Subgroups — #30864453

Order 17152.

1022 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863518	C₂	1	4	yes	maximal, complemented
2	#22568921	C₂	1	2	yes	maximal, complemented
3	#22561103	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×#14523953	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#3147	C₂	1	1	yes	maximal, characteristic
8	C₂×#14524050	C₂²	1	4	yes	—
9	#16863518	—	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×#14523953	C₂²	1	2	yes	—
17	C₂×#14523953	C₂²	1	2	yes	—
18	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
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	#14523953	—	2	4	no	—
40	#14523953	D₄	1	8	yes	complemented
41	C₄×C₆₇⋊C₈	D₄	1	4	yes	—
42	C₂×#41×C₆₇	C₂³	1	4	yes	—
43	#135×C₆₇	—	2	8	no	—
44	#135×C₆₇	—	2	8	no	—
45	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
46	C₂×SD₁₆×C₆₇	C₂×C₄	1	8	yes	—
47	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
49	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
50	#127×C₆₇	C₂×C₄	1	2	yes	—
51	#124×C₆₇	C₂×C₄	1	4	yes	—
52	#121×C₆₇	—	2	4	no	—
53	C₄×D₄×C₆₇	—	2	4	no	—
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
58	C₂²×Dic₆₇	—	2	2	no	—
59	#11761612	—	2	8	no	—
60	#11761612	—	4	8	no	—
61	C₂×C₆₇⋊C₈	—	2	2	no	—
62	C₂×C₆₇⋊C₈	—	2	4	no	—
63	C₂×C₆₇⋊C₈	—	4	8	no	—
64	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
65	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
66	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
67	C₄×Dic₆₇	—	2	2	no	—
68	C₄×Dic₆₇	—	4	4	no	—
69	C₄×Dic₆₇	C₂×D₄	1	4	yes	—
70	#41×C₆₇	—	2	8	no	—
71	#41×C₆₇	—	2	8	no	—
72	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
73	C₂×D₄×C₆₇	—	2	4	no	—
74	C₂×D₄×C₆₇	—	2	4	no	—
75	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₆₇	—	2	4	no	—
77	C₂²×C₄×C₆₇	#41	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
79	Q₁₆×C₆₇	—	2	8	no	—
80	SD₁₆×C₆₇	—	2	16	no	—
81	D₈×C₆₇	—	2	8	no	—
82	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
85	C₂×C₈×C₆₇	#41	1	4	yes	—
86	C₄⋊C₄×C₆₇	—	2	4	no	—
87	C₂²⋊C₄×C₆₇	—	2	8	no	—
88	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₆₇	C₂×D₄	1	2	yes	—
91	C₂³×C₆₇	—	2	4	no	—
92	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
93	Q₈×C₆₇	—	2	8	no	—
94	D₄×C₆₇	—	2	8	no	—
95	D₄×C₆₇	—	4	8	no	—
96	D₄×C₆₇	—	4	8	no	—
97	C₂×C₄×C₆₇	—	2	4	no	—
98	C₂×C₄×C₆₇	—	2	4	no	—
99	C₂×C₄×C₆₇	—	4	4	no	—
100	C₂×C₄×C₆₇	—	4	4	no	—
101	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
102	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
105	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
107	C₂×Dic₆₇	—	2	4	no	—
108	C₂×Dic₆₇	—	4	4	no	—
109	C₈×C₆₇	C₄×D₄	1	4	yes	—
110	C₈×C₆₇	C₄×D₄	1	4	yes	—
111	C₆₇⋊C₈	—	2	2	no	—
112	C₆₇⋊C₈	—	4	4	no	—
113	C₆₇⋊C₈	#127	1	4	yes	—
114	C₂²×C₆₇	—	2	2	no	—
115	C₂²×C₆₇	—	4	4	no	—
116	C₂²×C₆₇	—	4	8	no	—
117	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
118	C₂²×C₆₇	#532	1	2	yes	—
119	C₄×C₆₇	—	2	2	no	—
120	C₄×C₆₇	—	4	4	no	—
121	C₄×C₆₇	#444	1	4	yes	complemented
122	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
123	C₄×C₆₇	C₂×#127	1	1	yes	characteristic
124	C₄×C₆₇	#532	1	1	yes	characteristic
125	Dic₆₇	—	4	4	no	—
126	#21706	—	67	1	no	maximal, Sylow
127	C₂×C₆₇	—	2	2	no	—
128	C₂×C₆₇	—	8	4	no	—
129	C₂×C₆₇	C₄×#127	1	1	yes	characteristic
130	C₂×C₆₇	C₂×#444	1	2	yes	—
131	#3147	D₆₇	1	1	yes	characteristic
132	C₂×#444	—	67	4	no	—
133	#2931	—	67	2	no	—
134	#1975	—	67	4	no	—
135	#1962	—	67	2	no	—
136	#1848	—	67	1	no	—
137	C₄×#97	—	67	1	no	—
138	C₆₇	#21706	1	1	yes	characteristic, complemented, Sylow
139	C₂×#135	C₂×D₆₇	1	4	yes	—
140	C₂×#131	—	67	4	no	—
141	#532	C₂×D₆₇	1	2	yes	—
142	#495	Dic₆₇	1	4	yes	complemented
143	#494	Dic₆₇	1	1	yes	characteristic, complemented
144	#493	Dic₆₇	1	1	yes	characteristic, complemented
145	#492	Dic₆₇	1	2	yes	complemented
146	#444	—	134	16	no	—
147	#436	—	134	4	no	—
148	C₈×D₄	—	134	4	no	—
149	C₂×#108	—	67	4	no	—
150	#423	—	67	2	no	—
151	C₂×#104	—	67	8	no	—
152	#408	—	67	4	no	—
