Subgroups — #30864446

Order 17152.

1046 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×#16863513	C₂	1	4	yes	maximal, complemented
2	#22568921	C₂	1	2	yes	maximal, complemented
3	#22561098	C₂	1	4	yes	maximal, complemented
4	#22561038	C₂	1	2	yes	maximal, complemented
5	#22560381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#3147	C₂	1	1	yes	maximal, characteristic
8	C₂×#14524050	C₂²	1	4	yes	—
9	#16863513	—	2	16	no	—
10	#16863474	—	2	4	no	—
11	D₄×C₆₇⋊C₈	—	2	4	no	—
12	C₂×#14523979	C₂²	1	4	yes	—
13	#16863420	C₂²	1	2	yes	—
14	C₂×#14523970	C₂²	1	8	yes	—
15	#16863363	C₂²	1	4	yes	—
16	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
18	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
19	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
20	C₂×#135×C₆₇	C₂²	1	4	yes	—
21	#532×C₆₇	C₂²	1	2	yes	—
22	#495×C₆₇	C₄	1	4	yes	complemented
23	#494×C₆₇	C₄	1	1	yes	characteristic, complemented
24	#493×C₆₇	C₄	1	1	yes	characteristic, complemented
25	#492×C₆₇	C₄	1	2	yes	complemented
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	#14524050	—	2	16	no	—
28	C₂×#11761612	—	2	4	no	—
29	C₂×#11761612	C₂³	1	4	yes	—
30	C₂²×C₆₇⋊C₈	—	2	4	no	—
31	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
33	C₂×C₄×Dic₆₇	—	2	2	no	—
34	C₂×C₄×Dic₆₇	C₂³	1	2	yes	—
35	#14523979	—	2	8	no	—
36	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
37	#14523970	—	2	16	no	—
38	C₂²⋊(C₆₇⋊C₈)	—	2	8	no	—
39	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
40	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
41	C₈×Dic₆₇	—	2	4	no	—
42	C₈×Dic₆₇	D₄	1	8	yes	complemented
43	C₂×#41×C₆₇	C₂³	1	4	yes	—
44	#135×C₆₇	—	2	8	no	—
45	#135×C₆₇	—	2	8	no	—
46	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
47	C₂×SD₁₆×C₆₇	C₂×C₄	1	8	yes	—
48	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
51	#127×C₆₇	C₂×C₄	1	2	yes	—
52	#124×C₆₇	C₂×C₄	1	4	yes	—
53	#121×C₆₇	—	2	4	no	—
54	C₄×D₄×C₆₇	—	2	4	no	—
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
59	C₂²×Dic₆₇	—	2	2	no	—
60	#11761612	—	2	8	no	—
61	#11761612	—	4	8	no	—
62	C₂×C₆₇⋊C₈	—	2	2	no	—
63	C₂×C₆₇⋊C₈	—	2	4	no	—
64	C₂×C₆₇⋊C₈	—	4	8	no	—
65	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
66	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
67	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
68	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
69	C₄×Dic₆₇	—	2	8	no	—
70	C₄×Dic₆₇	—	2	2	no	—
71	C₄×Dic₆₇	C₂×D₄	1	4	yes	—
72	#41×C₆₇	—	2	8	no	—
73	#41×C₆₇	—	2	8	no	—
74	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
75	C₂×D₄×C₆₇	—	2	4	no	—
76	C₂×D₄×C₆₇	—	2	4	no	—
77	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₆₇	—	2	4	no	—
79	C₂²×C₄×C₆₇	#41	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
81	Q₁₆×C₆₇	—	2	8	no	—
82	SD₁₆×C₆₇	—	2	16	no	—
83	D₈×C₆₇	—	2	8	no	—
84	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
87	C₂×C₈×C₆₇	#41	1	4	yes	—
88	C₄⋊C₄×C₆₇	—	2	4	no	—
89	C₂²⋊C₄×C₆₇	—	2	8	no	—
90	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₆₇	C₂×D₄	1	2	yes	—
