13	#427×C₆₇	—	2	1	no	—
14	#427×C₆₇	—	2	1	no	—
15	#427×C₆₇	C₄	1	2	yes	—
16	#427×C₆₇	C₂²	1	2	yes	—
17	C₂×#107×C₆₇	C₄	1	1	yes	characteristic
18	C₂×#107×C₆₇	C₄	1	1	yes	characteristic
19	C₂×#106×C₆₇	—	2	1	no	—
20	C₂×#11761612	—	2	2	no	—
21	C₂×C₆₇⋊C₁₆	—	2	2	no	—
22	C₂×C₆₇⋊C₁₆	—	2	4	no	—
23	#14523979	—	2	4	no	—
24	C₂²⋊(C₆₇⋊C₈)	—	4	1	no	—
25	C₂²⋊(C₆₇⋊C₈)	—	4	1	no	—
26	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	—	2	4	no	—
28	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
34	#117×C₆₇	—	2	2	no	—
35	#117×C₆₇	—	2	2	no	—
36	#117×C₆₇	—	4	1	no	—
37	#117×C₆₇	—	4	1	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
39	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
40	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
41	C₂×C₄²×C₆₇	—	4	1	no	—
42	#107×C₆₇	—	2	1	no	—
43	#107×C₆₇	—	2	1	no	—
44	#107×C₆₇	C₂×C₄	1	2	yes	—
45	#107×C₆₇	C₂×C₄	1	2	yes	—
46	#106×C₆₇	—	2	2	no	—
47	#106×C₆₇	—	2	2	no	—
48	#11761612	—	4	2	no	—
49	#11761612	—	4	2	no	—
50	C₂×C₆₇⋊C₈	—	4	2	no	—
51	C₆₇⋊C₁₆	—	4	2	no	—
52	C₂⁴×C₆₇	SD₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	—	2	2	no	—
54	C₂²×C₄×C₆₇	—	2	2	no	—
55	C₂²×C₄×C₆₇	—	4	1	no	—
56	C₂²×C₄×C₆₇	—	4	1	no	—
57	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₆₇	—	2	2	no	—
64	C₂×C₈×C₆₇	—	2	2	no	—
65	C₂×C₈×C₆₇	—	2	2	no	—
66	C₂×C₈×C₆₇	—	2	2	no	—
67	C₂×C₈×C₆₇	—	2	2	no	—
68	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₆₇	#41	1	4	yes	—
73	C₄⋊C₄×C₆₇	—	2	2	no	—
74	C₄⋊C₄×C₆₇	—	2	2	no	—
75	C₄⋊C₄×C₆₇	—	4	1	no	—
76	C₄⋊C₄×C₆₇	—	4	1	no	—
77	C₂²⋊C₄×C₆₇	—	4	2	no	—
78	C₂²⋊C₄×C₆₇	—	4	2	no	—
79	C₄²×C₆₇	—	4	2	no	—
80	C₄²×C₆₇	—	4	2	no	—
81	C₂³×C₆₇	—	2	2	no	—
82	C₂³×C₆₇	—	2	2	no	—
83	C₂³×C₆₇	—	4	1	no	—
84	C₂³×C₆₇	#102	1	1	yes	characteristic
85	C₂³×C₆₇	#102	1	1	yes	characteristic
86	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	2	1	no	—
89	C₂×C₄×C₆₇	—	2	1	no	—
90	C₂×C₄×C₆₇	—	2	1	no	—
91	C₂×C₄×C₆₇	—	2	1	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	4	2	no	—
95	C₂×C₄×C₆₇	—	4	2	no	—
96	C₂×C₄×C₆₇	—	8	2	no	—
97	C₂×C₄×C₆₇	—	8	2	no	—
98	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	#123	1	2	yes	—
102	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₆₇	#137	1	2	yes	—
104	C₈×C₆₇	—	2	1	no	—
105	C₈×C₆₇	—	2	1	no	—
106	C₈×C₆₇	#117	1	2	yes	—
107	C₈×C₆₇	#123	1	2	yes	—
108	C₆₇⋊C₈	—	8	2	no	—
109	C₂²×C₆₇	—	2	1	no	—
110	C₂²×C₆₇	—	2	1	no	—
111	C₂²×C₆₇	—	2	1	no	—
112	C₂²×C₆₇	—	2	1	no	—
113	C₂²×C₆₇	—	2	1	no	—
114	C₂²×C₆₇	—	2	1	no	—
115	C₂²×C₆₇	—	2	1	no	—
116	C₂²×C₆₇	—	4	4	no	—
117	C₂²×C₆₇	#359	1	2	yes	—
118	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
119	C₂²×C₆₇	#483	1	2	yes	—
120	C₄×C₆₇	—	2	1	no	—
121	C₄×C₆₇	—	2	1	no	—
122	C₄×C₆₇	—	8	2	no	—
123	C₄×C₆₇	—	8	2	no	—
124	C₄×C₆₇	#381	1	1	yes	characteristic
125	C₄×C₆₇	#419	1	1	yes	characteristic
126	C₄×C₆₇	#480	1	2	yes	—
127	#12869	—	67	1	no	maximal, Sylow
128	C₂×C₆₇	—	2	2	no	—
129	C₂×C₆₇	—	2	2	no	—
130	C₂×C₆₇	—	4	1	no	—
131	C₂×C₆₇	#1823	1	1	yes	characteristic
132	C₂×C₆₇	C₂×#359	1	1	yes	characteristic
133	C₂×C₆₇	#2143	1	1	yes	characteristic
134	C₂×#427	D₆₇	1	1	yes	characteristic
135	C₂×C₂²⋊C₁₆	—	67	1	no	—
136	#1810	—	67	1	no	—
137	#1382	—	67	4	no	—
138	C₆₇	#12869	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
140	C₂×#117	—	2	1	no	—
141	C₂²×C₁₆	—	67	2	no	—
142	C₂×#108	—	67	2	no	—
143	#427	—	2	1	no	—
144	#427	—	2	1	no	—
145	#427	Dic₆₇	1	2	yes	—
146	#427	C₂×D₆₇	1	2	yes	—
147	C₂×#107	Dic₆₇	1	1	yes	characteristic
148	C₂×#107	Dic₆₇	1	1	yes	characteristic