153	C₂×#97	—	67	2	no	—
154	C₂×#97	—	67	2	no	—
155	C₂×C₄×C₈	—	67	1	no	—
156	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
157	C₄³	—	67	1	no	—
158	C₂×#41	C₂²×D₆₇	1	4	yes	—
159	#135	—	2	8	no	—
160	#135	—	2	8	no	—
161	C₂×Q₁₆	C₂×Dic₆₇	1	4	yes	—
162	C₂×SD₁₆	C₂×Dic₆₇	1	8	yes	—
163	C₂×D₈	C₂×Dic₆₇	1	4	yes	—
164	#131	—	134	16	no	—
165	C₂×M₁₆	—	67	4	no	—
166	C₂×M₁₆	—	134	4	no	—
167	C₂²×C₈	—	67	1	no	—
168	C₂²×C₈	—	67	1	no	—
169	C₂²×C₈	—	134	4	no	—
170	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
171	C₄⋊Q₈	C₂×Dic₆₇	1	2	yes	—
172	#127	C₂×Dic₆₇	1	2	yes	—
173	#124	C₂×Dic₆₇	1	4	yes	—
174	#121	—	2	4	no	—
175	C₄×D₄	—	2	4	no	—
176	C₂×C₄²	—	67	2	no	—
177	C₂×C₄²	—	134	2	no	—
178	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
179	#108	—	134	8	no	—
180	C₄⋊C₈	—	134	4	no	—
181	#104	—	134	16	no	—
182	C₂²⋊C₈	—	134	8	no	—
183	#97	—	67	8	no	—
184	#97	—	134	4	no	—
185	C₄×C₈	—	67	4	no	—
186	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
187	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
188	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
189	#41	—	2	8	no	—
190	#41	—	2	8	no	—
191	C₂×Q₈	C₂²×Dic₆₇	1	4	yes	—
192	C₂×D₄	—	2	4	no	—
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	C₂²×Dic₆₇	1	4	yes	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	134	2	no	—
197	C₂²×C₄	#11761627	1	2	yes	—
198	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
199	Q₁₆	—	2	8	no	—
200	SD₁₆	—	2	16	no	—
201	D₈	—	2	8	no	—
202	M₁₆	—	268	8	no	—
203	M₁₆	—	134	8	no	—
204	C₂×C₈	—	268	8	no	—
205	C₂×C₈	—	67	4	no	—
206	C₂×C₈	—	67	2	no	—
207	C₂×C₈	—	67	2	no	—
208	C₂×C₈	—	134	4	no	—
209	C₂×C₈	—	134	2	no	—
210	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
211	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
212	C₂×C₈	D₄×D₆₇	1	4	yes	—
213	C₂×C₈	#11761627	1	4	yes	—
214	C₄⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	—	2	8	no	—
216	C₄²	—	268	4	no	—
217	C₄²	—	67	4	no	—
218	C₄²	—	134	2	no	—
219	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
220	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
221	C₄²	D₄×D₆₇	1	2	yes	—
222	C₂³	—	2	4	no	—
223	C₂³	C₂×#11761627	1	1	yes	characteristic
224	Q₈	—	2	8	no	—
225	D₄	—	2	8	no	—
226	D₄	—	4	8	no	—
227	D₄	—	4	8	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	268	4	no	—
233	C₂×C₄	—	134	4	no	—
234	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
235	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
236	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
237	C₂×C₄	C₂×D₄×D₆₇	1	2	yes	—
238	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
239	C₂×C₄	C₂×#11761627	1	2	yes	—
240	C₈	—	268	4	no	—
241	C₈	—	67	4	no	—
242	C₈	—	134	2	no	—
243	C₈	D₄×Dic₆₇	1	4	yes	—
244	C₈	D₄×Dic₆₇	1	4	yes	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	8	no	—
247	C₂²	—	4	4	no	—
248	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
249	C₂²	#16863944	1	2	yes	—
250	C₄	—	2	2	no	—
251	C₄	—	4	4	no	—
252	C₄	—	268	4	no	—
253	C₄	#16863518	1	4	yes	complemented
254	C₄	C₂×D₄×Dic₆₇	1	2	yes	—
255	C₄	#127×D₆₇	1	1	yes	characteristic
256	C₄	#16863944	1	1	yes	characteristic
257	C₂	—	2	2	no	—
258	C₂	—	8	4	no	—
259	C₂	#127×Dic₆₇	1	1	yes	characteristic
260	C₂	C₂×#16863518	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	#21706
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×#16863518	C₂	semidirect
2. #22568921	C₂	semidirect
3. #22561103	C₂	semidirect
4. #22561038	C₂	semidirect
5. #22560381	C₂	semidirect
6. C₄×#14523953	C₂	semidirect
22. #495×C₆₇	C₄	semidirect
23. #494×C₆₇	C₄	semidirect
24. #493×C₆₇	C₄	semidirect
25. #492×C₆₇	C₄	semidirect
40. #14523953	D₄	semidirect
121. C₄×C₆₇	#444	semidirect
138. C₆₇	#21706	semidirect
142. #495	Dic₆₇	semidirect
143. #494	Dic₆₇	semidirect
144. #493	Dic₆₇	semidirect
145. #492	Dic₆₇	semidirect
253. C₄	#16863518	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863518	2
#22568921	2
#22561103	2
#22561038	2
#22560381	2
C₄×#14523953	2
C₆₇×#3147	2
#21706	67