93	C₂³×C₆₇	—	2	4	no	—
94	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
95	Q₈×C₆₇	—	2	8	no	—
96	D₄×C₆₇	—	2	8	no	—
97	D₄×C₆₇	—	4	8	no	—
98	D₄×C₆₇	—	4	8	no	—
99	C₂×C₄×C₆₇	—	2	4	no	—
100	C₂×C₄×C₆₇	—	2	4	no	—
101	C₂×C₄×C₆₇	—	4	4	no	—
102	C₂×C₄×C₆₇	—	4	4	no	—
103	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
107	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
109	C₂×Dic₆₇	—	2	8	no	—
110	C₂×Dic₆₇	—	2	4	no	—
111	C₈×C₆₇	C₄×D₄	1	4	yes	—
112	C₈×C₆₇	C₄×D₄	1	4	yes	—
113	C₆₇⋊C₈	—	2	2	no	—
114	C₆₇⋊C₈	—	4	4	no	—
115	C₆₇⋊C₈	#127	1	2	yes	—
116	C₆₇⋊C₈	#127	1	2	yes	—
117	C₂²×C₆₇	—	2	2	no	—
118	C₂²×C₆₇	—	4	4	no	—
119	C₂²×C₆₇	—	4	8	no	—
120	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
121	C₂²×C₆₇	#532	1	2	yes	—
122	C₄×C₆₇	—	2	2	no	—
123	C₄×C₆₇	—	4	4	no	—
124	C₄×C₆₇	#443	1	4	yes	complemented
125	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
126	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
127	C₄×C₆₇	#532	1	1	yes	characteristic
128	Dic₆₇	—	2	8	no	—
129	#21705	—	67	1	no	maximal, Sylow
130	C₂×C₆₇	—	2	2	no	—
131	C₂×C₆₇	—	8	4	no	—
132	C₂×C₆₇	C₄×#127	1	1	yes	characteristic
133	C₂×C₆₇	C₂×#443	1	2	yes	—
134	#3147	D₆₇	1	1	yes	characteristic
135	C₂×#443	—	67	4	no	—
136	#2931	—	67	2	no	—
137	#1974	—	67	4	no	—
138	#1962	—	67	2	no	—
139	#1848	—	67	1	no	—
140	C₄²×C₈	—	67	1	no	—
141	C₆₇	#21705	1	1	yes	characteristic, complemented, Sylow
142	C₂×#135	C₂×D₆₇	1	4	yes	—
143	C₂×#131	—	67	4	no	—
144	#532	C₂×D₆₇	1	2	yes	—
145	#495	Dic₆₇	1	4	yes	complemented
146	#494	Dic₆₇	1	1	yes	characteristic, complemented
147	#493	Dic₆₇	1	1	yes	characteristic, complemented
148	#492	Dic₆₇	1	2	yes	complemented
149	#443	—	134	16	no	—
150	#436	—	134	4	no	—
151	C₈×D₄	—	134	4	no	—
152	C₂×#108	—	67	4	no	—
153	#423	—	67	2	no	—
154	C₂×#104	—	67	8	no	—
155	#408	—	67	4	no	—
156	C₂×C₄×C₈	—	67	2	no	—
157	C₂×C₄×C₈	—	67	1	no	—
158	C₂×C₄×C₈	—	67	2	no	—
159	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
160	C₄³	—	67	1	no	—
161	C₂×#41	C₂²×D₆₇	1	4	yes	—
162	#135	—	2	8	no	—
163	#135	—	2	8	no	—
164	C₂×Q₁₆	C₂×Dic₆₇	1	4	yes	—
165	C₂×SD₁₆	C₂×Dic₆₇	1	8	yes	—
166	C₂×D₈	C₂×Dic₆₇	1	4	yes	—
167	#131	—	134	16	no	—
168	C₂×M₁₆	—	67	4	no	—
169	C₂×M₁₆	—	134	4	no	—
170	C₂²×C₈	—	67	1	no	—
171	C₂²×C₈	—	67	1	no	—
172	C₂²×C₈	—	134	4	no	—
173	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
174	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
175	#127	C₂×Dic₆₇	1	2	yes	—
176	#124	C₂×Dic₆₇	1	4	yes	—
177	#121	—	2	4	no	—
178	C₄×D₄	—	2	4	no	—
179	C₂×C₄²	—	67	2	no	—
180	C₂×C₄²	—	134	2	no	—
181	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
182	#108	—	134	8	no	—
183	C₄⋊C₈	—	134	4	no	—
184	#104	—	134	16	no	—
185	C₂²⋊C₈	—	134	8	no	—
186	C₄×C₈	—	67	8	no	—
187	C₄×C₈	—	67	2	no	—
188	C₄×C₈	—	67	2	no	—
189	C₄×C₈	—	134	4	no	—
190	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
191	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
192	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
