Subgroups — #30774092

Order 17152.

574 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863209	C₂	1	4	yes	maximal, complemented
2	C₄×#14523983	C₂	1	1	yes	maximal, characteristic, complemented
3	#22560381	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#493×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	C₂²	1	2	yes	—
6	C₂×#14523979	—	2	1	no	—
7	C₂×#14523979	C₂²	1	2	yes	—
8	#16863420	—	2	1	no	—
9	#16863209	—	2	8	no	—
10	#16863172	—	2	1	no	—
11	C₄×C₆₇⋊C₁₆	—	2	1	no	—
12	C₂²×D₈×C₆₇	—	2	1	no	—
13	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
14	C₂×#127×C₆₇	—	2	1	no	—
15	#493×C₆₇	—	2	1	no	—
16	#493×C₆₇	—	2	1	no	—
17	#493×C₆₇	C₄	1	2	yes	—
18	#493×C₆₇	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#11761612	—	2	2	no	—
21	#14523983	—	2	2	no	—
22	#14523983	D₄	1	4	yes	complemented
23	C₂×C₆₇⋊C₁₆	—	2	2	no	—
24	#14523979	—	2	4	no	—
25	#14523979	—	4	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
27	C₂²×D₄×C₆₇	—	2	2	no	—
28	C₂²×D₄×C₆₇	—	4	2	no	—
29	C₂×D₈×C₆₇	—	2	2	no	—
30	C₂×D₈×C₆₇	—	2	2	no	—
31	C₂×D₈×C₆₇	—	2	4	no	—
32	C₂×D₈×C₆₇	—	2	2	no	—
33	C₂×D₈×C₆₇	—	2	4	no	—
34	C₂×D₈×C₆₇	—	4	2	no	—
35	C₂×D₈×C₆₇	—	4	2	no	—
36	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	#127×C₆₇	—	2	2	no	—
40	#127×C₆₇	—	2	2	no	—
41	#127×C₆₇	—	4	2	no	—
42	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
47	#11761612	—	4	2	no	—
48	#11761612	—	4	2	no	—
49	C₂×C₆₇⋊C₈	—	4	2	no	—
50	C₆₇⋊C₁₆	—	2	4	no	—
51	C₂⁴×C₆₇	—	4	2	no	—
52	C₂×D₄×C₆₇	—	2	4	no	—
53	C₂×D₄×C₆₇	—	4	4	no	—
54	C₂×D₄×C₆₇	—	4	4	no	—
55	C₂×D₄×C₆₇	—	4	2	no	—
56	C₂×D₄×C₆₇	—	4	2	no	—
57	C₂×D₄×C₆₇	—	4	2	no	—
58	C₂×D₄×C₆₇	—	4	2	no	—
59	C₂×D₄×C₆₇	—	4	2	no	—
60	C₂×D₄×C₆₇	—	8	8	no	—
61	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	D₈×C₆₇	—	2	8	no	—
64	D₈×C₆₇	—	4	4	no	—
65	D₈×C₆₇	—	4	4	no	—
66	D₈×C₆₇	—	4	4	no	—
67	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₆₇	D₈	1	2	yes	—
70	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
71	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₆₇	#41	1	2	yes	—
75	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₆₇	D₈	1	2	yes	—
78	C₂³×C₆₇	—	4	4	no	—
79	C₂³×C₆₇	—	8	8	no	—
80	C₂³×C₆₇	—	8	2	no	—
81	C₂³×C₆₇	—	8	2	no	—
82	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
83	D₄×C₆₇	—	4	4	no	—
84	D₄×C₆₇	—	4	4	no	—
85	D₄×C₆₇	—	8	8	no	—
86	D₄×C₆₇	—	8	8	no	—
87	D₄×C₆₇	—	8	2	no	—
88	D₄×C₆₇	—	8	2	no	—
89	C₂×C₄×C₆₇	—	2	4	no	—
90	C₂×C₄×C₆₇	#102	1	2	yes	—
91	C₂×C₄×C₆₇	#102	1	2	yes	—
92	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
94	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
98	C₈×C₆₇	#102	1	2	yes	—
99	C₈×C₆₇	#102	1	2	yes	—
100	C₈×C₆₇	C₄×D₄	1	2	yes	—
101	C₈×C₆₇	#121	1	2	yes	—
102	C₆₇⋊C₈	—	8	2	no	—
103	C₂²×C₆₇	—	2	2	no	—
104	C₂²×C₆₇	—	8	8	no	—
105	C₂²×C₆₇	—	8	4	no	—
106	C₂²×C₆₇	—	16	4	no	—
107	C₂²×C₆₇	—	16	4	no	—
108	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
109	C₂²×C₆₇	#486	1	1	yes	characteristic
110	C₂²×C₆₇	#492	1	1	yes	characteristic
111	C₄×C₆₇	—	2	2	no	—
112	C₄×C₆₇	#361	1	4	yes	complemented
113	C₄×C₆₇	#390	1	1	yes	characteristic
114	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
115	C₄×C₆₇	#486	1	1	yes	characteristic
116	C₄×C₆₇	#493	1	1	yes	characteristic
117	#13213	—	67	1	no	maximal, Sylow
118	C₂×C₆₇	—	2	1	no	—
119	C₂×C₆₇	—	2	1	no	—
120	C₂×C₆₇	—	16	4	no	—
121	C₂×C₆₇	#1957	1	1	yes	characteristic
122	C₂×C₆₇	C₂×#361	1	2	yes	—
123	C₂×#493	D₆₇	1	1	yes	characteristic
124	C₂×#361	—	67	4	no	—
125	C₄×M₃₂	—	67	1	no	—
126	#1848	—	67	1	no	—
127	C₆₇	#13213	1	1	yes	characteristic, complemented, Sylow
128	C₂²×D₈	—	2	1	no	—
129	C₂²×D₈	C₂×D₆₇	1	2	yes	—
130	C₂×#127	—	2	1	no	—
131	C₂×M₃₂	—	67	2	no	—
132	#493	—	2	1	no	—
133	#493	—	2	1	no	—
134	#493	Dic₆₇	1	2	yes	—
135	#493	Dic₆₇	1	2	yes	—
136	C₂×#108	—	67	2	no	—
137	C₂×#108	—	134	1	no	—
138	#423	—	134	1	no	—
139	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
