Subgroups — #30826554

Order 17152.

590 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569144	C₂	1	2	yes	maximal
2	#22559060	C₂	1	8	yes	maximal, complemented
3	#22558968	C₂	1	1	yes	maximal, characteristic
4	#22558664	C₂	1	2	yes	maximal
5	C₂×#16863118	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#3170	C₂	1	1	yes	maximal, characteristic
7	#16863527	—	2	2	no	—
8	#16863527	C₂²	1	4	yes	—
9	Q₈×C₆₇⋊C₈	—	2	2	no	—
10	Q₈×C₆₇⋊C₈	C₂²	1	4	yes	—
11	#16863423	—	2	2	no	—
12	#16863420	C₂²	1	2	yes	—
13	#16863352	—	2	2	no	—
14	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
16	#16863149	C₂²	1	4	yes	—
17	Q₈⋊(C₆₇⋊C₈)	C₂²	1	8	yes	—
18	#16863118	C₂²	1	4	yes	—
19	#533×C₆₇	C₂²	1	2	yes	—
20	#500×C₆₇	C₄	1	1	yes	characteristic
21	#494×C₆₇	C₄	1	1	yes	characteristic
22	#487×C₆₇	C₄	1	2	yes	—
23	#467×C₆₇	C₄	1	4	yes	—
24	C₄×Q₁₆×C₆₇	C₂²	1	4	yes	—
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#11761612	—	2	2	no	—
27	C₂²×C₆₇⋊C₈	—	2	1	no	—
28	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
29	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
30	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
32	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
33	C₄⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
34	#14523954	—	2	4	no	—
35	C₄×C₆₇⋊C₈	—	2	4	no	—
36	C₄×C₆₇⋊C₈	—	2	1	no	—
37	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
38	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
39	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
40	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
41	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
42	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
43	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
44	#125×C₆₇	—	2	2	no	—
45	C₂²⋊Q₈×C₆₇	—	2	2	no	—
46	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
47	C₄×Q₈×C₆₇	—	2	2	no	—
48	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
49	#117×C₆₇	—	2	2	no	—
50	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	#107×C₆₇	C₂×C₄	1	4	yes	—
52	#103×C₆₇	C₂×C₄	1	8	yes	—
53	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₆₇	C₂³	1	2	yes	—
56	#11761612	—	4	4	no	—
57	C₂×C₆₇⋊C₈	—	2	2	no	—
58	C₂×C₆₇⋊C₈	—	2	4	no	—
59	C₂×C₆₇⋊C₈	—	2	2	no	—
60	C₂×C₆₇⋊C₈	—	4	2	no	—
61	C₂×C₆₇⋊C₈	Q₁₆	1	4	yes	complemented
62	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
63	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
64	C₂×Q₈×C₆₇	—	2	2	no	—
65	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
69	Q₁₆×C₆₇	—	4	4	no	—
70	C₂×C₈×C₆₇	—	4	2	no	—
71	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
73	C₄⋊C₄×C₆₇	—	2	2	no	—
74	C₄⋊C₄×C₆₇	—	2	2	no	—
75	C₄⋊C₄×C₆₇	—	2	2	no	—
76	C₄⋊C₄×C₆₇	—	2	4	no	—
77	C₄⋊C₄×C₆₇	—	2	4	no	—
78	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
79	C₂²⋊C₄×C₆₇	—	2	4	no	—
80	C₄²×C₆₇	—	2	4	no	—
81	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₆₇	#41	1	1	yes	characteristic
84	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
85	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
86	Q₈×C₆₇	—	2	8	no	—
87	Q₈×C₆₇	—	4	4	no	—
88	C₂×C₄×C₆₇	—	2	2	no	—
89	C₂×C₄×C₆₇	—	2	2	no	—
90	C₂×C₄×C₆₇	—	2	4	no	—
91	C₂×C₄×C₆₇	—	2	4	no	—
92	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
96	C₂×C₄×C₆₇	#135	1	2	yes	—
97	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
100	C₈×C₆₇	—	2	2	no	—
101	C₈×C₆₇	—	2	2	no	—
102	C₆₇⋊C₈	—	2	2	no	—
103	C₆₇⋊C₈	—	4	2	no	—
104	C₆₇⋊C₈	—	4	4	no	—
105	C₆₇⋊C₈	C₂×Q₁₆	1	4	yes	—
106	C₂²×C₆₇	—	2	1	no	—
107	C₂²×C₆₇	—	2	1	no	—
108	C₂²×C₆₇	C₄×Q₁₆	1	2	yes	—
109	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
110	C₄×C₆₇	—	2	1	no	—
111	C₄×C₆₇	—	2	1	no	—
112	C₄×C₆₇	—	4	4	no	—
113	C₄×C₆₇	—	4	4	no	—
114	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
115	C₄×C₆₇	#444	1	2	yes	—
116	C₄×C₆₇	#568	1	2	yes	—
117	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
118	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
119	#17977	—	67	1	no	maximal, Sylow
120	C₂×C₆₇	—	2	2	no	—
121	C₂×C₆₇	#2932	1	1	yes	characteristic
122	C₂×C₆₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