193	#41	—	2	8	no	—
194	#41	—	2	8	no	—
195	C₂×Q₈	C₂²×Dic₆₇	1	4	yes	—
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	C₂²×Dic₆₇	1	4	yes	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	134	2	no	—
201	C₂²×C₄	#11761627	1	2	yes	—
202	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
203	Q₁₆	—	2	8	no	—
204	SD₁₆	—	2	16	no	—
205	D₈	—	2	8	no	—
206	M₁₆	—	268	8	no	—
207	M₁₆	—	134	8	no	—
208	C₂×C₈	—	268	8	no	—
209	C₂×C₈	—	67	4	no	—
210	C₂×C₈	—	67	1	no	—
211	C₂×C₈	—	67	1	no	—
212	C₂×C₈	—	67	2	no	—
213	C₂×C₈	—	134	4	no	—
214	C₂×C₈	—	134	2	no	—
215	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
216	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
217	C₂×C₈	D₄×D₆₇	1	4	yes	—
218	C₂×C₈	#11761627	1	4	yes	—
219	C₄⋊C₄	—	2	4	no	—
220	C₂²⋊C₄	—	2	8	no	—
221	C₄²	—	67	4	no	—
222	C₄²	—	134	8	no	—
223	C₄²	—	134	2	no	—
224	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
225	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
226	C₄²	D₄×D₆₇	1	2	yes	—
227	C₂³	—	2	4	no	—
228	C₂³	C₂×#11761627	1	1	yes	characteristic
229	Q₈	—	2	8	no	—
230	D₄	—	2	8	no	—
231	D₄	—	4	8	no	—
232	D₄	—	4	8	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	134	8	no	—
238	C₂×C₄	—	134	4	no	—
239	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
240	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
241	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
242	C₂×C₄	C₂×D₄×D₆₇	1	2	yes	—
243	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
244	C₂×C₄	C₂×#11761627	1	2	yes	—
245	C₈	—	268	4	no	—
246	C₈	—	67	2	no	—
247	C₈	—	67	2	no	—
248	C₈	—	134	2	no	—
249	C₈	D₄×Dic₆₇	1	4	yes	—
250	C₈	D₄×Dic₆₇	1	4	yes	—
251	C₂²	—	2	2	no	—
252	C₂²	—	4	8	no	—
253	C₂²	—	4	4	no	—
254	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
255	C₂²	#16863944	1	2	yes	—
256	C₄	—	2	2	no	—
257	C₄	—	4	4	no	—
258	C₄	—	134	8	no	—
259	C₄	#16863513	1	4	yes	complemented
260	C₄	C₂×D₄×Dic₆₇	1	2	yes	—
261	C₄	#127×D₆₇	1	1	yes	characteristic
262	C₄	#16863944	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	8	4	no	—
265	C₂	#127×Dic₆₇	1	1	yes	characteristic
266	C₂	C₂×#16863513	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting, prime radical	C₆₇×#3147	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#3147	D₆₇
hypocenter	C₆₇	#21705
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#16863513	C₂	semidirect
2. #22568921	C₂	semidirect
3. #22561098	C₂	semidirect
4. #22561038	C₂	semidirect
5. #22560381	C₂	semidirect
6. C₄×C₈×Dic₆₇	C₂	semidirect
22. #495×C₆₇	C₄	semidirect
23. #494×C₆₇	C₄	semidirect
24. #493×C₆₇	C₄	semidirect
25. #492×C₆₇	C₄	semidirect
42. C₈×Dic₆₇	D₄	semidirect
124. C₄×C₆₇	#443	semidirect
141. C₆₇	#21705	semidirect
145. #495	Dic₆₇	semidirect
146. #494	Dic₆₇	semidirect
147. #493	Dic₆₇	semidirect
148. #492	Dic₆₇	semidirect
259. C₄	#16863513	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863513	2
#22568921	2
#22561098	2
#22561038	2
#22560381	2
C₄×C₈×Dic₆₇	2
C₆₇×#3147	2
#21705	67