149	C₂×#106	—	2	1	no	—
150	#407	—	134	1	no	—
151	C₂²⋊C₁₆	—	134	1	no	—
152	C₂²⋊C₁₆	—	134	1	no	—
153	#341	—	134	1	no	—
154	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
155	C₂×M₁₆	—	134	2	no	—
156	C₂²×C₈	—	2	4	no	—
157	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
158	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
159	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
160	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
161	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
162	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
163	#117	—	2	2	no	—
164	#117	—	2	2	no	—
165	#117	—	4	1	no	—
166	#117	—	4	1	no	—
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₂²⋊C₄	—	4	1	no	—
170	C₂×C₄²	—	4	1	no	—
171	C₂×C₁₆	—	134	2	no	—
172	C₂×C₁₆	—	134	4	no	—
173	#108	—	134	4	no	—
174	#107	—	2	1	no	—
175	#107	—	2	1	no	—
176	#107	C₂×Dic₆₇	1	2	yes	—
177	#107	C₂×Dic₆₇	1	2	yes	—
178	#106	—	2	2	no	—
179	#106	—	2	2	no	—
180	C₂²⋊C₈	—	268	1	no	—
181	C₂²⋊C₈	—	268	1	no	—
182	C₂⁴	#11761616	1	1	yes	characteristic
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	—	4	1	no	—
187	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
188	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
189	C₂²×C₄	#11761613	1	1	yes	characteristic
190	C₂²×C₄	#11761613	1	1	yes	characteristic
191	C₂²×C₄	#11761616	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
193	M₁₆	—	268	2	no	—
194	M₁₆	—	268	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	268	2	no	—
201	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
203	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
204	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
205	C₂×C₈	#11761628	1	4	yes	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	4	1	no	—
209	C₄⋊C₄	—	4	1	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₂²⋊C₄	—	4	2	no	—
212	C₄²	—	4	2	no	—
213	C₄²	—	4	2	no	—
214	C₁₆	—	268	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	4	1	no	—
218	C₂³	#14523964	1	1	yes	characteristic
219	C₂³	#14523964	1	1	yes	characteristic
220	C₂³	C₂×#11761616	1	1	yes	characteristic
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	#14523964	1	1	yes	characteristic
233	C₂×C₄	#14523964	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
235	C₂×C₄	#14524026	1	2	yes	—
236	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
237	C₂×C₄	#14524074	1	2	yes	—
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	536	2	no	—
241	C₈	#14524002	1	2	yes	—
242	C₈	#14524026	1	2	yes	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	4	4	no	—
251	C₂²	#16863203	1	2	yes	—
252	C₂²	C₂×#14523964	1	1	yes	characteristic
253	C₂²	#16863726	1	2	yes	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	8	2	no	—
257	C₄	—	8	2	no	—
258	C₄	#16863259	1	1	yes	characteristic
259	C₄	#16863405	1	1	yes	characteristic
260	C₄	#16863709	1	2	yes	—
261	C₂	—	2	2	no	—
262	C₂	—	2	2	no	—
263	C₂	—	4	1	no	—
264	C₂	#22560295	1	1	yes	characteristic
265	C₂	C₂×#16863203	1	1	yes	characteristic
266	C₂	#22561937	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#427×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂⁴×C₆₇	SD₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#427	D₆₇
hypocenter	C₆₇	#12869
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
3. #22558304	C₂	semidirect
138. C₆₇	#12869	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊(C₆₇⋊C₁₆)	2
#22560247	2
#22558304	2
C₂×#427×C₆₇	2
#12869	67