123	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
124	#3170	D₆₇	1	1	yes	characteristic
125	#2960	—	67	2	no	—
126	#1584	—	67	8	no	—
127	#1567	—	67	1	no	—
128	#1493	—	67	2	no	—
129	C₂×#322	—	67	1	no	—
130	C₆₇	#17977	1	1	yes	characteristic, complemented, Sylow
131	#533	C₂×D₆₇	1	2	yes	—
132	#500	Dic₆₇	1	1	yes	characteristic
133	#494	Dic₆₇	1	1	yes	characteristic
134	#487	Dic₆₇	1	2	yes	—
135	#467	Dic₆₇	1	4	yes	—
136	#446	—	67	4	no	—
137	#446	—	134	2	no	—
138	C₈×Q₈	—	67	4	no	—
139	C₈×Q₈	—	134	2	no	—
140	C₄×Q₁₆	C₂×D₆₇	1	4	yes	—
141	#424	—	134	2	no	—
142	#423	—	67	2	no	—
143	#405	—	134	2	no	—
144	C₂×C₄×C₈	—	67	1	no	—
145	C₂×C₄×C₈	—	67	1	no	—
146	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
147	#335	—	67	4	no	—
148	Q₈⋊C₈	—	67	8	no	—
149	#322	—	67	4	no	—
150	C₂×Q₁₆	C₂×Dic₆₇	1	4	yes	—
151	C₂×M₁₆	—	134	2	no	—
152	C₂²×C₈	—	67	1	no	—
153	C₂²×C₈	—	67	1	no	—
154	C₂²×C₈	—	134	1	no	—
155	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
156	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
157	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
158	#125	—	2	2	no	—
159	C₂²⋊Q₈	—	2	2	no	—
160	C₂²⋊Q₈	C₂×Dic₆₇	1	4	yes	—
161	C₄×Q₈	—	2	2	no	—
162	C₄×Q₈	C₂²×D₆₇	1	4	yes	—
163	#117	—	2	2	no	—
164	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
165	#107	C₂×Dic₆₇	1	4	yes	—
166	C₄⋊C₈	—	67	4	no	—
167	C₄⋊C₈	—	134	4	no	—
168	C₄⋊C₈	—	134	4	no	—
169	C₄⋊C₈	—	134	2	no	—
170	#103	C₂×Dic₆₇	1	8	yes	—
171	#97	—	134	4	no	—
172	C₄×C₈	—	67	2	no	—
173	C₄×C₈	—	67	2	no	—
174	C₄×C₈	—	67	2	no	—
175	C₄×C₈	—	134	4	no	—
176	C₄×C₈	—	134	1	no	—
177	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
178	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
179	C₄×C₈	C₂²×D₆₇	1	2	yes	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	C₂²×Dic₆₇	1	4	yes	—
182	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
183	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
184	C₂²×C₄	#11761627	1	1	yes	characteristic
185	Q₁₆	—	4	4	no	—
186	M₁₆	—	268	4	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	268	2	no	—
189	C₂×C₈	—	67	4	no	—
190	C₂×C₈	—	67	2	no	—
191	C₂×C₈	—	67	2	no	—
192	C₂×C₈	—	134	4	no	—
193	C₂×C₈	—	134	2	no	—
194	C₂×C₈	—	134	2	no	—
195	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
196	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	C₂²×Dic₆₇	1	4	yes	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₄²	—	2	4	no	—
205	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
206	C₄²	D₄×D₆₇	1	1	yes	characteristic
207	C₄²	#11761627	1	1	yes	characteristic
208	C₄²	C₂³×D₆₇	1	1	yes	characteristic
209	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
210	Q₈	—	2	8	no	—
211	Q₈	—	4	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
217	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
218	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
219	C₂×C₄	Q₁₆×D₆₇	1	2	yes	—
220	C₂×C₄	#14524066	1	2	yes	—
221	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
222	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
223	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	268	2	no	—
227	C₈	—	268	4	no	—
228	C₈	—	67	4	no	—
229	C₈	—	134	2	no	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	Q₁₆×Dic₆₇	1	2	yes	—
233	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	4	4	no	—
237	C₄	—	4	4	no	—
238	C₄	Q₁₆×Dic₆₇	1	2	yes	—
239	C₄	#16863518	1	2	yes	—
240	C₄	#16864163	1	2	yes	—
241	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
242	C₄	C₂×#14524066	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	#22568927	1	1	yes	characteristic
245	C₂	C₂×Q₁₆×Dic₆₇	1	1	yes	characteristic
246	C₂	C₂×#16863518	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#3170	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#3170	D₆₇
hypocenter	C₆₇	#17977
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #22559060	C₂	semidirect
61. C₂×C₆₇⋊C₈	Q₁₆	semidirect
130. C₆₇	#17977	semidirect

Maximal subgroups

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

subgroup	index
#22569144	2
#22559060	2
#22558968	2
#22558664	2
C₂×#16863118	2
C₆₇×#3170	2
#17977	67