140	#361	—	134	8	no	—
141	#346	—	134	1	no	—
142	C₄×C₁₆	—	134	1	no	—
143	C₂²×D₄	—	2	2	no	—
144	C₂²×D₄	—	4	2	no	—
145	C₂×D₈	—	2	2	no	—
146	C₂×D₈	—	2	2	no	—
147	C₂×D₈	—	2	4	no	—
148	C₂×D₈	—	2	2	no	—
149	C₂×D₈	—	2	4	no	—
150	C₂×D₈	—	4	2	no	—
151	C₂×D₈	—	4	2	no	—
152	C₂×D₈	C₂×Dic₆₇	1	4	yes	—
153	C₂×M₁₆	—	134	2	no	—
154	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
156	#127	—	2	2	no	—
157	#127	—	2	2	no	—
158	#127	—	4	2	no	—
159	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
160	M₃₂	—	67	4	no	—
161	M₃₂	—	134	2	no	—
162	C₂×C₁₆	—	134	2	no	—
163	#108	—	268	2	no	—
164	#108	—	134	4	no	—
165	C₄⋊C₈	—	268	2	no	—
166	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
167	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
168	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
169	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
170	C₂⁴	—	4	2	no	—
171	C₂×D₄	—	2	4	no	—
172	C₂×D₄	—	4	4	no	—
173	C₂×D₄	—	4	4	no	—
174	C₂×D₄	—	4	2	no	—
175	C₂×D₄	—	4	2	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂×D₄	—	8	8	no	—
180	C₂²×C₄	#11761616	1	2	yes	—
181	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
182	D₈	—	2	8	no	—
183	D₈	—	4	4	no	—
184	D₈	—	4	4	no	—
185	D₈	—	4	4	no	—
186	M₁₆	—	268	2	no	—
187	M₁₆	—	268	2	no	—
188	C₂×C₈	—	268	2	no	—
189	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
190	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
191	C₂×C₈	#11761613	1	2	yes	—
192	C₂×C₈	#11761616	1	2	yes	—
193	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
194	C₂×C₈	D₄×D₆₇	1	2	yes	—
195	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
196	C₂×C₈	#11761627	1	2	yes	—
197	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
198	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
199	C₄²	#11761613	1	2	yes	—
200	C₁₆	—	134	4	no	—
201	C₂³	—	4	4	no	—
202	C₂³	—	8	8	no	—
203	C₂³	—	8	2	no	—
204	C₂³	—	8	2	no	—
205	C₂³	C₂×#11761616	1	1	yes	characteristic
206	D₄	—	4	4	no	—
207	D₄	—	4	4	no	—
208	D₄	—	8	8	no	—
209	D₄	—	8	2	no	—
210	D₄	—	8	8	no	—
211	D₄	—	8	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	#14523964	1	2	yes	—
214	C₂×C₄	#14523964	1	2	yes	—
215	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
216	C₂×C₄	#14524030	1	1	yes	characteristic
217	C₂×C₄	#14524039	1	1	yes	characteristic
218	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
219	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
220	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
221	C₈	—	536	2	no	—
222	C₈	#14523964	1	2	yes	—
223	C₈	#14523964	1	2	yes	—
224	C₈	D₄×Dic₆₇	1	2	yes	—
225	C₈	#14524014	1	2	yes	—
226	C₂²	—	2	2	no	—
227	C₂²	—	8	8	no	—
228	C₂²	—	8	4	no	—
229	C₂²	—	16	4	no	—
230	C₂²	—	16	4	no	—
231	C₂²	C₂×#14523964	1	1	yes	characteristic
232	C₂²	#16863742	1	1	yes	characteristic
233	C₂²	#16863768	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	#16863209	1	4	yes	complemented
236	C₄	#16863299	1	1	yes	characteristic
237	C₄	C₂×#14523964	1	1	yes	characteristic
238	C₄	#16863742	1	1	yes	characteristic
239	C₄	#16863772	1	1	yes	characteristic
240	C₂	—	2	1	no	—
241	C₂	—	2	1	no	—
242	C₂	—	16	4	no	—
243	C₂	#22561016	1	1	yes	characteristic
244	C₂	C₂×#16863209	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14523964
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting, prime radical	C₂×#493×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#102
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#493	D₆₇
hypocenter	C₆₇	#13213
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. C₂×#16863209	C₂	semidirect
2. C₄×#14523983	C₂	semidirect
3. #22560381	C₂	semidirect
22. #14523983	D₄	semidirect
112. C₄×C₆₇	#361	semidirect
127. C₆₇	#13213	semidirect
235. C₄	#16863209	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863209	2
C₄×#14523983	2
#22560381	2
C₂×#493×C₆₇	2
#13213	67