Subgroups — #30774399

Order 17152.

338 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22570723	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#22558306	C₂	1	4	yes	maximal
4	C₄×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#16863953	—	2	1	no	—
6	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
7	#16863801	C₄	1	1	yes	characteristic
8	#16863801	C₄	1	1	yes	characteristic
9	#16863798	—	2	1	no	—
10	#16863798	C₄	1	2	yes	—
11	#16863794	—	2	1	no	—
12	#16863434	—	2	1	no	—
13	#16863434	C₂²	1	2	yes	—
14	C₄⋊(C₆₇⋊C₁₆)	—	2	1	no	—
15	C₄⋊(C₆₇⋊C₁₆)	—	2	1	no	—
16	C₂×#108×C₆₇	C₂²	1	2	yes	—
17	C₄×M₁₆×C₆₇	—	2	1	no	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#341×C₆₇	—	2	1	no	—
20	#14524043	—	2	1	no	—
21	#14524043	—	2	1	no	—
22	#14524035	—	2	2	no	—
23	#14524022	—	4	1	no	—
24	#14524022	—	4	1	no	—
25	C₄×Dic₁₃₄	—	4	1	no	—
26	C₄×Dic₁₃₄	—	4	1	no	—
27	#14524002	—	2	2	no	—
28	C₂×C₆₇⋊C₁₆	—	2	2	no	—
29	C₂×C₆₇⋊C₁₆	—	2	4	no	—
30	#14523977	—	2	1	no	—
31	#14523977	—	2	1	no	—
32	#14523977	C₂×C₄	1	2	yes	—
33	#14523977	C₂×C₄	1	2	yes	—
34	#14523975	—	2	2	no	—
35	#14523975	—	2	2	no	—
36	C₂×M₁₆×C₆₇	—	2	2	no	—
37	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
40	#108×C₆₇	—	2	2	no	—
41	#108×C₆₇	Q₈	1	4	yes	complemented
42	#97×C₆₇	—	4	1	no	—
43	C₄×C₈×C₆₇	—	4	1	no	—
44	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
47	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
48	C₂×Dic₁₃₄	—	4	1	no	—
49	C₂×Dic₁₃₄	—	4	1	no	—
50	C₄⋊Dic₆₇	—	2	2	no	—
51	C₄⋊Dic₆₇	—	2	2	no	—
52	C₄⋊Dic₆₇	—	4	1	no	—
53	C₄⋊Dic₆₇	—	4	1	no	—
54	Dic₆₇⋊C₄	—	4	2	no	—
55	Dic₆₇⋊C₄	—	4	2	no	—
56	C₂²⋊Dic₆₇	—	4	2	no	—
57	C₄×Dic₆₇	—	4	2	no	—
58	C₆₇⋊C₁₆	—	4	2	no	—
59	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
62	M₁₆×C₆₇	—	2	4	no	—
63	M₁₆×C₆₇	—	4	2	no	—
64	C₂×C₈×C₆₇	—	2	2	no	—
65	C₂×C₈×C₆₇	—	4	2	no	—
66	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
71	C₂×C₈×C₆₇	#41	1	2	yes	—
72	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₆₇	D₈	1	1	yes	characteristic
75	C₄²×C₆₇	D₈	1	1	yes	characteristic
76	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₆₇	—	2	2	no	—
78	C₂×C₄×C₆₇	—	4	1	no	—
79	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
80	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
81	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
82	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
85	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₆₇	#137	1	2	yes	—
87	C₂×Dic₆₇	—	4	2	no	—
88	C₂×Dic₆₇	—	4	2	no	—
89	Dic₁₃₄	—	8	2	no	—
90	Dic₁₃₄	—	8	2	no	—
91	C₈×C₆₇	—	2	1	no	—
92	C₈×C₆₇	—	2	1	no	—
93	C₈×C₆₇	—	4	4	no	—
94	C₈×C₆₇	C₄×Q₈	1	2	yes	—
95	C₈×C₆₇	#124	1	1	yes	characteristic
96	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
97	C₂²×C₆₇	—	2	1	no	—
98	C₂²×C₆₇	—	2	1	no	—
99	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
100	C₂²×C₆₇	#475	1	2	yes	—
101	C₄×C₆₇	—	2	1	no	—
102	C₄×C₆₇	—	2	1	no	—
103	C₄×C₆₇	—	2	1	no	—
104	C₄×C₆₇	#359	1	2	yes	—
105	C₄×C₆₇	#391	1	1	yes	characteristic
106	C₄×C₆₇	#419	1	1	yes	characteristic
107	C₄×C₆₇	#474	1	2	yes	—
108	Dic₆₇	—	8	2	no	—
109	Dic₆₇	—	8	2	no	—
110	#13274	—	67	1	no	maximal, Sylow
111	C₂×C₆₇	—	2	2	no	—
112	C₂×C₆₇	#1988	1	1	yes	characteristic
113	C₂×C₆₇	C₂×#359	1	1	yes	characteristic
114	C₂×C₆₇	#2143	1	1	yes	characteristic
115	#3160	—	67	1	no	—
116	C₂×C₄⋊C₁₆	—	67	1	no	—
117	C₄×#108	D₆₇	1	1	yes	characteristic
118	#1382	—	67	4	no	—
119	C₆₇	#13274	1	1	yes	characteristic, complemented, Sylow
120	#533	—	134	1	no	—
121	C₂²×C₁₆	—	67	2	no	—
122	#500	—	67	1	no	—
123	#500	—	67	1	no	—
124	#499	—	67	2	no	—
125	#499	—	134	1	no	—
126	#498	—	134	1	no	—
127	C₂×#108	C₂×D₆₇	1	2	yes	—
128	#427	—	67	2	no	—
129	#427	—	134	1	no	—
130	C₄×M₁₆	—	2	1	no	—
131	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
132	C₄⋊C₁₆	—	134	1	no	—
133	C₄⋊C₁₆	—	134	1	no	—
134	#341	—	2	1	no	—
135	C₂×M₁₆	—	2	2	no	—
136	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
137	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
138	C₄⋊Q₈	—	134	1	no	—
139	C₄⋊Q₈	—	134	1	no	—
140	#125	—	134	2	no	—
141	C₂²⋊Q₈	—	268	1	no	—
142	C₂²⋊Q₈	—	268	1	no	—
143	C₄×Q₈	—	268	1	no	—
144	C₄×Q₈	—	268	1	no	—
145	#117	—	134	2	no	—
146	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
147	C₂×C₁₆	—	134	2	no	—
148	C₂×C₁₆	—	134	4	no	—
149	#108	—	2	2	no	—
150	#108	Dic₁₃₄	1	4	yes	complemented
151	#107	—	67	2	no	—
152	#107	—	67	2	no	—
153	#107	—	134	1	no	—
154	#107	—	134	1	no	—
155	#106	—	134	2	no	—
156	#106	—	134	2	no	—
157	#97	—	4	1	no	—
158	C₄×C₈	—	4	1	no	—
159	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
160	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
161	C₄×C₈	D₂₆₈	1	1	yes	characteristic
162	C₄×C₈	D₂₆₈	1	1	yes	characteristic
163	C₂×Q₈	—	268	1	no	—
164	C₂×Q₈	—	268	1	no	—
165	C₂²×C₄	#11761615	1	1	yes	characteristic
166	C₂²×C₄	#11761615	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
168	M₁₆	—	2	4	no	—
169	M₁₆	—	4	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
173	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
174	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
175	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
176	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
177	C₂×C₈	#11761628	1	2	yes	—
178	C₄⋊C₄	—	268	2	no	—
179	C₄⋊C₄	—	268	2	no	—
180	C₄⋊C₄	—	268	1	no	—
181	C₄⋊C₄	—	268	1	no	—
182	C₄⋊C₄	—	134	2	no	—
183	C₄⋊C₄	—	134	2	no	—
184	C₂²⋊C₄	—	268	2	no	—
185	C₄²	—	268	2	no	—
186	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
187	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
188	C₄²	#11761613	1	1	yes	characteristic
189	C₄²	#11761613	1	1	yes	characteristic
190	C₁₆	—	268	2	no	—
191	C₂³	C₂×#11761615	1	1	yes	characteristic
192	Q₈	—	536	2	no	—
193	Q₈	—	536	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	268	2	no	—
197	C₂×C₄	—	268	2	no	—
198	C₂×C₄	#14523963	1	1	yes	characteristic
199	C₂×C₄	#14523963	1	1	yes	characteristic
200	C₂×C₄	#14523963	1	1	yes	characteristic
201	C₂×C₄	#14523963	1	1	yes	characteristic
202	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
203	C₂×C₄	#14524022	1	2	yes	—
204	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
205	C₂×C₄	#14524073	1	2	yes	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	4	4	no	—
209	C₈	C₄×Dic₁₃₄	1	2	yes	—
210	C₈	#14524031	1	1	yes	characteristic
211	C₈	#14524043	1	1	yes	characteristic
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	C₂×#14523963	1	1	yes	characteristic
215	C₂²	#16863681	1	2	yes	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	536	2	no	—
220	C₄	—	536	2	no	—
221	C₄	#16863202	1	2	yes	—
222	C₄	#16863302	1	1	yes	characteristic
223	C₄	#16863404	1	1	yes	characteristic
224	C₄	#16863676	1	2	yes	—
225	C₂	—	2	2	no	—
226	C₂	#22561161	1	1	yes	characteristic
227	C₂	C₂×#16863202	1	1	yes	characteristic
228	C₂	#22561936	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₄×#108×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂³×C₆₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₄×#108	D₆₇
hypocenter	C₆₇	#13274
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
41. #108×C₆₇	Q₈	semidirect
119. C₆₇	#13274	semidirect
150. #108	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
#22570723	2
C₂×C₄⋊(C₆₇⋊C₁₆)	2
#22558306	2
C₄×#108×C₆₇	2
#13274	67